Handbook of Philosophical Logic Vol 5

Handbook of Philosophical Logic 2nd Edition Volume 5

edited by Dov M. Gabbay and F. Guenthner

CONTENTS

Editorial Preface

vii

Dov M. Gabbay Intuitionistic Logic

1

Dirk van Dalen Dialogues as a Foundation for Intuitionistic Logic

115

Walter Felscher Free Logics

147

Ermanno Bencivenga Advanced Free Logic

197

Scott Lehmann Partial Logic

261

Stephen Blamey Index

354

PREFACE TO THE SECOND EDITION

It is with great pleasure that we are presenting to the community the second edition of this extraordinary handbook. It has been over 15 years since the publication of the rst edition and there have been great changes in the landscape of philosophical logic since then. The rst edition has proved invaluable to generations of students and researchers in formal philosophy and language, as well as to consumers of logic in many applied areas. The main logic article in the Encyclopaedia Britannica 1999 has described the rst edition as `the best starting point for exploring any of the topics in logic'. We are con dent that the second edition will prove to be just as good.! The rst edition was the second handbook published for the logic community. It followed the North Holland one volume Handbook of Mathematical Logic, published in 1977, edited by the late Jon Barwise. The four volume Handbook of Philosophical Logic, published 1983{1989 came at a fortunate temporal junction at the evolution of logic. This was the time when logic was gaining ground in computer science and arti cial intelligence circles. These areas were under increasing commercial pressure to provide devices which help and/or replace the human in his daily activity. This pressure required the use of logic in the modelling of human activity and organisation on the one hand and to provide the theoretical basis for the computer program constructs on the other. The result was that the Handbook of Philosophical Logic, which covered most of the areas needed from logic for these active communities, became their bible. The increased demand for philosophical logic from computer science and arti cial intelligence and computational linguistics accelerated the development of the subject directly and indirectly. It directly pushed research forward, stimulated by the needs of applications. New logic areas became established and old areas were enriched and expanded. At the same time, it socially provided employment for generations of logicians residing in computer science, linguistics and electrical engineering departments which of course helped keep the logic community thriving. In addition to that, it so happens (perhaps not by accident) that many of the Handbook contributors became active in these application areas and took their place as time passed on, among the most famous leading gures of applied philosophical logic of our times. Today we have a handbook with a most extraordinary collection of famous people as authors! The table below will give our readers an idea of the landscape of logic and its relation to computer science and formal language and arti cial intelligence. It shows that the rst edition is very close to the mark of what was needed. Two topics were not included in the rst edition, even though

viii

they were extensively discussed by all authors in a 3-day Handbook meeting. These are:



a chapter on non-monotonic logic



a chapter on combinatory logic and -calculus

We felt at the time (1979) that non-monotonic logic was not ready for a chapter yet and that combinatory logic and -calculus was too far removed.1 Non-monotonic logic is now a very major area of philosophical logic, alongside default logics, labelled deductive systems, bring logics, multi-dimensional, multimodal and substructural logics. Intensive reexaminations of fragments of classical logic have produced fresh insights, including at time decision procedures and equivalence with non-classical systems. Perhaps the most impressive achievement of philosophical logic as arising in the past decade has been the eective negotiation of research partnerships with fallacy theory, informal logic and argumentation theory, attested to by the Amsterdam Conference in Logic and Argumentation in 1995, and the two Bonn Conferences in Practical Reasoning in 1996 and 1997. These subjects are becoming more and more useful in agent theory and intelligent and reactive databases. Finally, fteen years after the start of the Handbook project, I would like to take this opportunity to put forward my current views about logic in computer science, computational linguistics and arti cial intelligence. In the early 1980s the perception of the role of logic in computer science was that of a speci cation and reasoning tool and that of a basis for possibly neat computer languages. The computer scientist was manipulating data structures and the use of logic was one of his options. My own view at the time was that there was an opportunity for logic to play a key role in computer science and to exchange bene ts with this rich and important application area and thus enhance its own evolution. The relationship between logic and computer science was perceived as very much like the relationship of applied mathematics to physics and engineering. Applied mathematics evolves through its use as an essential tool, and so we hoped for logic. Today my view has changed. As computer science and arti cial intelligence deal more and more with distributed and interactive systems, processes, concurrency, agents, causes, transitions, communication and control (to name a few), the researcher in this area is having more and more in common with the traditional philosopher who has been analysing 1 I am really sorry, in hindsight, about the omission of the non-monotonic logic chapter. I wonder how the subject would have developed, if the AI research community had had a theoretical model, in the form of a chapter, to look at. Perhaps the area would have developed in a more streamlined way!

PREFACE TO THE SECOND EDITION

ix

such questions for centuries (unrestricted by the capabilities of any hardware). The principles governing the interaction of several processes, for example, are abstract an similar to principles governing the cooperation of two large organisation. A detailed rule based eective but rigid bureaucracy is very much similar to a complex computer program handling and manipulating data. My guess is that the principles underlying one are very much the same as those underlying the other. I believe the day is not far away in the future when the computer scientist will wake up one morning with the realisation that he is actually a kind of formal philosopher! The projected number of volumes for this Handbook is about 18. The subject has evolved and its areas have become interrelated to such an extent that it no longer makes sense to dedicate volumes to topics. However, the volumes do follow some natural groupings of chapters. I would like to thank our authors are readers for their contributions and their commitment in making this Handbook a success. Thanks also to our publication administrator Mrs J. Spurr for her usual dedication and excellence and to Kluwer Academic Publishers for their continuing support for the Handbook.

Dov Gabbay King's College London

x

Logic

IT Natural language processing

Temporal logic

Expressive power of tense operators. Temporal indices. Separation of past from future

Modal logic. Multi-modal logics

generalised quanti ers

Action logic

Algorithmic proof

Discourse representation. Direct computation on linguistic input Resolving ambiguities. Machine translation. Document classi cation. Relevance theory logical analysis of language Quanti ers in logic

Montague semantics. Situation semantics

Nonmonotonic reasoning

Probabilistic and fuzzy logic Intuitionistic logic

Set theory, higher-order logic, calculus, types

Program control speci cation, veri cation, concurrency

Arti cial intelligence

Logic programming

Planning. Time dependent data. Event calculus. Persistence through time| the Frame Problem. Temporal query language. temporal transactions. Belief revision. Inferential databases

Extension of Horn clause with time capability. Event calculus. Temporal logic programming.

New logics. Generic theorem provers

General theory of reasoning. Non-monotonic systems

Procedural approach to logic

Loop checking. Non-monotonic decisions about loops. Faults in systems.

Intrinsic logical discipline for AI. Evolving and communicating databases

Negation by failure. Deductive databases

Real time systems

Semantics for logic programs

Constructive reasoning and proof theory about speci cation design

Expert systems. Machine learning Intuitionistic logic is a better logical basis than classical logic

Non-wellfounded sets

Hereditary nite predicates

-calculus ex-

Expressive power for recurrent events. Speci cation of temporal control. Decision problems. Model checking.

Negation failure modality

by and

Horn clause logic is really intuitionistic. Extension of logic programming languages tension to logic programs

PREFACE TO THE SECOND EDITION

xi

Imperative vs. declarative languages

Database theory

Complexity theory

Agent theory

Special comments: A look to the future

Temporal logic as a declarative programming language. The changing past in databases. The imperative future

Temporal databases and temporal transactions

Complexity questions of decision procedures of the logics involved

An essential component

Temporal systems are becoming more and more sophisticated and extensively applied

Dynamic logic

Database updates and action logic

Ditto

Possible tions

Multimodal logics are on the rise. Quanti cation and context becoming very active

Types. Term rewrite systems. Abstract interpretation

Abduction, relevance

Ditto

Agent's implementation rely on proof theory.

Inferential databases. Non-monotonic coding of databases

Ditto

Agent's reasoning is non-monotonic

A major area now. Important for formalising practical reasoning

Fuzzy and probabilistic data Database transactions. Inductive learning

Ditto

Connection with decision theory Agents constructive reasoning

Major now

Semantics for programming languages. Martin-Lof theories Semantics for programming languages. Abstract interpretation. Domain recursion theory.

Ditto

Ditto

ac-

area

Still a major central alternative to classical logic

More central than ever!

xii

Classical logic. Classical fragments

Basic ground guage

Labelled deductive systems

Extremely useful in modelling

A unifying framework. Context theory.

Resource and substructural logics Fibring and combining logics

Lambek calculus

Truth maintenance systems Logics of space and time

backlan-

Dynamic syntax

Program synthesis

Modules. Combining languages

A basic tool

Fallacy theory

Logical Dynamics Argumentation theory games

Widely applied here Game semantics gaining ground

Object level/ metalevel

Extensively used in AI

Mechanisms: Abduction, default relevance Connection with neural nets

ditto

Time-actionrevision models

ditto

Annotated logic programs

Combining features

PREFACE TO THE SECOND EDITION

Relational databases

Labelling allows for context and control. Linear logic

Linked databases. Reactive databases

Logical complexity classes

xiii

The workhorse of logic

The study of fragments is very active and promising.

Essential tool.

The new unifying framework for logics

Agents have limited resources Agents are built up of various bred mechanisms

The notion of self- bring allows for selfreference Fallacies are really valid modes of reasoning in the right context.

Potentially applicable

A dynamic view of logic On the rise in all areas of applied logic. Promises a great future

Important feature of agents

Always central in all areas

Very important for agents

Becoming part of the notion of a logic Of great importance to the future. Just starting

A new theory of logical agent

A new kind of model

DIRK VAN DALEN

INTUITIONISTIC LOGIC

INTRODUCTION Among these logics that deal with the familiar connectives and quanti ers two stand out as having a solid philosophical{mathematical justi cation. On the one hand there is a classical logic with its ontological basis and on the other hand intuitionistic logic with its epistemic motivation. The case for other logics is considerably weaker; although one may consider intermediate logics with more or less plausible principles from certain viewpoints none of them is accompanied by a comparably compelling philosophy. For this reason we have mostly paid attention to pure intuitionistic theories. Since Brouwer, and later Heyting, considered intuitionistic reasoning, intuitionistic logic has grown into a discipline with a considerable scope. The subject has connections with almost all foundational disciplines, and it has rapidly expanded. The present survey is just a modest cross-section of the presently available material. We have concentrated on a more or less semantic approach at the cost of the proof theoretic features. Although the proof theoretical tradition may be closer to the spirit of intuitionism (with its stress on proofs), even a modest treatment of the proof theory of intuitionistic logic would be beyond the scope of this chapter. The reader will nd ample information on this particular subject in the papers of, e.g. Prawitz and Troelstra. For the same reason we have refrained from going into the connection between recursion theory and intuitionistic logic. Section 8 provides a brief introduction to realizability. Intuitionistic logic is, technically speaking, just a subsystem of classical logic; the matter changes, however, in higher-order logic and in mathematical theories. In those cases speci c intuitionistic principles come into play, e.g. in the theory of choice sequences the meaning of the pre x 8 9x derives from the nature of the mathematical objects concerned. Topics of the above kind are dealt with in Section 9. The last sections touch on the recent developments in the area of categorical logic. We do not mention categories but consider a very special case. There has been an enormous proliferation in the semantics of intuitionistic second-order and higher-order theories. The philosophical relevance is quite often absent so that we have not paid attention to the extensive literature on independence results. For the same reason we have not incorporated the intuitionistic ZF-like systems.

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Intuitionistic logic can be arrived at in many ways|e.g. physicalistic or materialistic|we have chosen to stick to the intuitionistic tradition in considering mathematics and logic as based on human mental activities. Not surprisingly, intuitionistic logic plays a role in constructive theories that do not share the basic principles of intuitionism, e.g. Bishop's constructive mathematics. There was no room to go into the foundations of these alternatives to intuitionism. In particular we had to leave out Feferman's powerful and elegant formalisations of operations and classes. The reader is referred to Beeson [1985] and Troelstra and van Dalen [1988] for this and related topics. We are indebted for discussions and comments to C.P.J. Koymans, A.S. Troelstra and A. Visser. 1 A SHORT HISTORY Intuitionism was conceived by Brouwer in the early part of the twentieth century when logic was still in its infancy. Hence we must view Brouwer's attitude towards logic in the light of a rather crude form of theoretical logic. It is probably a sound conjecture that he never read Frege's fundamental expositions and that he even avoided Whitehead and Russell's Principia Mathematica. Frege was at the time mainly known in mathematical circles for his polemics with Hilbert and others, and one could do without the Principia Mathematica by reading the fundamental papers in the journals. Taking into account the limited amount of specialised knowledge Brouwer had of logic, one might well be surprised to nd an astute appraisal of the role of logic in Brouwer's Dissertation [Brouwer, 1907]. Contrary to most traditional views, Brouwer claims that logic does not precede mathematics, but, conversely, that logic depends on mathematics. The apparent contradiction with the existing practice of establishing strings of `logical' steps in mathematical reasoning, is explained by pointing out that each of these steps represents a sequence of mathematical constructions. The logic, so to speak, is what remains if on takes away the speci c mathematical constructions that lead from one stage of insight to the next. Here it is essential to make a short excursion into the mathematical and scienti c views that Brouwer held and that are peculiar to intuitionism. Mathematics, according to Brouwer, is a mental activity, sometimes described by him as the exact part of human thought. In particular, mathematical objects are mental constructions, and properties of these objects are established by, again, mental constructions. Hence, in this view, something holds for a person if he has a construction (or proof) that establishes it. Language does not play a role in this process but may be (and in practice: is) introduced for reasons of communication. `People try by means of sounds and symbols to originate in other copies of mathematical constructions and

INTUITIONISTIC LOGIC

3

reasonings which they have made themselves; by the same means they try to aid their own memory. In this way mathematical language comes into being, and as its special case the language of logical reasoning'. The next step taken by man is to consider the language of logical reasoning mathematically, i.e. to study its mathematical properties. This is the birth of theoretical logic. Brouwer's criticism of logic is two-fold. In the rst place, logicians are blamed for giving logic precedence over mathematics, and in the second place, logic is said to be unreliable (Brouwer [1907; 1908]). In particular, Brouwer singled out the principle of the excluded third as incorrect and unjusti ed. The criticism of this principle is coupled to the criticism of Hilbert's famous dictum that `each particular mathematical problem can be solved in the sense that the question under consideration can either be aÆrmed, or refuted' [Brouwer, 1975, pp. 101 and 109]. Let us, by way of example, consider Goldbach's Conjecture, G, which states that each even number is the sum of two odd primes. A quick check tells us that for small numbers the conjecture is borne out: 12 = 5 + 7, 26 = 13 + 13, 62 = 3 + 59, 300 = 149 + 151. Since we cannot perform an in nite search, this simple method of checking can at best provide, with luck, a counter example, but not a proof of the conjecture. At the present stage of mathematical knowledge no proof of Goldbach's conjecture, or of its negation, has been provided. So can we aÆrm G _ :G? If so, we should have a construction that would decide which of the two alternatives holds and provide a proof for it. Clearly we are in no position to exhibit such a construction, hence we have no grounds for accepting G _ :G as correct. The undue attention paid to the principle of the excluded third, had the unfortunate historical consequence that the issues of the foundational dispute between the Formalists and the Intuitionists were obscured. An outsider might easily think that the matter was a dispute of two schools{ one with, and one without, the principle of the excluded third (or middle), PEM for short. Brouwer himself was in no small degree the originator of the misunderstanding by choosing the far too modest and misleading title of `Begrundung der Mengenlehre unabhangig vom logischen Satz vom ausgeschlossenen Dritten' for his rst fundamental paper on intuitionistic mathematics. For the philosophical-mystical background of Brouwer's views, see [van Dalen, 1999a]; a foundational exposition can be found in [van Dalen, 2000]. The logic of intuitionism was not elaborated by Brouwer, although he proved its rst theorem: :' $ :::'. The rst mathematicians to consider the logic of intuitionism in a more formal way were Glivenko and Kolmogorov. The rst presented a fragment of propositional logic and the second a fragment of predicate logic. In 1928 Heyting independently formalised intuitionist predicate logic and the fundamental theories of arithmetic and `set

4

DIRK VAN DALEN

theory' [Heyting, 1930]. For historical details, cf. Troelstra [1978; 1981]. Heyting's formalization opened up a new eld to adventurous logicians, but it did not provide a `standard' or `intended' interpretation, thus lacking the inner coherence of a conceptual explanation. In a couple of papers (cf. [Heyting, 1934]), Heyting presented from 1931 on the interpretation that we have come to call the proof-interpretation (cf. [Heyting, 1956, Chapter VII]). The underlying idea traces back to Brouwer: the truth of a mathematical statement is established by a proof, hence the meaning of the logical connective has to be explained in terms of proofs and constructions (recall that a proof is a kind of construction). Let us consider one connective, by way of example: A proof of ' ! is a construction which converts any proof of ' into a proof of . Note that this de nition is in accord with the conception of mathematics (and hence logic) as a mental constructive activity. Moreover it does not require statements to be bivalent, i.e. to be either true or false. For example, ' ! ' is true independent of our knowledge of the truth of '. The proofinterpretation provided at least an informal insight into the mysteries of intuitionistic truth, but it lacked the formal clarity of the notion of truth in classical logic with its completeness property. An analogue of the classical notion of truth value was discovered by Tarski, Stone and others who had observed the similarities between intuitionistic logic and the closure operation of topology (cf. [Rasiowa and Sikorski, 1963]). This so-called topological interpretation of intuitionistic logic also covers a number of interpretations that at rst sight might seem to be totally devoid of topological features. Among these are the lattice (like) interpretations of Jaskowski, Rieger and others, but also the more recent interpretations of Beth and Kripke. All these interpretations are grouped together as semantical interpretations, in contrast to interpretations that are based on algorithms, one way or another. A breakthrough in intuitionistic logic was accomplished by Gentzen in 1934 in his system of Natural Deduction (and also his calculus of sequents), which embodied the meaning of the intuitionistic connectives far more accurately than the existing Hilbert-type formalizations. The eventual recognition of Gentzen's insights is to a large extent due to the eorts of Prawitz who reintroduced Natural Deduction, and considerably extended Gentzen's work [1965; 1971]. In the beginning of the thirties the rst meta-logical results about intuitionistic logic and its relation to existing logics appeared. Godel, and independently Gentzen, formulated a translation of classical predicate logic into a fragment of intuitionistic predicate logic, thus extending early work of Glivenko [Glivenko, 1929; Gentzen, 1933; Godel, 1932]. Godel also established the connection between the modal logic S4 and intuitionistic logic [Godel, 1932].

INTUITIONISTIC LOGIC

5

The period after the Second World War brought new researchers to intuitionistic logic and mathematics. In particular Kleene, who based an `effective' interpretation of intuitionistic arithmetic on the notion of recursive function. His interpretation is known as realizability (Kleene [1952; 1973]). In 1956 Beth introduced a new semantic interpretation with a better foundational motivation than the earlier topological interpretations, and Kripke presented a similar, but more convenient interpretation in 1963 [Kripke, 1965]. These new semantics showed more exibility than the earlier interpretations and lent themselves better to the model theory of concrete theories. General model theory in the lattice and topological tradition had already been undertaken by the Polish school (cf. [Rasiowa and Sikorski, 1963]). In the meantime Godel had presented his Dialectica Interpretation [1958], which like Kleene's realizability, belongs to the algorithmic type of interpretations. Both the realizability and the Dialectica Interpretation have shown to be extremely fruitful for the purpose of Proof Theory. Another branch at the tree of semantic interpretations appeared fairly recently, when it was discovered that sheaves and topoi present a generalisation of the topological interpretations [Goldblatt, 1979; Troelstra and van Dalen, 1988]. The role of a formal semantics will be expounded in Section 3. Its most obvious and immediate use is the establishing of underivability results in a logical calculus. However, even before a satisfactory semantics was discovered, intuitionists used to show that certain classical theorems were not valid by straightforward intuitive methods. We will illustrate the naive approach for two reasons. In the rst place it is direct and the rst thing one would think of, in the second place it has its counterparts in formal semantics and can be useful as a heuristics. The traditional counterexamples are usually formulated in terms of a particular unsolved problem. The problem in the following example goes back to Brouwer. Consider the decimal expansion of  : 3; 14 : : : , hardly anything is known about regularities in this expansion, e.g. it is not known if it contains a sequence of 9 nines. Let A(n) be the statement `the nth decimal of  is a nine and it is preceded by 8 nines'. 1. The principle of the excluded third is not valid. Suppose 9xA(x) _ :9xA(x), then we would have a proof that either provides us with a natural number n such that A(n), or that shows us that no such n exists. Since there is no such evidence available we cannot accept the principle of the excluded third. 2. The double negation principle is not valid. Observe that ::(9xA(x) _ :9xA(x)) holds. In general the double negation of the principle of the excluded third holds, since ::(' _ :') is equivalent to :(:' ^ ::') and the latter is correct on the intuitive interpretations.

6

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Since 9xA(x) _ :9xA(x) does not hold, we see that ::' ! ' is not valid. 3. One version of De Morgan's Law fails. The suspect case is :(' ^ ) ! :' _: , since its conclusion is strong and its premise is weak. Consider :(:9xA(x) ^ 9xA(x)) ! ::9xA(x) _ :9xA(x). The premise is true, but the conclusion cannot be asserted, since we do not know if it is impossible that there is no sequence of 9 nines or it is impossible that there is such a sequence. Counterexamples of the above kind show that our present state of knowledge does not permit us to aÆrm certain logical statements that are classically true. They represent evidence of implausibility, all the same it is not the strongest possible result. Of course we cannot expect to establish the negation of the principle of the excluded third because that is a downright contradiction. By means of certain strong intuitionistic, or alternatively algorithmic, principles one can establish a strongly non-classical theorem like :8x('(x) _ :'(x)) for a suitable '(x). We will now present an informal version of the proof interpretation. For convenience we will suppose that the variables of our language range over natural numbers. This is not strictly necessary, but it suÆces to illustrate the working of the interpretation. Recall that we understand the primitive notion `a is a proof of '', where a proof is a particular kind of (mental) construction. We will now proceed to explain what it means to have a proof of a non-atomic formula ' in terms of proofs of its components. (i)

a is a proof of ' ^ i a is a pair (a1 ; a2 ) such that a1 is a proof of ' and a2 is a proof of .

(ii)

a is a proof of ' _ i a is a pair (a1 ; a2 ) such that a1 = 0 and a2 is a proof of ' or a1 = 1 and a2 is a proof of .

(iii) a is a proof of ' ! i a is a construction that converts each proof b of ' into a proof a(b) of . (iv) nothing is a proof of ? (falsity). (v)

a is a proof of 9x'(x) i a is a pair (a1 ; a2 ) such that a1 is a proof of '(a2 ).

(vi) a is a proof of 8x'(x) i a is a construction such that for each natural number n, a(n) is a proof of '(n). Note that intuitionists consider :' as an abbreviation for ' ! ?. The clause that a trained logician will immediately look for is the one dealing with the atomic case. We cannot provide a de nition for that case since it

INTUITIONISTIC LOGIC

7

must essentially depend on the speci c theory under consideration. In the case of ordinary arithmetic the matter is not terribly important as the closed atoms are decidable statements of the form 5 = 7 + 6, 23:16 = 5(3 + 2:8), etc. We can `start' the de nition in a suitable fashion. Remark. If one wishes to preserve the feature that from a proof one can read o the result, then some extra care has to be taken, e.g. according to clause (iii) (0; p) proves ' _ for all possible , where ' is a proof of '. One may beef up the `proof' by adding the disjunction to it: replace (0; p) by (0; p; ' _ ), etc. The above version is due to Heyting (cf. [Heyting, 1956; Troelstra, 1981]). Re nements have been added by Kreisel for the clauses involving the implication and universal quanti cation [Kreisel, 1965]. His argument being: the de nition contains a part that is not immediately seen to be of the ultimate simple and lucid form we wish it to be. In particular one could ask oneself `does this alleged construction do what it purports to do?' For this reason Kreisel modi ed clause (iii) as follows: a is a proof of ' ! i a is a pair (a1 ; a2 ) such that a1 is a construction that converts any proof b of ' into a proof a1 (b) of , and a2 is a proof of the latter fact. A similar modi cation is provided for (vi). The situation is akin to that of the correctness of computer programs. In particular we use Kreisel's clause if we want the relation `a is a proof of '' to be decidable. Clauses (iii) and (vi) clearly do not preserve decidability, moreover they do not yield `logic free' conditions. It must be pointed out however that the decidability of the proof-relations has been criticised and that the `extra clauses' are not universally accepted. Sundholm [1983] contains a critical analysis of the various presentations of the `proof interpretation'. In summing up the views of Brouwer, Heyting and Kreisel, he notes a certain confusion in terminology. In particular he points out that constructions (in particular proofs) can be viewed as processes and dier from the resulting construction-object. The latter is a mathematical object, and can be operated upon, not so the former. The judgements at the right-hand side, explaining the meaning of the logical constants, are taken by Kreisel to be mathematical objects, a procedure that is objected to by Sundholm. indeed, on viewing the judgement `a converts each proof of ' into a proof of ' as extra-mathematical, the need for a second clause disappears. In Beeson [1979] a theory of constructions and proofs is presented violating the decidability of the proof relation. Troelstra and Diller [1982] study the relation between the proof interpretation and Martin-Lofs's type theory. The proofs inductively de ned above are called canonical by Martin-Lof, Prawitz and others. Of course there are also non-canonical proofs, and some of them are preferable to canonical ones. Consider, e.g. 1011 + 1110 = 1110 + 1011 in arithmetic. One knows how to get a canonical proof: by simply carrying out the addition according to the basic rules (x + 0 = x

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DIRK VAN DALEN

and x + Sy = S (x + y), where S is the successor function). An obvious non-canonical (and shorter) proof would be: rst show 8xy(x + y = y + x) by mathematical induction and then specialise. We will now proceed to illustrate the rules in use. (1) (' ^

! ) ! (' ! ( ! )): Let a be a proof of ' ^ ! , i.e. a is a construction that converts any proof (b; c) of ' ^ into a proof a((b; c)) of . We want a proof of ' ! ( ! ). So let p be a proof of ' and q a proof of . De ne a construction k such that k(p) is a proof of ! , i.e. (k(p))(q) is a proof of . Evidently we should put (k(p))(q) = a((p; q)); so, using the functional abstraction operator, k(p) = q:a((p; q)) and k = p:q:a((p; q)). The required proof is a construction that carries a into k, i.e. apq:a((p; q)). (2)

:(' _

) ! (:' ^ : ):

Let a be a proof of :(' _ ), a construction that carries a proof of ' _ into a proof of ?. Suppose now that p is a proof of ', then (0; p) is a proof of ' _ , and hence a((0; p)) is a proof of ?. So p:a((0; p)) is a proof of :'. Likewise q:a((1; q)) is a proof of : . By de nition (p:a((0; p)); q:a((1; q))) is a proof of :' ^ : . So the construction that carries a into (p:a((0; p)), q:a((1; q)), i.e. a:(p:a((0; p)); q:a((1; q))), is the required proof.

9x:'(x) ! :8x'(x): Let (a1 ; a2 ) be a proof of 9x:'(x), i.e. a1 is a proof of '(a2 ) ! ?. Suppose p is a proof of 8x'(x), then in particular p(a2 ) is a proof of '(a2 ), and hence a1 (p(a2 )) is a proof of ?. So p:a1 (p(a2 )) is a proof of :8x'(x)). Therefore (3)

(a1 ; a2 )p:a1 (p(a2 )) is the required proof. The history of intuitionistic logic is not as stirring as the history of intuitionism itself. The logic itself was not controversial, Heyting's formalization showed it to be a subsystem of classical logic. Moreover, it convinced logicians that there was a coherent notion of `constructive reasoning'. In the following sections we will show some of the rich structure of this logic. One problem in intuitionistic logical theories is how to codify and exploit typically intuitionistic principles. These are to be found in particular in the second-order theories where the concepts of set (species) and function play a role. Despite Brouwer's scorn for logic, some of the ner distinctions that are common today were introduced by him. In his thesis we can already nd the fully understood notions of language, logic, metalanguage, metalogic, etc. (cf. Brouwer [1907; 1975]). The Brouwer{Hilbert controversy seems from our present viewpoint to be one of those deplorable misunderstandings. Hilbert wanted to justify by

INTUITIONISTIC LOGIC

9

metamathematical means the mathematics of in nity with all its idealizations. He considered mathematics as based on the bedrock of its nitistic part, which is just a very concrete part of intuitionistic mathematics. The latter transcends nitism by its introduction of abstract notions, such as set and sequence. 2 PROPOSITIONAL AND PREDICATE LOGIC The syntax of intuitionistic logic is identical to that of classical logic (cf. Wilfrid Hodges' chapter in Volume 1 of this Handbook). As in classical logic, we have the choice between a formalisation in a Hilbert-type system or in a Gentzen-type system. Heyting's original formalisation used the rst kind. We will exhibit a Hilbert-type system rst.

2.1 An Axiom System for Intuitionistic Logic Axioms 1. ' ! ( ! ') 2. (' ! ) ! ((' ! (

! )) ! (' ! ))

3. ' ! ( 4. 5. 6. 7. 8. 9. 10.

!'^ ) '^ !' '^ ! ' !'_ !'_ (' !  ! (( ! ) ! (' _ ! )) (' ! ) ! ((' ! : ) ! :') '(t) ! 9x'(x) 8x'(x) ! '(t) ' ! (:' ! )

Rules Modus Ponens

' '!

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DIRK VAN DALEN

Quanti er rules

' ! (x) ' ! 8x (x) '(x) ! 9x'(x) ! The quanti er axioms and rules are subject to the usual variable conditions: t is free for x and x does not occur free in . The deducibility relation, `, is de ned as in Hodges' chapter (Vol. 1) of the Handbook. As in classical logic, we have the Deduction theorem: 1; : : :

; n`',

1; : : :

; n

1

`

n

! ':

If we add to the axioms the principle of the excluded third, ' _ :', or the double negation principle, ::' ! ', we obtain the familiar classical logic. We should note that the axioms contain all connectives, and not, as in classical logic, just _; : and 9 (or whatever your favourite choice may be). The reason is that the de nability of the connectives in terms of some of them (Hodges Chapter in Volume 1 of this Handbook) fails, as we will see later. Since intuitionistic logic is more of an epistemic than of an ontological nature, we will study it mainly by means of Gentzen's Natural Deduction, as this latter system re ects the speci c constructive reasoning of the intuitionist best. This particular system has only rules and no axioms. The simplest rules have the form :::::: ' , and are to be read as ' follows (immediately) from the premises above the line. Some of the rules, however, involve manipulations with the so-called assumptions. The prime example is the rule that corresponds to the deduction theorem in Hilbert-type systems. Suppose we can derive by means of a derivation D from a number of assumptions among which is a formula ', then we can derive ' ! from the mentioned assumptions without '. We denote this by [']

D

'! we say that the assumption ' is cancelled, this is indicated by the use of square brackets.

INTUITIONISTIC LOGIC

11

It appears to be convenient to employ a choice of connectives that includes

? and excludes :. Of course :' can be introduced as an abbreviation for ' ! ?. We will also use the traditional abbreviation ' $ . The rules come in two kinds, Introduction rules and Elimination rules. Introduction rules

^I _I

!I

' '^

Elimination rules

^E

'

'_

[']

D

'!

8I

'(x) 8x'(x)

9I

'(t) 9x'(x)

'^ '

'^ ['] [ ]

'_

_E !E ? 8E

D1 D2

'_







' '!

? '

8x'(x) '(t)

['(y)]

9E

9x

D 

 For the quanti er rules we have to add a few conditions: in the rules 9I and 8E , t has to be `free for x'. An application of 8I is allowed only if the variable x does not occur in any of the assumptions in the derivation of '(x). Similarly the free variable y in the cancelled formula '(y) may not occur free in  or any of the assumptions in the right-hand derivation of  (in 9E ). The rules of Gentzen's system of Natural Deduction are intended to represent the meaning of connectives as faithfully as possible (cf. [Gentzen, 1935] or [Szabo, 1969, p. 74]). Gentzen's goals have recently been made more precise in [Dummett, 1973] and [Prawitz, 1977]. We will set ourselves a speci c goal by showing that the natural deduction rules are in accordance with the meaning of the logical connectives as put forward in Heyting's proof interpretation.

12

DIRK VAN DALEN

We will consider a few representative cases.

'1 '2

^I : ' ^ ' : 1 2 Let proofs pi of 'i be given. Then we can form the ordered pair (p1 ; p2 ) which is a proof of '1 ^ '2 . This is the step that, given canonical proofs of the conjuncts, provides the canonical proof of the conjunction. ' ^' ^E : 1 2 : 'i Given a canonical proof p of '1 ^ '2 , we know that it must be an ordered pair (p1 ; p2 ). The projection i yields the required canonical proof of 'i .

!I :

[']

D : '!

Suppose that we have a proof of under a number of assumptions, including '. Then this proof, when supplemented by a proof of ' yields a proof of , i.e. we have a construction that transforms any proof of ' into a proof of , but that means that we have a proof of ' ! .

8I :

'(x) : 8x'(x)

Suppose that we have a proof of '(x), i.e. a proof schema, that for each instance '(n) of '(x) yields a proof of it. Since x does not occur in the assumptions, the proof is uniform in x, i.e. it is a method for converting n into a proof of '(n). Again we have found a proof of 8x'(x), along the lines of Heyting's interpretation. The reader will now be able to continue this line of argument. We will only dwell for a moment on the ex falso rule.

?:

?

: ' The justi cation in terms of constructions is not universally accepted, e.g. [Johansson, 1936] rejected the rule and formulated his so-called minimal logic, which has the same rules as intuitionistic logic with deletion of the ex falso rule. Now, ? has, in the intuitionistic conception, no proof. What we have to provide is a construction that automatically yields for every proof of ? a

INTUITIONISTIC LOGIC

13

proof of '. Nothing is simpler; take for example the identity construction i : p 7! p, i promises to give a proof of ' as output as soon as it gets a proof of ? as input. Obviously, i keeps its promise because it is never asked to ful ll it. Note that there is an alternative way of looking at the Natural Deduction system, we could consider it as a concrete illustration of Heyting's proof interpretation. For instance, the actual formal derivations are the proofs and/or constructions. In that sense they realized Heyting's clauses. Let us, by way of illustration, make a few derivations. 1. (' ! ) ! ((

! ) ! ' ! )) 1 ['] [' ! ]3 !E [ ! ]2 !E  (1) ' !  ! I (2)

(3)

(' !

!I ( ! ) ! (' ! ) ) ! (( ! ) ! (' ! ))

!I

2. By substitution of ? for  we obtain the law of contraposition (' ! ) ! (: ! :'). 3. ' ! ::'

1 [:']

(1) (2) 4.

[']2

? ::'

!E

' ! ::'

(recall that :' stands for ' ! ?)

!I

:::' ! :' 1 [:']

(1) (2) (3)

[']2

? ::'

[:::']3

? :' :::' ! :'

14

DIRK VAN DALEN

5. From 3. we get :' ! :::', combining this with 4. we have :' $ :::'. 6.

::8x'(x) ! 8x::'(x)

1 [8x'(x)]

'(x)

(1) (2) (3)

[:'(x)]2

? :8x'(x) [::8x'()]3 ? ::'(x) 8x::'(x) ::8x'(x) ! 8x::'(x)

7. :(' _ ) $ (:' ^ : ) [']1 '_ [:(' _ )]3 (1) :?' (3)

:(' _

[ ]2 ' _ [:(' _ )]3 (2) :?

:' ^ : ) ! :' ^ :'

The arrow from right to left is trivial. 8. ' _ :' and ::' ! ' are equivalent as schema's, i.e. all instances of PEM follow from all instances of the double negation principle and vice versa. We will consider one direction. the proof requires a number of derivations, each of which is simple. (a) (b)

` ::(' _ :') (use (7)) ::(' _ :') ! (' _ :') ::('D_ :') ' _ :' where D is a derivation obtained in (a).

The other direction is left to the reader. The following list of provable statements will come in handy (relevant variables are shown) 1. ' ! ::'

2. :' $ :::' 3. :(' ^ :')

INTUITIONISTIC LOGIC

4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

15

::(' _ :') :(' _ ) $ :' ^ :' (' _ :') ! (::' ! ') (' ! ) ! :(' ^ : ) (' ! : ) $ :(' ^ ) (::' ^ :: ) $ ::(' ^ ) (::' ! :: ) $ ::(' ! ) (::' ! ) ! (: ! :') 9x:'(x) ! :8x'(x) :9x'(x) $ 8x:'(x) ' _ 8x (x) ! 8x(' _ (x)) 8x(' ! (x)) $ (' ! 8x (x)) 8x('(x) ! ) $ (9'(x) ! ) 9x(' ! (x)) ! (' ! 9x (x)) ::8x'(x) ! 8x::'(x).

Furthermore, conjunction and disjunction have the familiar associative, commutative and distributive properties. For counterexamples to invalid propositions and sentences see Section 3.11. The systems of intuitionistic propositional and predicate (or quanti cational) logic are, without consideration of their formalisations, denoted by IPC and IQC. Derivability will pedantically be denoted by j IPC ' (resp. j IQC '), or IPC ` ' (resp. IQC ` '), for empty . When no confusion arises, we will however delete the subscripts. The derivations are in tree form, but one can easily represent them in linear form (cf. [Prawitz, 1965, p. 89 ]). The present form, however, is more suggestive and since there is nothing sacrosanct about linearity we will stick to Gentzen's notation. There is, nonetheless, a good reason for a more complete notation that makes the cancellation of assumptions explicit. As usual, we write ` ' for `there is a derivation of ' from uncancelled assumptions that belong to the set '. The rules of natural deduction can be formulated in terms of `. For convenience we write ; '1 ; : : : ; 'n for [ f'1 ; : : : ; 'n g and ,
for [
.

16

DIRK VAN DALEN

The following facts follow immediately from our rules: 1. ` ' if ' 2 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

` ' and
` ) ;
` ' ^ `'^ ) `' `'^ ) ` `') `'_ ` ) `'_ ` ' _ and
; ' `  and
0 ; `  ) ;
;
0 `  ;' ` ) ` ' ! ` ' and
` ' ! ) ;
` `?) `' ` '(x) ) ` 8x'(x), where x is not free in ` 8x'(x) ) ` '(t) ` '(t) ) ` 9x'(x) ` 9x'(x) and
; '(y) `  ) ;
` , where y is not free in
and

. The above presentation of natural deduction can be viewed as a kind of sequent calculus, cf. [Troelstra and Schwichtenberg, 1996, x2.1.4] We can now turn the tables and de ne ` ' inductively by the preceding clauses. D is the least class of pairs ( ; ') (denoted by ` ') such that

` ' 2 D if ' 2 ` ' 2 D;
` 2 D ) ;
` ' ^ 2 D .. .

` 9x'(x) 2 D;
; '(y) `  2 D ) ;
`  2 D; where y is not free in
and .

Observe that a derivation in D corresponds to a derivation in tree form, as presented before. The linearisation of natural deduction derivations that some authors have practised obscures the perspicuity of the derivations and we will stick to the tree form (remember what Frege said about `the convenience of the printer'). EXAMPLE 1. Take the string

'; ;  ` ' (by 1) ; ` ' ! ' (by 6) `  ! (' ! ') (by 6)

INTUITIONISTIC LOGIC

It shows that form from it:

17

`  ! (' ! ') an we can recover the derivation in tree

rst derivation

'

second derivation [']

third derivation

[']

'!'

'!'  ! (' ! ')

All this calls for some clari cation. 1. The matter of cancellation is somewhat delicate, you don't have to cancel all occurrences of the relevant formula, not even any occurrence. This is made explicit in, e.g. rule 6, ; ' ` ) ` ' ! . may still contain '. 2. The tree derivation shows only the assumptions that actually play a role, but in ` ' there may be lots of super uous assumptions (in nitely many if you wish!). It is for example quite simple to show, on the basis of the rules 1{12

` ' ) ;
` '.

Natural Deduction, or for that matter its sister system of the Sequent Calculus, lends itself well to study derivations for their own sake. This particular branch of logic has in the case of Natural Deduction been rigorously practised and promoted by Dag Prawitz, who established the main facts of the system and who demonstrated its exibility and usefulness (cf. Prawitz [1965; 1971]). The fundamental theorem in the subject is concerned with derivations without super uous parts. The following is evidently awkward.  !' !E ' [ ] ^I '^ ^E ' !I !' We have introduced the super uous conjunction ' ^ eliminate it again. A more eÆcient proof is  !' !E ' !I !'

only in order to

18

DIRK VAN DALEN

We have eliminated the introduction followed by an elimination, thus simplifying the derivation. A derivation in which an introduction is never followed by an elimination is called normal. Here it has to be explained what `follow' means. For this purpose a special partial ordering is introduced; e.g. in ! I ' ! ' follows after , in _E ' follows after ' _ ', etc. See [van Dalen, 1997, p. 199,203]. Prawitz proved the THEOREM 2 (Normal Form Theorem). If ` ', then there is a normal derivation of ' from (cf. [Prawitz, 1965]). There is a better result called the THEOREM 3 (Normalisation Theorem). Any derivation reduces to a normal derivation. Here a reduction step consists in the removal of a super uous introduction followed by an elimination (cf. [Prawitz, 1971]). There is even a still stronger form, the THEOREM 4 (Strong Normalisation Theorem). Every sequence of reduction steps terminates in a normal form. The whole tradition of normalisation and reduction is traditionally a part of combinatory logic and -calculus, a systematic account is given in [Klop, 1980] and [Barendregt, 1984]. There is an interesting interplay between natural deduction derivations and -terms, and hence between normalisation in natural deduction and in -calculus (cf. [Gallier, 1995; Howard, 1980; Pottinger, 1976; Troelstra and van Dalen, 1988]). One of the pleasant corollaries of the normal form of a derivation is the PROPERTY 5 (Subformula property). In a normal derivation of ` ' only subformulas of and ' occur. In particular only connectives from and ' can occur. As a consequence we have THEOREM 6. Intuitionistic predicate logic is conservative over intuitionistic propositional logic. Let ` ' where ' is a proposition. Consider a normal derivation  of '. By the subformula property only propositional connectives can occur, hence we have a derivation using only propositional rules.  Proof.

Natural deduction was given an interesting extension by Schroeder-Heister, [1984]; an exposition and applications can be found in [Negri and von Plato, 2001].

INTUITIONISTIC LOGIC

19

3 PROOF TERMS AND THE CURRY{HOWARD ISOMORPHISM Since natural deduction is so close in nature to the proof interpretation, it is perhaps not surprising that a formal correspondence between a term calculus and natural deduction can be established. We will rst demonstrate this for a small fragment, containing only the connective `!'. Consider an ! introduction: [']

D

'!

[x : ']

D

t: x
t : ' !

We assign in a systematic way proof-terms to formulas in the derivation. Since ' is an assumption, it has a hypothetical proof term, say x. On cancelling the hypotheses; we introduce a x in front of the (given) term t for . By binding x, the proof term for ' ! no longer depends on the hypothetical proof x of '. Note that this corresponds exactly to our intuitive proof interpretation. The elimination runs as follows:

'!

'

t:'! t(s) :

s:'

Observe the analogy to the proof interpretation. Let us consider a particular derivation. [']

[x : ']

!' y
x : ! ' ' ! ( ! ') x
y
x : ' ! ( ! ') Thus the proof term of ' ! ( ! ') is xy:x, this is Curry combinator K . A cut elimination conversion now should give us information about the conversion of the proof term. x: D0 D s: t:' D0 reduces to D [s=x] x
t : ! ' s: t [ s=x ]:' (x
t)(s) : ' The proof theoretic conversion corresponds to the -reduction of the calculus.

20

DIRK VAN DALEN

In order to deal with full predicate logic we have to introduce speci c operations in order to render the meaning of the connectives and their derivation rules:

 

p p0 ; p1 D k

pairing projections discriminator (\case dependency") case obliteration

E { witness extractor ? { ex falso operator

^I

t0 : '0 t1 : '1 p(t0 ; t1 ) : '0 ^ '1

^E

t : '0 ^ '1 (i = 0; 1) pi (t) : 'i

_I

t : 'i (i = 0; 1) ki (t) : '0 _ '1

_E

t : ' _ t0 [x' ] :  t1[x ] :  Du;v (t; t0 [u]; t1 [v]) : 

!I

t[x' ] : y'
t[y'] : ' !

!E

t : ' ! t0 : ' t(t0 ) :

8I

t[x] : '(x) y
t[y] : 8y'(y)

8E

t : 8x'(x) t(t0 ) : '(t0 )

9I

t1 : '(t0 ) p(t0 ; t1 ) : 9x'(x)

9E

t : 9x'(x) t1 [y; z '(y)] :  Eu;v (t; t1 [u; v]) : 

There are a number of details that we have to mention. (i) In ! I the dependency on the hypothesis has to be made explicit in the term. We do this by assigning to each hypothesis its own variable. E.g. x' : '. (ii) In _E (and similarly 9E ) the dependency on the particular (auxilliary) hypotheses ' and disappears. This is done by a variable binding technique. In Du;v the variables u and v are bound. (iii) In the falsum rule the result, of course, depends on the conclusion '. So ' has its own ex falso operator ?'.

INTUITIONISTIC LOGIC

21

Now the conversion rules for the derivation automatically suggest the conversion for the term. We have seen that the term calculus corresponds with the natural deduction system. This suggests a correspondence between proofs and propositions on the one hand and elements (given by the terms) and types (the spaces where these terms are to be found). This correspondence was rst observed for a simple case (the implication fragment) by Haskell Curry, [Curry and Feys, 1958], ch. 9, x E, and extended to full intuitionistic logic by W. Howard, [Howard, 1980]. Let us rst look at a simple case, the one considered by Curry. Since the meaning of proposition is expressed in terms of possible proofs | we know the meaning of ' if we know what things qualify as proofs | one may take an abstract view and consider a proposition as its collection of proofs. From this viewpoint there is a striking analogy between propositions and sets. A set has elements, and a proposition has proofs. As we have seen, proofs are actually a special kind of constructions, and they operate on each other. E.g. if we have a proof p : ' ! and a proof q : ' then p(q) : . So proofs are naturally typed objects. Similarly one may consider sets as being typed in a speci c way. If ' and are typed sets then the set of all mappings from ' to is of a higher type, denoted by ' ! or ' . Starting from certain basic sets with types, one can construct higher types by iterating this `function space'-operation. Let us denote `a is in type '' by a 2 '. Now there is this striking parallel. Propositions a:' p : ' ! ;q : ' ) p(q) : x : ' ) t(x) : then x
t : ' !

Types a2' p 2 ' ! ;q 2 ' ) p(q) 2 x : ' ) t(x) 2 then x
t 2 ' !

It now is a matter of nding the right types corresponding to the remaining connectives. For ^ and _ we introduce a product type and a disjoint sum type. For the quanti ers generalizations are available. The reader is referred to the literature, cf. [Howard, 1980], [Gallier, 1995]. The main aspect of the Curry-Howard isomorphism, (also known as \proofs as types"), is the faithful correspondence: proofs elements propositions = types with their conversion and normalization properties.

22

DIRK VAN DALEN

The importance of the connection between intuitionistic logic and type theory was fully grasped and exploited by Per Martin-Lof. Indeed, in his approach the two are actually merge into one master system. His type systems are no mere technical innovations, but they intend to capture the foundational meaning of intuitionistic logic and the corresponding mathematical universe. Expositions of `proofs as types' and the Martin-Lof type theories can be found in e.g. [Gallier, 1995], [Girard et al., 1989], [Martin-Lof, 1977], [Martin-Lof, 1984], [Troelstra and van Dalen, 1988], [Sommaruga, 2000]. 4 SEMANTICS The intended interpretation of intuitionistic logic as presented by Heyting, Kreisel and others so far has proved to be rather elusive, in as much that the completeness properties that are on every logicians shopping list, have not (yet) been established. Even in the case of the interpretation of arithmetic the results are far from nal. The Curry{Howard isomorphism, also known by the name `formulas as types', in a sense ful lls the promise of the proof interpretation for intuitionistic logic, in the sense that there is a precise correspondence between natural deductions and proof terms, [Troelstra and van Dalen, 1988, p. 556]. However, ever since Heyting's formalisation, various, more or less arti cial, semantics have been proposed. In the thirties the topological interpretation was introduced by Tarski, and in the fties and sixties Beth and Kripke formulated two closely related semantics. We will rst consider the topological interpretation. DEFINITION 7. A topological space is a pair hX; Oi where O  P (X ) such that 1. ;; X 2 O

2O !U \V 2O Ui 2 O(i 2 I ) ! [fIi j i 2 I g 2 O.

2. U; V 3.

In plain words, a topological space is a set that comes with a family O of open subsets that is closed under arbitrary unions and nite intersections and that contains ; and X . A familiar example is the Euclidean plane, where O consists of unions of open discs. In general we can de ne a topological space when a basis is given, i.e. a collection B of subsets such that 1. Ai 2 B; p 2 Ai (i = 1; 2) ) 9A 2 B(p 2 A  A1 \ A2 ) 2. 8p 2 X 9A 2 B(p 2 A).

INTUITIONISTIC LOGIC

23

We now de ne open sets as arbitrary unions of basis-elements. It is a simple exercise to show that the open sets, thus introduced, indeed satisfy the condition of De nition 7. The open discs of the Euclidean plane evidently form a basis for the natural topology. We call U a neighbourhood of a point p if U is open and p 2 U , and if, for a given basis B; U 2 B, we say that U is a basic neighbourhood of p. Now we will interpret sentences as open subsets (opens, for short) of a topological space. In order to motivate the interpretation we recall that, when a xed basis B is given, the evidence for p 2 U is a basic neighbourhood A of p such that A  U . Let us now assign to each statement ' an open subset [ '] of X . We will try to motivate the topological operations that accompany the connectives. Let us say that a basic neighbourhood U proves ' if U  [ '] . Suppose that Ui proves 'i then by the de nition of basis we can nd U 2 B such that U  U1 \ U2 ; U proves both '1 and '2 . The union of all those U 's that prove both '1 and '2 is [ '1 ] \ [ '2 ] , so let us put [ '1 ^ '1 ] := [ '1 ] \ [ '2 ] . Similarly we put [ '1 _ '2 ] := [ '1 ] [ [ '2 ] . Since ? should not have a proof, we put [ ?] := ;. Note that this leaves ; as a proof of ?, therefore we consider ; as the empty proof (or a kind of degenerate proof that carries no evidence). The interesting case is the implication. [ '1 ]

[ '2 ]

U

U2

U1

A proof of '1 ! '2 should give us a method to convert a proof of '1 into a proof of '2 . Therefore we take a basic neighbourhood U in [ '1 ] c [ [ '2 ] , now for any proof U1 that intersects U we can nd a proof U2 of '2 in U \ U1: So U indeed provides the required method. The U 's with that property make up the largest open subset of [ '1 ] c [ [ '2 ] , which we call the interior of that set. So let us put [ '1 ! '2 ] := Int ([['1 ] c [ [ '2 ] ) (= Intfx j x 2 [ '1 ] ) x 2 [ '2 ] g):

24

[ '1 ]

DIRK VAN DALEN

[ '2 ]

U1

U In order to interpret quanti ed statements we assume that a domain A of individuals is given. Then we put [ 9x'(x)]] := [f[ '(a)]] j a 2 Ag [ 8x'(x)]] := Int \ f[ '(a)]] j a 2 Ag:1 Let us now accept the above as an inductive de nition of the value [ '] X of ' in X under a given assignment of open sets to atomic sentences. When no confusion arises we will delete the index X . The notation suppresses O, a better notation would be [ '] O , but the reader will have no diÆculty nding the correct meaning. A formula ' is said to be true in the topological space X , notation X ', if for all valuations [ cl(')]] = X , where cl(') is the universal closure of '. ' is true,  ', if ' is true in all topological spaces. For the consequence relation, , we de ne X ' := Int \ f[ ] X j 2 g  [ '] X and  ' i X ' for all X . Observe that for nite (= f 1 ; : : : ; n g);  ' , 1 ^ : : : ^ n ! '. Observe that nothing has been said about the topological space X , in particular X could be the onepoint space with a resulting two-valued, classical logic! This shows that the above motivation has not enough special assumptions on `constructions', or `evidence' to lead to a speci cally intuitionistic logic. The explanation is too liberal. The topological interpretation is complete in the following sense: THEOREM 8. ` ' ,  '. The implication from left to right (the soundness with respect to the topological interpretation) is easily veri ed by the reader. Just check all the axioms of the Hilbert-type system and show that the derivation rules preserve truth, or do the latter for the rules of natural deduction. 1 For convenience we will abuse notation and use the same symbol for the individual and its name.

INTUITIONISTIC LOGIC

25

We will treat the ! I rule. S Let us abbreviate [ ] X ; [ '] X and [ ] X as U; V; W (where [ ] X = f[ ] X j 2 g). Then the induction hypothesis is U \ V  W (note that we use the formulation of p. 20). Since U is open, U  Int(V c [ W ) , U  V c [ W . Now it is a matter of elementary set theory to show U \ V  W , U  V c [ W. The implication from right to left will follow from a later result. EXAMPLE 9. [ :'] = [ ' ! ?] = Int[['] c . Let ' be an atom and assign to it the complement of a point p (in the plane), then [ :'] = ; and [ ' _:'] = X fpg 6= X . By the soundness of the logic we have 6` ' _ :'. The topological interpretation is extensively studied in [Rasiowa and Sikorski, 1963] (cf. also [Schutte, 1968; Dummett, 1977]). We will move on to a semantics that belongs to the same family as the topological interpretation but that has certain advantages. Beth and Kripke have each introduced a semantics for intuitionistic logic and shown its completeness. The semantics that we present here is a common generalisation introduced for metamathematical purposes in [van Dalen, 1984]. The underlying heuristics is based on the conception of mathematics (and hence logic) as a mental activity of an (idealised) mathematician (or logician if you like). Consider the mental activity of this person, S , as structured in linear time of type !, i.e. time t runs through 0; 1; 2; 3; : : : . At each time t S has acquired a certain body of facts, knowledge. It seems reasonable to assume that S has perfect memory, so that the body of facts increases monotone in time. Furthermore S has at each time t, in general, a number of possibilities to increase his knowledge in the transition to time t +1. So if we present `life' graphically for S , it turns out to fork. However, S not only collects, experiences or establishes truths, but he also constructs objects, the elements of his universe. Here also is considerable freedom of choice for S , going fromptime t to t + 1 he may decide to construct the next prime, or to construct 2. This yields a treelike picture of S 's possible histories. Each node of the tree represents a stage of knowledge of S and a stage in his construction of his universe. So to each node i we have assigned a set of sentences Si and a set of objects Ai , subject to the condition that Si and Ai increase, i.e.

i  j

) Si  Sj and Ai  Aj :

Given this picture of S 's activity, let us nd out how he interprets the logical constants. First, two auxiliary notions: a path through  is a maximal linearly ordered subset, a bar for  is a subset B such that each path through  intersects B . It is suggestive to picture bars above i , i.e. to situate them in the future. It is no restriction to restrict ourselves to this kind of bars we will see. Now let ' be an atomic sentence. How can S know ' at state ? He could

26

DIRK VAN DALEN

7

6

5

4 1



A5 S5

2 0



3





A3 S3

A2 S2

A0 S0

path

B bar



INTUITIONISTIC LOGIC

27

require that ' were then and there given to him. That however seems a bit restrictive. He might know how to establish ', but need more time to do so. In that case we say that S knows ' at stage  if for each path through  (so to speak each `research') there is a stage such that at ' is actually established (or, maybe, experienced). In other words, if there is a bar B for  such that at each 2 B ' is given. The following clauses x the knowledge of S concerning composite statements. . S knows ' ^ at stage  if he knows both ' and at stage

Conjunction

.

. For S to know that ' _ holds at stage  he need not know right away which one holds, he may again need a bit more time. All he needs to know is that eventually ' or will hold. To be precise, that there is a bar B for  such that for each 2 B S knows ' at stage or he knows at stage . Disjunction

. For S to know ' ! at stage , he need not know anything about ' or at stage , all he must be certain of is that if he comes to know ' in any later stage , he must also know at that stage. Implication

. S , being an idealised person, never establishes a falsity.

Falsity

. For S to know 8x'(x) at stage  it does not suÆce to know '(a) for al objects a that exist at stage , but also for all objects that will be constructed in the future. Universal Quantification

. S knows 9x'(x) at stage  if eventually he will construct an element a such that he knows '(a). To be precise, if there is a bar B for  such that for each 2 B there exists an element a at stage such that S knows '(a) at that stage. Existential Quantification

Examples.

';

'; ; 







';



 



 knows ' !

';



at 

28

DIRK VAN DALEN

'(0) '(2) '(7)



'(0)

'(1)













 



 knows 9x'(x) at  We will now give a formal de nition of a model for a given similarity type (without functions). DEFINITION 10. 1. A model is a quadruple M = hM; ; D; i where M is partially ordered by  , and D is a function that assigns to each element of M a structure of the given type, such that for ; 2 M;   ) D()  D( ). Warning: we mean literally `subset', not `substructure'. D()  D( ) is used as a shorthand for `the universe of D() is a subset of that of D( ), and the relations of D() are subsets of the corresponding relations of D( )'. We write a 2 D() for `a is in the universe of D()'. 2. The relation between elements of M and sentences, called the forcing relation is inductively de ned by (a)  ', for ' atomic, if there is a bar B for  such that 8 2 B; D( )  ' (b)  ' ^ if  ' and  (c)  ' _ if there is a bar B for  such that 8 2 B; ' or (d)  ' ! if 8  ; ' ) (e)  8x'(x) if 8  8b 2 D( ); '(b) (f)  9x'(x) if there is a bar B for  such that 8 2 B; 9b 2 D( ); '(b). Observe that for no ;  ?, so by de ning :' := ' ! ? we get

3.  :' if 8  ; 6 ' (where 6 ' stands for ').

INTUITIONISTIC LOGIC

29

Our de nition used the approach with auxiliary names for elements of the structures D(). The alternative approach with assignments works just as well. We say that a formula ' holds (is true)in a model M if  cl(') for all  2 M . If we also allow for the language to contain proposition letters, then the interpretation of propositional logic is contained as a special case. The following lemma is rather convenient for practical purposes LEMMA 11. 1.   ;  ' ) ' 2.  6 ' , there is a path P through  such 8 2 P ( 6 ') 3.  ' , there is a bar B for  such that 8 2 B ( ').

Induction on '. Note that (2) is obtained from (3) by negating both sides.  Proof.

For sentences we have LEMMA 12 (Soundness).

` ' ) '.

' stands for `for each M and each  2 M;  for all 2 )  ''. The proof proceeds by induction on the derivation of ` '. We consider one case: ; ' ` ! ` ' ! . Let, in a model M;   for all  2 . Suppose that  6 ' ! , then there is a   such that ' but 6 . This con icts with the induction hypothesis ; ' . Hence ' ! .  Proof.

We obtain the Beth models and Kripke models by specialisation: DEFINITION 13. 1. 2.

M is a Beth model if jD()j is a xed set D for all . M is a Kripke model if in (a), (c) and (f) B = fg. To spell it out: (a0 )  ' if D()  ' (c0 )  ' _ if  ' or  (f0 )  9x'(x) if 9a 2 D();  '(a). For a Beth model we can simplify clause 5: (a0 ) 8x'(x) , 8a 2 D;  '(a) (repeat the proof of Lemma 11(a)).

30

DIRK VAN DALEN

Generally speaking, Kripke models are somewhat superior to Beth models. A small example may serve to illustrate this. We will summarily present models by a simple diagram. For each node we list the propositions that are forced by it.

'





' 3 3 



 Kripke model

' 2 2 

' 1 1 

' 0 0 

Beth Model The Kripke model is a counter-example to ' _ :', and so is the Beth model. Note that the Beth model has to be in nite in order to refute a classical tautology, since in a well-founded model all classical tautologies are true. One sees this by observing that in a well-founded model (i.e. there are no in nite ascending sequences; if we had turned the model upside down, we would have had the proper well-foundedness) there is a bar of maximal nodes. Now consider a maximal node , if  6 ', then  :'. So  ' _ :'. So ' _ :' is forced on the bar B and hence in each node of the model. B

INTUITIONISTIC LOGIC

31

So, as a rule, we have simpler Kripke models for our logical purposes than Beth models. A Beth model is a special case of our model, so we automatically have soundness for Beth models. For Kripke models, however, we have to show soundness separately. Each class of models is complete for intuitionistic logic. This can be shown as follows, rst show the Model Existence Lemma for Kripke semantics, then modify a Kripke model into a model and nally a model into a Beth model. LEMMA 14 (Model Existence Lemma for Kripke Semantics). If 6` ' then there is a Kripke model K with a bottom node 0 such that 0 for all 2 and 0 6 '. Proof. We'll use a Henkin-style proof after Aczel, Fitting and Thomason. For simplicity's sake we'll treat the case of a denumerable language, i.e. we have denumerably many individual variables and individual constants. A set of sentences is called a prime (also, saturated) theory if 1. it is closed under derivability 2. ' _ 3.

2 ) ' 2 or 2 9x'(x) 2 ) '(c) 2 for some constant c.



The fundamental fact about prime theories is the following: LEMMA 15. If 6` ' then there is a prime theory p  such that ' 62 p . We have to make a harmless little assumption, namely that there are enumerably many constants ci , not in . We approximate the p , as in the case of the Hintikka sets. To start, we add enumerably many new constants to the language of ; '. Since we have a countable language, we may assume that the sentences are given in some xed enumeration. We will treat these sentences one by one. This `treatment' consists of adding witnesses (as in the case of the Hintikka set) and deciding disjunctions. We, so to speak, approximate the required p . Proof.

step 0

0

=

step k + 1 k is even. Let 9x (x) be the rst existential sentence such that k ` 9x (x), that has not been treated, and let c be the rst fresh constant not in k . then put k+1 = k ; (c). k is odd. Let 1 _ 2 be the rst disjunction that has not been treated, such that k ` 1 _ 2 . Pick an i such that k ; i 6` ', then put k+1 = k ; i . By 2. below, at least one of 1 ; 2 will do.

32

DIRK VAN DALEN

The prime theory we are looking for is p

=

[

k0

k:

We will check the properties. 1. 2.

 p , trivially. p 6` '. This amounts to

6` ' for all k. We use induction on k. 2k ; (c). Assume 2k 6` '. If 2k+1 ` ' then by k

Case 1. 2k+1 = 9E; 2k ` '. Contradiction. Case 2. We have to show that 2k+1 ; 1 6` ' or 2k+1 ; 2 6` '. Suppose both are false, then by (_E ) 2k+1 ` '. Contradiction. So, we proved k 6` ' for all k.

3. p is a prime theory.

_ 2 2 p , then 1 _ 2 2 k for some k, and hence h ` 1 _ 2 for all h  k . Now look for the rst h such that 1 _ 2 is treated at step h; then by de nition 1 2 h+1 or p 2 2 h+1 . And so at least one of the i 's is in . 9x (x) 2 p implies by a similar argument that (c) 2 p for

(a) Let

(b)

1

some c. (c) If p ` , then p `

_

and, as in (1),

2

p.



We now can construct the required Kripke model. In order to obtain elements for the various domains we consider denumerably many disjoint sets Vi of denumerably many constants fcim j m  0g. By joining these Vi 's we get a denumerable family of languages Li partially ordered by inclusion. The nodes of our Kripke model are prime theories  0 , which are prime with respect to some Li , and the partial ordering is the inclusion relation. The domain of such a is the set of constants of its language Li . The forcing relation is de ned by

, 2

for atomic :

Claim: , 2 holds for all sentences . We use induction on . For 1 _ 2 ; 9x (x) we apply the prime property of . Consider 1 ! 2 , if 1 ! 2 62 , then ; 1 6` 2 , so we can nd
 ; 1 such that
6` 2 and
is prime with respect to Li+1 (where Li belongs to ). So, by induction hypothesis,
1 and
6 2 . Contradiction. Hence 1 ! 2 2 . The converse is simple. A similar argument is used for 8x (x). Let 8x (x), i.e. 8
 ; 8c 2 D(
);
(c), and by induction hypothesis (c) 2
. Now if

INTUITIONISTIC LOGIC

33

8x (x) 62

, then 6` 8x (x) and hence 6` (c) for a fresh constant c of the next language Li . But then we can nd a prime theory
with respect to Li that contains and
6` (c), so (c) 62
. Contradiction. So 8x (x) 2 . Again the converse is simple. We now nally can nish our proof: the model that we have constructed satis es the requirements. To be precise, we rst extend to a prime theory 0 and then construct the model with 0 as bottom node. As a corollary we have the THEOREM 16 (Strong Completeness Theorem for Kripke Semantics). ` ' , '. Proof. ) is the soundness property. ( If 6` ', then we have a Kripke model such that its bottom node 0 6 ' and 0 for all 2 . Hence 6 '.  In order to carry the result over to the other two semantics it suÆces to modify a Kripke model so that we obtain a (Beth) model that does the trick of Lemma 14. Kripke has indicated how to do this. In one step we obtain a general model, and in one more step a Beth model. We will indicate only the rst modi cation









02 01

3

12

11

11

02

01

2 10 00

10 1

00

0 We basically repeat each node in nitely often, complete with its domain. If we look at the Kripke model and its modi cation below, then we see that

34

DIRK VAN DALEN

each i forces the same atoms as  in the Kripke model, since any bar intersects the path 0 1 2 : : : . An inductive argument shows that Æ K ' , Æ0 '; where Æ = ; ; and Æ0 is one of the indexed Æ's below and where K stands for Kripke-forcing, and for general forcing. In order to make the procedure general, we introduce nite sequences h1 ; 2 ; : : : ; n i of nodes of the Kripke model, with i  i+1 , as nodes of the new model. Put D(h1 ; : : : ; n i) = D(n ). It is a simple exercise to show that the new model serves to establish Lemma 14. This suÆces to show the completeness of our semantics. In order to obtain a Beth model we have to collect everything into one domain. This is worked out in [Kripke, 1965, p. 112 ] or [Schutte, 1968]. As a result we have

` ' , K ' , ' , B '; where K and B stand for Kripke and Beth forcing. Let us nally return to the topological interpretation. We wills how that each Beth model can be viewed as a topological model. Consider a Beth model hB; ; D; i, the poset B gives rise to a topological space as follows: the points of TB are paths in B . We de ne a topology by indicating the basic open sets U , where U = fP j 9  ; P passes through g. The opens (short for `open sets') of TB are unions of U 's. In the terminology of topology: fU j  2 B g is a basis for the topology on TB . We check the properties of a basis 4: 1. P 2 U \ U , then there are   and Æ  such that ; Æ 2 P . Let Æ  then P 2 UÆ and UÆ  U \ U . 2. For any path P and any  2 P we have P

2 U .

We next turn to the de nition of the truth values. Put [ '] = [fU j  'g for atomic '. We thus obtain a canonical topological model TB . THEOREM 17. For the topological model TB the identity [ '] = [fU j  'g holds for all sentences '. Induction on '. For atoms the identity holds by de nition. _ and ^ are simple. Consider !. We must show U  [ ' ! ] ,  ' ! . We use a small topological lemma: U  Int(V c [ W ) , U \ V  W , cf. the proof of Theorem 8. Proof.

So, U  [ ' ! ] , U  Int([['] c [ f ] ) , U \ [ ']  [ ] :

INTUITIONISTIC LOGIC

35

We want to show ' ) for all  . So let '. then by induction hypothesis, U  [ '] , and by   U  U . Therefore U \ [ '] = U  [ ] , i.e. . Conversely we have to show U \ [ ']  [ ] . since the U 's form a basis it suÆces to show U  U \ [ '] ! U  [ ] , but U  [ '] implies ', and hence , which in turn implies, U  [ ] . The quanti er cases are simple, we leave them to the reader.  COROLLARY 18. For the topological interpretation the completeness theorem holds, i.e. ` ' , '. Soundness is shown by a routine induction. Completeness follows from the completeness of the Beth semantics and Theorem 17.  Proof.

We have introduced a number of semantics each of which has certain drawbacks. For designing counterexamples and straightforward theoretical applications the Kripke semantics is the most convenient one. We will demonstrate this below in a few examples. EXAMPLE 19. The following, classically valid, sentences are not derivable. 1. ' _ :' (principle of the excluded middle, PEM) 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

::' ! ' (double negation principle) :(' ^ ) ! :' _ : (De Morgan's Law) :' _ ::' (' ! ) _ ( ! ') (Dummett's axiom) (::' ! ') ! ' _ :' (:' ! : ) ! ( ! ') (' ! ) ! :' _ :8x'(x) ! 9x:'(x) 8x::'(x) ! ::8x'(x) (double negation shift, DNS) 8x(' _ (x)) ! ' _ 8x (x) (constant domain axiom) (' ! 9x (x)) ! 9x(' ! (x)) (independence of premiss principle, IP)

13. (8x'(x) ! ) ! 9x('(x) ! ) 14. 8x('(x) _ :'(x)) ^ ::9x'(x) ! 9x'(x)

36

15. 16.

DIRK VAN DALEN

::8xy(x = y _ x 6= y) ::8xy(:x 6= y ! x 6= y)

Consider the following Kripke models (where the nodes are labelled with the forced atoms and forced formulas). Proof.

'

 ' a:





 ';

 c:

b:





1 and 2 are refuted by model a. 4 and 6 are refuted by model b. (forget about the ). 3 and 5 are refuted by model b. 7 is refuted by model c. 8 is refuted by model a (take := '). For the quanti ed sentences we need to indicate universes.

'(0) 0_ 1_ a.

'; (0) 0_ 1_

0_ 1_ 2_ 3_

b. 0_

0_

'; (1) 0_ 1_

'(1) 0_ 1_

c.

d. 0 0

0_

9 and 13 are refuted in model a. 10 is refuted in model e. 11 is refuted in model b.

(0)

0_ 1_ 2_ 0_ 1_ e.

0_

'(0) '(1) '(2) '(0) '(1) '(0)

INTUITIONISTIC LOGIC

f.

0_ 1_ 2_ 3_ 4_

i = j ; i; j  3

0_ 1_ 2_ 3_

i = j ; i; j  2

0_ 1_ 2_

i = j ; i; j  1

37

0_ 1_ 12 is refuted in model c. 14 is refuted in model d. 15 and 16 are refuted in model f. The identity relation satis es the obvious axioms of re exivity, symmetry, transitivity and compatibility with basic relations. Model f clearly satis es these axioms. Observe that we could have refuted 9, 11, 12, 13, 14 by the familiar reduction of a quanti ed statement to a proposition mimicking a nite domain. Sentence 10 is of a dierent ilk, we can even show that 10 is true in all nite Kripke models (i.e. with a nite tree). In a nite tree each node is dominated by an end (or top) node. Suppose that 8x::'(x) holds, then in an end node  we have  8x::'(x), i.e. 8a 2 D;  ::'(a). But, since  is an end node, this implies  '(a) hence  8x'(x). As a result we get 0 ::8x'(x) for the bottom node. As we will show in the next section, IPC is complete for nite Kripke models, so IQC essentially needs a wider class of partially ordered sets for its Kripke semantics.  Heyting algebras, the common generalization of the preceding semantics.

Boole's discovery of the algebraic nature of the logical laws and operations was repeated for the case of intuitionistic logic by McKinsey, Stone, Tarski and others. The resulting algebra has been called closure algebra, Brouwerian algebra, pseudo-Boolean algebra, but nowadays the term Heyting algebra is generally accepted.

38

DIRK VAN DALEN

There are various axiomatisations for the theory of Heyting algebras (cf. [Rasiowa and Sikorski, 1963; Johnstone, 1982]), we will use one that stays very close to the axioms of IPC. For the formulation it is convenient to use the notion of lattice. DEFINITION 20. hA; i is a lattice if it is a poset in which each pair of elements has a sup and an inf. We denote the sup and inf of x and y by x t y and x u y. by de nition u and t satisfy

x u y  x; y  x t y x; y  z ! x t y  z z  x; y ! z  x u y: We can alternatively obtain a lattice from a structure hA; t; ui satisfying

xty = ytx xuy = yux x t (y t z ) = (x t y) t z x u (y u z ) = (x u y) u z x u (x t y) = x x t (x u y) = x: We de ne the relation `' by x  y := x u y = x. It is a simple exercise to show that  de nes a lattice (cf. [Rasiowa and Sikorski, 1963, pp. 35,36]). A lattice with top > and bottom ? is a lattice with two elements > and satisfying ?  x  > for all x. Note that we can show x u y = x , x  y $ x t y = y, so the ordering can also be expressed by t. DEFINITION 21. A Heyting-algebra is a structure hA; u; t; ); >; ?i such that 1. it is a distributive lattice with respect to u; t and with top and bottom. 2. x u (x ) y) = x u y 3. (x ) y) u y = y 4. (x ) y) u (x ) z ) ) (x ) (y u z ) 5. 6.

?ux =? ? ) ? = >.

Any Boolean algebra obviously is a Heyting algebra. The paradigm of a Heyting algebra is O(X ), the set of opens of a topological space X , where U ) V is de ned as in Section 3: Int(U c [ V ).

INTUITIONISTIC LOGIC

39

We have the following key properties LEMMA 22. 1. x ) x = > 2. x u y  z $ x  y ) z .

We de ne the complement by x := x ) ?. The obvious connection with logic is via the Lindenbaum algebra of a theory. Consider some theory T in IPC, then

'  := T `IPC ' $ is a congruence relation, as one easily shows. On the equivalence classes we de ne a Heyting algebra, by putting

'=  u =  := (' ^ )=  '=  t =  := (' _ )=  '= ) =  := (' ! )=  ? := ?=  > := (? ! ?)=  : It is a routine matter to show that one thus obtains a Heyting algebra, the so-called Lindenbaum algebra of T . Examples of Heyting algebras

1. 0 1 2 3 ::: ::: !     ::: :::  Consider the set of natural numbers with a sup ! (i.e. the ordinal ! + 1) and de ne n u m := min(n; m); n t m := max(n; m),



if n > m n ) m := m ! if n  m; for n; m  ! ? := 0; > := !: The ordering is the natural one. In this Heyting algebra the excluded third fails:



? if n 6= 0 > else: For n 6= ?, > we get n t n = n t ? = n = 6 >. n=n)?=

40

DIRK VAN DALEN

2. From the diagram below we can read o the operations. The nontrivial one is the `implication' (relative complement).

>  c a

b  ?

The relation x  y ) z , x u y  z tells us that y ) z is the greatest element x such that x u y  z , so we can write down the table for ).

) ? a b c > ? > > > > > a b > b > > b a a > > > c ? a b > > > ? a b c > The rst column yields the negation. One can view the Heyting algebras as a suitable generalisation of the classical truth table. In this form Heyting algebras occur already in Heyting's paper of 1930. Truth tables also occur in [Jaskowski, 1936]. 3. The Rieger{Nishimura lattice [Nishimura, 1966] In the diagram below one of the two points immediately above the bottom, is the complement of the other. If we call the right hand one p, we can compute the remaining elements. We enumerate the points as indicated. We put

'0 := ? '1 := p '2 := p '2n+3 := '2n+1 t '2n+2 '2n+4 := '2n+2 ) '2n+1 : The operations on the lattice follow from its order. The Rieger{ Nishimura lattice is the free Heyting algebra with one generator, i.e. in logical terms it is the Lindenbaum algebra of IPC with just one atom.

INTUITIONISTIC LOGIC

41

>

11 10

9 7 5

p ) p6 pt p p

8

2 0

pt

3

4 1

p p

p

?

There are two things to be shown: (a) each proposition in p is one of the 'i 's; (b) the dependencies between the 'i 's are as shown in the diagram. (a) is shown by induction on '. We'll do one case. Let ' = ^ . By induction hypothesis ` $ 'i ; `  $ 'j for some i; j . If the elements i; j are comparable, then we immediately see that ' is a 'k . So the interesting cases are i = 2n + 1; j = 2n + 2 and i = 2n + 3; j = 2n + 4. In the rst case ` ' $ '2n 1 , in the second case ` ' $ '2n+1 . The proof of (b) is a matter of tedious bookkeeping. Given the dependencies between '0 ; '1 ; '2 , one checks the dependencies for higher 'n 's inductively. Consider for example '2n+3 and '2n+4 . 2n + 4

2n + 3



 

2n + 1



2n + 2

('2n+3 ! '2n+4 ) $ [('2n+1 _ '2+2 ) ! ('2n+2 ! '2n+1 )] $ '2n+2 ! '2n+1 :

42

DIRK VAN DALEN

So, from the induction hypothesis 6` '2n+2 ! '2n+1 , we obtain 6` '2n+3 ! '2n+4 , i.e. '2n+3= ) '2n+4= 6= >, i.e. '2n+3= 6 '2n+4= . An interpretation of IPC in a Heyting algebra A is given by a mapping h from the atoms into A. h is then extended to all propositions in the canonical way i.e. h(' ^ ) = h(') u h( ); h(' _ ) = h(') t h( ); h(' ! ) = h(') ) h( ); ' is true in A if for all interpretations h; h(') = >. A simple inductive proof shows the Soundness Theorem IPC

` ' ) ' is true in all Heyting algebras.

The converse also holds, for consider the Lindenbaum algebra of IPC and interpret each proposition canonically: h() = = , then IPC `  , h() = >. So ' is true in the Lindenbaum algebra. Hence we have the .

Completeness Theorem for Heyting Algebras

true in all Heyting algebras.

IPC

` ' , ' is

There is a simple connection between Kripke models and Heyting algebras. We can associate to a Kripke model a topological space as follows. The points of the space are the nodes of the poset; the opens are the sets U with the property  2 U ^   ) 2 U . As in the case of the topological model associated to a Beth model over a tree, the sets U = f j  g form a basis for this topology. For atoms we de ne [ '] = f j  'g(). One shows by induction on ' that () holds for all propositions (cf. also [Fitting, 1969, p.23]). Thus we have associated to each Kripke model an interpretation in the Heyting algebra of the opens of the associated topological space. Instead of considering Kripke or Beth models with a prescribed interpretation (forcing) of the atoms, we can also consider the underlying poset only. We then speak of a Kripke (Beth) frame. A frame is thus turned into a model by assigning structures to the nodes. There is an alternative formulation of Kripke (Beth, etc.) models, that sticks closer to the language. Instead of assigning classical structures to nodes, one can just as well assign sets of atoms to nodes, e.g. think of V () as the set of atomic sentences that are true in D(). So V is a function from M to the power set of the set of closed atoms, subject to the condition that   ) V ()  V ( ). Alternatively one can de ne a binary interpretation function i : At  M ! f0; 1g (where At is the set of closed atoms), such that   and i('; ) = 1 ) i('; ) = 1 (think of i('; ) = 1 as D()  ').

INTUITIONISTIC LOGIC

43

4.1 An External View of Kripke Models If one looks at a Kripke model from the outside, then it appears as a complicated concoction of classical structures, and hence as a classical structure itself. Such a structure has its own language and we can handle it by ordinary, classical, model-theoretical means. What is involved in this `master structure' of K? (i) the partially ordered set of nodes, (ii) the relations between these structures. We can simply describe this master structure K by a language, containing two sorts of individuals (or alternatively one sort, but two predicates N (x) and E (x), for `x is a node' and `x is an element'). Let us use ; ; ; : : : for the `node-sort' and x; y; z; : : : for the `element sort'. Then we add  to the original language, and replace each predicate symbol P by P  with one more argument than P and add a domain predicate symbol. The structure K validates the following laws (referred to by ):

 ^  !   ^  != 8 ~x(  ^ P  (; ~x) ! P  ( ; ~x)) 8 x(D(; x) ! D( ; x)) Now we can mimic the forcing clauses in the extended language. Consider the translation of  ' given by the inductive de nition: 1. ( P (~t)) := P  (; ~t) and ( ?) := ?. 2. ( ' ^ ) := ( ') ^ ( ) . 3. ( ' _ ) := ( ') _ ( ) . 4. ( ' ! ) := 8  (( ') ! ( ) ). 5. ( 9x'(x)) := 9x(D(; x) ^ ( '(x)) ).

6. ( 8x'(x)) := 8  8x(D( ; x) ! ( '(x)) ). It is obvious that: 1.  ' , K  ( ') 2. each model of  corresponds uniquely to a Kripke model. Now we can apply the full force of classical model theory to the models of  in order to obtain results about Kripke models. For example, one gets for free the ultraproduct theorem and the Hilbert{Bernays completeness theorem (consistent RE theories have
02 models, cf. [Kleene, 1952, Ch XIV]. Similar `translations' can be applied to Beth semantics or the general semantics (cf. [van Dalen, 1978] for an application to lawless sequences).

44

DIRK VAN DALEN

4.2 Model theory of intuitionistic logic in an intuitionistic setting If one is willing to give up the strong results of all the arti cial semantics (completeness, Skolem{Lowenheim, etc.), there is no reason why one should not practise model theory of intuitionistic theories as an ordinary part of intuitionistic mathematics. That is to say, to adopt an intuitionistic variant of the Tarskian semantics. A number of interesting results have been obtained for speci c theories and structures, e.g., the continuum and the irrationals are elementarily equivalent for the theories of equality, apartness and linear order. Note that even a seemingly trivial theory, such as that of equality, turns out to be highly complicated|in contrast to the classical case. Also, strong classical theorems cannot always be upheld in an intuitionistic setting. e.g. the existence of winning strategies for Ehrenfeucht{Frasse games implies elementary equivalence (cf. [van Dalen, 1993]), but the converse fails (cf. [Veldman and Waaldijk, 1996]). The last mentioned paper contains a wealth of interesting methos and results, it is recommended for getting acquainted with the eld. 5 SOME METALOGICAL PROPERTIES OF IPC AND IQC Intuitionistic logic is in a sense richer in metalogical properties than classical logic. There are common properties, such as completeness, compactness and deduction theorem, but soon the logics start to diverge. Classical logic has phenomena such as prenex normal forms, Skolem form, and Herbrand's theorem which are absent in intuitionistic logic. Intuitionistic logic on the other hand is more blessed with derived rules. The rst example is the .

Disjunction Property, DP

` ' _ ) ` ' or `

.

Clearly, the nature of is relevant, for if contains all instances of PEM, then DP is false, since in CPC ' _ :' is a tautology, but neither ', nor :' needs to be a tautology. A suÆcient condition on is that it exists of Harrop formulas, i.e. formulas without dangerous occurrences of _ or 9. To be precise, the class of Harrop formulas is inductively de ned by 1. ' 2 H for atomic '

2H !'^ 2H ' 2 H ) 8x' 2 H 2 H ) ' ! 2 H.

2. '; 3. 4.

INTUITIONISTIC LOGIC

45

2

1 0

THEOREM 23. The disjunction property holds for sets of Harrop formulas. For a proof using natural deduction, see [Prawitz, 1965, p. 55], [van Dalen, 1997, p. 209]. In Aczel [1968] a proof is given using a metamathematical device `Aczel's slash'. See also [Gabbay, 1981, Ch. 2, Section 3]. The intuitionistic reading of the disjunction property is: given a proof of '_ we can eectively nd a proof of ' or a proof of . The proof theoretical demonstrations of DP have this intuitionistic character, not however the model-theoretic proof below. The proof uses classical meta-theory, to be speci c, it uses reductio ad absurdum. To demonstrate the use of Kripke models, we give the proof for a simple case, = ;. Let ` ' _ and suppose 0 ' and 0 . Then there are Kripke models K1 and K2 with bottom node 1 and 2 such that 1 1 ' and 2 6 . We construct a new Kripke model K by taking the disjoint union of K1 and K2 and placing an extra node 0 at the bottom, see gure above. We stipulate that nothing is forced at 0 . Clearly, the result is a Kripke model. 0 ' _ , so 0 ' or 0 . If 0 ', then 1 '. Contradiction. And if 0 , then 2 . Contradiction. Hence we have ` ' or ` . For predicate logic we can also establish the Existence Property: ` 9x'(x) ) ` '(t) for a chosen term t, where consists of Harrop formulas (9x'(x) is closed). See [Prawitz, 1965; Aczel, 1968; Gabbay, 1981; van Dalen, 1997]. Since the only closed terms in our present approach are constants, we can replace the conclusion of EP by ` ` '(c) for a constant c'. In the case that there are no constants at all the conclusion is rather surprising: ` 8x'(x). Like its classical counterpart IPC is decidable; there are various proofs for this fact. In [Kleene, 1952, Section 80], [Troelstra and van Dalen, 1988, p. 541] and [Szabo, 1969, p. 103], a sequent calculus is used. The use of normal derivations in natural deduction likewise yields a decision procedure. In [Rasiowa, 1974, p. 266] decidability is derived from the completeness of IPC for nite Heyting algebras. We will use a similar argument based on Kripke models.

46

DIRK VAN DALEN

Our rst step is to reduce Kripke models for IPC to nite models, following [Smorynski, 1973]. We consider a Kripke model K with a tree as its underlying poset such that K 6` '; a suitable re ning will yield a `submodel' K , such that 1. K is nite 2.  

,

, for all subformulas of '.

Let S be the set of subformulas of ', and put S = f 2 S j g. We de ne a sequence of sets Kn : K0 = f0 g (0 is the bottom node of K). Let Kn be de ned, and 2 Kn . We consider sets fÆ1; : : : ; Æk g  K such that 1.  Æi

2. S 6= SÆi 3. the SÆ jumps only once between and Æi , i.e. SÆ = S or SÆ = SÆi for  Æ  Æi 4. SÆi = 6 SÆj for i 6= j .

Since there are only nitely many SÆ 's we can nd a maximal such set say f 10 ; : : : ; k0 g , if there are such Æ's at all. De ne K1 = f00;1; : : :S; 00;k g [ f0 g Kn+1 = Kn [ ff 20 ; : : : ; k0 g j 2 Kn Kn 1 g; n  1: As the S 's increase, and there are only nitely many subformulas, the sequence Kn stops eventually. Clearly each Kn is nite, hence K  = [Kn is nite. Claim: K  with its inherited  is the required nite submodel. Property (2) is shown by induction on . For atomic (2) holds by de nition. For _ and ^ the result follows immediately. Let us consider 1 ! 2 . Suppose that for 2 K  ; 6 1 ! 2 , then there is a  in K such that 1 and 6 2 . If 1 2 S we are done. Else we nd by our construction a Æ 2 K  with < Æ  such that 1 2 SÆ and 2 62 SÆ , hence 6  1 ! 2 . The converse is simple. We now may conclude. THEOREM 24. IPC is complete for nite Kripke models over trees. Proof.

By the above and Lemma 14.

As a consequence we get COROLLARY 25. IPC is decidable.



INTUITIONISTIC LOGIC

47

We can eectively enumerate all nite Kripke models over trees, and hence eectively enumerate all refutable propositions. By enumerating all proofs in IPC we also obtain an eective enumeration of all provable propositions. By performing these enumerations simultaneously we obtain an eective test for provability in IPC.  Proof.

Theorem 24 is also paraphrased as `IPC has the Finite Model Property (FMP)', i.e. IPC 6` ' ) ' is false in a nite model. The FMP is the key concept in our decidability proof. Note that the decision procedure of Corollary 25 is horribly ineÆcient. The procedures based on sequent calculus or natural deduction are much more practical. Corollary 25 can be considerably improved, in the sense that narrower classes of Kripke models can be indicated for which IPC is complete. Examples.

1.

is complete for the Jaskowski sequence Jn . The sequence Jn is de ned inductively. J1 is the one point tree. Jn+1 is obtained from Jn by taking n +1 disjoint copies Jn and adding an extra bottom node. IPC

 

J1

  

J2

  

  

 



J3 Cf. [Gabbay, 1981, p. 70 .]. The Jaskowski sequence is the Kripke model version of Jaskowski's original sequence of truth tables, [Jaskowski, 1936]. 2. IPC is complete for the full binary tree (cf. [Gabbay, 1981, p. 72]. Strictly speaking we have given classes of Kripke frames, where completeness with respect to a class K of frames means `completeness with respect to all Kripke models over frames from K'. During the early childhood of intuitionism and its logic it was put forward by some mathematicians that intuitionistic logic actually is a three-valued logic with values true, false, undecided. This proposal is wrong on two counts, it is philosophically wrong and by a result of Godel no nite truth table completely characterizes intuitionistic logic (see Section 5). Our comments on the failure of the double negation shift, DNS, (Section 3.11-10) have already made it clear that IQC is not complete for nite

48

DIRK VAN DALEN

Kripke frames. The usual re nement of the completeness proof tells us that (for a countable language) IQC is complete for countable Kripke models over trees. Intuitionistic predicate calculus diers in a number of ways from its classical counterpart. Although both IQC and CQC are undecidable, monadic IQC is undecidable (Kripke) (cf. [Gabbay, 1981, p. 234]), whereas the monadic fragment of CQC is decidable (Behmann). Another remarkable result is the decidability of the prenex fragment of IQC, which implies that not every formula has a prenex normal form to which it is equivalent in IQC. We will consider the class of prenex formulas below. LEMMA 26. IQC ` 9y'(x1 ; : : : ; xn ; y) ) IQC ` 8x1 ; : : : ; xn '(x1 ; : : : ; xn ; t), where all variables in ' are shown, and where t is either a constant or one of the variables x1 ; : : : ; xn . Add new constants a1 ; : : : ; an , then IQC ` 9y'(a1 ; : : : ; an ; y) and apply EP.  Proof.

We now get the following intuitionistic version of the Herbrand Theorem. THEOREM 27. Let Q1 x1 ; : : : ; Qn xn ' be a prenex sentence, then IQC ` Q1 x2 ; : : : ; Qn xn ' i IPC ` '0 , where '0 is obtained form ' by replacing the universally quanti ed variables by distinct new constants, and the existentially quanti ed variables by suitable old or new constants. Proof.

Induction on n. Use EP and Lemma 26.



As a corollary of Theorem 27 and Corollary 25 we get THEOREM 28. The prenex fragment of IQC is decidable. and COROLLARY 29. There is not for every ' a prenex such that IQC ` '$ . Among the properties that classical and intuitionistic logic share is the so- called THEOREM 30 (Interpolation Theorem). If IQC ` ' ! , then there exists a , called an interpolant of ' ! , such that 1.

IQC IQC

`'! `!

and

2. all non-logical symbols in  occur in ' and in .

The interpolation theorem was established by proof theoretical means by [Schutte, 1962] and [Prawitz, 1965]. Gabbay [1971] proved the theorem by

INTUITIONISTIC LOGIC

49

model theory, he also established a suitable form of Robinson's consistency theorem. For proofs and re nements the reader is referred to [Gabbay, 1981, Chapter 8], and [Troelstra and Schwichtenberg, 1996, x4.3], whereas in CPC the interpolation theorem holds in all fragments. Zucker has shown this not to be the case for IPC (cf. [Renardel de Lavalette, 1981]).

5.1 Independence of the Propositional Connectives Whereas in classical logic the propositional connectives are interde nable, this is not the case in IPC, a fact already known to McKinsey [1939]. There are a number of ways to show the independence of the intuitionistic connectives. A proof theoretical argument, based on the normal form theorem, is given by [Prawitz, 1965, p. 59 ]. We will use some ad hoc considerations. 1. The independence of _ from !; ^; :; ? is clear, since !; ^; :; ? are preserved under the double negation translation (up to provable equivalence), but _ is not. 2. 3. 4.

: is independent from _; !; ^ already in CPC, so let alone in IPC. ! is independent from ^; _; :. We use the simple fact that for !-free ', ` (p ! q) ! ' )` (p ! ::q) ! '. De nability of ! would yield ` (p ! ::q) ! (p ! q). ^ is independent of _; !; :; ?. Consider the Kripke model p; q



 p

 q

A simple inductive argument shows that the ^-free formulas are either equivalent to ? or are forced in at least one of the lower nodes. Although even the traditional de nability result fail in intuitionistic logic, there is a completeness of the sets f!; ^; _; ?g for IPC or f!; ^; _; ?; =;9; 8g for IQC under special assumptions. Zucker and Tragesser [1978] showed that logical constants, given by Natural Deduction rules are de nable in the above sets. A similar result is to be found in [Prawitz, 1979]. In view of the incompleteness of the intuitionistic connectives there have been a number of de nitions of new connectives, e.g. by model theoretic means (cf. Gabbay [1977; 1981, p. 130 ], Goad [1978] and de Jongh [1980]). Kreisel introduced the connective  by a second-order propositional condition: (' := 9 (' $ : _ :: ). Matters of de nability, etc. of  have been extensively investigated in [Troelstra, 1980].

50

DIRK VAN DALEN

a=c b=d a_ b_ c_ d_

a=b a_ b_ c_

a_ 

b_ c_

d_

d_

Figure 1.

5.2 The Addition of Skolem Functions is not Conservative It is a fact of classical logic that the extension of a theory by Skolem functions does not essentially strengthen T (Vol 1, p. 89), i.e. (a simple case) if T ` 8x9y'(x; y) then we may form T S by adding a function symbol f and the axiom 8x'(x; f (x)) and T S is conservative over T : if T S `  where  does not contain f , then T ` . In general this is not true in intuitionistic logic [Minc, 1966]. We will show this by means of a simple counter example of Smorynski [1978]. Consider the theory T of equality EQ plus the extra axiom 8x9y(x 6= y), and its Skolem extension T S = EQ + 8x(x 6= f (x)) ^ 8xy(x = y ! f (x) = f (y)), then T S is not conservative over T . It suÆces to nd a statement  in the language of EQ such that T S `  and T 6` . We take  := 8x1 9y1 8x29y2 [x1 6= y1 ^ x2 6= y2 ^ (x1 = x2 ! y1 = y2)]. Clearly T S ` . The Kripke model of gure 1 establishes T 6` . Clearly  8x9y(x 6= y). Now suppose  . Take a; b for x1 ; x2 then we must take d; c for y1 ; y2 (in that order). However  6 a = b ! c = d. The equality fragment of T S is axiomatised in [Smorynski, 1978].

5.3 Fragments of

IPC

The situation in intuitionistic logic radically changes if one leaves out some connectives. We mention the following result: (Diego, McKay) there are only nitely many non-equivalent propositions built from nitely many atoms in the _-free fragment (cf. [Gabbay, 1981, p. 80]).

5.4 Some Remarks on Completeness and Intuitionistically Acceptable Semantics This section uses notions of later sections, in particular Section 9. the reader is suggested to consult those sections.

INTUITIONISTIC LOGIC

51

As we have argued in Section 1, an interpretation of the logical constants based on intuitionistic principles must somehow exploit the notion of construction. This has been proposed by Heyting, and extended by Kreisel. It has not (so far), however, led to a exible semantics that provided logic with completeness. The more successful semantics have provided completeness theorems, but at the price of importing classical metamathematics. This is a matter of considerable philosophical interest. As Intuitionism is a legitimate, well-motivated philosophy, it should at least have a semantics for its logic that stands up to the criteria of the underlying philosophy; unless one adopts Brouwer's radical view that `mathematics is an essentially languageless activity'. The traditional semantics lend themselves perfectly well to an intuitionistic formulation. One has to select among the various classically equivalent formulations the intuitionistically correct one (e.g. in the topological interpretation [ ' ! ] = Intfx j x 2 [ '] ! x 2 [ ] g and not Int(([['] c [ [ ] )). Soundness does not present problems, so independence results can usually be obtained by Intuitionisitc means. For the more sophisticated applications of semantics one usually needs completeness, and the original completeness proofs relied heavily on classical logic. For propositional logic the problem is relatively simple. The rst positive result was provided by Kreisel, who in [Kreisel, 1958] interpreted IPC by means of lawless sequences, and showed by intuitionistic means IPC to be complete for this particular interpretation. The basic idea is to relate Beth models (which are special cases of topological models) to lawless sequences, considered as paths through the underlying trees; one assigns sets of lawless sequences to propositions, ' 7! [ '] , cf. Theorem 17, such that the logical operations correspond to the Heyting algebra operations. Since one can restrict oneself to nitely branching trees in this context, one can show completeness for the topological space of lawless sequences using only the simple properties of lawless sequences (including the fan theorem). Kripke [1965] indicates a similar procedure on the basis of Kripke models. A more serious matter is the completeness of predicate calculus. The plausible approach, i.e. to interpret `validity' as `validity in structure a la Tarski', called internal validity by Dummett [1977, p. 215], led to an unexpected obstacle. Kreisel [1962], following Godel, established the following result: if IQC is complete for internal validity, then 8 ::9x'(; x) ! 8 9x'(; x) holds for all primitive recursive predicates '. So validity of the above kind would give us Markov's `Principle (cf. Section 6.5.3), a patently non-intuitionistic principle. It does not do any good to consider Beth semantics, for one can obtain the same fact for validity in all Beth models [Dyson and Kreisel, 1961]. Even worse, under the assumption of Church's Thesis (i.e. all functions from N ! N are recursive, cf. Chapter 4 of Vol. 1 of this Handbook) IQC is incomplete in the sense that the set

52

DIRK VAN DALEN

of valid formulae is not recursively enumerable, as established by [Kreisel, 1970] (cf. [van Dalen, 1973; Leivant, 1976]). The strongest result so far is McCarty's theorem; constructive validity is nonarithmetic, [McCarty, 1988]. This bleak situation in semantics for IQC changed when Veldman in 1974 introduced a technical device that allowed for a modi ed Kripke (and similarly, Beth) semantics for which the completeness of IQC can be established in an intuitionistically acceptable manner. Although Veldman's proposal can be implemented in more than one way, its main feature is relaxation of the forcing conditions for atoms:  ? is in general allowed. For these more general models intuitionistic completeness proofs have been give for the Kripke version by [Veldman, 1976], and for the Beth version by [Swart, 1976]. Extensive discussions of the aspects of intuitionistic completeness of IQC are to be found in [Dummett, 1977] and [Troelstra, 1977]. H. Friedman [1977; 1977a] has sketched intuitionistically correct completeness proofs for MQC and the ? (and :)-free part of IQC. The details of a slightly upgraded version can be found in [Troelstra and van Dalen, 1988, x13.2], there the result is cast in the form of a universal Beth model:

1. There is a Beth model M such that M ' , IQC ` ' for all ?-free formulas '. 2. There is a Beth model MQC ` ' for all '.

M for minimal logic such that M ' ,

3. there is a modi ed Beth model for all '.

M such that M ' , IQC ` '

5.5 The Intuitionistic View of Non-intuitionistic Model Theoretic Methods It should not come as a surprise that for intuitionists such semantical proofs as employed, e.g. in the case of DP (cf. Theorem 23) do not carry much weight. After all, one wants to extract a proof of either ' or from a proof of ' _ , and the gluing proof doe not provide means for doing so. There is however a roundabout way of having one's cake and eating it. For example, in the case of the proof of DP one shows classically that `' has no proof in IQC' ` has no proof in IQC' then `' _ has no proof in IQC', and hence (classically ) IQC ` ' _ ) IQC ` ' _ IQC ` . One formalizes this statement in Peano's Arithmetic, so PA

` 9xPrIQC (x; p' _ q) ! 9yPrIQC (y; p'q) _ 9z PrIQC (z; p q)

or PA

` 8x9yz (PrIQC (x; p' _ q) ! PrIQC (u; p'q) _ PrIQC (z; p q)):

INTUITIONISTIC LOGIC

53

Now one uses the fact that PA is conservative over HA for 02 statements, ). so that HA ` 8x9yz ( This shows that DP is intuitionistically correct. In [Smorynski, 1982] problems of this kind is considered in a more general setting. Of course, one might wonder why go through all this rigmarole when direct proofs (e.g. via natural deduction, or slash operations) are available. A matter of taste maybe. 6 INTERMEDIATE LOGICS By adding the principle of the excluded middle to IPC we obtain full classical propositional logic. It is a natural question what logics one gets by adding other principles. We will consider extensions of IPC by schemas, e.g. IPC + (' ! ) _ ( ! '). First we remark that all such extensions are subsystems of CPC, for let T be such an extension and suppose that T 6 CPC, then there is a ' such that T ` ' (and hence all substitution instances) and ' is not a tautology. but then we nd by substituting, say p0 ^ :p0 and p0 ! p0 for suitable atoms of ' an instance '0 which is false. therefore CPC ` :'0 and, by Glivenko's theorem (Corollary 51) 0 0 IPC ` :' . This contradicts T ` ' . So there are only logics between IPC and CPC to consider. The study of intermediate logics is mainly a matter for pure technical logic, dealing with completeness, nite model property, etc. There are however certain intermediate logics that occur more or less naturally in real life (e.g. in the context of Godel's Dialectica interpretation, or of realizability), so that their study is not merely l'art pour l'art. One such instance is Dummett's logic LC, which turns up in the provability logic of Heyting's arithmetic (cf. [Visser, 1982]). One of the most popular topics in intermediate logic was the investigation of classes of semantics for which various logics are complete. Furthermore there is the problem to determine the structure of the family of all intermediate logics under inclusion. The eld has extensively been studied and an even moderately complete treatment is outside the scope of this chapter. the reader is referred to [Rautenberg, 1979] and [Gabbay, 1981].

6.1 Dummett's Logic

LC

DEFINITION. LC = IPC + (' ! ) _ ( Theorem.

LC

! ').

is complete for linearly ordered Kripke models.

One direction is simple, one just checks that (' ! ) _ ( ! ') holds in all linearly ordered Kripke models. For the converse, consider the model,

54

DIRK VAN DALEN

obtained in the Model Existence Lemma 14, consisting of prime theories, ordered by inclusion. The bottom node 0 forces all instances of the schema (' ! ) _ ( ! '). Consider 1 ; 2 with ' 2 1 2 for some '. We will show that 2  1 . Let 2 2 . Since 0 ' ! or 0 ! ' and 0  i (i = 1; 2) we have 2 1 or ' 2 2 . As the latter is ruled out we nd 2 1 . Hence for any two 1 ; 2 , we have 1  2 or 2  1 . This establishes the semantic characterisation of LC.

6.2 Filtration and Minimalisation Some models are needlessly complicated because some of their nodes are in a sense redundant. A simple case is a model with two nodes  < , which force exactly the same formulas. The idea to collapse nodes that force the same formulas presents itself naturally. Scott and Lemmon introduced such a procedure in modal logic under the name of ltration [Lemmon and Scott, 1966], and Smorynski did something similar in intuitionistic logic under the name of minimalisation [Smorynski, 1973; Segerberg, 1968]. Let a Kripke model K = hK; ; i be given. We consider forcing on K for a class of formulas closed under subformulas. For  2 K de ne [] := f' 2 j  'g. Put K = f[] j  2 K g; []  [ ] if []  [ ] and [] ' if ' 2 [] for atomic '. Observe that the mapping  ! [] is a homomorphism of posets. Obviously K = hK ;  i is a Kripke model. THEOREM 31. [] ' ,  ' for ' 2 . Induction on '. The only non-trivial case is the implication. (i)  6 ' ! , 9   ' and 6 , (induction hypothesis) , 9  ([ ] ' and [ ] 6 ). Since   implies [ ]  [] , we have  6 ' ! . (ii)  ' ! . Let []  [ ] and [ ] '. By induction hypothesis ' and hence ' 2 [ ] . But ' ! 2 []  [ ] , so and again by induction hypothesis [ ] . This shows [] ' ! .  Proof.

Observe that this procedure does not preserve all desirable properties, e.g. being a tree. EXAMPLE.

';

';

Æ

Æ

Æ

'

Æ

Æ

';

!

Æ

'

Æ

Æ Æ

INTUITIONISTIC LOGIC

55

Gabbay has re ned the notion of ltration in order to obtain models with special properties. For this selective ltration cf. [Gabbay, 1981, p. 87 .].

6.3 The Finite Model Property, FMP An intermediate logic is said to have the Finite Model Property if it is complete for a class of nite models. We have already seen the importance of the FMP for logic: if T is eectively axiomatised (RE will do) and has the FMP, then T is decidable [Harrop, 1958]. The following facts may be helpful in establishing the FMP in some cases. THEOREM 32 (Smorynski [1973]). 1. Let T be complete for a class of Kripke models with posets characterised by positive sentences in a language extended by individual constants, then T has the FMP. 2. as (1) but with universal sentences and nitely many constants. Proof.

1. Let 0 6 ' for 0 bottom node of K. Apply ltration to K and call the result K0 . K0 is a homomorphic image of K and since positive sentences are preserved under homomorphic images (a simple fact of model theory), K0 belongs to the given class. Since we only have to consider subformulas of '; K0 evidently is nite. 2. Use the fact that universal sentences are preserved under substructures (cf. [van Dalen, 1997, p. 141, ex. 3]) and apply the construction given in the proof of Theorem 24.  COROLLARY 33. LC has the FMP and is decidable.

6.4 The `Bounded Height' models A Kripke frame (model) is said to have height n if the maximum length of its chains is n. If the length of the chains is unbounded, we say that the height is !. Can we nd an intermediate logic such that it is complete for all frames of height at most n? We de ne a sequence of propositions 'i by: '1 := p1 _ :p1 'n+1 := pn+1 _ (pn+1 ! 'n ); where pn is the nth atom. Let BHn = IPC + 'n , where we take 'n to be a schema (i.e. we add all substitution instances of 'n to IPC). THEOREM 34. BHn is complete for all Kripke frames of height  n.

56

DIRK VAN DALEN

Suppose that K has height  n and for some 0 2 K; 0 6 'n . So 0 6 n ! 'n 1 ; 0 6 n for some n . by de nition of forcing we nd an 1 > 0 such that 1 n an 1 6 'n 1 . By iterating this step we nd an increasing sequence 0 < 1 < : : : < n . This contradicts the condition on the heights, so K BHn . Conversely, we have to show that if BHn 6  then there exists a model of height  n which falsi es . So let K be a Kripke model of BHn and not of . We obtain K0 from it by ltration. It remains to show that K0 has height  n. Suppose K0 has a chain 0 < 1 < : : : < n . Since K0 is ltrated we can nd atoms pi (i = 1; : : : ; n), such that n i+1 pi and n i 6 pi . So j pi if and only if j > n i. Claim: n i 6 'i . We show this by induction on i. By de nition n 1 6 '1 . n i 1 'i+1 , n i 1 pi+1 _ (pi+1 ! 'i ). Now n i 1 6 pi+1 , and n 1 pi+1 but n 1 6 'i , by induction hypothesis; so n i 1 6 'i+1 . We now may apply the induction principle: 0 6 'n . Contradiction. So K0 has height  n.  Proof.

COROLLARY 35. Proof.

BHn

has the FMP and is decidable.

The posets of height  n are axiomatised by

8x0 : : : xn

n^1 ^ i=0

xi  xi+1 !

n_1 _ i=0

!

xi = xi+1 :

Apply Theorem 32.



It is obvious that BH! coincides with IPC, so only the nite BHn 's are relevant here for us. Another approach to the bounded height logics is via a sequence of generalisations of Peirce's law: 1 = ((p1 ! p0 ) ! p1 ) ! p1 n+1 = ((pn+1 ! n ) ! pn+1 ) ! pn+1 : Put LPn = IPC + n ; LP! = IPC. Ono [1972] and Smorynski [1973] showed that BHn = LPn . The notion of nth slice was introduced by [Hosoi, 1967] to capture logic of exact height n : Sn is the class of logics that are complete for models of height n, but not for models of smaller height.

6.5 Cardinality Conditions Consider the statement

Cn :=

W p $p ; i j

0i
INTUITIONISTIC LOGIC

57

if Cn holds in a model with bottom node then two atoms must be forced on exactly the same set of nodes. So if we have n modes, then there are 2n subsets and hence C2n holds in all models of  n elements. This bound is in general too crude. Let us therefore specialise the class of models to linearly ordered frames. Put Sn = LC + Cn+1 (as a schema). THEOREM 36. Sn is complete for all linear models with n nodes. If the model has n nodes then by a simple inspection one sees that Cn+1 holds. Conversely, if a model of K of Cn+1 , obtained by ltration, has at least k + 1 nodes 0 < 1 < : : : < n , then by ltration we can nd '1 ; : : : ; 'n such that i 6 'i+1 and i+1 'i+1 . Putting '0 := p ! p and 'n+1 := p ^ :p, we obtain an instance of Cn+1 that is not forced by 0 . This shows that Sn is complete for models with linear poset of length  n. But, since we can always add some nodes for free, it is also complete for all linear models with exactly n nodes. Converting the linear, nite Kripke frames into truth tables one obtains Godel's many-valued logic, used to establish the fact that IPC is not a ( nite) many-valued logic [1932].  Proof.

6.6 Some More Intermediate Logics A number of intermediate logics have found their way into the literature. We will mention some of them, with their main properties.

KC

is axiomatised by :' _ ::'.

1.

KC

is strongly complete for the class of directed Kripke frames. (A poset is direct if it satis es 8 9 (  ^  ).)

2.

KC

is strongly complete for the class of Kripke frames with a maximum.

For proofs see [Gabbay, 1981, p. 66 .], [Smorynski, 1973]. KC can alternatively be axiomatised by (:' ! : ) _ (: ! :'). This shows that LC is an extension of KC. The Kreisel{Putnam system KP is axiomatised by (: ! ' _ ) ! [(: ! ') _ (: ! )] [Kreisel and Putnam, 1957]. KP has the DP and FMP, and Gabbay has shown it to be complete for the class of Kripke models satisfying the condition # below [Gabbay, 1970].

58

DIRK VAN DALEN

For a poset hK; i with bottom element we de ne for a subset E of K :

E + = fp j 9q 2 E (q  p)g; E = fp j 9q 2 E (p  q)g. #

For every E element.

P

the set P

(E + ) is either empty or has a rst

For a proof see [Gabbay, 1981, p. 96 .]. Notions of width

The width of a frame can be conceived in various ways. One can look for the length of maximal anti-chains, or the maximal number of successors of individual nodes (say in nite frames). We will consider some notions below. (a) Anti-chain width. A Kripke frame K has a.c. width n if it has an anti-chain length n, but no anti-chain of length n + 1. De ne

'n =

0 n _ _ @ i=0

1 _ _ A pi ! pj and BAn = IPC + 'n (as a schema): j 6=i

THEOREM. [Smorynski, 1973] frames of a.c. width at most n.

BAn

is strongly complete for the class of

The proof is routine, use ltration. COROLLARY. BAn has the FMP and is decidable. (b) Top-width. In nite frames one can just count the number of top nodes, this gives a maximum width for trees, but not for posets in general. De ne the top-width of a frame as the number of maximal nodes and put

! n W W Æn := :pi ! :pj ; i=0

n

:=

j 6=i

! V :(:p ^ :p ) ! Æ i j n

0i
and BTWn := IPC + n (as a schema). THEOREM. [Smorynski, 1973] BTWn is complete for the class of all frames of top-width at most n. COROLLARY. BTWn has the FMP and is decidable.

INTUITIONISTIC LOGIC

59

(c) Local width [Gabbay and de Jongh, 1974]. We consider nite trees and de ne the local width of a tree frame as the maximum number of successors of its nodes. De ne

n :=

"

nV +1 i=0

pi !

W

i6=j

pj

!

W

! pj i6=j

!#

n+1 ! W pi i=0

and LWn = IPC + n (as a schema). Theorem. LW n is complete for tree frames of local width at most n. Corollary. LW n has the FMP and is decidable. Furthermore one can show that 1. 2. 3.

LW n

has the DP (use the gluing trick)

\LWn = IPC LW n = 6 LWn+1 .

For proofs see [Gabbay, 1981, p. 83 .].

6.7 The Lattice of Intermediate Logics Intermediate logics constitute a poset under the natural order of inclusion. Let us agree to consider intermediate logics as being given by axiom schemata. Observe that any such consistent extension of IPC is a subsystem of CPC. Hence we can safely form the meet and join of intermediate logics as follows: let Ti be axiomatised by the schemas 'ij (j  0), then T1 u T2(T1 t T2 ) are axiomatised by 'j1 _ 'k2 ('j1 ^ 'k2 ). It immediately follows that the intermediate logics constitute a distributive lattice. This lattice has extensively been investigated. We have already met some properties: e.g. there is a descending sequence of logics with intersection IPC (Section 5.6.3 (c)). Further properties are: There are 2@0 many intermediate logics [Jankov, 1968]. There are intermediate logics, that are not nitely axiomatisable. There exists a sequence of formulas 'i such that the logics TA axiomatised by f'i j i 2 Ag for A  !, satisfy TA = TB , A = B . Such a string is called strongly independent (cf. [Gabbay, 1981, p. 73 .]). There are intermediate logics without the FMP (cf. [Gabbay, 1981, p. 103 ]). There exists a strictly increasing chain of intermediate logics [Jankov, 1968; Fine, 1970]. There are exactly eight intermediate logics with the interpolation theorem [Maximova, 1977]. For more information cf. [Rautenberg, 1979, p. 288 ].

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6.8 Extensions of

IQC

The study of intermediate predicate logics has not yet made advances comparable to those in propositional logic. We refer the reader to, e.g. [Ono, 1973]. The best known extension of IQC is the logic of constant domains introduced by [Grzegorczyk, 1964], and axiomatised and shown complete by [Gornemann, 1971]. Put CD = IQC + 8x(' _ (x)) ! (' _ 8x (x)). Theorem. [G ornemann] CD is complete for the class of Kripke models with constant domain. Proof.

See [Gabbay, 1981, p. 50 ].



has somewhat unpleasant features as it is not closed under relativisation, i.e. if  (x) is some suitable formula then we may have CD `  but  (x) , where   (x) is the sentence obtained by relativizing all quantiCD 6`  ers. The reason being that although the domain is xed, predicates need not be constant in Kripke models for CD. The dierence between IQC and CD disappears when we restrict ourselves to formula without 8 [Fitting, 1969] or without _ and 9 [Gabbay, 1981]. Another noteworthy principle is the double negation shift DNS: CD

8x::'(x) ! ::8x'(x): Put MH := IQC + DNS. MH turns out to be complete for Kripke models with the property that each node is below some maximal node [Gabbay, 1981, p. 57 ]. Keeping the proof of Glivenko's theorem in mind, it is not surprising that it holds for MH. Actually MH is the smallest such extension of IQC [Gabbay, 1981, p. 14]. To nish this section, let us mention a rather dierent enterprise. Ono and Komori [1985] studied intuitionistic propositional calculus in the Gentzen sequent formalisation, without the contraction rule. They generalise Kripke models to monoid-Kripke models and establish the completeness theorem. Ono [1985] extends this study to predicate calculus. The study of logics without structural rules is the subject of Girard's linear logic, cf. [Girard et al., 1989]. 7 FIRST-ORDER THEORIES A number of basic notions of intuitionistic mathematics can faithfully be studied in the framework of intuitionistic rst-order logic. Although the

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61

situation is similar to that in classical logic, there is one disturbing aspect peculiar to intuitionistic logic (or rather its semantics): the absence of natural (or standard) models. Let us compare the state of aairs with classical logic. For theories in CQC we have not only the traditional notion of model, as presented by Tarski (cf. [van Dalen, 1997]), but also the notion of Boolean-valued model [Rasiowa and Sikorski, 1963]. The reader who is not familiar with the theory of Boolean valued models, may think of a topological model over a discrete space, i.e. with O(X ) = P (X ). Or he may think of Heyting algebras with the extra condition x = x for all x (or x t x = T ). The truth values [ '] are simply elements of a Boolean algebra B . There is among the Boolean algebras a canonical one that is contained in all Boolean algebras, the two-element algebra 2 = f0; 1g, with operations given by the traditional truth tables. Now there is for each Boolean-valued model a (truth preserving homomorphism onto a Boolean-valued model over 2, i.e. an ordinary model. Hence truth in all Boolean-valued models is equivalent to truth in all ordinary models. So the notion of truth according to ordinary model theory coincides with that of Boolean-valued model theory (cf. [Rasiowa and Sikorski, 1963, p. 295]). The ordinary models can thus be considered as the real (or standard) models among the Boolean-valued ones. This relation does not exist for intuitionistic semantics (say Heytingvalued, to take the most general one). For although 2 is contained in (and can be obtained as homomorphic image of) all Heyting-algebras, truth in all Heyting value model is certainly not the same as truth in all 2-valued (i.e. classical) models. The fact that for intuitionistic rst-order theories there does not exist a canonical model notion in the various semantics that we have exhibited is one that we have to accept, unpleasant as it may be. Philosophically speaking there are two ways to open to use| (1) look for a codi cation of the Brouwer{Heyting{Kreisel notion of proof- interpretation, (2) give up the notion of `standard' truth, or intended model. We will discuss the problem of semantics later, but not without pointing out that the absence of a standard model for arithmetic in any of the semantics introduced earlier, is rather embarrassing (see below). We will discuss a number of basic rst-order theories below. The most fundamental is the theory of equality.

7.1 The Theory of Equality

EQ

The axioms for EQ are the familiar ones: the universal closures of 1. x = x 2. x = y ! y = x

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DIRK VAN DALEN

3. x = y ^ y = z ! x = z (whenever we list axioms we will tacitly presuppose the clause `the universal closure of'). As in classical logic one shows by induction on ' x = y ! ('(x) ! '(y)). Let us consider a Kripke model for EQ. We will denote the binary relation that interprets = in  by  . One easily sees that  is an equivalence relation in (the domain of) each node. In general, however,  is not the identity. For, suppose that  were the identity in each node, then if for a; b 2  a 6 b, we see that for all  ; a 6 b in . Hence  a = b _ a 6= b. Conversely, if K 8xy(x = y _ x 6= y), then we can construct from K a Kripke model K0 with  the identity in each node. For, since  is an equivalence relation we can form equivalence classes [a] for each a in . De ne  0 a = b := [a] = [b] (, a  b). Claim:  ' ,  0 ' for all  2 K . Induction on '. The de nition of 0 takes care of the atomic '. ^ and _ are immediate,  6 ' ! , 9  ; ' and 6 , (induction hypothesis) 9  ; 0 ' and 6 0 ,  6 0 ' ! .  8x'(x) , 8a 2 ;  '(a) , 8[a] 2 ;  0 '(a) ,  0 8x'(x) (where a is the name of a in K and of [a] in K0 ). A similar argument handles 9. A slight boost of the argument yields the same result for arbitrary languages. Summing up: IQC with decidable equality is complete for normal Kripke models (i.e. with = interpreted by real equality). The theory EQ is of interest since the basic theories of real life depend heavily on equality. In general T c will denote the classical theory T + ' _ :'; T d will denote the theory T + decidability for atoms. We will keep superscripts for `logical' and pre xes for `mathematical' variants of theories. The following facts have been proved: d = EQc EQ Proof.

decidable [Lifschitz, 1969] + [Lifschitz, 1969] [Lifschitz, 1969], EQ where EQs is the theory of stable inequality: 8xy(::x = y ! x = y). d s EQ EQ

7.2 The Theory of Apartness,



AP

For practical purposes one needs in intuitionistic mathematics a strong inequality relation. For example, in the theory of the reals one needs a prop-

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erty like `x has a positive distance to 0 (9k(j x j> 2 k ), to make sure that x has an inverse. A mere inequality would not do. The positive inequality relation was introduced by Brouwer in 1918 and axiomatized by Heyting. Notation x#y read x is apart from y. AP has the axioms of EQ plus the following ones:

:x#y $ x = y x#y ! x#z _ y#z: One easily derives the following: FACT 37. The following are derivable in AP. x#y ! y#x x#y ! x 6= y ::x = y ! x = y: In particular AP has a stable equality. Most theories that occur in basic mathematics have an apartness relation. Combinatory logic, however, does not allow an apartness relation since its equality is not stable. A theory with decidable equality trivially has an apartness relation, namely the inequality. One and the same structure may, however, carry more apartness relations. EXAMPLE 0.

0

1.

1#0 0#1 0=0 1=1

0

1

1#0 0#1 0=0 1=1

1

0=0 1=1 0#1 1#0

0

1

0=0 1=1

The above models carry the same, decidable, equality, but distinct apartness relation. The apartness relation in uences the equality relation, the question is does it stop at the stability axiom or does it carry stronger conditions? The answer is provided in [van Dalen and Statman, 1979] where the axiomatisation of the equality fragment of AP is studied. Consider the following sequence of inequalities = 6 n x 6=0 y := :x = y x 6=n+1 y := 8z (x 6=n z _ y 6=n z ):

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DIRK VAN DALEN

For each n we formulate a stability axiom

Sn := :x 6=n y ! x = y: FACT.

` Sn ! Sm ` Sn AP ` x#y ! x 6=n y EQ

AP

for n > m for all n for all n:

Consider the !-stable theory of equality EQs! = EQ + fSn j n 2 !g s

EQ!

turns out to be the equality fragment of AP:

THEOREM. is conservative over EQs! .

1.

AP

2.

EQ!

s

is not nitely axiomatisable.

Van Dalen and Statman proved the theorem by means of a normal form theorem for AP. There is however a short and elegant proof by Smorynski using model theory [Smorynski, 1977], that we will reproduce here. s Proof of 1. Suppose EQ! 6` '. Consider the Kripke model K obtained in the model existence lemma. De ne a#b := a 6=n b for all n. Claim # is an apartness relation. We will only consider 8xy(:x#y ! x = y) (everything else is trivial). Suppose the bottom node, 0 , does not force it, then 1 :a#b and s 1 6 a = b for some 1 . Since K is a model of EQ! we have 1 6 :a 6=n b for all n. Now 1 [fa 6=n b j n 2 !g is consistent, for else 1 [fa 6=n b j n 2 !g ` ? and hence 1 ` :a 6=m b, i.e. 1 :a 6=m b, for some m. Therefore there exists a prime theory
 1 with a 6=n b 2
for all n, so
a#b. Contradiction. Hence 0 8xy(:x#y ! x = y). 2. is shown by constructing suitable Kripke models  As a corollary we obtain the unde nability of # in terms of =. For, if AP ` x#y $ '(x; y ) for a suitable equality formula ', then we would have a nite axiomatisation of EQs! (note that the above example also establishes the same fact). Observe that we could accept the apartness relation as basic and de ne equality by x = y := :x#y if we replace :x#y $ x = y by :x#x and x#y ! y#x.

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65

Further facts: AP is undecidable [Smorynski, 1973a], [Gabbay, 1981, p. 258]. There are Kripke models of EQs! that do not carry any apartness relation at all [van Dalen and Statman, 1979]. For a treatment of apartness in a sequent calculus setting, see [Negri and von Plato, 2001].

7.3 The Theory of Order,

LO

In classical logic linear order is singled out from the partial orders by requiring any two elements to be comparable, i.e. x < y _ x = y _ y < x. This axiom would be excessively strong in an intuitionistic context, since not even the reals would be ordered. Therefore Heyting proposed another axiom, that we shall adopt. The language of LO contains the predicate symbols < and =. The axioms of LO are those of EQ, plus

x
+ 8xy(x < y _ x = y _ y < x) ` (x < y _ :x < y) ^ (x = y _ :x = y)

we call this system decidable linear order LOd. Conversely, the decidability of < and = implies the comparability of x and y. The theory of dense linear ordering, DLO, is obtained by adding

9z (x < y ! x < z < y); 9y(x < y); 9y(y < x): Variants of DLO are considered by Smorynski [1977].

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DIRK VAN DALEN

Not only can we de ne a canonical apartness relation in LO, but also show that the theory containing apartness and order (AP + LO + x#y ! x < y _ y < x) is conservative over AP (and hence over EQs! ) [Smorynski, 1977; Hartog, 1978]. The theories LO and DLO are undecidable [Smorynski, 1977] cf. [Gabbay, 1981], but DLOd is decidable and coincides with its classical counterpart DLOc , [Smorynski, 1973a]. In [Gabbay, 1981] a number of re nements of the above results are treated.

7.4 Logic with Operations The set theoretical view of operations (functions)is that they are a special kind of relations, so we could do without the complications of introducing function symbols. However, the circumvention of function symbols is most unnatural, and, when we come to choice sequences, disastrous. The syntactic aspects of a rst-order language with function symbols are strictly analogous to the classical ones. We will therefore look into the semantic aspects. Consider a Kripke model K with a functional relation R(x; y), i.e. K 8x9!yR(x; y). The properties of forcing tell us that for each  we have that for each a 2 D() there is a unique b 2 D() such that  R(a; b). That is R is a function on D(). hence we de ne for each function symbol f an n-ary function f : D()n ! D(), for each . The monotonicity condition is not quite obvious since elements of jD()j are determined up to the relation  . The simplest solution is to put:   ) fa  f . There is another solution, however, that modi es the concept of Kripke model in the spirit of category theory, where one de nes a Kripke model as a pre-sheaf over P , where P is a poset (cf._ [Goldblatt, 1979, p. 256]). Before formulating the modi ed notion of Kripke model, we recall the notion of homomorphism for (classical) structures. f : A ! B is a homomorphism from structure A to B if f is a function from the universe of A into the universe of B such that f preserves all relations and functions, i.e.,

RA (a1 ; : : : ; an ) ) RB (f (a1 ); : : : ; f (an )) for all relations R; and

f (F A (a1 ; : : : ; an )) = F B (f (a1 ); : : : ; f (an )) for all functions F :

INTUITIONISTIC LOGIC

67

EXAMPLE

Æ1

a

Æ

bÆ

Æc

!f

Æ d

2Æ 3Æ

Æ4

wheref (a) = 1 f (b) = f (c) = 2 f (d) = 3

f is a homomorphism of the poset A into the poset B. We now come to our DEFINITION 38. A Kripke model is a quintuple K = hK; ; d; f; i where (K; ) is a poset, D assigns to each  2 K a structure D(), f assigns to each pair ; with   a homomorphism f : D() ! D( ) such that f = idD() , for all ; f Æ f = f for all    . The forcing relation is de ned as in De nition 13. Furthermore, equality is always interpreted as real identity. We can always associate a model in the new sense to a model in the old sense by lumping together elements in equivalence classes under  . In dealing with concrete Kripke models we will act broad-mindedly and choose whichever notion is most convenient, or even use the old notion and think of a new one.

7.5 Heyting's Arithmetic,

HA

The language of arithmetic contains, =; +;
; S; 0; 1 (and, when convenient, as many primitive recursive functions as we wish). The axioms of HA are those of Peano's arithmetic plus the axioms of EQ. 1. x = y $ Sx = Sy 2. :Sx = 0 3. x + 0 = x x + Sy = S (x + y) 4. x
0 = 0 x
Sy = x
y + x 5. '(0) ^ 8x('(x) ! '(Sx)) ! 8x'(x). Number 5 is the schema of mathematical (or complete) induction. It can

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DIRK VAN DALEN

also be presented in the form of a natural deduction rule ['(x)] .. . '(0) '(Sx) '(x) It is a simple exercise to show the decidability of the equality relation. THEOREM 39. HA HA

` 8xy(x = y _ x 6= y) ` 8x(x 6= 0 ) 9y(x = Sy)):

For a number of formal proofs, see [Kleene, 1952]. Arithmetic has correctly received a considerable amount of attention. It is the theory of the hard core of intuitionistic mathematics, put forward by Brouwer in his First Act of Intuitionism (cf. [Brouwer, 1975, pp. 509], [Brouwer, 1981, p. 4], [Heyting, 1956, p. 13 ]). Apart from foundational motivations for studying HA, there is a pragmatic argument for investigating arithmetic. It is, so to speak, a showroom of metamathematical tools and results. We will only be able to discuss a minute part of the material that is available. The reader is referred to Troelstra [1973]. Since HA is a subsystem of PA (Peano's arithmetic) we cannot expect to nd theorems contradicting the classical practice. We will have to look for metamathematical methods that capitalise on the constructive nature of intuitionistic logic. HA has the properties EP and DP, that are popularly considered to be the hallmark of constructive theories. We will rst show EP and return to the signi cance of EP and DP later. THEOREM 40. DP: HA ` ' _ ) HA ` ' or HA ` (disjunction property) EP: HA ` 9x'(x) ) HA ` '(n) for some n. (existence property) DP follows immediately from EP, since disjunction can be de ned in terms of the existential quanti er: HA ` (' _ ) $ 9x((x = 0 ! ') ^ (x 6= 0 ! )). therefore we will consider EP. Let HA ` 9x'(x), but HA 6` '(n) for all n. Then, by the completeness theory, there are Kripke models K0 ; K1 ; K2 ; : : : such that Kn 6 '(n). We form a new Kripke model K. Proof.

INTUITIONISTIC LOGIC

K0

K1

K2

0

1

2

69

We take the disjoint union of the models Ki and add an extra bottom node . The structure belonging to is the standard model of (classical) arithmetic. Since N is contained in all domains of the Ki 's, the resulting model satis es the conditions of the Kripke semantics. We will show that K HA. The only non- trivial axiom is mathematical induction. So we must show

(0), 8x( (x) ! (Sx)) ) 8x (x), for all . For 6= this is so by hypothesis, so consider (0); 8x( (x) ! (Sx)). We must show 8  8c 2 D( ); (c). Again the only case that must be taken care of is itself. So we must show (n) for all n. but we know (0) and (n) ! (Sn). Hence, by induction in the metalanguage we get (n) for all n. Now HA, so 9x'(x), and hence '(n) for some n. Contradiction with Kn 6 '(n). Therefore HA ` '(n) for some n.  Although EP seems to be stronger than DP, a result of Friedman shows this not to be the case for a large class of extensions of HA. THEOREM 41 ([Friedman, 1975]). For all RE extensions of HA EP follows from DP. It seems attractive to consider EP as the characteristic of constructivity; if we can show the existence of an object with a property ', then we can eectively indicate such an object. This is the constructive counterpart of the classical notion of `pure existence'. Kreisel has shown, however, that the possession of EP is neither necessary nor suÆcient for constructive theories. The following example (due to Kreisel, cf. [Troelstra, 1973, p. 91]) may illustrate the matter. Let Prf be the proof predicate of HA (cf. [Kleene, 1952, p. 254]). De ne '(x) := Prf(x; p0 = 1q) _ 8y:Prf(y; p0 = 1q). As HA is consistent on the intended interpretation, 8y:Prf(y; p0 = 1q) is true, so evidently 9x'(x) is true. Moreover, :Prf(n; p0 = 1q) is true for each n, and (Prf being primitive recursive) provable. Hence, for any n HA ` '(n) $ 8y:Prf(y; p0 =

70

DIRK VAN DALEN

1q). By de nition HA ` 9x'(x) $ [9yPrf(y; p0 = 1q) _ 8y:Prf(y; p0 = 1q)]. Now HA ` 9x'(x) ! '(n) yields HA ` [9yPrf(y; p0 = 1q) _ 8y:Prf(y; p0 = 1q)] ! 8y:Prf(y; p0 = 1q), hence HA ` 9yPrf(y; p0 = 1q) ! 8y:Prf(y; p0 = 1q), and so HA ` 8y:Prf(y; p0 = 1q), or HA proves its own consistency, contradicting Godel's second theorem. Note that T = HA + 9x'(x) is an intuitionistically true theory and T ` 9'(x), but by the above argument 6` '(n) for all n. So T does not have EP. The proof of EP above was itself not constructive, we have used a proof by contradiction. So we cannot actually exhibit the promised number n. There are various proofs that do provide the required instances. For example, by means of the Kleene slash [Troelstra, 1973, p. 177], q-realisability [Troelstra, 1973, p. 189 ], and normalisation in Gentzen systems [Minc, 1974]. An interesting feature is the stability of the instantiation member in various methods. That is, quite dierent techniques for converting a provable 9-statement into its instantiation yield the same number (cf. [Stein, 1980]). It is to be noted that all proofs of DP (or EP) for HA go essentially beyond the means of HA. Actually one can make this precise (Myhill) in the following form: let T be an r.e. extension of HA then there are sentences ' and such that if T ` Pr(' _ ) ! Pr(') _ Pr( ), then T ` Pr(p0 = 1q) where Pr(x) is the provability predicate for T . In words, the price for `provable DP' is that T proves its own inconsistency (cf. [Leivant, 1985]). Closure under rules

'1 ; : : : ; 'n R we automatically get that provFor a given derivation rule ability of the premises yields provability of the conclusion. We say that a theory is closed under a rule R if T ` '1 ; : : : ; T ` 'n ) T ` . Intuitionistic systems tend to be closed under various rules that are themselves not correct. We will list a few cases below. Consider the following principle. Markov's Principle, MP

8x('(x) _ :'(x)) ^ ::9x'(x) ! 9x'(x): MP plays an important role in metamathematics. It naturally turns up in certain interpretations. Markov postulated it in the context of recursion theory in the form `if it is impossible that a Turing machine does not halt, then it must halt', in the formalism of recursion theory: ::9zT (e; n; z ) ! 9zT (e; n; z ) (cf. [Troelstra, 1973]). Thus Markov's formulation can be taken to deal with a primitive recursive '(x). THEOREM 42. MP is not derivable in HA.

INTUITIONISTIC LOGIC

71

[Smorynski] Let '(x) be a primitive recursive formula such that

Proof.

9x'(x) is independent of HA (e.g. :ConHA , the inconsistency of HA). Let K be a Kripke model of HA + 9x'(x). We put an extra node a at the bottom of K, with D() the standard model for PA.

K  

Suppose HA ` ::9x'(x) ! 9x'(x). () Since the new model is a model of HA (cf. the proof of Theorem 40), we have  ::9x'(x) ! 9x'(). But evidently  ::9x'(x), therefore  9x'(x) and hence  '(n) for some n. '(x) being primitive recursive and '(n) being true, a theorem from arithmetic tells us that HA ` '(n), so HA ` 9x'(x). This contradicts the independence of 9x'(x). Therefore () is false.  Next we will show that HA is closed under Markov's rule. THEOREM 43. HA ` 8x('(x) _ :'(x)); HA ` ::9x'(x) ) HA ` 9x'(x); for F V (') = fxg: From HA ` ::9x'(x), we conclude PA ` 9x'(x) and so 9x'(x) is true in the standard model. Therefore, '(n) is true for some n. Now using HA ` '(n) _ :'(n) and DP we immediately get HA ` '(n), and thus HA ` 9x'(x).  Our next principle is the Independence of Premise Principle, IP (:' ! 9x (x)) ! 9x(:' ! (x)): The heuristic argument against IP is as follows: :' ! 9x (x) may be seen to hold by constructing an instance n that depends on the proof of :'. In 9x(:' ! (x)), however, we are required to construct the instance n beforehand. This evidently is a stronger requirement. Formal independence proofs are given in [Troelstra, 1973, pp. 179, 369]. THEOREM 44. HA ` :' ! 9x (x) ) HA ` 9x(:' ! (x)) (F V ( ) = fxg): Proof.

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DIRK VAN DALEN

Proof.

See below.



The case of Church's Thesis will be considered in Section 8. We return to the closure under Markov's Rule, to demonstrate an extremely elegant proof by H. Friedman [1977]. First we introduce the Friedman translation ' ! ' : replace in ' each atomic subformula by _  (where  is a formula of HA). The translation has the following properties. LEMMA 45. 1. 2. 3.

` ' )  ` ' and  ` ' .  HA ` ' ! HA ` ' . For any term t; HA ` ::9x(t(x; y) = 0) ) HA ` 9x(t(x; y) = 0).

(1) and (2) are easily shown by a suitable induction. (3) HA ` (9xt(x; y) = 0 ! ?) ! ?. We apply the Friedman Translation with respect to

Proof.

 := 9x(t(x; y) = 0); then ((9xt(x; y) = 0 ! ?) ! ?) = [9x(t(x; y) = 0 _ 9x(t; (x; y) = 0)) ! ? _ 9x(t(x; y) = 0)] ! (? _ 9x(t(x; y) = 0)): The latter formula is equivalent to 9x(t(x; y) = 0). Now apply (2).



So for the special case of t(x; y) = 0, closure under Markov's Rule has been established (i.e. in particular for primitive recursive functions f (x; y)). The general closure result is obtained by an application of closure under Church's Rule (cf. Section 8), i.e. if HA ` 8x9y'(x; y), then HA ` 8x'(x; fegx) for some e (index of total recursive function). One easily derives HA ` '(x; y)_:'(x; y) ) HA ` '(x; y) $ feg(x; y) = 0, for a suitable index e. We can replace feg(x; y) = 0 by 9z (T (e; x; y; z ) ^ U (z ) = 0) (cf. van Dalen's Algorithms chapter in Volume 1 of this Handbook). The matrix of the latter expression is primitive recursive, so we may conservatively extend HA by adding a symbol f for its characteristic function. Hence we get HA0 ` '(x; y) $ 9z (f (x; y; z ) = 0), where HA0 is the extension by f and its de ning equations. Now we may apply Lemma 45(3): 0 HA ` ::9x'(x; y ) ) HA ` ::9xz (f (x; y; z ) = 0) ! (Lemma 45 carries 0 0 over to HA ) HA ` 9xz (f (x; y; z ) = 0) ) HA ` 9x'(x; y). Observe that the above argument yields closure under Markov's Rule for formulas with parameters. We now apply the Friedman translation to the rule of independence of premises (A. Visser). For convenience we write ``' for `HA `'.

INTUITIONISTIC LOGIC

73

Let ` :' ! 9x (x). We apply the Friedman translation with respect to ::'. By Lemma 45(2) ` (:')::' ! (9x (x))::' : : : (1). Observe that ` : ! ( $ ) for any  and  : : : (2),

as one can easily show by induction on . Therefore also `  ! (: ! ) : : : (3). From(1) we get ` (:')::' ! 9x( (x)::' ), and an application of (3) yields ` (:')::' ! 9x(:' ! (x)). (:')::' = (' ! ?)::' = '::' ! ::'. Now apply (3) with  = ' and  = ::', then '::' ! (:' ! '), hence '::' ! ::'. Hence ` 9x(:' ! (x)). Friedman's translation is closely related to a straightforward translation of intuitionistic into minimal logic, cf. [Leivant, 1985] for details and also for syntactic criteria for closure under Markov's rule. Closure under Markov's rule is exactly what one needs for identifying provably recursive functions in classical and intuitionistic arithmetic. Using the notion of Ch. 4 of Vol. 1 of this Handbook, we can say that the recursive function with index e is provably recursive in a theory S if S ` 8x9yT (e; x; y) (for each input x the computation provably halts). Closure under Markov's rule tells us that PA and HA have exactly the same provably recursive functions (Kreisel). In other words, by restricting our arguments to intuitionistic logic we do not lose any recursive functions. Friedman extended this result to classical and intuitionistic set theory ZF ([Friedman, 1977], cf. also [Leivant, 1985]). 8 RELATION WITH OTHER LOGICS First we consider a sub-logic of intuitionistic logic. Minimal logic was proposed by Johansson in reaction to the role of negation, in particular the Ex falso sequitur quodlibet rule (our falsum rule). His critique resulted in a rejection of the rule `?'. As a result, in his system of minimal logic, ? cannot properly be distinguished from other atoms. This is re ected in the Kripke semantics for minimal logic. DEFINITION 46. A Kripke model for MQC is obtained from De nitions 10 and 13 by deleting the condition on ? (i.e.  ? is allowed). By a proof that is completely similar to that of Lemma 14 we get THEOREM 47 (Completeness for MQC).

j MQC ' , '

(where is understood in the sense of De nition 46). It now follows immediately that MQC is a proper subsystem of IQC (similarly for MPC and IPC), for MQC 6 ? ! '. Consider the one point model in which ? is forced.

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Although minimal logic is strictly weaker than intuitionistic logic, they are in a sense of the same strength. To be precise, each can faithfully be interpreted in the other. DEFINITION 48. The translations  and y are de ned by ' := ' _ ? for atomic ' (' ^ ) := ' ^  (' _ ) := ' _  (' ! ) := ' !  (8x') := 8x' (9x') := 9x'

'y := '[p=?]: where p is a propositional letter not occurring in '. Observe that the translation y eliminates ?, so for 'y we cannot use the falsum rule in IQC. That makes it plausible that 'y behaves in IQC as ' does in MQC. THEOREM 49. 1. IQC ` ' , MQC ` ' 2.

MQC

` ' , IQC ` 'y ,

(1) ( is trivial. For ) use induction on the derivation of

Proof.

` ' and observe

` ! ' , ` ': (2) ? behaves in the semantics for minimal logic like any atom, so validity of ' in all `minimal' Kripke models is equivalent to validity of 'y in all `intuitionistic' Kripke models. Alternatively, a simple proof theoretic argument based on the normal form theorem will do.  Minimal logic enjoys most metalogical properties that can be expected, e.g. there is a normal form theorem for natural deduction derivations, normal derivations have he subformula property, etc. Its propositional calculus, MPC, is decidable (use Theorem 49). For more information the reader is referred to [Johansson, 1936; Prawitz, 1965; Prawitz and Malmnas, 1968]. Our next goal is to investigate the relation between classical logic and intuitionistic logic. The rst results antedate Heyting's formalisation. Kolmogorov already in 1925 established a translation procedure [Kolmogorov, 1925], and in [Glivenko, 1929] a similar result is to be found. The next to investigate the relation between classical and intuitionistic logic (in the wider

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context of arithmetic) were Godel and Gentzen ([Godel, 1933; Gentzen, 1933]). for more information on translations the reader is referred to [Leivant, 1985], [Troelstra and van Dalen, 1988]. We rst present the result of Glivenko: THEOREM 50. CPC

` ' , IPC ` ::':

Recall that IPC is complete for nite Kripke models. Since in each maximal node all tautologies are forced, we see that ::' is valid in all nite Kripke models if ' is a classical tautology. This shows ). The converse is trivial.  Proof.

COROLLARY 51. CPC

` :' , IPC ` :':

There are a number of translations from IQC into CQC that have roughly similar properties. The main feature is the elimination of _ and 9. Let us call a formula negative if it does not contain _ and 9 and if all its atoms are negated. The translation below assigns to each formula a negative formula. Negative formulas have the following convenient property LEMMA 52. IQC ` ' $ ::' for negative '. An exercise in plain old logic. DEFINITION 53. The translation Æ is given by Proof.

'Æ (' ^ )Æ (' ! )Æ (' _ )Æ (8x')Æ (9x')Æ



:= ::' for atomic ' := 'Æ ^ Æ := 'Æ ! Æ := :(:'Æ ^ Æ ) := 8x'Æ := :8x:'Æ

THEOREM 54. 1. CQC ` ' , IQC ` 'Æ 2. CQC ` ' $ 'Æ . (2) is routine. For (1) we consider instead `c ' and Æ `i 'Æ (where `c; `i stand for classical and intuitionistic derivability, and Æ is the set of translated 's from ). Proof by induction on the derivation of ` ', use Lemma 52.  Proof.

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Since negative formulas are invariant under the translation we get COROLLARY 55. 1.

CQC

` ' , IQC ` ' for negative '.

2. For theories with decidable atoms (i.e. T have T `c ' , T Æ `i ' for negative '.

` ' _ :' for atomic ') we

The latter is the case for arithmetic, HA. The above result tells us that PA and HA are relatively consistent, so intuitionistic arithmetic is, from a foundational point of view, just as much in need of a consistency proof as PA. There are some special results in the area, we list some. FACTS 56. 1. 2. 3.

` ' , IQC ` ' for ' a negation of a prenex formula (Kreisel). CQC ` ' , IQC ` ::', and CPC ` :' , IQC ` :', for ' without 8. [Fitting, 1969, p. 52]. If 8 does not occur negatively in ' then CQC ` :' , IQC ` :' CQC

[Smorynski, 1973].

4. If ' is a 02 -sentence (i.e. of 89 form), then PA ` ' (Kreisel, cf. [Troelstra, 1973, Ch. 3, S. 8], lemma 46.)

, HA ` '.

Intuitionistic logic seen from the modal viewpoint As we have sketched earlier, intuitionistic logic has certain strong intensional aspects, in particular the meaning of the connectives| expressed in terms of proofs and construction, or of knowledge|has an intensional ring. This has been observed by Godel, who proposed a translation of intuitionistic logic into modal logic [Godel, 1932]. The `necessity' operator cold be read here as `I have a proof' or `I know that'. DEFINITION 57. The translation m is de ned by

(' _ (' ^ (' !

'm )m )m )m

:= ' for atomic ' := 'm _ m := 'm ^ m := ('m ! m )

We will establish the relation between S4 and IPC. Observe that in S4

? $ ?;

so (:')m $ :'m :

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We borrow from modal logic the fact that S4 is complete for Kripke models hK; R; i with a re exive, transitive R (cf. [Hughes and Cresswell, 1968; Schutte, 1968] and Chapter II.1 of the Handbook). The forcing notion is de ned by

 6 ?  ' ^ ,  ' and   ' _ ,  ' or   ' ! ,  6 ' or   ' , for all with R ; ': Observe that the propositional fragment is classical. Further, an intuitionistic Kripke model may be viewed as a modal Kripke model (not always conversely). LEMMA 58. Let hK; ; i be an intuitionistic Kripke model. De ne a modal Kripke model with the same underlying poset and the same forcing for atoms denoted by m . Then for all ';  ' ,  m 'm , Proof.

Induction on '.



` ' , S4 ` 'm . Proof. (. Let IPC 6` ', then there is a Kripke model with  6 ' for the bottom node . Now apply Lemma 58, then S4 6` 'm . ). Let S4 6` 'm , then there is a Kripke model such that 0 6 m 'm THEOREM 59.

IPC

for the bottom node : we turn this model into an intuitionistic Kripke model by rst collapsing the model, i.e. we consider the equivalence relation   0 :=   0 ^ 0   and introduce a new underlying set of nodes = . For =  we put =  ' :=  m 'm , for atomic '. This relation is obviously well-de ned, so is the forcing relation for all formulas. An argument similar to that of Lemma 58 establishes =  ' ,  m 'm . we now conclude 0 = 6 ', so IPC 6` '.  Artemov has picked up Godel's thread and designed logic which incorporates both `proof' and `modality', [Artemov, 2001].

8.1 Strong Negation Intuitionistic negation does not conform to the classical laws of double negation, De Morgan, etc. This is mainly so because negation is a rather weak connective. Think of its interpretations in a Kripke model: it is not decided on the spot. Or one may think of inequality versus apartness, `being unequal' carries so much less information than `being apart'. Could we possibly strengthen negation, so that the classical rules would e obeyed? As a matter of fact, this is what Nelson [1949] and Markov [1950] have done.

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The new connective can more or less be viewed as an attempt to save the classical laws by brute fore. Let us write ` '' for the strong negation of '. The axioms for the logic with strong negation are those of IQC plus the following

 (' ! ) $ '^   (' ^ ) $ '_   (' _ ) $ '^  '^' !  9x'(x) $ 8x  '(x)  8x'(x) $ 9x  '(x)  :' $ ';  ' $ ';  ' ! :': One obtains a Kripke model for a logic with strong negation by incorporating a strong falsity which is veri ed at the spot. DEFINITION 60. A Kripke model is a quadruple hK; ; D; ii where K;  and D are as in De nition 10, i is the interpretation map which assigns 1; 0; 1 to atoms and nodes such that   ; i('; ) 6= 0 ) i('; ) = i('; ). We de ne ['] for formulas ' and nodes : ['] := i('; ) for atomic ' with parameters in D(); where [?] 6= 1 [' ^ ] := min (['] ; [ ] ) [' _ ] := max(['] ; [ ] )

8 < [' ! ] := : 8 <1 [ '] = : 01

if 8  ; ['] = 1 ) [ ] = 1 1 if ['] = 1 and [ ] = 1 0 otherwise 1

if ['] = 1 if ['] = 1 otherwise

8 < 1 if 8  ; ['] 6= 1 ['] = 1 [:'] = : 0 1 ifotherwise 8 < 1 if 8  ; 8b 2 D( )['(b)] = 1 [8x'(x)] = : 1 if 9a 2 D()['(a)] = 1 0

otherwise

8 < 1 if 9a 2 D(); ['(a)] = 1 8  ; 8b 2 D( )['(b)] = 1 [9x'(x)] = : 0 1 ifotherwise :

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It is a matter of routine to show this semantics sound for logic with strong negation, IQCsn , in the sense that IQCsn ` ' ) ['] = 1 in all Kripke models K and nodes  2 K . There are alternative ways to present the same model. One can assign to each node two sets of atomic statements: + = f' j i('; ) = 1g and  = f' j i('; ) = 1g, and extend those sets as shown above to the strongly veri ed and strongly falsi ed sentences. Some authors write  ' for ['] = 1 and  k ' for ['] = 1. These notations are purely a matter of convenience. The completeness of IQCsn can be shown by the usual Henkin- or tableaux-technique (cf. [Thomason, 1969]) but also by a reduction to IQC. We will indicate the steps. (1) Observe that all strong negations can be driven in, so each ' is provably equivalent to a ' with all strong negations in front of atoms. (2) We want to consider strongly negated atoms as atoms in their own right, so we double the language by adding a predicate P^ for each predicate P . Indicate the new atoms by '^. At the very least the strongly negated atoms should imply the negated atoms. Put = f'^ ! :' j ' atomic sentenceg, and let  be the formula one obtains by replacing   (atomic ') by '^ in  . Claim: `IQC () ! : for all . Prove this by induction on . (3) allows us to reduce IQCsn to IQC in the following sense: IQCsn `  , IQC + `  . Proof: Induction on the proof length (or on the derivation in natural deduction). (4) We can now apply the completeness theorem for IQC. Let IQCsn 6`  then IQC + 6`  , so there is an ordinary Kripke model K in which all axioms of are valid, but not so  . Turn K into a strong negation model K0 , putting  k ' snif  '^ for atomic sentences '. Now one shows that K0 is a model of IQC , but not of . Since an ordinary model is trivially a strong negation model, it is immediately seen that IQCsn is conservative over IQC. sn has some unusual properties, e.g. ` ' $ IQC does not imply ` ' $ (consider :' $ ' ! ?). For more information, cf. [Gabbay, 1981, p. 124 ], [Rasiowa, 1974, Ch. XII] and [Rautenberg, 1979, p. 305 ].

8.2 The Connections with -Calculus and Combinatory Logic Already in 1958 Curry pointed out that there is a remarkable correspondence between the implication fragment of IPC and combinatory logic, CL.

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In particular the axioms correspond to the CL axioms as follows: '!' Ix = x ' ! ( ! ') Kxy = x ( ! (' ! )) ! (( ! ') ! ( ! )) Sxyz = xz (yz ): Howard extended the correspondence in his `Formulas as types' paper (1969, published in [Howard, 1980]). Once such a correspondence exists one is almost forced to look at the reduction processes in CL or -calculus and in natural deduction systems (cf. [Prawitz, 1971]). In Pottinger [1976] the isomorphism between natural deduction derivations and -terms has been exploited to obtain alternative proofs of the normalisation theorem for IPC. In Martin{Lof's type theory the parallelism between types and formulas is a key feature. For more information the reader is referred to Martin-Lof [1977; 1984] and Troelstra and van Dalen [1988]. 9 THE ALGORITHMIC TRADITION Intuitionistic logic was intended to codify constructive reasoning. The proofinterpretation expresses the meaning of the logical constants in terms of constructions. It seems plausible to try to delimit the class of constructions involved. Stephen Kleene conjectured in 1940 that in particular for a statement of the form 8x9y'(x; y) provability in HA should entail the existence of a recursive function f that acts as a choice function: 8x'(x; f (x)) (cf. [Kleene, 1973]). This led Kleene to the notion of statements as `incomplete communications', taking his cue from Hermann Weyl, see [van Dalen, 1995] e.g. 9x'(x) is an incomplete communication of a fuller statement giving an object x such that '(x). Likewise the other composite statements can be considered as incomplete statements, to e supplemented by extra information. The result was the so-called 1945-realizability or recursive realizability, a notion that we will formulate in the framework of HA. The sentence ' is realized by the number n; nr', must, be thought of as n codes `the necessary information to establish ''. DEFINITION 61. x r ' is a formula of HA with at most one free variable x, associated to the sentence ' (we use notation from Ch. 4 of Vol. 1 of this Handbook).

xr' x r (' ^ ) x r (' _ ) x r (' ! ) x r 9y'(y) x r 8y'(y)

:= := := := := :=

' for atomic ' (x)0 r ' ^ (x)1 r ((x)0 = 0 ! (x)1 r ') ^ ((x)0 6= 0 ! (x)1 r ) 8y(y r ' ! fxgy # ^fxgy r ) (x)1 r '((x)0 ) 8y(fxgy # ^fxgy r '(y))

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Explanation: The rst clause tells us that any number realizes a true atomic sentence. However, no number realizes a false atomic sentence. The second clause is self-evident. The clause for the disjunction exhibits the eective nature of the disjunction. We can eectively test if (x)0 = 0 or (x)0 6= 0. Hence, the `realizer' of a disjunction contains enough information to indicate the desired disjunct. The implication clause shows a resemblance to the proof interpretation of !; x is the index of a partial recursive function that transforms any realizer of ' into a realizer of . In the case of 8 a similar resemblance can be observed. The clause for 9 tells us that a realizer of 9y'(y) contains the required instance and the information that realizes it. Note that xr' is a formula of HA, so it makes sense to ask for the truth of an instance nr', or its derivability in HA. EXAMPLES 62.

1. x r (2 = 1 + 1) :, 2 = 1 + 1($ >).

2. x r 8z 9y(z = y) :, 8z (fxgz # ^fxgz r 9y(z = y)) :, 8z (fxgz # ^(fxgz )1 r (z = (fxgz )0 )) :, 8x(fxgz # ^z = (fxgz )0. So if we take the index e of the identity function z 7! z , then (fxgz )0 = fegz and we can put fxgz = hfegz; 0i. This is a (total) recursive function, so it has an index, say e0 . The number e0 realizes 8z 9y(z = y). Kleene's realizability can be considered as an interpretation of HA in HA bringing out the constructive character of HA. This interpretation is sound in the following sense: THEOREM 63. HA ` ' ) HA ` n r ' for some n. The proof is mainly a matter of perseverance (cf. [Kleene, 1952, p. 504], [Troelstra, 1973, p. 189]). A consequence of this theorem is the fact that (assuming the consistency of HA) a realizable sentence ' is consistent with HA. For suppose that nr' and HA + ' is inconsistent, then HA ` :', and hence HA ` mr(:') for some m. But mr(:') is equivalent to 8y(yr' ! fmgy # ^fmgyr?), and since ? is not realisable, neither is '. Contradiction. The most striking application of this procedure for establishing consistency is: THEOREM 64. Church's Thesis is consistent with HA. We have in mind a special form of Church's Thesis, namely one that can be formulated in HA. We choose the following form: Proof.

CT0

8x9y'(x; y) ! 9z 8x(fz gx # ^'(x; fz gx)):

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Observe that we can avoid the abbreviation fz gx #: 8x9y'(x; y) ! 9z 8x9u(T (z; x; u) ^ '(x; Uu)) where T is Kleene's T predicate (cf. van Dalen's Algorithms Chapter in Volume 1 of this Handbook), and U is the output-extraction function. For convenience we suppose that ' has only the variables x and y free. We will need the following notation: if t is a term for a partial recursive function, then x:t is the index of the partial recursive function given by t depending on x (if there are more variables we consider them as parameters; strictly speaking the notation is based on the Snm -theorem, Cf. Vol. 1 of this Handbook, p. 275, or [Kleene, 1952, p. 344]). For example, x:x + y is the index of the unary function that adds y. We will sketch the proof in such a way that the reader, if he wishes to do so, can provide the full details himself. Let u r 8x9y'(x; y), then 8x(fugx # ^fugx r 9y'(x; y)), i.e.

8x(fugx # ^(fugx)1 r '(x; (fugx)0 )) : : : (0): Put t := fugx, and a = x:(t)0 ; b = wT (a; x; w); Æ(u) = ha; x:hb; ho; (t)1 iii. Claim: Æ(u) r 9z 8x9v(T (z; x; v) ^ '(x; Uv)) : : : (1). We carry out the steps as given in the de nition.

x:hb; h0; (t)1 ii r 8x9v(T (a; x; v) ^ '(x; Uv)) : : : (2): hb; h0; (t)1 ii r 9v(T (a; x; v) ^ '(x; Uv)) : : : (3): h0; (t)1 i r T (a; x; b) ^ '(x; Ub) : : : (4): or

T (a; x; b) ^ (t)1 r '(x; Ub) : : : (5): Now observe that by the de nition of a and b; T (a; x; b) is true for all x. So 0 realizes it (where for convenience T (a; x; b) has been taken to be atomic; this is achieved by a simple conservative extension of HA). Furthermore, Ub is the output of fag on input x, which is (t)0 , so (t)1 r '(x; Ub) can be read as (fugx)1 r '(x; (fugx)0 ). This holds by (0). The passage from (0) to (1) tells us that u:Æ(y) r CT0 .  Almost the same argument establishes ECT0 (see below) [Troelstra, 1973, p. 195]. Troelstra has investigated the theory of the realizable sentences of arithmetic. It turns out that this fragment has a simple axiomatization (cf. [Troelstra, 1998, p. 416]). HA HA

+ ECT0 ` ' $ 9x(x r ') + ECT0 ` ' $ HA ` 9x(x r ');

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where ECT0 is the Extended Thesis of Church:

8x(' ! 9y xy) ! 9u8x(' ! (fugx # ^ (x; fugx))) for almost negative ' (i.e. ' does not contain _, and 9 only in front of atoms). HA + CT0 has been studied in detail by David McCarty, for his results see [McCarty, 1988]. Perhaps the most striking fact established by him is the categoricity of the theory: HA + CT0 has no non-standard models. Since Kleene's pioneering papers there has been a proliferation of notions. The reader is referred to [Troelstra, 1973] and [Troelstra, 1998] for the major notions in the context of arithmetic. There are also extensions to higher theories (e.g. set theory- like ones) (cf. [Feferman, 1979], [Beeson, 1985]). In the fties Godel proposed a new interpretation of HA (and extensions) based on functionals of all nite types (cf. [Godel, 1958; Kreisel, 1959; Troelstra, 1973; Avigad and Feferman, 1998]). The basic idea is to reduce the logical complexity of sentences at the cost of increasing the types of the objects. Kreisel proposed the notion of `modi ed realizability' (cf. Troelstra [1973; 1998]); Kleene transferred realizability to analysis by means of `continuous function application'; in the context of rst-order logic we mention Lauchli's `abstract realizability. A systematic and unifying treatment of various realizabilities has been given (cf. [Stein, 1980]). The above-mentioned interpretations have led to a wealth of proof theoretic results, such as conservative extensions, and closure under rules. the reader is referred to [Troelstra, 1973; Troelstra, 1998] for detailed information. The Russian school of A. A. Markov has made the algorithmic tradition the guideline for its actual mathematical practice. Its members consider mathematics as dealing with concrete, constructive objects. In particular they adhere to Church's thesis, so that, e.g. real numbers in their approach are given by recursive Cauchy sequences (hence the name `recursive analysis'). Following Markov, they accept the principle ::9x'(x) ! 9x'(x) for primitive recursive '(x)|Markov's Principle. For a survey, cf. [Demuth and Kucera, 1979]. For a long time the `algorithmic' interpretations have withstood attempts of unifying treatment together with the semantic interpretations. Recently, however, the framework of topos theory has provided a more semantic treatment of, e.g. realisability interpretations. In particular work of Hyland, Johnstone and Pitts [1980] on tripos theory and Hyland [1982] on the eective topos has provided a semantical home for the above kind of interpretations.

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10 SECOND-ORDER LOGIC Whereas rst-order intuitionistic logic and its prominent theories, such as arithmetic, are just subtheories of the corresponding classical ones, the notions of second-order logic seem to dictate their own laws in the light of intuitionistic conceptions. Traditionally, second-order logic is concerned with individuals, sets (and relations) and the only non-logical principle that is considered is the socalled comprehension axiom. Most studies are centered around secondorder arithmetic, and extensions of it. We will rst discuss second-order logic. The language of intuitionistic second-order logic IQC2 contains variables and constants for individuals x0 ; x1 ; x2 ; : : : ; c0 ; c1 ; c2 ; c3 ; : : : n-ary relations X0n; X1n; X2n ; : : : ; C0n ; C1n ; C2n ; : : : ; where n  0. 0-ary variables (constants) are called propositional variables (constants), 1-ary variables (constants) are called set (species) variables (constants). The atoms of IQC2 are of the form X 0 ; C 0 for 0-ary second-order terms, or X n (t1 ; : : : ; tn ); C n (t1 ; : : : ; tn ) for n-ary second-order terms X n ; C n and rst-order terms t1 ; : : : ; tn (i.e. individual variables or constants). In classical logic one thinks of 0-ary terms as denoting the truth values `true', `false'. In our case we may think of truth values in a Heyting- algebra. Formulas are de ned as usual by means of the connectives ^; _; !; ?; 8x; 8X n; 9x; 9X n. The rules of derivation (in Natural Deduction) are extended by the following quanti er rules:

'

82I 8X n' 92 I

' 9X n '

82E

8X n' '

9X n' ['] 92 E

.. .

where ' is obtained from ' by replacing each occurrence of X n(t1 ; : : : ; tn ) by (t1 ; : : : ; tn ), for a certain , such that no free variable among the ti becomes bound after substitutions. Observe that 92 I takes the place of the traditional Comprehension Principle (cf. [van Dalen, 1997, Chapter 4]) 9X n 8x1 ; : : : ; xn ['(x1 ; : : : ; xn ) $ X n (x1 ; : : : ; xn )]:

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The surprise of second-order logic is the fact that the usual connectives are de nable in terms of 8 and !, this in sharp contrast to IQC (Prawitz). Given the rules for 8 and ! we can de ne the connectives as follows. DEFINITION 65. 1. 2. 3. 4. 5.

? := 8X 0:X 0 ' ^ := 8X 0[(' ! ( ! X 0 )) ! X 0] ' _ := 8X 0[(' ! X 0 ) ! (( ! X 0) ! X 0)] 9x' := 8X 0[8x(' ! X 0) ! X 0] 9X n ' := 8X 0[8X n(' ! X 0) ! X 0].

To be precise: given the rules for 8 and ! we can prove the rules for the de ned connectives (cf. [Prawitz, 1965, p. 67], [van Dalen, 1997, p. 152]). For proof theoretical purposes the reduction of the number of connectives turns out to be an asset (cf. [Tait, 1975; Prawitz, 1971]), The semantics for second-order logic are relatively straightforward generalizations of the existing semantics for rst-order logics (cf. [Prawitz, 1970; Takahashi, 1970; Fourman and Scott, 1979]).

10.1 Second-order Arithmetic,

HAS

The simplest formalisations of HAS (Heyting's second-order arithmetic with set variables), is obtained by adding the axioms for HA to secondorder logic (in an extended language containing the obligatory operations and relations for arithmetic). Observe that, as a schema, the induction axiom is de ned for the full language. The traditional issue in second-order arithmetic concerns the Comprehension Principle, CA. Should it have the full strength or should it be restricted to the predicative case? This topic has never been really central in intuitionistic considerations on higher-order objects. There certainly is not much to go on in Brouwer's writings. If we embrace the viewpoint that a set X is given when we know what it means to prove n 2 X , then it is still not obvious to decide between the predicative and the impredicative viewpoint. Since the matter of predicativity is an issue in it own right, we bypass the topic. Even at a quite low level sets of natural numbers turn out to be rather elusive. If we consider

fn j

the nth decimal of  is preceded by 20 ninesg

then we do not know whether it is empty or not. So, even sets that have simple de nitions may be rather wild (although not surprisingly so, as recursion theory has already shown us).

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The universe of sets diers in an essential way from the universe of natural numbers. Whereas the latter are discretely given and completely determined with respect to each other, the rst are pretty undetermined in the extensional sense, i.e. considered as being determined by their elements. This undeterminedness is brought out in the following uniformity principle, formulated by Troelstra

UP

8X 9x'(X; x) ! 9x8X'(X; x):

In words: if for each set X one can nd a natural number x such that '(X; x) then there is already one number x0 that satis es '(X; x0 ) for all X . Surprising as this may seem, the almost immediate counter-examples from classical logic are seen not to work. For example, consider 8X (X = ;_ X 6= ;)), which can be written as 8X 9x((x = 0 ! X = ;) ^ (x 6= 0 ! X 6= ;)). Classically, this statement is true, but intuitionistically is in general not decidable whether a set is empty, cf. the set de ned above. The uniformity principle is consistent with HAS + AC-NS, where the axiom or choice from number to species reads

8x9X'(X; x) ! 9Y 8x'((Y )x ; x) (where y 2 (Y )x , hx; yi 2 Y ) [Troelstra, 1973a; van Dalen, 1974]. AC-NS

has been studied via Kripke semantics in [Jongh and Smorynski, ] 1976 ). They interpreted the rst-order part as usual and took for sets of natural numbers growing families of sets (just like unary predicates in an ordinary Kripke model). EXAMPLE 66. HAS



 N





?

Take in  and the standard model of (classical) arithmetic and let S = ?; S = N . Then  ::8x(x 2 S ), i.e. ::S = N , but  6 8x(x 2 S ) and even  6 9x(x 2 S ). A number of proof theoretic results are obtained by semantic means, e.g. HAS has the disjunction and the existence property, but also the existence property for 9X : HAS

` 9X'(X ) ) HAS ` '(fx j (x)g);

for a suitable (x);

i.e. if `there (provably) exists a set' X , then `there already exists a de nable set'. We list a few closure properties.

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is closed under the Uniformity Rule, URc HAS ` 8X 9'(X; x) ) HAS ` 9x8X'(X; x) (where F V (') = fX; xg).

1.

HAS

2.

HAS

3.

HAS

4.

HAS

is closed under Markov's Rule, MR HAS ` 8x('(x) _ :'(x)) ^ ::9x'(x) ) HAS ` 9x'(x). is closed under Church's Rule, CR HAS ` 8x9y'(x; y ) ) HAS ` 9e8x'(x; fegx). HAS

is closed under the Rule of Choice, RC-NS

` 8x9X'(X; x) ) HAS ` 9X 8x'((X )x; x).

Intuitionistic second-order arithmetic has been extensively studied by prooftheoretical means, e.g. [Martin-Lof, 1971; Prawitz, 1971; Girard, 1971] and Troelstra [1973; 1973a].

10.2 Choice Sequences Whereas in classical mathematics one can de ne functions in terms of sets and vice versa, we here treat functions and sets more or less independently. Philosophically speaking this is rather obvious; the two notions are radically dierent. A set (say of natural numbers) is given to us as a property of natural numbers (cf. Brouwer [1918; 1981a]), whereas a function (say from natural numbers to natural numbers) is given as a process of assigning values to arguments. Interde nability of these notions would be an unexpected coincidence. One can, of course, consider a function as a set of pairs, conversely one cannot, in general, give a set by a characteristic function. For let f : N ! f0; 1g and n 2 A , f (n) = 1, then it follows from 8m(m = 1 _ m 6= 1) that 8n(n 2 A _ n 62 A), i.e. A is decidable (mind you, not recursive, but decidable in the sense that `n belongs to A or does not belong to A'). Since there is in intuitionistic mathematics an abundance of undecidable sets, we must conclude that the characteristic function approach to sets does not work. For the sake of perspicuity we will in the following restrict ourselves to functions from N to N . The nineteenth century had already brought us the immense progress of widening the function concept, in the form of `a function is a law that assigns a natural number to each natural number', thus doing away with conditions of analyticity, etc. However, the discussions at the beginning of this century made it clear that on a reasonable reading of `law', one would end up with a countable universe of functions, with all its mathematical drawbacks. Or, even worse, with the de nability paradoxes (Richard and Berry).

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In order to overcome these diÆculties Brouwer introduced in 1918 a more liberal notion of function, which identi ed functions with choice processes. To quote from Brouwer [1981a]: \Admitting two ways of creating new mathematical entities: rstly in the shape of more or less freely proceeding in nite sequences of mathematical entities previously acquired; : : : " These sequences were introduced in 1918 as choice sequences (Wahlfolgen). In a later stage Brouwer spoke of arrows. One has to think an idealised mathematician who at consecutive stages chooses natural numbers. This (mental) choice process may be highly involved, e.g. the subject may be in the course of the process put all kinds of restrictions on future choices. He may, for example, at a certain stage give up all freedom and follow a given law, or he may decide at the beginning that he will never completely give up his freedom of choice. The matter of higher-order restrictions on future choices (i.e. restrictions on restrictions, etc.) has sparked some debate. Indeed, Brouwer himself has questioned their usefulness (cf. Brouwer [1981a, p.13]; [1975, p. 511]). Once choice sequences were introduced, Brouwer was faced with the nontrivial problem of how to exploit them in mathematics. Put otherwise, what properties can one extract from the basic conception of a choice process? In the very rst paper on the subject Brouwer laid down the following continuity principles: A law that assigns to each choice sequence  a natural number n must completely determine n after a nite initial segment of  has been determined (cf. Brouwer [1918, p. 13]; [1975, p. 160]). The matter of establishing the basic properties of choice sequences calls for `informal rigour' (a term introduced by Kreisel [1967], referring to a precise non-formal analysis of certain conceptually given concepts, leading to more or less basic axioms (principles)). A general analysis of this kind is not within the scope of the present chapter. The reader is referred to [Troelstra, 1977; Dummett, 1977] and [van Atten and van Dalen, forthcoming]. Without going into all details we will indicate a language for a theory of choice sequences (also called intuitionistic analysis). Add to the language of arithmetic, function variables 1 ; 2 ; 3 ; : : : and suitable function constants (e.g. the primitive recursive functions). The result is a two sorted language. We add all axioms of IQC for both sorts. In general one ads the (rather weak) comprehension principle

8x9!y'(x; y) ! 9 8x'(x;  (x)): What more is to be added depends on the notion under consideration. The comprehension principle is, e.g., correct for general choice sequences, but not in general for lawlike or lawless sequences. We will use roman symbols f; g; h; : : : for lawlike sequences (i.e. choice sequences given by a law). One may use various sorts of choice sequences in one and the same context (cf. [Kreisel and Troelstra, 1970; Troelstra, 1977]). We will, however, consider

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a particularly perspicuous kind of choice sequence, introduced by Kreisel. The choice sequences we have in mind constitute, so to speak, a limiting case where no restrictions whatsoever will be placed on future choices. The resulting notion is that of `lawless sequence'. In what way does the lawlessness of a sequence manifest itself? Let us do a thought experiment: we make successive choices  (0);  (2);  (2); : : : , such that at each stage there is a complete freedom for the next choice (think of the throws of a die, where there is however an overall restriction to numbers  6), we add the successive values and we nd that the sum of a certain initial segment is a prime number. For example,  (0) = 4;  (1) = 2;  (2) = 2;  (3) = 0;  (4) = 1;  (5) = 8;  (6) = 2; : : : , and 4 + 2 + 2 + 0 + 1 + 8 = 17, which is a prime. So for this  we have established `There is an initial segment of  such that its sum is a prime number'|abbreviated by '( ). It is , however, immediately clear that any lawless sequence  that starts with the same initial segment h4; 2; 2; 0; 1; 8i also satis es '. This is an instance of the general principle of open data:  = x ! '()); '( ) ! 9x8(x  = h (0);  (1); : : : ;  (x 1)i, i.e. the coded (cf. Algorithms Chapwhere x ter, volume 1 of this Handbook) initial segment of length x of  . In words: if ' holds for the lawless sequence  then there is an initial segment of  such that all lawless continuations of it also satisfy '. Or less precise,  having the property ' is determined by a suitable initial fragment. The principle can be justi ed as follows: the idealized mathematician establishes '( ) after a nite number of values of  has been chosen, because at any time that is all the available information on  he has. but, therefore, the continuation of this particular initial segment is irrelevant, i.e. all continuations  also have the property '. It is quite often helpful to think of a choice sequence (function) as a path in the full tree of all nite sequences of natural numbers. So suppose '( ) holds on the basis of the information of segment h0; 2; 3; 0; 1i, then ' holds for all lawless sequences (paths) that pass through the node h0; 2; 3; 0; 1i in the tree. but in the tree topology this is a (basic) open. Hence if ' holds for  , it holds for an open neighbourhood of  (this explains the name `open data'). There are two more basic principles: LS1 8x9 ( 2 x), where x is a coded initial segment and  2 x stands for  (0) = (x)0 ; : : :  (k 1) = (x)k 1 , where k is the length of x. In words each initial segment can be extended into a lawless sequence. This principle is harder to justify, if no restrictions at all are allowed, how does one make certain that the rst k choices can be made to conform to

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a given segment? It is best to view this principle as slightly modifying the original notion: we allow at the beginning the speci cation of an arbitrary nite initial segment. In this way al nite sequences actually occur as initial segments. If we make no assumptions about initial segments then the sequences oer a less satisfactory mathematical theory. Troelstra [1983] introduced this weaker notion (without LS1) under the name of proto-lawless sequences. Finally, if the idealized mathematician considers two (mental) lawless choice processes, then he knows if the processes are identical or not, so we have

8(   _ :  ), where  is the intensional identity between sequences, considered as mental LS2

choice processes. The principle of open data is formulated as LS3 '( ) ! 9x( 2 x ^ 8 2 x'()).

Actually, Kreisel's notion of `lawlessness' requires also a certain independence of sequences, so that the sequences are also lawless with respect to each other. An example: say we generate a lawless sequence  (0);  (1);  (3); : : : and we drop the rst value, is the remaining sequence  (1);  (2);  (3); : : : lawless? Individually viewed, yes, but in conjunction with the original sequence, no. This leads us to extend LS3 as follows: LS3n '(; 1 ; : : : ; n )^ 6 (; 1 ; : : : ; n ) 9x( 2 x ^ 8 2 x(6 (; 1 ; : : : ; n ) ! '(; 1 ; : : : ; n ))),

Vn

where ! (; 1 ; : : : ; n ) stands for  6 i . i=1 Without the extra clause 6 (; 1 ; : : : ; n ) the principle is false. For example, consider '(;  ) :=    . Suppose we could apply LS31 , then

   ! 9x( 2 x ^ 8 2 x(  )); i.e. there is an initial segment of  such that every extension of it coincides with  . This plainly contradicts LS1. From these principles we can already derive a number of unusual results. THEOREM 67. 1.    $ 8x(x = x) 2. 8 ::9x(x = 0)

3. 8 :98x( (x + 1) = (x))

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91

:9 ( = f ), for a lawlike function f .

(a) ! is trivial. For we can use a proof by contradiction by LS2. So let  6 , we may apply LS31 to '(; ) := 8y( (y) = (y)): Proof.

9x( 2 x ^ 8 2 x( 6  ! 8y( (y) = (y)): Let, therefore, the initial segment h (0); : : : ;  (k)i be such that for  6  and  (0) =  (0); : : : ;  (k) =  (k) it follows that 8y( (y) = (y)). Since  (k + 1)

can be chosen freely, we choose  (k + 1) = (k + 1) + 1 (i.e. we apply LS2 to h (0); : : : ;  (k); (k + 1) + 1i to obtain a  . But this contradicts 8y( (y) = (y)). Hence   . (b) Suppose :9x(x = 0), or by logic, 8x(x 6= 0). Apply LS3 and add a zero to the initial segment of  that exists according to LS3. (c) Apply LS31 to 8x( (x + 1) = (x)). (d)  = f is an abbreviation for 8x( (x) = f (x)) (extensional equality). Apply LS3 to the latter formula. 

Could we do better than (b) and even show 8 9x( (x) = 0)? The answer is no, but we need a strengthening of the system to show this. We have already de ned what a bar is (cf. p. 25. Now we can formalise  2 B ). it in analysis: B is a bar (in the tree of all nite sequences) if 8 9x(x Such a bar is a denumerable set of sequences. Can we present such a bar by a convenient function? The technique is not diÆcult, we consider a function e : N ! N , such that e(x) = 0 if x is a (coded) sequence above the bar, and e(x) > 0 for x on or below the bar. More formally:  ) > 0) ^ 8xy(e(x) > 0 e 2 K0 := 8 9x(e(x ^y  x ! e(x) = e(y)) (where y  x stands for `the sequence y extends x'). K0 is the class of neighbourhood functions (or moduli of continuity, also called Brouwer operations. Note that the part of the tree above B is a well-founded tree. Kreisel and Troelstra have considered K0 as an inductively de ned class of lawlike functions (cf. [Troelstra and van Dalen, 1988, p. 223 .]). The neighbourhood functions have been introduced with respect to lawless sequences, i.e. if e gives a bar B then each lawless sequence  hits the

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bar. It is however plausible to widen the scope of such e, such that each sequence (not necessarily lawless) hits the bar. This extension principle which states that for each e 2 K0 we have 89x(e(x) 6= 0) (where  ranges over all sequences), is required for certain applications of the theory. A justi cation of the extension principle by means of an abstraction operator is put forward by Troelstra [1977, p. 20]. Here we will for simplicity use K0 as given above. We will now formulate a strong continuity principle: if 8 9x'(; x), than x can be found for a given  by a neighbourhood function e from K0 as follows: look for the rst  such that e(y  ) = k + 1 > 0 and put x = k. Let us initial segment y agree to write e( ) = k when we follow the above procedure, then we get the following principle: LS4

8 9x'(; x) ! 9e 2 K0 8'(; e( )).

By the extension principle e operates on all possible sequences. We will use this to show :8 9x( (x) = 0). Suppose 8 9x( (x) = 0), then by LS4 8 ( (e( )) = 0), i.e. e picks a zero of  . Now consider f such that 8x(f (x) = 1). Determine e(f ), say k. f `hits' the bar determined by e in a node m, which hence is an initial segment of f . Now extend this segment m with enough 1's such that the total length of the resulting n exceeds k. By LS1 there is a lawless sequence  with initial segment n. By de nition, however, e( ) = e(f ) and 0 =  (e( )) = f (e(f )) = 1. Contradiction. Hence :8 9x( (x) = 0). The above lines may serve to illustrate the highly unusual character of lawless sequences and the extraordinary richness of the intuitionistic universe of functions. Kreisel and Troelstra have established for a certain system elimination theorems, i.e. translations that eliminate the choice sequences (cf. [Kreisel and Troelstra, 1970]; [Troelstra and van Dalen, 1988, 12.3.1]). This may, with due caution, be viewed as evidence for the viewpoint that choice sequences are only a facon de parler. Note however that the evidence is rather incomplete in the sense that the theorems range over a few formal theories. Moreover such a viewpoint would violently con ict with the ontological status of the mentally generated objects of intuitionism. Choice sequence have the didactic disadvantage that one cannot show an isolated copy, unless it happens to be given by a law. This situation changed when Joan Moschovakis adapted a topological model of Scott for the reals to choice sequences. A similar interpretation was presented by the author in the framework of Beth models. Since the latter approach allows one a nice visualisation we will sketch it here (cf. [van Dalen, 1978]). In order to facilitate the presentation, we consider models with the universal tree (the tree of all nite sequences of natural numbers) as underlying poset. We will also denote the nite sequences (n0 ; : : : ; nk 1 ) by ~n.

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Whereas in a Kripke model the condition 8x9!y (x)y forces us to interpret a sequence (function) in each node as a total function, in a Beth model ~n 8x9!y (x) = y only tells us that eventually on a bar B , the outputs y for an input x will be determined, so the natural interpretation of a sequence  is a growing family ~n of partial functions with the property that along a path the union of all these ~n 's yield a total function. A concept that con rms rather well to the heuristic notion of `choices being made in time'. In the nodes the choice function is only partially determined, but the whole model allows us to view the choice sequences, as it were from a higher viewpoint, as completed. So we take a Beth model of arithmetic (containing only standard numbers) and consider as the universe of choice sequences all such growing families of partial functions. Examples

(1) De ne ~n := ~n for each ~n, i.e. in each nite sequence the partial function is just this sequence. Evidently the conditions are satis ed. (2) De ne  h i = h i and  hii = x:i, i.e. at the bottom node we take the empty function, and at its immediate successors hii we take the constant functions with value i. Observe that h i 9x8y( (y) = x) (a simple exercise in Beth semantics), so the model tells us that the sequence is constant, although externally it is not. The particular model with underlying tree of all nite sequences of natural numbers validates a list of principles: AC - NF

8x9'(x;  ) ! 98x'(x; ()x );

where ()x (y) = (hx; y i), the axiom of choice from numbers to functions. SC!

8 9!x'(x;  ) ! 9 2 K0 8'( ( );  );

the strong continuity principle with uniqueness restriction. BIM monotone bar induction, a principle that can be considered as an intuitionistic version of induction over well-founded relations, or of trans nite induction (cf. [Troelstra and van Dalen, 1988; Kleene and Vesley, 1965]). KS

9 (' $ 9x (x) 6= 0);

Kripke's Schema, a principle which will be discussed in the next section. The validity of Kripke's Schema is fairly simple to establish. For the remaining principles we refer to [van Dalen, 1978]. The idea is to go up in the tree and in each node to check if ' has been forced. So we de ne ~n to be a nite sequence of the same length as ~n. Let ~n have length k, then we put ~n (k) = 0 if ~n 6 ' and ~n (k) = 1 if ~n '. this de nes a proper choice sequence. Clearly, for any ~n, if ~n ' then ~n 9x (x) 6= 0.

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Conversely, suppose that ~n 9x (x) 6= 0, then there is a bar B for ~n such for each m ~ 2 B; m ~  (n) 6= 0 for some n. This implies  m~ (n) = 1, which by de nition means that m ~ '. So ' is forced on a bar for ~n. Hence ~n  ' (cf. Lemma 11). Therefore h i 9 (' $ 9x (x) 6= 0). Although Kripke semantics has shown itself superior to Beth semantics in many respects, the latter is the more convenient one to treat functions. For, in the Kripke model, a function must in each world be interpreted by a total function (just evaluate  8x9y(f (x) = y); in a Beth model however one can `postpone' the assignment of outputs to inputs, and this allows for a particularly simple model for analysis. In order to use Kripke models for dealing with analysis one has to exploit the expanding of the domains, and this calls, in the case of arithmetic, immediately for non-standard numbers. Hardly natural! The above model for analysis has the drawback that its rst-order theory is classical, i.e. each classically true sentence of rst-order arithmetic holds in the model. Therefore the model cannot play the role of `standard model' of analysis. The model shows, however, that it is consistent to put an intuitonistic second-order theory on top of a classical rst-order theory. The problem of the `standard model' occurs already for rst order arithmetic. In HA one can show not only 8xy(x = y _ x 6= y), but also m = n , HA ` m  = n . So in a topological model for arithmetic, with only standard numbers we have [ t = s] = X or ?, i.e. atoms take only the values > or ?. Now a simple induction shows that [ '] takes the values > or ? for all sentences '. So we get full true, classical arithmetic. Therefore, in order to obtain intuitionistic features, say the failure of the principle of the excluded middle, we have to assume the presence of non-standard numbers. This all points towards serious limitations of the present semantic treatment of intuitionistic theories.

10.3 The Disjunction Property for Analysis We have seen that Kripke models may be `put together' as a means for proving metalogical results, e.g. the disjunction property (cf. p. 44). In view of the usefulness of Beth models for interpreting analysis, it would be convenient to have a similar operation in Beth semantics. Roughly speaking, one takes the disjoint union of two Kripke models and adds one bottom node (left-hand gure). The domain of the bottom node is contained in all domains of the Kripke models A and B. In the case of Beth models one would place the models A and B alternatingly on top of the linearly ordered set of natural numbers (right-hand gure). Here we run into diÆculties; what should the domain be in the nodes 0, 1, 2, : : : ? The semantics does not allow for non-constant domains, so we reach a deadend. Now our generalised semantics comes in handy, if we allow expanding domains then we can use this `gluing' technique. Let us outline how to

INTUITIONISTIC LOGIC



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4 A

B



 

95



A

3



B

2



A

1



0 obtain DP. Suppose T ` ' _ and T 6` '; T 6` , and B 6 . Find a domain that is contained in the bottom nodes of A and B (this is not always possible, it depends on the class of models of T !), and place this in the nodes 0; 1; 2; : : : of the co-called spine. Now 0 ' _ , so there is a bar B such that for each 2 B; ' or . The bar intersects the spine, say in n. If n ' then there is a copy of A above n in which ' is forced. Contradiction. Similarly for n . Hence T ` ' or T ` . The above gluing construction is particularly fruitful for analysis, (cf. Dalen [1984; 1986]). It yields simple proofs of the disjunction and existence property for various systems of analysis. The main problem is the de nition of the universe of sequences in the resulting model. One assigns to the nodes n on the spine, the set of nite sequences of length n. The sequences (in the sense of the model) in node n are then all possible extensions of these sequences to partial functions in higher nodes. J. Moschovakis had already established DP and EP for some systems by proof theoretical means in [Moschovakis, 1967]. The reason for dwelling on arithmetic and its extensions, in particular analysis, is that this hard core of mathematical logic is the ultimate testing ground for logical methods. Analysis is extremely important because it illustrates the typical consequences of intuitionism. In analysis one can most clearly see the con ict between the classical and the intuitionistic approaches. Brouwer used the following theorem to illustrate the typical

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properties of intuitionism: Each function from the closed interval [0,1] to R is uniformly continuous [1923]. From the intuitionistic principles we can derive similar results. The continuity principle SC! tells us that each function from sequences to numbers is continuous. however, classically one can de ne the function F such that

F ( ) =



0 if  contains a 0 1 otherwise:

This F cannot be continuous, because continuity would mean that the occurrence of a 0 in a sequence  can be predicted on the basis of an initial segment of  . Quod non. The con ict with classical mathematics is here particularly striking. In general, analysis and higher-order systems are the perfect grounds for demonstrating the proper character of intuitionism; it distinguishes itself from narrow constructivism and nitism by its embracing abstract notions. A topic that has been omitted altogether is second-order propositional logic. Whereas in classical logic this is a subject of great dullness (think of quanti cations over a set of two truth-values), it is not so in the intuitionistic version. In contrast to the classical system, the intuitionistic one is undecidable (cf. [Gabbay, 1981] for a systematic treatment of this topic).

10.4 Remarks on the Axiom of Choice In classical mathematics the axiom of choice used to be considered as something that, if it should hold at all, should hold globally (this is not to say that no re nements of AC have been considered, but that the foundational evidence seems to point that way (cf. [Shoen eld, 1967, p. 253]). In intuitionism there is fairly solid evidence for the validity of the principle of countable choice; let 8x9y'(x; y) be given for, say x and y ranging over natural numbers, then we have a proof of 8x9y'(x; y), i.e. a construction that provides for each x 2 N a proof of 9y'(x; y). this, in turn, means that we have a construction that yields a y and a proof of '(x; y). So we have a construction that for each x yields a y such that '(x; y) holds (i.e. has a proof). This construction provides a choice function f , such that 8x'(x; f (x)) holds. So AC-NN is valid on an intuitionistic interpretation of the logic. (In Martin-Lof's type theory is is actually provable.) Let us now consider AC-RN. The following is true: for each real number x there is a natural number y such that x < y (recall that x is given by a Cauchy sequence). If there were a choice function f such that x < f (x), then|by Brouwer's theorem (cf. [Heyting, 1956, p. 46], [Brouwer, 1981a, p. 80])|f has to be continuous. But a continuous function from R to N is constant. Contradiction.

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An inspection of the proof, as given for AC-NN, will show what has gone wrong. Our assumption reads in full: 8x 2 R ; 9y 2 N (x < y), so the proof interpretation tells us that for each choice sequence x of rational numbers (x) is a proof of `x is a Cauchy sequence ! 9y 2 N (x < y)', i.e. (x) applied to a proof of `x is a Cauchy sequence' yields a proof of

9y 2 N (x < y): Now, nishing the argument, we nd a choice function that depends on x and on a proof that x is a Cauchy sequence. But now we see that f is not extensional, i.e. x1 = x2 ! f (x1 ; 1 ) = f (x2 ; 2 ) fails, where i is a proof that xi is a Cauchy sequence. Therefore the continuity theorem was not applicable. The moral of this digression is that one has to spell out the assumption of AC in full. In case of AC-NN we are on safe ground because a natural number by virtue of its mode of generation carries its own proof that it is a natural number. The general axiom of choice is intuitionistically out of the question, as Diaconescu (cf. [Goldblatt, 1979]) has shown that it implies the excluded third. The following simple argument, due to Goodman and Myhill, proves Diaconescu's result. Let ' be any statement. Form the sets

A := fn 2 N j n = 0 _ (n = 1 ^ ')g; B := fn 2 N j n = 1 _ (n = 0 ^ ')g: We have 8X 2 fA; B g9y 2 N (y; X ). AC would supply us with a function f such that 8X 2 fA; B g(f (X ) 2 X ). Since f (X ) is a natural number, we get f (A) = f (B ) _ f (A) 6= f (B ). If F (A) = f (B ), then ' holds, and if f (A) 6= f (B ), then :' holds. For suppose ', then A = B (extensionally) so f (A) = f (B ). Contradiction. So the validity of AC for this particularly simple case implies ' _ :'. 11 THE CREATING SUBJECT In Brouwer's writings some explicit references to the agent of mathematical activity occur (1948) (cf. [Brouwer, 1975, p. 478]). The basic ideas were already present in his lectures in the late 1920s, cf. [Brouwer, 1992]; the publication was, however, postponed until after the second world war. In due time this practice has become known under the name `theory of the creating subject'. Brouwer introduced the creating subject for the purpose of establishing some stronger results in the area of so-called negative predicates. In particular he showed that inequality on the reals is strictly weaker than apartness: :8xy(x 6= y ! x#y).

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Kreisel, Kripke and others have analyzed the principles involved in those proofs (cf. [Kreisel, 1967]). The creating subject is assumed to operate in linear time of order-type !. It `experiences the truth' of statements ' at stages 0; 1; 2; : : : The exact nature of `experiencing the truth' is, of course, left open. One may think of `proving', `observing' or `knowing', etc. By suitably idealising the creating subject we may assume that: 1. it retains truths that have been experienced; 2. at each stage it knows ' or it does not know '. that is, `knowing ' at stage n' is decidable; 3. ' holds if it has been `experienced', by the creating subject. This is in perfect accordance with the intuitionistic dogma that mathematics has its seat in the human mind, and that the only way to establish something is to have a mental `proof' or `experience' for it. The converse can be defended under the purely solipsistic view that what is the case solely depends on mental experience of the (unique) creating subject. Then, if ' holds it follows that the creating subject has come to know it at some stage. If one allows for an intersubjective viewpoint, then the matter is less clear. The statement ' may hold without the creating subject (one of many) having established it. In this case it seems plausible that it is impossible that the creating subject will never experience '. The theory of the creating subject has been formalized by Kreisel [1967], in a theory containing at least (a fragment of) arithmetic, and a tensed modal operator x, to be read as `the creating subject knows (has evidence, a proof for, : : : etc.) at time x'. The principles under (1), (2) and (3) can now be formulated as 1. x ' ! x+y ' 2. x ' _ :x ' 3. ' $ 9xx '. In the intersubjective case (3) splits into the following parts x' ! ' and ' ! ::9xx ': For the applications that Brouwer had in mind the weaker reading suf ces. The justi cation for the solipsistic version, however, is more convincing. In principle there is no objection to iterate the operator , and in Brouwer's consistent view that re ection on one's own mental activity is possible, or even necessary it seems quite correct to do so. However, all problems that arise in and around predicativity, reappear here as well. In the following pages we will look at the full theory as given by (1), (2) and (3).

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11.1 Kripke's Schema If we have function variables available, then we can eliminate the modal operator and retain all its bene ts by keeping track of the knowledge of the creating subject by means of a function that registrates if the creating subject knows ' at stage x. De ne

 (x) =



0 if :x' 1 if x';

then by (3) ' ! 9xx ', so ' ! 9x (x) 6= 0. conversely 9x (x) = 6 0 9xx ' and hence 9x (x) 6= 0 ! '. This proves Kripke's Schema. KS

!

9 (' $ 9x (x) 6= 0):

By a similar argument one obtains the weak Kripke's Schema in the intersubjective case KS

9 ((9x (x) 6= 0 ! ') ^ (:' ! 8x (x) = 0)):

Kripke's Schema is used, for instance, for the construction of certain strong counterexamples (cf. [Hull, 1969]). We list some of them: The statements refer to the intuitionistic reals. 1. 2. 3.

:8xy((:x < y ^ x 6= y) ! y < x) :8xy(x 6= y ! x#y) :8xy(x = 6 y ! :x < y _ :y < x)

4. not every bounded set without points of accumulation is bounded in number (refutation of the Bolzano{Weierstrauss theorem). Note that these results are strong in comparison to the older results which only yielded `we cannot prove that : : : '. The rst of these strong counterexamples was presented by Brouwer in 1949. Myhill has shown that KS is inconsistent with the continuity principle for functions, which is a generalisation of SC:

8 9'(; ) ! 9F 8'(; F ( )); where F is a continuous function (cf. [Troelstra, 1969]). KS is however consistent with SC (Kroll (cf. [Grayson, 1981; Scowcroft, 1999])).

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11.2 Kriple's Schema and the Continuum Brouwer's strong refutations already show that the creating subject re nes our insight into the structure of the continuum. Further results in this area have been obtained in [van Dalen 1999A]. Brouwer had already shown that the continuum is indecomposable, in the sense that if R = A [ B and A \ B = ;, then A = R or A = ;. On the basis of KS it can be shown that all negative, dense subsets of R are likewise indecomposable (where X  R is negative if ::x 2 X ! x 2 X ). Hence, e.g., the irrationals and the notnot-rationals are indecomposable. These subsets are therefore connected in the topological sense, and they have dimension 1. This is in sharp contrast to te classical theory, where the irrationals are zero-dimensional. Kripke's schema also allows us to show a kind of converse to Brouwer's indecomposability theorem: KS + R is indecomposable ) there are no discontinuous functions on R. We will, by way of illustration, sketch the proof: let f : R ! R be discontinuous. It is no restriction to assume that f (0) = 0 and that f is discontinuous in 0. So there is a k and there are xn such that jxn j < 2 n and jf (xk )j > 2 k . Now consider the statement r 2 Q for an r 2 R. We apply KS to r 2 Q _ r 62 Q : 9 (9x (x) 6= 0 $ r 2 Q _ r 62 Q ). For convenience we assume that  is positive at most  once, with value 1. ( (k) = 0) De ne an = xxn ifif 8p knnand  (p) = 1 p

Clearly (an ) converges, say lim an = a. Now jf (an )j < 2 k or jf (an )j > 0, hence a 6= xn for all n, or a#0. The rst is impossible, since it would imply :(r 2 Q _ r 62 Q ). The latter inplies 9n (n) = 1, and hence r 2 Q _ r 62 Q . Since this holds for arbitrary r, we have got a decomposition of R. Contradiction. Therefore there are no discontinuous functions on R. Although the theory of the creating subject has a richer language it is actually conservative over the theory with Kripke's Schema ([van Dalen, 1978]).

11.3 The Interpretation of the Creating Subject We have already seen how to validate KS in the Beth model for analysis. A slight adaptation will provide an interpretation of the tensed modal `knowledge' operator. We de ne ~n k ' if for all m ~ on the bar for ~n of nodes of length k; m ~ ' (note that this bar may be below ~n). We'll check the axioms (2) and (3).

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n

 (2) For a given ' we have for each m ~ of length k; m ~ ' or m ~ 6 '. For such a m ~ we can conclude 8p~  m ~ ; p~ 6 n ' from m ~ 6 ', i.e. m ~ 6 ' ) m ~ :k '. So for each m~ on the bar of nodes of length k we have m~ k ' or m ~ :k '. Hence h i k ' _ :k '. (3) If ~n ' and with lth(~n) = k, then ~n k ' and hence ~n 9xx '. Conversely, if ~n 9xx ' then there is bar B for ~n such that for each m ~ 2B there is an k(m ~ ) with m ~ k '. Applying the above de nition and Lemma 11 we conclude m ~ ' for each m ~ 2 B. Applying Lemma 11 once more we get ~n '.

11.4 Kripke's Schema and a Representation of Sets of Natural Numbers Although we cannot use characteristic functions to represent sets, we can use Kripke's Schema to obtain a substitute. Let X be a set of natural numbers, then by KS

8x9 [x 2 X $ 9y(y = 0)]: (switching = and 6= is a harmless act). Applying the axiom of choice from numbers to functions, AC-NF, we get

98x[x 2 X $ 9y(hx; y i = 0]: So each set X can be represented by a sequence. This allows for a translation of second-order arithmetic into analysis with KS. Using this representation, there is a simple argument that deduces the Uniformity Principle,

8X 9x'(X; x) ! 9x8X'(X; x); from the Weak Continuity Principle,  = y ! '(; x)); 8 9x'(; x) ! 8 9xy8(y in the presence of KS, cf. [van Dalen, 1977].

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The theory of the creating subject has remained controversial until this day. The introduction of an element of subjectivity runs counter to the tradition of the exact sciences. It is, however, an unavoidable step in representing some of Brouwer's arguments. Evidently the theory, as it stands, is far from complete. Questions concerning the number of conclusions per step, or regarding disjunction (e.g. is n (' _ ) ! n ' _ n a valid principle?), remain unsettled, and evidence seems to be scarce. Dummett has investigated the subject [Dummett, 1977] and Posy has applied the theory in a case study of Brouwer's paper on virtual order [Posy, 1980], cf. also [Posy, 1976]. 12 THE LOGIC OF EXISTENCE For the practice of classical logic and mathematics it suÆces to consider only total operations. for example, consider the inverse-operation, a 1 . for real numbers a 1 exists if a#0, so we cannot apply the classical trick of de ning a 1 := 0 for the remaining a's. This should leave us with a partial function, since # is not a decidable relation on R .

a

1

=



1=a if a#0 0 if a = 0

(note that it could not possibly be total, (for then it had to be uniformly continuous on [0,1]). One could avoid the problem by only discussing multiplication, but that would be a sin against the time-honoured practice of mathematics. So we would prefer to have a 1 , even if it means allowing partially de ned terms and problems of existence. Traditionally the matter is dealt with in free logic (cf. Bencivenga's chapter on Free Logics in this Handbook); we will, however brie y discuss existence here since it comes up naturally in intuitionistic logic, and since it has surprising semantic aspects. The semantic aspects of partial elements can conveniently be demonstrated in any of the models introduced earlier. We will rst consider Kripke models. Elements occur in certain domains and not in others, so they have a natural mode of existence in Kripke models. We de ne  Ea i a 2 D(). E behaves as an ordinary predicate and we can handle it as usual. We may explicitly introduce the extent of a as follows. [ Ea] = f j a 2 D()g, clearly [ Ea] is an open set in the canonical topology. [ Ea] is that part of the underlying topological space where a exists; this explains the name partial element: [ Ea] need not be all of the space. Since in Beth models elements exist always (likewise in topological models), one has to consider the general models of Section 3 in order to introduce partial elements. There is a kind of paradigm for the semantics of partial elements: sheaves over topological spaces. Without going into technical details we will sketch

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b

a

a

b [[a = b]]

[[Ea]] the approach so that the reader can form an impression. Consider two topological spaces X and Y and continuous maps into Y de ned on open subsets of X . The reader may take R for X and Y . We de ne for such a map (section) a : [ Ea] = domain a. In order to get a reasonable realistic theory we also want to interpret equality of partial elements. A natural choice is [ a = b] = Intft 2 X j a(t) = b(t)g. Note that [ a = a] = [ Ea] and also [ a = b]  [ Ea] \ [ Eb] . Using our knowledge of the topological interpretation (i.e. the interpretation of the connectives), we see that a = b ! Ea ^ Eb is true. Equality satis es the laws a = b $ b = a and a = b ^ b = c ! a = c, since [ a = b] = [ b = a] and [ a = b] \ [ b = c]  [ a = c] , but not a = a. For [ a = a] 6= X , in general, we see that the presence of partial elements aects the theory of identity (cf. [Scott, 1979]). Of course propositional logic is not aected by the introduction of partial elements. It is predicate logic that requires attention. Turning Quine's dictum `existence = being quanti ed over' around, we stipulate that one can only quantify over existing elements. For existential quanti cation this makes sense, 9x'(x) means that there exists an element that satis es '. Adding `but it need not exist' would be plain cheating. For universal quanti cation we read 8x'(x) as `for any a picked from the domain '(a) holds', which commits us to existing elements (note that in classical logic 9 decides the matter for 8). The above is re ected in the axioms and rules of quanti cation.

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[Ex] .. . 8I '(x) 8x'(x)

8E

'(t) Et 9I 9x'(x)

8x'(x) Et '(t)

9E 9x'(x)

['(x); Ex] .. .  

(with the obvious restrictions). In a Hilbert-type system we retain the rules 8I and 9E in the form

^ Ex ! '(x) '(x) ^ Ex !  ! 8x'(x) 9x'(x) ! 

and add the axioms

8x'(x) ^ Et ! '(t) '(t) ^ E (t) ! 9x'(x): As sketched above one also has to revise the identity rules. There are two possible notions of identity, a strong one, where one requires both elements to exist, and a weaker one, where one automatically equates elements there where they do not exist. The above equality, =, is the strong one. The weaker one can be de ned by a  b := Ea _ Eb ! a = b. The notions are interde nable as is shown by the following fact

a = b $ a  b ^ Ea ^ Eb: this provides us with the following axioms

x = x $ Ex x=y!y=x x = y ^ y = z ! x = z: One has to select carefully the right formulation in cases involving equivalence or existence. For example, x  x ! Ex is false, but 8x(x  x ! Ex) is correct, for it is equivalent on logical grounds to 8x(Ex ! (x  x ! Ex)). The theory of partial elements is the ideal setting for the introduction of a description operator. For Ix:'(x) is just a term, it has no existential import; it has to satisfy a certain formula when and where it uniquely exists. For instance, it is axiomatized by

8y[y = Ix:'(x) $ 8x('(x) $ x = y)]:

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In our model we just look for those nodes where a unique element is forced to satisfy '(x) and we put them together to one partial element. Let  '(0)^9!x'(x); '(1)^9!'(x); '(0)^'(1), then Ix:'(x) is interpreted in the model as being 0 in  and 1 in and unde ned (non-existent) in . So Ix:'(x) is locally equal to given elements. This being `locally' equal to something, or `locally' true, etc. is a characteristic consequence of the forcing conditions of Beth semantics (cf. Section 3 above). In a theory with identity we want to be able to replace equals by equals. Should one restrict this to the strong equality? For general (extensional) formulas this seems too restrictive, even weak equality would preserve properties, so we formulate the axiom as

x  y ^ '(x) ! '(y): Following Scott [1979] we call `=' identity and `' equivalence. The reader is referred to this basic paper, and to [Troelstra and van Dalen, 1988, p. 50], for more information on existence and partial elements. We add one more remark: if the theory has function symbols, then the following can be said: if one gets an output, then there must have been an input, or more formally Ef (x) ! Ex. Reversing the arrow we get the condition for a total function: Ex ! Ef (x). One should observe that quanti ers aect existence; they are not neutral ' is `for as one would maybe expect. the interpretation of `8x all x that exist '. One can actually prove 8x'(x) $ 8x(Ex ! '(x)). We now return to the model of continuous maps from R to R . We can operate on these functions in a pointwise manner, e.g. add multiply etc. By de nition we have [ f  g] := Intfx 2 R j f (x) = g(x) _ f (x) and g(x) are unde nedg: For quanti cation we put: [ 8x'(x)]] = Int \f [ Ef ! '(f )]] [ 9x'(x)]] = [f [ Ef ^ '(f )]]: Now it is a matter of simple veri cation to check the axioms listed above. Let us look at the description operator in this model.

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b a

c c

[ a = b] [ Ea]

1

[ Eb]

c

1

The elements of the model have the convenient property, peculiar to sheaves, that they can be glued together if they coincide on overlapping domains (elementary analysis). Moreover, one can always restrict a function to a smaller open domain. We use these properties to interpret Ix:'(x) as the union of all continuous functions restricted to the part where they uniquely satisfy '(x), more formal [f f  [ 8x('(x) $ f = x)]], where f  U stands for the subfunction of f obtained by restriction of f to U . Applying this to the inverse, we put h 1 = Ix:(xh = 1). Putting together all small functions, that locally act as in inverses, we obtain a function de ned on the subdomain of h obtained by leaving out the zero's of h. Note that the model once more demonstrates the necessity of strengthening the equality relation for the existence of inverses. Note that [ f 6= 0]] = Int[[f = 0]]c = Int(Intfx j f (x) = 0g)c = R but [ 9x(xf = 1)]] = R nf0g. So [ f 6= 0 ! 9x(xf = 1)]] 6= R .

y=x f Therefore we use the apartness relation, #, [ f #g] := fx j f (x) 6= g(x) andf (x) and g(x) are de nedg: Now we get [ f #0 ! 9x(xf = 1)]] = R (observe that this amounts to [ f #0]]  [ 9x(xf = 1)]]). After Fourman had developed the sheaf interpretation for the case of topological spaces, Fourman and Scott generalised the approach to sheaves over complete Heyting algebra's (so-called -sets), this approach is to be found in their paper of 1979. At that time there had already been done

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a great deal in categorical logic, after the pioneering work of W. Lawvere. This topic has reached a size and technical re nement that places it utterly beyond the present book. For an introduction the reader is referred to Goldblatt's book [1979], Fourman and Scott [1979], Grayson [1984], [MacLane and Moerdijk, 1992; McLarty, 1992]. The generalisations of sheaves over topological spaces (in particular over sites) have provided models for various kinds of choice sequences (cf. [Hoeven and Moerdijk, 1984]).

Recommended Reading Beeson [1985], Bishop and Bridges [1985], Brouwer [1975; 1981], van Dalen [1973; 1999b], Dummett [1977]. Fraenkel et al. [1973], Goldblatt [1979], Heyting [1956], Troelstra [1969; 1977] and Troelstra and van Dalen [1988], Philosophica Mathematica 6, 1998. University of Utrecht.

BIBLIOGRAPHY [Aczel, 1968] P. Aczel. Saturated intuitionistic theories. In [Schmidt, Schutte and Thiele, 1968, pp. 1{11]. [Artemov, 2001] S. Artemov. Explicit provability: the intended semantics for intuitionistic and modal logic. Bull. Symb. Logic, 7, 1{36, 2001. [van Atten and van Dalen, forthcoming] M. van Atten and D. van Dalen. Arguments for Brouwer's continutity principle. Forthcoming. [Avigad and Feferman, 1998] Avigad and S. Feferman. Godel's functional (`Dialectica') interpretation. In Handbook of Proof Theory, S. R. Buss, ed. pp. 337{406. Elsevier, Amsterdam, 1998. [Barendregt, 1984] H.P. Barendregt. The Lambda Calculus. Its Syntax and Semantics. North-Holland, Amsterdam, 1984, 2nd reprint edition in paperback, 1997. [Beeson, 1979] M. Beeson. A theory of constructions and proofs. Preprint No 134. Dept of Maths, Utrecht University, 1979. [Beeson, 1985] M. Beeson. Foundations of Constructive Mathematics. Metamathematical Studies. Springer Verlag, Berlin, 1985. [Bishop and Bridges, 1985] E. Bishop and D. Bridges. Constructive Analysis, Springer, Berlin, 1985. [Brouwer, 1907] L.E.J. Brouwer. Over de Grondslagen der Wiskunde. Thesis, Amsterdam. Translation `On the foundations of mathematics', in [Brouwer, 1975, pp. 11{101]. New edition in [Brouwer, 1981]. [Brouwer, 1908] L.E.J. Brouwer. De onbetrouwbaarheid der logische principes. Tijdschrift voor wijsbegeerte, 2, 152{158. Translation `The unreliability of the logical principles' in [1975, pp. 107{111]. Also in [1981]. [Brouwer, 1918] L.E.J. Brouwer. Begrundung der Mengenlehre unabhangig vom logischen Satz vom augsgeschlossenen Dritten. I. Koninklijke Nederlandse Akademie van Wetenschappen Verhandelingen le Sectie 12, no 5, 43 p. Also in [Brouwer, 1975, pp. 150{190]. [Brouwer, 1975] L.E.J. Brouwer. Collected Works, I. A. Heyting, ed. North Holland, Amsterdam, 1975.

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[Brouwer, 1981] L.E.J. Brouwer. Over de Grondslagen der Wiskunde, Aangevuld met Ongepubliceerde Fragmenten, Correspondentie met D.J. Korteweg. Recensies door G. Mannoury, etc. D. van Dalen, ed. Mathematisch Centrum, varia 1, Amsterdam, 1981. [Brouwer, 1981a] L.E.J. Brouwer. Brouwer's Cambridge Lectures on Intuitionism. D. van Dalen, ed. Cambridge University Press, Cambridge, 1981. [Brouwer, 1992] L.E.J. Brouwer. Intuitionismus, D. van Dalen, ed. BI-Wissenschaftsverlag, Mannheim, 1992. [Burgess, 1981] J.P. Burgess. The completeness of intuitionistic propositional calculus for its intended interpretation. Notre Dame J. Formal Logic, 22, 17{28, 1981. [Chang and Keisler, 1973] C.C. Chang and H.J. Keisler. Model Theory. North-Holland, Amsterdam, 1973. [Curry and Feys, 1958] H.B. Curry and R. Feys. Combinatory Logic I. North-Holland, Amsterdam, 1958. [van Dalen, 1973] D. van Dalen. Lectures on intuitionism. In [Mathias and Rogers, 1973, pp. 1{94]. [van Dalen, 1974] D. van Dalen. A model for HAS. A topological interpretation of the theory of species of natural numbers. Fund Math, 82, 167{174, 1974. [van Dalen, 1977] D. van Dalen. The use of Kripke's Schema as a reduction principle. Journal of Symbolic Logic, 42, 238{240, 1977. [van Dalen, 1978] D. van Dalen. An interpretation of intuitonistic analysis. Ann. Math. Logic, 13, 1{43, 1978. [van Dalen, 1984] D. van Dalen. How to glue analysis models. Journal of Symbolic Logic, 49, 1339{1349, 1984. [van Dalen, 1986] D. van Dalen. Glueing of analysis models in an intuitionistic setting. Studia Logica, 45, 181{186, 1986. To appear. [van Dalen, 1993] D. van Dalen. The continuum and rst-forder logic. JSL, 57, 1417{ 1424, 1993. [van Dalen, 1995] D. van Dalen. Hermann Weyl's intuitionistic mathematics. Bull. Symb. Logic, 4, 145{169, 1995. [van Dalen, 1997] D. van Dalen. Logic and Structure, 3rd edn. Springer Verlag, Berlin, 1997. [van Dalen, 1999a] D. van Dalen. From Brouwerian counterexamples to the creating subject. Studia Logica, 62, 305{314, 1999. [van Dalen, 1999b] D. van Dalen. Mystic, Geometer and Intuitionist. The Life of L.E.J. Brouwer. Volume 1: The Dawning Revolution, Oxford University Press, 1999. [van Dalen, 2000] D. van Dalen. Development of Brouwer's Intuitionism. In proof Theory. History and Philosophical Signi cance, V. F. Hendricks, S. A Pedersen and K. F. Jrgensen, eds. pp. 117{152. Kluwer, Dordrecht, 2000. [van Dalen and Statman, 1979] D. van Dalen and R. Statman. Equality in the presence of apartness. In [Hintikka, Niiniluoto and Saarinen, 1979, pp. 95{118]. [Davis, 1965] M. Davis, ed. The Undecidable. Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions. Raven Press, New York, 1965. [Demuth and Kucera, 1979] O. Demuth and A. Kucera. Remarks on constructive mathematical analysis. In Logic Colloquium `78, M. Boa, D. van Dalen and K. McAloon, eds. pp. 81{130. North Holland, Amsterdam, 1979. [Dummett, 1973] M. Dummett. The philosophical basis of intuitionistic logic. In Logic Colloquium `73, H. E. Rose and J. C. Shepherdson, eds. pp. 5{40, North-Holland, Amsterdam, 1973. Also in M. Dummett, Truth and Other Enigmas, pp. 215{247, Duckworth, London, 1978. [Dummett, 1977] M. Dummett. Elements of Intuitionism. Oxford University Press, Oxford, 1977. [Dyson and Kreisel, 1961] V.H. Dyson and G. Kreisel. Analysis of Beth's Semantic Construction of Intuitionistic Logic. Technical Report no. 3. Appl. math. and statistics lab. Stanford University, 65 pp. 1961. [Feferman, 1979] S. Feferman. Constructive theories of functions and classes. In Logic Colloquium `78, M. Boa, D. van Dalen and K. McAloon, eds. pp. 159{224. North Holland, Amsterdam, 1979.

INTUITIONISTIC LOGIC

109

[Fenstad, 1971] J.E. Fenstad, ed. Proceedings of the Second Scandinavian Logic Symposium, North- Holland, Amsterdam, 1971. [Fine, 1970] K. Fine. An intermediate logic without the nite model property. Unpublished, 1970. [Fitting, 1969] M.C. Fitting. Intuitionistic Logic, Model Theory and Forcing. NorthHolland, Amsterdam, 1969. [Fourman, 1982] M. Fourman. Notions of choice sequence. In [Troelstra and van Dalen, 1982, pp. 91{106]. [Fourman and Scott, 1979] M.P. Fourman and D. S. Scott. Sheaves and logic. In [Fourman, Mulvey and Scott, 1979, pp. 302{401]. [Fourman, Mulvey and Scott, 1979] M.P. Fourman, C.J. Mulvey and D.S. Scott. Applications of Sheaves. Proceedings Durham 1977. Lecture Notes No 753, Springer Verlag, Berlin, 1979. [Fraenkel et al., 1973] A. Fraenkel, Y. Bar-Hillel, A. Levy and D. van Dalen. Foundations of Set Theory. North-Holland, Amsterdam, 1973. [Friedman, 1975] H. Friedman. The disjunction property implies the numerical existence property. Proc Nat Acad. Sci., 72, 2977{2878, 1975. [Friedman, 1977] H. Friedman. The intuitionistic completeness of intuitionistic logic under Tarskian semantics. Abstract, SUNY at Bualo, 1977. [Friedman, 1977a] H. Friedman. New and old results on completeness of HPC. Abstract, SUNY at Bualo, 1977. [Friedman, 1977b] H. Friedman. Classically and intuitionistically provably recursive functions. In Higher Set Theory, pp. 21{27, Springer Verlag, Berlin, 1977. [Gabbay, 1970] D.M. Gabbay. Decidability of the Kreisel{Putnam system. Journal of Symbolic Logic, 35, 431{437, 1970. [Gabbay, 1971] D.M. Gabbay. Semantic proof of the Craig Interpolation theorem for intuitionistic logic and extensions I, II. In Logic Colloquium 69, (eds R.O. Gandy and C.M.E. Yates), pp. 391{410. North-Holland, Amsterdam, 1971. [Gabbay, 1977] D.M. Gabbay. On some new intuitionistic propositional connectives. Studia Logica, 36, 127{139, 1977. [Gabbay, 1981] D.M. Gabbay. Semantical Investigations in Heyting's Intuitionistic Logic. D. Reidel, Dordrecht, 1981. [Gabbay and de Jongh, 1974] D.M. Gabbay and D.H. de Jongh. Sequences of decidable nitely axiomatisable intermediate logics with the disjunction property. Journal of Symbolic Logic, 39, 67{79, 1974. [Gallier, 1995] J. Gallier. On the correspondence between proofs and -terms. In The Curry{Howard Isomorphism, (ed. Ph. de Groote), pp. 55{138. Cahiers de centre de Logique, 8. Academia Louvain-la-Neuve, 1995.  [Gentzen, 1933] G. Gentzen. Uber das Verhaltnis zwischen intuitionistischer und klassischer Arithmetik. English translation in [Szabo, 1969, pp. 53{67]. [Gentzen, 1935] G. Gentzen. Untersuchen uber das logische Schliessen I, II. Math. Zeitschrift, 39, 176{210, 405{409, 1935. [Girard, 1971] J.Y. Girard. Une extension de l'interpretation de Godel a l'analyse et son application a l'elimination des coupures dans l'analyse et lat theorie des types. In [Fenstad, 1971, pp 63{92]. [Girard et al., 1989] J.Y. Girard, P. Taylor and Y. Lafont. Proofs and Types. Cambridge University Press, Cambridge, 1989. [Glivenko, 1929] V. Glivenko. Sur quelques points de la logique de M. Brouwer. Academie Royale de Belgique, Bull. de la class des science, (5), Vl. 15, pp. 183{188, 1929. [Goad, 1978] C.A. Goad. Monadic in nitary propositional logic: a special operator. Repts Math. Logic, 10, 43{50, 1978. [Godel, 1932] K. Godel. Zum intuitionistischen Aussagenkallkul. Akademie der Wissenschaften in Wien. Math. naturwiss. Klasse. Anzeiger, 69, 65{66, 1932. Also in Ergebnisse eines Math. Koll., 4, 42. [Godel, 1933] K. Godel. Zur intuitionistischen Arithmetik und Zahlentheorie. Ergebnisse eines mathematischen Kolloquiums, 4, 34{38, 1933. English translation in [Davis, 1965, pp. 75{81]. Cf. Journal of Symbolic Logic, 31, 484{494, 1966.

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 [Godel, 1958] K. Godel. Uber eine bisher noch nicht benutzte Erweiterung des niten Standpunktes. Dialectica, 12, 280{287. English translation in Journal of Philosophical Logic, 9, 133{142, 1980. [Goldblatt, 1979] R. Goldblatt. Topoi. The Categorical analysis of Logic. North Holland, Amsterdam, 1979. Revised edition, 1984. [Gornemann, 1971] S. Gornemann. A logic stronger than intuitionism. Journal of Symbolic Logic, 36, 249{261, 1971. [Grayson, 1981] R. Grayson. Concepts of general topology in constructive mathematics and in sheaves. Ann. Math. Logic, 20, 1{41, 1981. [Grayson, 1984] R. Grayson. Heyting-valued semantics. In Logic Colloquium 1982, G. Lolli, G. Longo and A. Marcja, eds. pp. 181{208. North Holland, Amsterdam, 1984. [Grzegorczyk, 1964] A. Grzegorczyk. A philosophically plausible interpretation of intuitionistic logic. Indagationes Mathematicae, 26, 569{601, 1964. [Harrop, 1958] R. Harrop. On the existence of nite models and decision procedures for propositional calculi. Proc. Camb. Phil. Soc, 54, 1{13, 1958. [Hartog, 1978] W. den Hartog. A Proof Theoretic Study. The Theory of Pseudo-order as a Conservative Extension of the Theory of Apartness. Dept of Math. Rijksuniversiteit Utrecht, Preprint no. 77, 1978. [Heijenoort, 1967] J. van Heijenoort. From Frege to Godel, a Source Book in Mathematical Logic 1879- -1931. Harvard University Press, Cambridge, MA, 1967. [Heyting, 1930] A. Heyting. Die formalen Regeln der intuitionistischen Logik. Sitzungsberichte der preussischen Akademie von Wissenschaften, pp. 42{56, 1930. `Die formalen Regeln der intuitionistischen Mathematik', Ibid. pp. 57{ 71, 158{169. Also (partly) in Two Decades of Mathematics in the Netherlands, Amsterdam, 1978. [Heyting, 1934] A. Heyting. Mathematische Grundlagenforschung. Intuitionismus Beweistheorie. Springer, Berlin, 1934. [Heyting, 1956] A. Heyting. Intuitionism. An Introduction. North Holland, Amsterdam, 1956. [Hintikka, Niiniluoto and Saarinen, 1979] J. Hintikka, I. Niiniluoto and E. Saarinen, eds. Essays on Mathematical and Philosophical logic, D. Reidel, Dordrecht, 1979. [Hoeven and Moerdijk, 1984] G. van Hoeven and I. Moerdijk. Sheaf models for choice sequences. Annals of Pure and Applied Logic, 27, 63{107, 1984. [Hosoi, 1967] T. Hosoi. On intermediate logics, I. J. Fac. Sci. Univ. of Tokyo, 14, 293{ 312, 1967. [Howard, 1980] W. Howard. The formulae-as-types notion of construction. In [Seldin and Hindley, 1980, pp. 479{490]. [Hughes and Cresswell, 1968] G.E. Hughes and M.J. Cresswell. An Introduction to Modal Logic. Methuen, London, 1968. [Hull, 1969] R.C. Hull. Counterexamples in intuitionistic analysis using Kripke's schema. Z. Math. Logik und Grundlagen der Math, 15, 241{246, 1969. [Hyland, 1982] M. Hyland. The eective topos. In [Troelstra and van Dalen, 1982, pp. 165{216]. [Hyland, Johnstone and Pitts, 1980] M. Hyland, P.T. Johnstone and A.M. Pitts. Tripos theory. Math. Proc. Cambr. Phil. Soc., 88, 205{232, 1980. [Jankov, 1968] V.A. Jankov. Constructing a sequence of strongly independent superintuitionistic propositional calculi. Soviet Math. Dok, 9, 806{807, 1968. [Jaskowski, 1936] S. Jaskowski. Recherches sur le systeme de la logique intuitioniste.

Actes du Congres Intern. de Phil. Scienti que. VI. Phil des mathematiques, Act. Sc. et Ind 393. pp. 58{61. Paris, 1936.

[Johansson, 1936] I. Johansson. Der Minimalkalkul, ein reduzierter intuitionistischer Formalismus. Compositio Math, 4, 119{136, 1936. [Johnstone, 1982] P. Johnstone. Stone Spaces. Cambridge University Press, Cambridge, 1982. [Jongh, 1980] D.H. de Jongh. A class of intuitionistic connectives. In The Kleene Symposium, J. Barwise, H. J. Keisler and K. Kunen, eds. pp. 103{112. North Holland, Amsterdam, 1980. [Jongh and Smorynski, 1976] D.H. de Jongh and C. Smorynski. Kripke models and the intuitionistic theory of species. Ann. Math. Logic, 9, 157{186, 1976.

INTUITIONISTIC LOGIC

111

[Kino, Myhill and Vesley, 1970] A. Kino, J. Myhill and R.E. Vesley. Intuitionism and Proof Theory. Proceedings of the Summer Conference at Bualo, New York, 1968. North Holland, Amsterdam, 1970. [Kleene, 1952] S.C. Kleene. Introduction to Meta-mathematics. North Holland, Amsterdam, 1952. [Kleene, 1973] S.C. Kleene. Realisability: A retrospective survey. In [Mathias and Rogers, 1973, pp. 95{112]. [Kleene and Vesley, 1965] S C. Kleene and R.E. Vesley. The Foundations of Intuitionistic Mathematics, especially in Relation to Recursive Functions. North Holland, Amsterdam, 1965. [Klop, 1980] J.W. Klop. Combinatory Reduction Systems. Thesis, Rijks Universiteit Utrecht. Also MC Tract 127. Math. Centre Amsterdam, 1980. [Kolmogorov, 1925] A.N. Kolmogorov. On the principle of the excluded middle, Russian. Matematiceski Sbornik, 32, 646{667, 1925. English translation in [Heijenoort, 1967, pp. 414{437]. [Kreisel, 1958] G. Kreisel. A remark on free choice sequences and the topological completeness proofs. J. Symbolic Logic, 23, 369{388, 1958. [Kreisel, 1959] G. Kreisel. Interpretations of analysis by means of constructive functionals of nite type. In Constructivity in Mathematics, A. Heyting, ed. pp. 101{ 128. North Holland, Amsterdam, 1959. [Kreisel, 1962] G. Kreisel. On weak completenes of intuitionistic predicate logic. JSL, 27, 139{158, 1962. [Kreisel, 1965] G. Kreisel. Mathematical Logic. In Lectures on Modern Mathematics III, T. L. Saaty, ed. pp. 95{195. Wiley & Sons, New York, 1965. [Kreisel, 1967] G. Kreisel. Informal rigour and completeness proofs. In Problems in the Philosophy of Mathematics, I. Lakatos, ed. North Holland, Amsterdam, 1967. [Kreisel, 1970] G. Kreisel. Church's Thesis, a kind of reducibility axiom of constructive mathematics. In [Kino, Myhill and Vesley, 1970, pp. 121{150]. [Kreisel and Putnam, 1957] G. Kreisel and H. Putnam. Eine Unableitbarkeitsbeweismethode fur den intuitionistischen Aussagenkalkul. Archiv. f. math. Logik, 3, 74{78, 1957. [Kreisel and Troelstra, 1970] G. Kreisel and A.S. Troelstra. Formal systems for some branches of intuitonistic analysis. Ann. Math. Logic, 1, 229{387, 1970. [Kripke, 1965] S. Kripke. Semantical analysis of intuitionistic logic I. In Formal Systems and Recursive Functions, J. Crossley and M. Dummett, eds. pp. 92{129. North Holland, Amsterdam, 1965. [Leivant, 1976] D. Leivant. Failure of completeness properties of intuitionistic predicate logic for constructive models. Ann. Sc. Univ. Clermont. Ser Math., 13, 93{107, 1976. [Leivant, 1985] D. Leivant. Syntactic translations and provably recursive functions. Journal of Symbolic Logic, 50, 682{688, 1985. [Lemmon and Scott, 1966] E. Lemmon and D.S. Scott. Intensional logics. Published in 1977 as An Introduction to Modal Logic, K. Segerberg, ed. Blackwell, Oxford, 1966. [Lifschitz, 1969] V.A. Lifschitz. Problems of decidability for some constructive theories of equalities. In Studies in Constructive Mathematics and Mathematical Logic, A.O. Slisenko, ed. Consultants Bureau, New York, 1969. [MacLane and Moerdijk, 1992] S. MacLane and I. Moerdijk. Sheaves in Geometry and Logic, Springer, Berlin, 1992. [McCarty, 1988] D.C. McCarty. Constructive validity is nonarithmetic. JSL, 53, 1036{ 1041, 1988. [McKinsey, 1939] J.C.C. McKinsey. Proof of the independence of the primitive symbols of Heyting's calculus of propositions. JSL, 4, 155{158, 1939. [McLarty, 1992] C. McLarty. Elementary Categories, Elementary Toposes. Oxford University Press, 1992. [Markov, 1950] A.A. Markov. Konstruktivnaja logika. Usp. Mat. Nauk, 5, 187{188, 1950. [Martin-Lof, 1971] P. Martin-Lof. Hauptsatz for the theory of species. In [Fenstad, 1971].

112

DIRK VAN DALEN

[Martin-Lof, 1977] P. Martin-Lof. Constructive mathematics and computer programming. In Logic, Methodology and the Philosophy of Science VI, L.J. Cohen, J. Los, H. Pfeier and K.P. Podewski, eds. pp. 153{179. North Holland, Amsterdam, 1982. [Martin-Lof, 1984] P. Martin-Lof. Intuitionistic Type Theory. Notes by G. Sambin of a series of lectures given in Padova, June 1982. Bibliopolis, Naples, 1984. [Mathias and Rogers, 1973] A.R.D. Matias and H. Rogers, Jr. Cambridge Summer School in Mathematical Logic, Springer Lecture Notes, Vol. 337, Berlin, 1973. [Maximova, 1977] L.L. Maximova Craig interpolation theorem and amalgamable varieties. Soviet Math. Dok., 18, 1550{1553, 1977. [Minc, 1966] G. Minc. Skolem's method of elimination of positive quanti ers in sequential calculi. Dokl. Akad. Nauk SSR, 169, 861{864, 1966. [Minc, 1974] G. Minc. On E-theorems. (In Russian). Investigation in constructive mathematics VI. Zapisky Nauk. Sem. Leningrad Steklov Inst., 40, 110{118, 1974. [Moschovakis, 1967] J. Moschovakis. Disjunction and existence in formalised intuitionistic analysis. In Sets, Models and Recursion Theory, J. Crossley, ed. pp. 309{31. North Holland, Amsterdam, 1967. [Moschovakis, 1973] J. Moschovakis. A topological interpretation for second-order intuitionistic arithmetic. Comp. Math., 26, 261{276, 1973. [Moschovakis, 1980] J. Moschovakis. A disjunctive decomposition theorem for classical theories. In Constructive Mathematics, F. Richman, ed. pp. 250{259. Springer Verlag, Berlin, 1980. [Negri and von Plato, 2001] S. Negri and J. von Plato. Structural Proof Theory, Cambridge University Press, Cambridge, 2001. [Nelson, 1949] D. Nelson. Constructible falsity. Journal of Symbolic Logic, 14, 16{26, 1949. [Nishimura, 1966] T. Nishimura. On formulas of one variable in intuitionistic propositional calculus. Journal of Symbolic Logic, 25, 327{331, 1966. [Ono, 1972] H. Ono. Some results on intermediate logics. Publ. RIMS, pp. 117{130, Kyoto University 9, 1972. [Ono, 1973] H. Ono. A study of intermediate predicate logics. Publ. RIMS, pp. 619{649. Kyoto University 8, 1973 [Ono, 1985] H. Ono. Semantical analysis of predicate logics without the contraction rule. Studia Logica, 44, 187{196, 1985. [Ono and Komori, 1985] H. Ono and Y. Komori. Logics without the contraction rule. Journal of Symbolic Logic, 50, 169{201, 1985. [Posy, 1976] C.J. Posy. Varieties of indeterminacy in the theory of general choice sequences. J. Phil. Logic, 5, 91{132, 1976. [Posy, 1977] C.J. Posy. The theory of empirical sequences. J. Phil. Logic, 6, 47{81, 1977. [Posy, 1980] C.J. Posy. On Brouwer's de nition of unextendable order. History and Phil. Logic, 1, 139{149, 1980. [Pottinger, 1976] G. Pottinger. A new way of normalising intuitionist propositional logic. Studia Logica, 35, 387{408, 1976. [Prawitz, 1965] D. Prawitz. Natural Deduction. A Proof Theoretical Study. Alqvist & Wiksell, Stockholm, 1965. [Prawitz, 1970] D. Prawitz. Some results for intuitionistic logic with second-order quanti cation rules. In [Kino, Myhill and Vesley, 1970, pp. 259{270]. [Prawitz, 1971] D. Prawitz. Ideas and results in proof theory. In [Fenstad, 1971]. [Prawitz, 1977] D. Prawitz. Meanings and proofs: on the con ict between classical and intuitionistic logic. Theoria, 43, 2{40, 1977. [Prawitz, 1979] D. Prawitz. Proofs and the meaning and the completeness of the logical constants. In [Hintikka, Niiniluoto and Saarinen, 1979, pp. 25{40]. [Prawitz and Malmnas, 1968] D. Prawitz and P. Malmnas. A survey of some connections between classical, intuitionistic and minimal logic. In [Schmidt, Schutte and Thiele, 1968, pp. 215{229]. [Rasiowa, 1974] H. Rasiowa. An algebraic Approach to Non-Classical Logics. North Holland, Amsterdam. 1974. [Rasiowa and Sikorski, 1963] H. Rasiowa and R. Sikorski. The Mathematics of Metamathematics, Panstwowe Wydawnictwo Naukowe Warszawa, 1963.

INTUITIONISTIC LOGIC

113

[Rauszer, 1980] C. Rauszer. An algebraic and Kripke-style approach to a certain extension of intuitionistic logic. Dissertationes Mathematicae, CLXVII, Warszawa, 67 pp. 1980. [Rautenberg, 1979] W. Rautenberg. Klasische und nichtklassische Aussangenlogik, Vieweg & Sohn, Braunschweig/Wiesbaden, 1979. [Renardel de Lavalette, 1981] G.R. Renardel de Lavalette. The interpolation theorem in fragments of logic. Indag. Math., 43, 71{86, 1981. [Schmidt, Schutte and Thiele, 1968] H. A. Schmidt, K. Schutte and H.J. Thiele, eds. Contributions to Mathematical Logic. North Holland, Amsterdam, 1968. [Schroeder-Heister, 1984] P. Schroeder-Heister. A natural extension of natural deduction. Journal of Symbolic Logic, 49, 1284{1300, 1984. [Schutte, 1962] K. Schutte. Der Interpolationssatz der intuitionistischen Predikatenlogik. Math. Ann., 148, 192{200, 1962. [Schutte, 1968] K. Schutte. Vollstandige Systeme modaler und intuitionistischer Logic, Ergebnisse der Mathematik und ihrer Grenzgebiete, 42, Springer Verlag, Berlin, 1968. [Scott, 1968] D.S. Scott. Extending the topological interpretation to intuitionistic analysis II. In [Kino, Myhill and Vesley, 1970, pp. 235{255]. [Scott, 1979] D.S. Scott. Identity and existence in intuitionistic logic. In [Fourman, Mulvey and Scott, 1979, pp. 660{696]. [Scowcroft, 1999] P. Scowcroft. Some purely topological models for intuitionistic analysis. Ann. Pure Appl. Logic, 98, 173{216, 1999. [Segerberg, 1968] K. Segerberg. Propositional logics related to Heyting's and Johannsson's. Theoria, 34, 26{61, 1968. [Seldin and Hindley, 1980] J.P. Seldin and J.R. Hindley. To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, Academic Press, London, 1980. [Shoen eld, 1967] J.R. Shoen eld. Mathematical Logic, Addison Wesley, Reading, MA, 1967. Re-issued in paperback by A.K. Peters, 2001. [Smorynski, 1973] C. Smorynski. Investigation of Intuitionistic Formal Systems by Means of Kripke Models. Diss. Univ. of Illinois, 1973. [Smorynski, 1973a] C. Smorynski. Elementary intuitionistic theories. Journal of Symbolic Logic, 38, 102{134, 1973. [Smorynski, 1977] C. Smorynski. On axiomatising fragments. Journal of Symbolic Logic, 42, 530{544, 1977. [Smorynski, 1978] C. Smorynski. The axiomatisation problem for fragments. Ann. Math. Logic, 14, 193{221, 1978. [Smorynski, 1982] C. Smorynski. Non-standard models and constructivity. In [Troelstra and van Dalen, 1982, pp. 459{464]. [Sommaruga, 2000] G. Sommaruga. History and Philosophy of Constructive Type theory. Kluwer, Dordrecht, 2000. [Stein, 1980] M. Stein. Interpretations of Heyting's arithmetic. An analysis by means of a language with set symbols. Ann. Math. Logic, 19, 1{31, 1980. [Stein, 1981] M. Stein. A general theorem on existence Aheorems. Zeit. Math. Logik Grundl. Math, 27, 435{452, 1981. [Sundholm, 1983] G. Sundholm. Constructions, proofs and the meanings of the logical constants. J. Phil. Logic, 12, 151{172, 1983. [Swart, 1976] H.C.M. de Swart. Another intuitionistic completeness proof. Journal of Symbolic Logic, 41, 644{662, 1976. [Szabo, 1969] M.E. Szabo, ed. The Collected Papers of Gerhard Gentzen. North Holland, Amsterdam, 1969. [Tait, 1975] N. Tait. A realisability interpretation of the theory of species. In Logic Colloquium, R. Parikh, ed. pp. 240{251. Springer Verlag, Berlin, 1975. [Takahashi, 1970] M. Takahashi. Cut-elimination theorem and Brouwerian-valued models for intuitionistic type theory. Comment. Math. Univ. St Pauli, 19, 55{72, 1970. [Thomason, 1969] R. H. Thomason. A semantical study of constructible falsity. Z. Math. Logik und Grundlagen der Mathematik, 15, 247{257, 1969. [Troelstra, 1969] A.S. Troelstra. Principles of Intuitionism. Springer Verlag, Berlin, 1969.

114

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[Troelstra, 1973] A.S. Troelstra. Metamathematical Investigation of Intuitionistic Arithmetic and Analysis. Springer Verlag, 1973. [Troelstra, 1973a] A.S. Troelstra. Notes on intuitionistic second-order arithmetic. In [Mathias and Rogers, 1973, pp. 171{ 205]. [Troelstra, 1977] A.S. Troelstra. Choice Sequences. A Chapter of Intuitionistic Mathematics. Oxford University Press, Oxford, 1977. [Troelstra, 1978] A.S. Troelstra. Commentary on Heyting [1930]. In Two Decades of Mathematics in the Netherlands, Part 5, Math. Centre, Amsterdam, 1978. [Troelstra, 1979] A.S. Troelstra. A Supplement to Choice Sequences. Report 79-04 of the Math Institute University of Amsterdam. 22 pp. 1979. [Troelstra, 1980] A.S. Troelstra. The interplay between logic and mathematics: intuitionism. In Modern Logic|A Survey. E. Agazzai, ed. pp. 197{221. D. Reidel, Dordrecht, 1980. [Troelstra, 1981] A.S. Troelstra. On a second-order propositional operator in intuitionistic logic. Studia Logica, 40, 113{140, 1981. [Troelstra, 1981a] A.S. Troelstra. Arend Heyting and his contribution to intuitionism. Nieuw Archief voor Wiskunde, 24, pp. 1{23, 1981. [Troelstra, 1983] A.S. Troelstra. Analysing choice sequences. Journal of Philosophical Logic, 12, 197{260, 1983. [Troelstra, 1998] A.S. Troelstra. Realizability. In Handbook of Proof Theory, (ed. S.R.Buss). pp. 407{474. Elsevier, Amsterdam, 1998. [Troelstra and van Dalen, 1982] A. Troelstra and D. van Dalen, eds. The L.E.J. Brouwer Centenary Symposium, North Holland, Amsterdam, 1982. [Troelstra and van Dalen, 1988] A. Troelstra and D. van Dalen. Constructivism in Mathematics, I, II, Elsevier, Amsterdam, 1988. [Troelstra and Schwichtenberg, 1996] A. Troelstra and H. Schwichtenberg. Basic proof Theory. Cambridge University Press, Cambridge, 1996. [Veldman, 1976] W. Veldman. An intuitionistic completeness theorem for intuitionistic predicate logic. J. Symbolic Logic, 41, 159{166, 1976. [Veldman and Janssen, 1990] W. Veldman and M. Janssen. Some observations on intuitionistically elementary properites of linear orderings. Arch. Math. Logic, 29, 171{187, 1990. [Veldman and Waaldijk, 1996] W. Veldman and F. Waaldijk. Some elementary results in intuitionistic model theory. JSL, 61, 745{767, 1996. [Visser, 1982] A. Visser. On the completeness principle: a study of provability in Heyting's Arithmetic. Annals Math. Logic, 22, 263{295, 1982. [Zucker and Tragesser, 1978] J. T. Zucker and R. S. Tragesser. The adequacy problem for inferential logic. J. Philosophical Logic, 7, 501{516, 1978.

WALTER FELSCHER

DIALOGUES AS A FOUNDATION FOR INTUITIONISTIC LOGIC

SUMMARY OF CONTENTS The principal content of this article is a (new) foundation for intuitionistic logic, based on an analysis of argumentative processes as codi ed in the concepts of a dialogue and a strategy for dialogues. This work is presented in Section 3. A general historical introduction is given in Section2. Since already there the reader will need to know exactly what a dialogue and a strategy shall be, these basic concepts are de ned in the (purely technical) Section 1.

1 BASIC CONCEPTS: DIALOGUES AND STRATEGIES I consider a rst-order language, built with variables x; y; : : : and terms t ; formulas shall be constructed from atomic formulas with the propositional connectives ^; _; !; : and the quanti ers 8; 9 ; I shall also consider _; ^1 ; ^2 ; 9 as special symbols in their own right. By an expression I understand either a term or a formula or a special symbol. I introduce two further symbols P and Q ; taking two new (and disjoint) copies of the set of expressions, I form for every expression e two new expressions P e and Qe, the P -signed and the Q-signed version of the expression e. The symbols P; Q shall symbolise two persons engaged in an argument or in a dialogue; I shall use X; Y as variables for P; Q and shall assume X 6= Y . An argumentation form is a schematic presentation of an argument, concerning a logically composite assertion; it describes how a composite assertion made by C may be attacked by Y and how, if possible, this attack may be answered by X . As the logical form of the composite assertion shall completely determine the argument, each of the four propositional connectives and each of the two quanti ers determines an argumentation form:

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^:

assertion: Xw1 ^ w2 attack: Y ^i answer: Xwi

_:

assertion: Xw1 _ w2 attack: Y_ answer: Xwi

!:

assertion: attack: answer: : : assertion: attack: answer:

(i.e., Y chooses i = 1 or i = 2)

(i.e., X chooses i = 1 or i = 2)

Xw1 ! w2 Y w1 Xw2 x:w Yw no answer possible

8:

assertion: X 8xw attack: Yt answer: Xw(t)

9:

assertion: X 9xw attack: Y9 answer: Xw(t)

(i.e., Y chooses the term t)

(i.e., X chooses the term t).

In the last two answers I have written w(t) for the substitution instance obtained from w if the term t is substituted for the variable x. A dialogue shall be a ( nite or in nite) sequence Æ of statements, i.e., signed expressions, stated alternatingly by P and Q and progressing in accordance with the argumentation forms; I shall consider only such dialogues which are begun by P . Since it is necessary to distinguish carefully between attacks, answers and the assertions they refer to, I shall introduce besides Æ an accompanying sequence  of references, and there I shall use the symbols A for attack and D for answer (defense). For notational convenience, I shall assume that a natural number is the set of all smaller natural numbers (whence 0 is the rst natural number), and a sequence shall always be a function, de ned on either a natural number or on the set ! of all natural numbers. The precise de nition then reads as follows: A dialogue Æ;  consists of two sequences such that

Æ is a sequence of signed expressions,  is a function de ned on the positive members of def(Æ), and if n in def() is an ordered pair [m; Z ] such that m is a natural number less than n and Z is either A or D , satisfying the properties (D00){(D02):

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(D00) Æ(n) is P -signed if n is even and Q-signed if n is odd; Æ(0) is a composite formula. (D01) If (n) = [m; A ] then Æ(m) is a composite formula and Æ(n) is attack upon Æ(m) according to the appropriate argumentation form. (D02) If (p) = [n; D ] then (n) = [m; A ] and Æ(p) is the answer to the attack Æ(n) according to the appropriate argumentation form. The signed formulas occurring as values of Æ are called the assertions of the dialogue while the remaining values of Æ are symbolic statements or, more correctly, symbolic attacks. The numbers in def(Æ) are called the positions or places of the dialogue. If P v is the assertion Æ(0), the dialogue is said to be a dialogue for the formula v (or, sometimes, for P v). Assume now that a particular class H of dialogues is given, de ned maybe by additional conditions, which has the property that, for every position n of an H -dialogue Æ; , the restrictions of Æ;  to positions i such that i  n form an H -dialogue again. Assume further hat a subclass of H has been de ned, consisting of certain nite H -dialogues which then are said to be the H -dialogues won by P . Let v be a composite formula; to say that P has an H -strategy shall mean that P is in possession of a system of information, consisting of possible choices of P -statements in dialogues, such that every H -dialogue for v is won by P if only P chooses, after every statement made by Q, its own statement from this system of information. In order to formulate a more precise de nition, recall that a tree S is a partially ordered set of elements called nodes with the following properties: there exists a largest element eS (the top node), and for every node e the number kek of nodes f such that e  f < eS is nite; every node except eS has exactly one upper neighbour but may have arbitrarily many lower neighbours (i.e., the tree is branching downwards). A path in S is a linearly ordered subset of nodes which, together with each of its elements e, contains all the preceding nodes f with e  f ; a branch is a path which is maximal. If A is a branch of S , let A be the unique order-preserving bijection which maps either a natural number or all of ! onto A, i.e. kA(i)k = i holds for every node A (i) in A. Consider now a tree S and functions Æ;  where Æ is de ned on all nodes of S and  on the nodes dierent from eS ; for every branch A de ne ÆA = Æ
A ; A = 
A . The triplet S; Æ;  then is an H -strategy for v if (S 0)

For every branch A of S the pair ÆA ; A is an H -dialogue for v which is won by P .

(S 1)

For every node e of S the following is the case. If kek is odd then S does not branch at e. If kek is even then e has as many lower neighbours as Q has possibilities to extend, by adding a new position, to an H -dialogue the (restricted) dialogue leading to e,

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and Æ;  assign these lower neighbours the values which realise these possibilities. The general de nitions having been established, particular classes of dialogues can be introduced. To do so, I shall need the following terminology. Let Æ;  be a dialogue, and let Æ(n) be one of its attacks. The attack Æ(n) will be said to be open at a position k with n
For every n in def(Æ): if n is odd then Æ(n) is either attack upon Æ(n 1) or answer to Æ(n 1) .

An E -dialogue is said to be won by P if, again, it is nite, ends with an even position and if now the rules for E -dialogues do not permit Q to continue

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with either an attack or an answer. There will be occasion to refer to the following result which is auxiliary to the proof of the Equivalence Theorem. EXTENSION LEMMA. There is a recursive algorithm by which every E -strategy can be embedded into a D-strategy. It follows from this lemma that the Equivalence theorem holds also for E strategies in place of D-strategies. Readers not familiar with the use of dialogues may appreciate the following examples in which a; b; : : : are assumed to be atomic formulas. (1a)

0. 1. 2. 3. 4. 5. 6. 7. Q^1 8. P a

(1b)

3. 4. 5. 6.

P (a ^ b) ! (a ^ b) Q(a ^ b) P ^1 Qa P ^2 Qb P (a ^ b) [6,Q] 7. [7,D] 8.

0. P (a ^ b) ! (a ^ b) 1. Q(a ^ b) 2. P (a ^ b) Q^1 [2,A] 3. P ^1 [1,A] 4. Qa [4,D] 5. P a [3,D] 6.

[0,A] [1,A] [2,D] [1,A] [4,D] [1,D] Q^2 [6,Q] P b [7,D] [0,A] [1,D] Q^2 P ^2 Qb Pb

[2,A] [1,A] [4,D] [3,D]

Here we have two dierent D-strategies for the same formula. (2a)

(2b)

0. 1. 2. 3. 4.

P (a ! ::a) Qa P ::a Q:a Pa

0. 1. 2. 3.

P (::a ! a) Q::a [0,A] P :a [1,A] Qa [3,A]

[0,A] [1,D] [2,A] [3,A]

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The rst example is a D-strategy. In the second example, P cannot win if (D11) shall not be violated. (3)

0. 1. 2. 3. 4. 5. 6.

P ((a ^ :a) ! b) Q(a ^ :a) P ^1 Qa P ^2 Q:a Pa

[0,A] [1,A] [2,D] [1,A] [4,D] [5,A]

This is a D-strategy. The same reasoning holds for P :(a ^ :a). (4)

3. 4. 5. 6.

0. P ((a ! a) ! b) ! b 1. Q(a ! a) ! b [0,A] 2. P (a ! a) [1,A] Qb [2,D] 3. Qa [2,A] P b [1,D] 4. P a [3,D] Qa [2,A] 5. Qb [2,D] P a [5,D] 6. P b [1,D]

This is a D-strategy. If we omit positions 5 and 6, we still obtain an E -strategy. (5a)

0. P ((a ! b) ! a) ! a 1. Q(a ! b) ! a [0,A] 2. P (a ! b [1,A] 3. Qa [2,D] 3. Qa [2,A] 4. P a [1,D] The left E -dialogue is won by P but not the right one. There is no strategy as long as (D11) shall not be violated.

(5b)

0. P ::(((a ! b) ! a) ! a) 1. Q:(((a ! b) ! a) [0,A] 2. P ((a ! b) ! a) ! a) [1,A] 3. Q(a ! b) ! a [2,A] 4. P (a ! b) [3,A] 5. Qa [4,D] 5. Qa 6. P a [3,D] 6. P ((a ! b) ! a) ! a 7. Q(a ! b) ! a 8. P a

[4,A] [1,A] [6,A] [7,D]

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This an E -strategy and is easily extended to a D-strategy. (6)

7. 8. 9. 10. 11. 12.

0. P ((a ! b) ! (a ! c)) ! (a ! (b ! c)) 1. Q(a ! b) ! (a ! c) 2. P (a ! (b ! c)) 3. Qa 4. P (b ! c) 5. Qb 6. P (a ! b) Qa [6,A] 3 7. Q(a ! c) Pb [7,D] 2 8. P a Q(a ! c) [6,D] 1 9. Qc Pa [9,A] 2 10. P c Qc [10,D] 1 11. Qa Pc [5,D] 0 12. P b

0 [0,A] 1 [1,D] 0 [2,A] 1 [3,D] 0 [4,A] 1 [1,A] 2 [6,D] [7,A] [8,D] [5,D] [6,A] [11,D]

1 2 1 0 3 2

This is again a D-strategy. If we omit positions 9{12 on the left branch and positions 11{12 on the right branch then we obtain an E -strategy. The numbers appearing to the right of the values of  are the orders of the respective assertions as they will be de ned in Section 3.3. (7)

Let d0 be the formula f ^ :f for some (atomic) f . 0. P (a ^ ((b ! a) ! d0 )) ! c 0 1. Q(a ^ ((b ! a) ! d0 )) [0,A] 1 2. P ^1 [1,A] 3. Qa [2,D] 1 4. P ^2 [1,A] 5. Q(b ! a) ! d0 [4,D] 1 6. P (b ! a) [5,A] 2 7. Qd0 [6,D] 1 7. Qb [6,A] 3 8. P a [7,D] 2 9. Qd0 [6,D] 1 This can be completed so as to become a D-strategy; Qd0 can be handled as in example (3), and on the left branch we then will have to add the steps occurring in the right branch as positions 7 and 8.

(8)

Let d0 be as in (7) and de ne recursively for i = 0; 1; : : :

ei = a ^ ((b ! a) ! di ) ! c ; di+1 = ei ! d0 :

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Example (7) then gives a d-strategy for e0 and, moreover, shows that there is a D-strategy for each ei+1 with Qdi appearing in the positions 7 and 9 respectively. The D-strategy for ei then contains assertions of orders up to i +3. (9)

0. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

P ((a ! b) ! v) ! (c ! (b ! a)) Q(a ! b) ! v [0,A] P (c ! (b ! a)) [1,D] Qc [2,A] P (a ! b) [1,A] Qa [4,A] 5. Qv [4,D] P (b ! a) [3,D] Qb [6,A] Pa [7,D] Qv [4,D] Pb [5,D]

The left-dialogue satis es (D10), (D12), (D13), and observing these rules Q has no possibility to continue it. The rule (D11) is violated at place 6. If the formula v is chosen suitably then the dialogues can be extended so as to obtain a strategy, e.g., if v is a ^ b or c ! a or (a ! d) ! d. However, for these choices of v already a D-strategy for the initial contention can be found. (10)

0. 1. 2. 3. 4. 5. 6. 7.

P (:(a ! b)) ! (a _ d) Q:(a ! b) P (a _ d) Q_ P (a ! b) Qa Pa Qc

[0,A] [1,D] [2,A] [1,A] [4,A] [3,D] [4,D]

Also this dialogue satis es (D10), (D12), (D13) and violates (D11). (11)

0. 1. 2. 3. 4. 5. 6.

P (a ^ ((:a ! b) ! c)) ! c Q(a ^ ((:a ! b) ! c)) P ^1 Qa P ^2 Q(:a ! b) ! c P (:a ! b)

[0,A] [1,A] [2,D] [1,A] [4,D] [5,A]

DIALOGUES AS A FOUNDATION FOR INTUITIONISTIC LOGIC

7. 8. 9. 10.

Q:a Pa Qc Pc

[6,A] [7,A] [6,D] [1,D]

7. 8. 9. 10.

Qc Pc Q:a Pa

123

[6,D] [1,D] [6,A] [9,A]

In the left dialogue, Q violates (D11) at place 9. If this place and the following place 10 are omitted, a D-strategy remains. (12)

0. 1. 2. 3. 4. 5. 6. 7. 8.

P (:(b ! (c _ d))) ! (a ! b) Q:(b ! (c _ d)) Pa ! b Qa P (b ! (c _ d)) Qb Pc _ d Q_ Pb

[0,A] [1,D] [2,A] [1,A] [4,A] [5,D] [6,A] [3,D]

P violates (D11) at place 8. 2 THE LITERATURE ON DIALOGUES

2.0 It was P. Lorenzen who, in addresses in 1958 and 1959, published as Lorenzen [1960; 1961], proposed the idea that an autonomous foundation of intuitionistic logic should be based on the concepts of a dialogue and of a strategy for dialogues. Emphasizing the autonomy of such a foundational approach, Lorenzen preferred to speak of a constructive or eective logic and avoided the more familiar name of intuitionistic logic. While the rst descriptions of dialogues seemed to use only the properties named here (D00){(D02), it soon became clear that additional rules would be required if only intuitionistically provable formulas should be those which could be secured by strategies for dialogues. Such additional rules were formulated by Lorenz [1961] who de ned (among other types) the kind of dialogues called D-dialogues here; they appear in Lorenzen [1962; 1967] and in Kamlah and Lorenzen [1967]. While these presentations attempted to arrive at an appropriate de nition for a dialogue by specializing the general notion, a dierent approach was taken in [Lorenz, 1973] and [Lorenzen and Schwemmer, 1973] where there is considered at rst a very narrow type of dialogue (permitting both P and Q to react only upon the immediately preceding step) which then is liberalised to types of dialogues which are, essentially, the E -dialogues of Section 1.

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Lorenzen's basic idea is indeed a very attractive one. However, although described in a variety of articles and books, its presentations have been marred by a recurrence of ambiguous de nitions and incomplete, if not erroneous proofs. And all these elaborations so far have suered from two major defects.

2.1 The rst defect is of a technical-mathematical character. If a new development of intuitionistic logic, based on the concepts of dialogues and strategies, shall be given, then one expects an equivalence theorem to be established which states that provability by strategies coincides with provability by one of the known calculi for intuitionistic logic. The proof of such a theorem remained missing for many years. A rst attempt to prove an equivalence theorem was made in Lorenz's dissertation [Lorenz, 1961]; it was repeated in [Lorenz, 1968]. A certain part of Lorenz's dissertation was corrected in [Stegmuller, 1964]; some claims made in other parts were proved while other ones were refuted in the Diplomarbeit of W. Kindt [1970]. Kindt's refutations were acknowledged in footnote 12 of [Lorenz, 1968] where it is said that a correction of the erroneous statements in [Lorenz, 1961] would require \ein paar detaillierte technische Vorbereitungen" (cf. also a similar remark in footnote No. 16); unfortunately, these few, detailed technical preparations have never been presented to the public, and the gaps in Lorenz's attempt still appear to be un lled. (It is somewhat distressing that in the presumably authoritative collection of [Lorenzen and Lorenz, 1978] the article [Lorenz, 1968] is simply reprinted together with its footnotes; the part of [Lorenz, 1961] to which footnote No.12 refers has been omitted altogether.) A rst correct proof of an equivalence theorem was given by Kindt [1972]; however, the dialogues studied by Kindt are not D-dialogues but employ instead of (D11) a dierent rule. In [Lorenzen and Schwemmer, 1973, pp. 59 and 71] it is observed that the E -strategies (in the sense of Section 1) considered there give rise to a calculus of `Dialogstellungen' which (at least in the propositional case) may be transformed into a calculus of Beth-tableaux such that provability by E -strategies implies intuitionistic provability. A new, and simpler approach to an equivalence theorem for D-dialogues was developed by Haas [1980] and it seems that the technical gaps contained in this work are only minor and can actually be lled. An attempt to prove an equivalence theorem for E -dialogues is contained in [Mayer, 1981] and Dr E. C. W. Krabbe informs me that an equivalence theorem for E -dialogues is contained in [Krabbe, 1982] and in [Barth and Krabbe, 1982]. The equivalence theorem stated in Section 1 was presented in [Felscher, 1981] and in a revised form in [Felscher, 1985].

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2.2 The second defect from which the elaborations of Lorenzen's idea have suffered concerns the matter of foundations. The rules (D00){(D01) are just precise descriptions of the intention that dialogues should proceed through applications of the argumentation forms. But when additional rules (the `Dialog-Rahmenregeln' in the terminology of the Lorenzen school) had to be imposed, the question arose whether such rules could be explained as natural codi cations of principles of argumentation. Such principles, if foundationally sound, would have to be based on an analysis of the use of dialogues as a means to establish a systematical, indisputable and convincing conduction of formal arguments. When Lorenzen [1960; 1961] wrote about `Sprachspiele', i.e., language games, he used this word in the sense of the ancients'  ! , referring to a regulated (linguistic) process, the rules of which were to be governed by an insight which, although not further explained, was clearly assumed to be present. This attitude changed with [Lorenz, 1961] who, attempting a mathematical formalisation, began to make use of the concepts of a mathematical discipline known as the Theory of Games. Games there are mathematical objects, describing procedures as varied as whist and rummy at the one end and the games invented by warriors and economists at the other end. The rules then may be arbitrary: what matters is that they are adhered to; and the convention that a game is won because the other player can't draw any more may be brought about by rather odd rules of the game (such as, e.g., the categorical application of an equaliser). Matters were not improved when Lorenz [1961] observed that a change of dialogue rules would give rise to a type of dialogue the strategies for which would prove precisely the classically provable formulas. For this situation made it perfectly clear that the mathematical arbitrariness of the Theory of Games, being a tool to describe formally such dierent ways of reasoning as are classical logic and intuitionistic logic, could not possibly produce a philosophical foundation for either one of them. The mathematical apparatus for the Theory of Games was used heavily in the mathematical work of Kindt [1970; 1972]. On the other hand, phrases referring to `dialogue games' have spread through a certain kind of literature where a mathematical terminology is borrowed in order to give at least the appearance of conceptual precision. It appears that Lorenzen himself did not follow the fashion of a gametheoretical reduction. However the foundational discussions presented, e.g. in [Lorenzen and Schwemmer, 1973] are not based on an argumentative analysis; in particular, atomic statements and their negations are discussed with respect to a semantical distinction of true and false, and the dierence between classical and intuitionistic logic is made to depend on the decidability (`Wahrheits-De nitheit': de niteness with respect to truth) of atomic statements. As crown's evidence for the generally unsatisfactory state of

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a foundational discussion I may quote Kambartel [1979] who, in the very conclusion of this article, writes in respect to the problem of justi cation: . . . Rechtfertigungsproblem. Dieses besteht darin, die schematischen Dialogspiele selbst von einem argumentativen Gebrauch der logischen Partikeln her zu begrunden. Die dialogische Logik hat dieses Prolbem bisher dadurch uberspielt, dass sie die Dialogspiele methodisch als `erste' Festlegung des Gebrauchs der logischen Partikeln behandelt. Das dabei vernachlassigte, uber die schematische Ebene hinausfuhrende Rechtfertigungsproblem schlagt dann spatestens in der Rahmenregeldiskussion wieder durch. In der Tat werden dort `Rechtfertigungen' fur die Wahl solcher Regeln, z.B. neuerdings immament schematisch oder, wie zunachst geschehen, im eher intuitiven Ruckgri auf halbschematisch analysierte Beispiele, beigebracht. Weder `technische' Kriterien noch die Verallgemeinerung von Beispielen stellen aber bereits einen im engeren Sinne normativen Zugang zur Logik dar . . .

2.3 Lorenzen's argumentation forms have also been put to use by K. J. J. Hintikka, but this with quite dierent intentions. Hintikka, beginning with [Hintikka, 1968], developed what he calls a game-theoretical semantics, and a more recent series of articles on this topic have been collected in [Saarinen, 1979]. A semantical game in Hintikka's terminology may indeed be viewed as a dialogue in the sense of Section 1 although Hintikka restricts his attention to the single argumentation forms and nowhere cares to formulate game rules proper (such that the implied reference to mathematical games remains but an incantation). The point, however, is that Hintikka is concerned with a linguistical analysis of natural languages and not with a foundation of (classical or intuitionistic) logic. For this purpose, argumentation forms and dialogues are used as tools for the semantical evaluation of logically composite expressions (which may be more complex than rstorder logic would permit to express) in domains governed by classical logic. There is, therefore, no connection of Hintikka's work with that discussed in the present article. 3 FOUNDATIONS OF DIALOGUES In this Section I shall develop an argumentative foundation for the use of particular types of dialogues, the D-dialogues, as a basis for intuitionistic logic. That such a foundation is wanted was outlined in subsection 2.2.

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3.1 The Argumentative Interpretation of Logical Operators There exists a well known provability interpretation of the logical operations (connectives and quanti es) which then may be considered as being represented by Gentzen's calculi of proofs, i.e., natural deduction and the sequent calculus LJ; for details, cf. van Dalen's article in this volume. In the same manner, the argumentation forms may be viewed as expressing an argumentative interpretation of the logical operations, and as far as the positive connectives and the quanti ers are concerned this interpretation is obvious enough. Concerning implication, Y attacks Xw1 ! w2 by oering Y w1 as an admission (or local hypothesis) and X may react by either answering with Xw2 or attacking Y w1 (provided w1 is composite). Concerning negation, the situation is the same as in the case of the provability interpretation: if external, semantical references to truth and falsity shall be avoided, we must enrich the basic concept of provability by adding either refutability or absurdity as a primitive notion. Since we are aiming for intuitionistic logic, we introduce a constant
symbolising absurdity and then understand :w as an abbreviation of w !
. The principle of ex absurdo quodlibet takes as its rst form that he, X , who is forced to assert
then must concede, without further argument, any assertion made by Y . A speaker professing
thus brings himself into a position precluding any further debate, and so we may just as well omit this fatal step and state, as the second form of ex absurdo quodlibet, that
must not be asserted. This then explains why an attack Y w upon Xw !
, i.e., X :w , cannot be answered.

3.2 Basic Principles for Dialogues Gentzen's calculi of proofs are easily explained in that they represent the weakest consequence relation for which the provability interpretation is valid. The connection between dialogues and the argumentative interpretation of logical operations is (not only more complicated but also) located on a dierent level: it is not the dialogues but the strategies for dialogues which will correspond to proofs. I thus formulate the basic purpose for the use of dialogues: (A0 )

Logically provable assertions shall be those which, for purely formal reasons, can be upheld by a strategy covering every dialogue chosen by Q.

The dialogue rules (D00){(D02), describing the use of argumentation forms, simply produce the (linguistic) material of the dialogue which then will have to be organised by the dialogue rules proper. Extending the intentions expressed in the formulation of the argumentation forms, I formulate the argumentative intent in the pursuit of a single dialogue:

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WALTER FELSCHER

A dialogue is, on the part of Q an attempt to put into doubt (to refute) the initial assertion made by P ; it is, on the part of P , an attempt to uphold this assertion, and if P succeeds in doing so this will mean that P wins the dialogue.

In the light of these intentions we now will have to clarify (b0 )

how to determine the dialogue rules proper,

(b1 )

the notion that P wins a dialogue.

It must be emphasised that the concepts occurring here cannot be studied separately but must be analysed simultaneously and in constant regard of the purpose (A0 ).

3.3 Dependence, Positive Dependence and Order The notions to be discussed in this subsection are auxiliary. Let Æ;  be a dialogue. I shall say that a statement depends directly on an earlier statement if it is either an attack upon or an answer to that statement; I de ne dependence to be the transitive and re exive relation generated by direct dependence. Dependence, therefore, is an order relation, contained in the linear order given by Æ ; since every statement, dierent from the initial one, depends directly on exactly one earlier statement, it follows that dependence de nes the ordering of a tree on the set of all statements, i.e., on im(Æ) with Æ(0) as its top node. I de ne a chain to be a sequence of statements in im(Æ) such that each of its members, except the rst one, depends directly on its predecessor in that sequence; every chain  thus arises from a path in the dependence tree by removing the nodes above (0) from the path. Every chain is a subsequence of Æ and Æ itself is pieced together from various chains, some of which may only have one member. While dependence is a relation de ned between arbitrary statements, a second relation will be de ned only between assertions of a dialogue Æ;  . An assertion Xv is an immediate positive dependent of an earlier assertion Xw if it is an answer to an attack upon Xw ; I de ne positive dependence to be the transitive and re exive relation generated by immediate positive dependence. The relation of positive dependence leads to the following classi cation of assertions. The initial assertion and its positive dependence shall be of order 0 ; if Xv ! w or X :v is of order n then an attack Y v shall be of order n+1 , and so shall be the positive dependence of this attack. It follows that the P -assertions are exactly those of even order and the Q-assertions are exactly those of odd order.

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3.4 Contentions and Hypotheses When asserting the initial statement of a dialogue, P contends it to be defensible. In this sense, the initial assertion is contended, and in the same manner assertions of order 0 are contended. Assertions of order 1 , if they do not arise as positive dependents of earlier ones, are attacks Qw1 upon assertions P w1 ! w2 or P :w1 ; hence they are (global) hypotheses oered by Q, and then also their positive dependents are (particularisations of ) hypotheses. Assertions of order 2 , if they do not arise as positive dependents of earlier ones, are attacks P w2 upon hypotheses Qw1 ! w2 or Q:w1 ; here P takes up the hypotheses by admitting P w1 as a (higher order) contention. Consequently, also these assertions are contended by P , and so are their positive dependents. Repeating this argument, it follows that all P -assertions are contended and that all Q-assertions are hypothetical. It will be advisable to observe the distinction made between global hypotheses in a dialogue as discussed here, and local hypotheses occurring as admissions in instances of argumentation forms. Applying the argumentation forms, assertions of logically composite formulas are dissolved into assertions of lesser complexity: contentions are upheld and hypotheses are developed. Obviously, contentions P w1 ^w2 ; P w1 _w2 ; P 8xw ; P 9xw are upheld by holding up the immediate positive dependents (as chosen by P or prescribed by Q), and the same holds for the development of the analogous hypotheses. Consider now contentions P w1 ! w2 where we include the case P :w1 by writing it as P w1 !
; when attacked by Qw1 then P may either uphold the answer P w2 or attack Qw1 in order to force Q into a further development of this hypothesis. Similarly, if a hypothesis Qw2 ! w2 is attacked by P w1 then Q may develop it into the answer Qw2 or Q may attack the contention P w1 . The process of dissolving logically composite assertions comes to an end once atomic assertions have been reached. Atomic formulas asserted by Q are hypotheses, intended by Q as describing particular situations which serve to refute P 's contention to have a defensible initial assertion; they do not need further justi cation. Atomic formulas asserted by P , however, remain contentions in need of justi cation. Securing them by material insight, such as, e.g., illumination or revelation, is unacceptable (anyway and in particular) if purely formal defensibility has been claimed by P . There remains, therefore, only one possibility for P to assert an atomic formula P a for purely formal reasons: (c0 )

P may assert P a only if Q admits Qa as a hypothesis relevant to the position of P a.

For in that case P confronts Q with its own hypothesis to which Q cannot possibly object (in [Krabbe, 1982] this principle is mentioned with the ap-

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propriate name of ipse dixisti). This principle (c0 ) is purely descriptive in that it refers to a given, completed dialogue; it does not provide the means in order to enforce that, already during the performance of the dialogue, assertions P a are made only in observance of (c0 ). Making reference to the linear structure of a dialogue, we therefore strengthen (c0 ) to (c1 )

P may assert P a only after Q has admitted Qa at an earlier, relevant position.

It appears that we now would have to clarify the notion of relevance, and a rst attempt to do so would consist in producing a de nition which describes, for every position of a dialogue, the set of hypotheses relevant for this position. I shall not proceed in this manner; rather, I shall introduce additional restrictive rules for dialogues with the eect that the family of all hypotheses occurring in a dialogue becomes coherent in the sense that its members, being admitted simultaneously, do not create distinctions of relevance: each of them is relevant for all atomic contentions asserted afterwards. The additional rules will, obviously, restrict the amount of information analysed in a dialogue. But the principal objects for us are strategies, not methods for winning a single dialogue, and no information will be lost if it only remains available within the system of dialogues belonging to a strategy.

3.5 How to Win a Dialogue P wins a dialogue if it succeeds in holding up its contentions. During the

course of a dialogue, the initial contention is dissolved into more and more specialised subcontentions, and this specialisation comes to an end with the contention of atomic formulas as regulated by (c1 ). It is implicit in this conception that no composite contention is accepted as being upheld without further dissolution, for Q may always challenge it with an attack. Consider now a dialogue containing an atomic contention P a which, for the moment, we assume as being of order 0 . We then nd a unique sequence of contentions, beginning with the initial contention P v and ending with P a , each of which (except the rst one) is an immediate positive dependent of its predecessor. Consequently, P a is the nal step of a process by which P v is narrowed down to more special contentions | and in view of (c1 ) this nal step may be asserted for purely formal reasons. Of course, other sequences of specialisations of P v, ending with other atomic formulas, may be possible, but since we are considering provability by strategies, these other possibilities are covered by other dialogues which a strategy will have to take into account. We thus arrive at (W A) The initial contention is considered as having been upheld successfully if P has asserted an atomic contention of order 0.

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Obviously, the same reasoning can be used to say that a contention of order n can be considered as having been upheld successfully if P has narrowed it down to an atomic contention of order n | but this observation remains without consequences. There is, however, another way to consider the initial contention as having been upheld successfully; it rests on the principle of ex absurdo quodlibet applied to dialogues (and not only to argumentation forms). As a simplest situation, consider a hypothesis Qw1 of order 1 , arising as an attack upon P w1 ! w2 of order 0 , and assume that, after a sequence of positive dependents of Qw1 , the only possibility left to Q would be the assertion of an absurdity Q
. In that case, the hypothesis Qw1 has been developed (and we may assume: by P 's prodding) into an absurdity and, therefore, has itself been shown as untenable. But as this hypothesis had been granted in the attack Qw1 , we conclude by ex absurdo quodlobet that P w1 ! w2 can be upheld without any further argument. More generally, a hypothesis Qr1 of order n+1 ; n> 0, arising as an attack upon r1 ! r2 of order n , is itself a development of the earlier hypothesis Qs1 ! s2 of order n 1which gave rise to an attack P s1 of which P r1 ! r2 is a positive dependent. Consequently, if Qr1 can be shown as leading to an absurdity, then also Qs1 ! s2 must be considered as leading to an absurdity. Descending from n 1 to 1 , we conclude that already the rst hypothesis Qw1 of order 1 , giving rise to the higher order hypotheses resulting in Qs1 ! s2 , leads to an absurdity, and thus again P w1 ! w2 can be upheld without further argument. We thus arrive at (W B ) The initial contention is considered as having been upheld successfully if (P has not asserted an atomic contention of order 0 but) Q has been brought into a position where its only possibility to continue would be the assertion of an absurdity. Of course, at the present stage of our discussion no reason is visible why Q may become so restricted in its possibilities as is supposed in (W B ); this will become clear after the following sections. What can be said already here is that if a dialogue is won by P then its last position is even (i.e., the last move is P 's), and if the initial contention does not contain a negation then the dialogue can be won only according to (W A).

3.6 Rami cations I shall now discuss how to avoid the distinctions of relevance as they appear in condition (c0 ). Let us begin by considering more closely the situation described there: let Æ;  be a dialogue with positions j; k such that j < k ; Æ(j ) = Qa ; Æ(k) = P a where a is atomic. Since both these assertions depend on Æ(0), there exist chains 0 ; 1 from Æ(0) to Æ(j ) and to Æ(k), and it follows from j
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WALTER FELSCHER

either 0 is an initial part of 1 or 0 ; 1 ramify at a position preceding Æ(j ). The rst case is unproblematical. For let n be the last argument of 0 such that 0 (n) = Æ(j ); being an atomic hypothesis, o(n) must be an attack upon 0 (n 1) = P a ! w which then is answered by 1 (n +1) = P w, for this is the only way in which the atomic hypothesis Qa can have dependents in 1 . Thus P a is P w or a dependent of P w, and Qa must certainly be considered as relevant for P a. In the second case, however, the rami cation may cause distinctions of relevance. I shall say that there is a rami cation at n for two chains 0 ; 2 of a dialogue if both chains coincide for all i such that i  n, if further both chains are de ned (at least) for n+1 and if 0 (n+1) is dierent from 1 (n+1) , i.e., if 0 (n +1); 1(n +1) are stated at dierent positions of the dialogue. A rami cation then arises in one of the three following ways: (1) 0 (n +1); 1(n +1) are dierent attacks upon 0 (n); (2) 0 (n+1); 1(n+1) are dierent answers to the attack 0 (n) upon 0 (n 1);

(3) 0 (n) is an attack Y w1 upon 0 (n 1) = Xw1 ! w2 ; i (n +1) is an attack upon 0 (n) and 1 i(n+1) is the answer Xw2 to 0 (n) ; i = 0; 1. The attacks in (1) shall be called distinct if not only their positions but also the statements made by them are dierent; otherwise they are only repeated attacks. In the same way, I shall speak of distinct and of repeated answers. In this subsection I shall be concerned with rami cations of the rst two types. Distinct attacks are possible upon assertions Xw1 ^ w2 ; X 8xw. If P contends, say, P w1 ^ w2 then P certainly should be able to contend both P w1 and P w2 . But a hypothesis arising during the analysis of P w1 (e.g., if w1 is a ! a) will not be relevant during the analysis of P w2 (e.g., if w2 is b ! a) and vice versa: such hypotheses are admitted for one but not for the other subcontention. Distinct attacks upon contentions, therefore, do cause distinctions of relevance. In a strategy, however, the possibility of distinct attacks by Q is already taken into consideration in that it leads to dierent dialogues which all have to be won by P . Consequently, there will be no loss of information if such distinct attacks are excluded from every single dialogue. On the other hand, a hypothesis Qw1 ^ w2 is present already before any rami cation caused by dierent attacks and it should remain in eect with its complete content also after the rami cation. If, for instance, during the development of Qw1 the moves of P lead Q to admit Qa then this hypothesis should be considered as coherent with Qw1 as well as with Qw1 ^ w2 and also with Qw2 : Qa should be relevant also for any development of Qw2 . Distinct answers can be given to attacks upon assertions Xw1 _w2 ; X 9xw. If P contends, say, P w1 _ w2 then it will have to contend only one of

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P w1 ; P w2 . Again now, hypotheses arising during the analysis of one of these answers (e.g., if w1 is :a) will not be relevant during the analysis of the other (e.g., if w2 is a). Distinct answers to attacks upon contentions, therefore, do cause distinctions of relevance. However, if P has answered an attack with, say, P w1 then a second, later answer with P w2 would be useful only if P w1 could not be upheld successfully and if P now would try a second attempt. But in a strategy we can demand that P knows what it is doing and does not proceed by trial and error, and there will be no loss of information if we exclude distinct answers to attacks upon contentions. On the other hand, if Q were to answer an attack upon, say Qw1 _ w2 with both Qw1 and Qw2 then it would grant not only the content of Qw1 _ w2 but actually that of Qw1 ^ w2 . In a strategy, however, P has to provide dialogues for all possible answers, and there will be no loss of information if we also exclude distinct answers to attacks upon hypotheses. In order to discuss repeated attacks and repeated answers, we may now assume that the two chains 0 ; 1 ramifying at n, are chosen as being maximal, i.e., as branches of the dependence tree. I now de ne inductively the notion of corresponding couples: 0 (0); 1 (0) form a corresponding couple; if 0 (i); 1 (i) form a corresponding couple then 0 (i+1); 1(i+1) shall form a corresponding couple if they either appear at the same position of the dialogue or if they are (at least) identical as signed expressions and also are identical in their mode within the dialogue, i.e., as attacks or answers referring to 0 (i); 1 (i). Let us assume now that the rami cation at n arises under repetitions. Then 0 (n +1) ; 1 (n +1) still form a corresponding couple, and we may look for all corresponding couples 0 (n + i); 1 (n + i) . If these couples exhaust already one of the two chains, say 0 , then that part of 0 which starts at n +1 does not contain any information which is not available in the corresponding part of 1 . No information will be lost if that part of 0 (or the corresponding part of 1 ) is omitted from the dialogue, and so we may exclude the repetition at n+1 which gave rise to the twofold presence of that part. It remains to consider the case that there exists a common argument m of 0 ; 1 ; n +1 < m, such that 0 (m); 1 (m) is not a corresponding couple. Let m be minimal for this property and observe that either P or Q acts at m in both of 0 ; 1 . If P attacks at m, say Qw1 ^ w2 with P w1 in the one and with P w2 in the other chain, then these distinct attacks could have been carried out without the repetitive rami cation at n (and with an acceptable rami cation at the position of Qw1 ^ w2 instead). If P answers at m, say Qv with P w1 and P w2 , then the repetitive rami cation at n just hides the fact that there is actually a rami cation caused by distinct answers. The same types of rami cations occur if Q in both 0 ; 1 attacks m or answers at m. Finally, if either P or Q attacks m in the one and answers m in the other chain then the repetitive rami cation at n delays an actual rami cation of type (3). Again now, either this rami cation of type (3) could have been

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carried out already at n or there are restrictions on the execution of such rami cations (as they actually will be discussed in the next section) and the repetitive rami cation at n is employed in order to circumvent these restrictions. Consequently, in all cases repetitive rami cations either can be avoided immediately or lead to distinctions of relevance of the sort discussed already for distinct rami cations. In any case, therefore, they may be excluded without loss of information. The conditions eecting the exclusion of dierent answers and of dierent attacks upon contentions are precisely (D12) and (D13).

3.7 Nested Attacks and Nested Answers There are good reasons why, in a dialogue, a certain answer upon an attack by Q is not stated immediately but only after some delay. The necessity to observe (c1 ) will cause such situations if a contention P w1 ! w2 has been attacked and the admitted hypothesis Qw1 needs further elaboration in order to permit P either to assert P w2 or to force Q into an absurdity, or a contention, itself stated already under a hypothesis, has been attacked and now the earlier hypothesis needs further elaboration. Delayed answers, therefore, cannot be excluded. It is such delays which cause a nesting of attacks and, thereby, a nesting of answers. Let Æ;  be a dialogue and let Æ(m); Æ(n) be assertions such that m n  0 (mod 2), let Æ(i) be an attack upon Æ(m) and let Æ(j ) be an attack upon Æ(n) (whence also i j  0 (mod 2)). I shall call Æ(j ); Æ(i) a pair of nested attacks if j
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(or, for that matter, even P
) under the hypothesis Qw1 . Therefore, the hypothesis Qw1 should be counted as being relevant if P actually ful ls the obligation to assert P W2 (or P
: : :): Qw1 together with P w2 preserves the complete content of Æ(m). We thus arrive at the principle (d0 ) For every pair of H -nested attacks: the hypothesis granted by the inner attack may be used as relevant in order to contend the answer to the outer attack provided P also contends the answer to the inner attack. The dialogue won by P in example (9) satis es (d0 ) for the H -nested attacks with n = 1 ; m = 4 ; j = 3 ; i = 5 ; the hypothesis Qw1 = Qa is used in order to contend the answer Æ(6) to Æ(j ) at 8, and conversely also this answer itself is employed in order to develop the hypothesis Qb which is needed when contending the answer Æ(10) to Æ(i). The principle (d0 ) is purely descriptive; it does not provide the means in order to enforce that only the situation described as desirable occurs. If the inner attack is upon P :w1 then P will never be able to ful l the obligation expressed in (d0 ), and if we wish to avoid an additional label declaring Qw1 to be irrelevant for P s then another formulation is wanted. But also if Æ(m) is P w1 ! w2 with w2 6=
, a dierent formulation would be useful: the example (10) shows a pair of H -nested attacks violating (d0 ) , but P has won the dialogue in accordance with (W A) and Q , if it is left to respect (D12), (D13), cannot continue: there just is no position left for P to state the answer to the inner attack. We thus formulate a rule which forces P to contend this answer before it makes use of the hypothesis: (d1 ) For every pair of H -nested attacks: the inner attack must have been answered before the outer attack may be answered. In this manner, we now have also a well-de ned nesting of answers. Consider now a pair of nested attacks upon contentions which is not H nested; this means that the inner attack is symbolic. The statement of such an attack does not create any hypotheses possibly needed for the contention of the answer P s to the outer attack. Still, there may be various reasons for P not to answer the outer attack Æ(j ) immediately at j + 1 , but to delay this answer to a position following that of the inner attack: actually the observance of (d1 ), together with that of (c1 ) may be one such reason. The example (12) communicated to me by Dr. E. C. W. Krabbe, shows a dialogue satisfying (D10), (D12), (D13) and (d1 ), but the outer attack at 3 is answered at 8 while the inner attack at 7 remains open. Given the outer attack 3, also the hypothesis Æ(5) arises as an inner attack, and observance of (d1 ) forces P to state the contention Æ(6) before Æ(5) can be used in order to state the answer P b to the outer attack. In avoiding the answer to the inner attack at 7 , P now fails to uphold the contention Æ(6). Thus (d1 ) has

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been observed, but the promise to uphold Æ(6) is given lip service only. It follows from this example that, in order to observe the full meaning of (d0 ) we have to strengthen the rule (d1 ) to (d2 ) For every pair of nested attacks upon contentions: the inner attack must have been answered before the outer attack may be answered. As for nested attacks upon hypotheses, it follows from the de nition of a strategy that, if P has a strategy at all, then P a fortiori has a strategy respecting (d02 ) For every pair of nested attacks upon hypotheses: the inner attack must have been answered before the outer attack may be answered. There is reason to conjecture that also the converse implication holds, and a proof should be related to that of the Extension Lemma mentioned in Section 1. The idea of such a proof is illustrated by example (11): violating (D11), Q may try to withhold a certain hypothesis (e.g., absurdity), but P knows from the strategy how the answer to the outer attack had to be treated if it was stated immediately after this attack.

3.8 D-Dialogues It was the purpose of the last two subsections to look for restrictions on dialogues which would permit us to avoid distinctions of relevance. We thus arrived at the rules (D12), (D13) in subsection 3.6 and at rule (D11) which is the conjunction of (d1 ) and (d02 ) in subsection 3.7. Having imposed these rules, let us look once more at the family of hypotheses occurring in a dialogue. Clearly, every positive dependent of a hypothesis Qw is nothing but a speci cation or an instantiation of Qw ; and dependents of higher order, coming from intermediary contents, keep this character as well. Passing through a chain of dependents, we see that the hypotheses occurring there form a coherent set, i.e., a set of simultaneously admitted assumptions without distinctions of relevance. Dierent chains of dependents are joined together with the help of delayed attacks and answers, and the three types of rami cations which thus may arise were described at the beginning of subsection 3.6. The presence of our rules now insures that all those rami cations are excluded which would cause distinctions of relevance. What remains permitted are rami cations of type (3) with properly nested answers and rami cations of type (1) caused by dierent attacks upon a hypothesis. If P attacks a hypothesis Qw a rst time, it forces from Q a certain system of speci cations of the hypothesis Qw asserted by Q in the beginning; if P attacks Qw a second time then it may obtain a dierent system of speci cations which, nevertheless, still is a system of speci cations of this same hypothesis Qw : once Q has admitted Qw then it must

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bear the consequences of being forced into speci cations, and the speci ed hypotheses brought forward under the dierent answers given by Q to the same attack of P upon Qw express, when considered simultaneously, more than is contained in Qw alone, and thus they do create distinctions of relevance. It must be emphasised that this dierence in the eects of attacks and answers is fundamental. We thus nd that, in the presence of (D11), (D12), (D13), the set of all hypotheses occurring in a dialogue is coherent in the sense that no distinctions of relevance appear. Consequently, the condition (c1 ) becomes (D10), and our dialogues are D-dialogues. At this point, a methodological observation appears to be appropriate. In the preceding two subsections I have presented arguments resulting in the introduction of additional rules with the purpose to avoid distinctions of relevance ((D12), (D13), (d0 )). In subsection 3.6 the exclusion of certain moves in a dialogue was supported with the observation that no information will be lost if the possibilities, excluded from single dialogues, remain present in strategies; in subsection 3.7 the introduction of (d0 ) was explained with the necessity to preserve the complete content. It should be noticed very clearly that these argumentations are based on an informal understanding of purposes; they are not justi cations based on mathematical theorems. As a matter of fact, as long as we abstain from a formal de nition of relevance, we cannot even formulate a theorem saying that strategies for dialogues with precautions on relevance (?) prove the same formulas as do strategies for dialogues with (D12), (D13), (d0 ).

3.9 How to Win a D-dialogue I have de ned in subsection 3.5 what it means that P wins a dialogue. On the other hand, there is the purely formalist de nition, taken from the literature and mentioned in Section 1, that a D-dialogue is won by P if Q has no way to continue. It remains to be shown that both notions are equivalent with respect to strategies. If a D-dialogue has been won according to (W B ) then it also has been won in the formal sense. If a D-dialogue has been won according to (W A) then it contains an atomic contention P a of order 0 which, therefore, must be an answer. Consequently, Q cannot continue the dialogue at this position by referring to P a. We now can show: If for some formula v; P has a strategy to win with (W A), (W B ) all D-dialogues for v then P also has a strategy to win in the formal sense all D-dialogues for v. For consider a strategy the branches of which are won with (W A), (W B ). These branches could not be continued by Q if Q would have to respect the rules for E -dialogues. We thus obtain an E -strategy if, where necessary, we cut o ends of branches when Q begins to violate the

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rule (E ). It then follows from the Extension Lemma that this E -strategy may be extended again to a D-strategy. For the converse implication, it can even be shown that a D-dialogue Æ;  which is won in the formal sense is won also according to (W A), (W B ). As this is clear if, during the dialogue, P has stated an atomic contention of order 0, we may assume now that P has not stated an atomic contention of order 0; (W B ) then will hold if we can prove that Q could continue if it were to state an absurdity: there still should exist a (last) open attack of the form P :w. Before continuing the proof, it will be useful to prepare some auxiliary notions. Let Æ;  be a dialogue; let n be an odd position and assume that Q has already stated Æ(n). Then 0 (n) shall be the number of contentions which, after this statement, still may be attacked by Q at later positions, and 1 (n) shall be the number of contentions which, being attacks, still may be answered by Q at later positions | and here I expressly include the unspeakable answers Q
to attacks P w upon some Q:w. The number  (n) = 0 (n)+ 1 (n) is called the degree (of freedom) of n. The following characterisation will be useful:  (n) is the dierence a(n) d(n) where a(n) is the number of attacks upon hypotheses Qw1 ! w2 ; w1 not atomic, which are contended before n , and d(n) is the number of atomic P -answers contended before n . This follows from the following observations in which 2i +1 is a position of Æ;  . If Æ(2i) is an attack then  (2i 1)   (2i + 1)   (wi 1) + 1, and the right inequality becomes equality if, and only if, Æ(2i) is an attack upon a hypothesis Qw1 ! w2 such that w1 is not atomic (whereas w2 may be
). If Æ(2i) is an answer which is not atomic then  (2i 1) =  (2i +1). If Æ(2i) is an answer which is atomic then  (2i 1) must be positive (for otherwise Q could not act at 2(i +1)), and  (2i +1) =  (2i 1) 1. Consider now a D-dialogue. It follows from (D12) (D13) that every assertion has at most one atomic immediate positive dependent; consequently, every assertion has at most one atomic positive dependent. Let now P a be an atomic contention of positive order; there then exists a rst (highest) contention P wa of which P a is a positive dependent, and P a; P wa have the same order. By the preceding remark, P wa is unique, and as it has no positive predecessor, it must be an attack upon a hypothesis Qwa ! ua (where ua may be
). I now resume the proof where it was interrupted. Let 2i be the last position of the dialogue. Then Æ(2i) cannot be a composite formula (since that could be attacked by Q) nor can it be a symbolic attack (since that could be answered); thus it must be an atomic contention P a, necessarily of positive order. Let P wa be the unique, highest contention determined by P a as above. If P wa is P a then ua must be
(for otherwise Q could answer), and thus (W B ) holds. Assume now that P wa is dierent from P a; then wa is not atomic. It now will be suÆcient to show that  (wi 1) is positive

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| because the possibilities numbered by  (i 1) cannot comprise attacks or answers which actually could be stated (for in that case Q could still continue), and thus there must be at least one open attack to be answered only by absurdity. I thus have to prove that the dierence a(2i 1) d(2i 1) is positive. Let P b be an atomic answer which contributes to d (2i 1); it is of positive order and thus determines the unique attack P wb . Since P b is an answer and P wb is an attack, P b must be a proper dependent of P wb . Thus wb is not atomic and, therefore, the attack P wb contributes to a(2i 1). It follows that the map ' sending b into wb is an injection of the set of contentions contributing to d(2i 1) into the set of contentions contributing to a(2i 1). Since wa is not atomic, the latter set contains P wa ; the former set, however, does not contain P a. Consequently, a(wi 1) is strictly larger than d(2i 1).

3.10 Intuitionistic versus Classical Logic As was mentioned in Section 2, Lorenz [1961] has observed that a change in the rules for D-dialogues produces a class of dialogues which I shall call C -dialogues, such that the formulas provable by C -strategies are precisely the classical provable formulas. The change leading from D-dialogues to C -dialogues consists in cancelling (D11) and (D12) for P , but leaving them in eect for Q. (If I understand Lorenz's and Lorenzen's writings correctly then they seem to demand the cancellation for P of (D12) only; the examples (2b) and (5a) show that this would not suÆce.) It is not hard to see that C -strategies prove only classically provable formulas. For the case of propositional logic, the converse implication (i.e., every classically provable formula can be proved by a C -strategy) can be seen as follows. Observe rst that an intuitionistically provable formula, being provable by a D-strategy, is trivially provable by a C -strategy. It is well known that if w is a classically provable formula then ::w is intuitionistically provable; assume now that w is not intuitionistically provable. Every D-dialogue Æ;  for ::w, won by P , begins with attacks Æ(1) = Q:w; Æ(2) = P w, and since we assume that the part beginning at position 2 is not a D-dialogue for w won by P , there must be positions below 2 at which P attacks Æ(1) again. If we compare the branches in the dependence tree and look for the rst positions at which they dier, we will nd, re ning the discussion in subsection 3.6, that this happens at contentions which could be obtained without the repetitive rst part of the branch if repeated answers or answers in disregard of (D11) were permitted to P . Permitting such answers in C -dialogues, it can be shown that a D-strategy for ::w can be rebuilt into a C -strategy for w.

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The mathematical fact that C -strategies can be used for classical logic is, in principle, not surprising; other proof-theoretical systems, e.g., those of Hilbert-type, can also be used for many varieties of logics. Rather than leading to amazement over the universal applicability of a mathematical tool (trees and strategies), this situation should teach us to emphasise the fundamental dierences between intuitionistic and classical logic. For the provability interpretation, as represented by Gentzen's calculi, Curry [1963, p. 260] has attempted a provability explanation of classical negation with help of his concept of complete absurdity, but this hardly will be considered to be a conceptual foundation. For the argumentative approach presented here, classical logic cannot be given a foundation by simply changing formal details of a foundation for intuitionistic logic. If we want to explain the rules governing classical negation then there appears to be no way to avoid the semantical notions of true and false: without these notions we cannot explain why distinctions of relevance may be discarded as it is done when P is permitted to repeat answers and to disregard (D11). Thus, for classical logic, the entire conceptual frame employed for the foundation of intuitionistic strategies, has to be abandoned: there is no use for contentions and hypotheses, for defendability by purely formal reasons and for considerations of relevance. What is required, is a completely dierent conceptual framework, based on the notions of true and false and on the distribution of truth-values under logical operations. The foundation of classical logic within such a framework is well known, and the elegant formulation of classical tableaux due to Smullyan [1968] may easily be read as to depict a dialogue-strategy leading to a failure of the attempt to falsify a formula. Again, the argumentative explanation of winning a dialogue according to (A); (W B ) is only formally related to the closure of branches in Smulluyan's tableaux which always means the advent of absurdity. 4 APPENDIX: CONCEPTS CONNECTED WITH THE EQUIVALENCE THEOREM The equivalence theorem, formulated in Section 1, states the existence of certain transformations between strategies and proofs in the calculus LJ; the proof of this theorem cannot be presented here. It may, however, be instructive for the reader to become familiar with some concepts which originally were developed for this proof. For details which have to be suppressed here I refer to Felscher [1981; 1985]. The reader will have noticed that among the examples, listed at the end of Section1, there is none which treats a formula with quanti ers. But this is no serious omission since the argumentation forms for quanti ers are, so to speak, the direct generalisations of the forms for conjunction and disjunction to the in nite case. For strategies, however, this treatment of quanti ers has

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the eect that there may occur in nite rami cations: if S; Æ;  is a strategy and if a node e carries as Æ(e) either a formula P 8xw or an attack P 9 upon a formula Q9xw then the tree S has an in nite rami cation at e | every term t determines a lower neighbour of e, carrying either an attack Qt or an answer Qw(t). Although the branches of S must be nite (as follows from (S 0)), the strategy itself is an in nite object. It is obvious that this is a clear disadvantage of strategies as compared to the more usual notions of proof. I now shall abstract a nite object from a strategy, its skeleton. Let H be a class of dialogues as in Section 1. An H -skeleton for a formula v is a triplet S; Æ;  with the same properties as an H -strategy for v except that in (S 1) certain nodes e are exceped and, instead, are covered by (S 1e ) If Æ(e) is P 8xw then only one lower neighbour of e carries an attack upon Æ(e), and this attack is Qy where y is a variable not occurring free in any expression Æ(h) with h  e . If Æ(e) is an attack P 9 upon Q9xw then only one lower neighbour of e carries an answer, and this answer is Qw(y) where y is a variable not occurring free in any expression Æ(h) with h  e . As is usual, the variable y will be called the eigenvariable in these situations. It is clear that every H -strategy contains various H -skeletons, and it is not hard to see that, conversely, every H -skeleton can be extended to an H strategy. This observation has the important consequence that it suÆces to consider H -skeletons which, having nite trees, are more easily handled in induction proofs. For instance, the Extension Lemma of Section 1 is proved in the form that every E -skeleton can be extended to a D-skeleton. Unfortunately, E -skeletons still have certain undesirable properties. Consider the example of a formula 9xa ! 9xa where a is atomic; there are two E -strategies, viz. 0. 1. 2. 3. 4. 5. 6.

P 9xa ! 9xa Q9xa P9 Qa(y) P 9xa Q9 P a(y)

[0,Q] [1,A] [2,D] [1,D] [4,A] [5,D]

0. 1. 2. 3. 4. 5. 6.

P 9xa ! 9xa Q9xa P 9xa Q9 P9 Qa(y) P a(y)

[0,A] [1,D] [2,A] [1,A] [4,D] [3,D]

In the right skeleton, the attack at 3 is answered with the substitution term y at 6; this answer must be delayed because the choice of the substitution term depends on the eigenvariable y appearing at 5. There are no phenomena of an analogous type in, say, the sequent calculus; in Lorenzen and Schwemmer [1973] and in Haas [1980], where an informal use of E -skeletons is made, the possibility that this situation might occur has been overlooked.

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In order to circumvene this diÆculty, I introduce the concepts of a formal dialogue and of a formal strategy, making use of the formal argumentation forms for 8 and 9: Q8: assertion: Q8xw P 8: assertion: P 8xw attack: P t attack: Qy (eigenvariable) answer: Qw(t) answer: P w(y)

Q9: assertion: Q9xw P 9: assertion: P 9xw attack: P 9 attack: Qt answer: Qw(y) (eigenvariable) answer: P w(t). I de ne a formal E -dialogue in exactly the same way in which I de ned an E dialogue, only now in (D01), (D02) the formal argumentation forms are used for quanti ers and the eigenvariable condition is imposed at the position indicated. The adjective formal then refers to the fact that, contrary to the intuitive understanding, in the attack Qt the term t is stated already by Q; eigenvariables chosen at a later position then must respect these expressions Qt. I de ne a formal E -strategy in the same way in which I de ned an E strategy, but now with formal dialogues instead and with the following changes: there is only one possibility for Q taken into account for answering an attack P 9 (case Q9), making an attack Qy (case P 8), making an attack Qt (case P 9). It then is obvious that every formal E -strategy can be transformed into an E -skeleton; it can be shown that, conversely, every E -skeleton can be transformed into a formal E -strategy. It is the formal E -strategies which can be set into correspondence with LJ-proofs. It follows from these observations that the disadvantage of dialogues consisting in 1. the treatment of quanti ers as in nite conjunctions and disjunctions disregarding Frege's discovery of nitary quanti er rules made possible by the use of free variables, and 2. the ensuing appearance of in nite strategies is only apparent. It arose because we wanted to use the same argumentation forms (concerning quanti ers) for both P and Q; it could have been avoided if, from the outset, we would have studied strategies instead of dialogues. This illustrates once more the diÆculty, mentioned at the beginning of Section3.2, that it is not the dialogues but the strategies which correspond to proofs: working with dialogues, we have to describe in advance the branches

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of strategies which themselves are to be de ned only with the help of these dialogues. It also should be observed that, in contrast to Gentzen's calculi for provability, strategies and dialogues do not appear as natural representations of the relation of provability from hypotheses but only as those of the relation of absolute provability. Of course, if M is a nite set of sentences and m is a conjunction of these sentences then, for every sentence w, the sequents

M

)w

and

)m!w

are simultaneously derivable in LJ, and this permits us to reduce the provability of w from the hypotheses M to the absolute provability of m ! w. It also is obvious that a dialogue, discussing the derivability of M ) w, should begin with an initial list of the Q-formulas determined by M , followed (or preceded) by the P -formula P w. But no general rule on how to proceed from this initial list can be stated as long as we want to keep the alternation between P and Q during the progress of our dialogue. If a is atomic, a sequent such as a ) a _ w produces the initial list Qa; P a _ w which must be followed by an attack of Q ; on the other hand, a sequent such as a ^ w ) a produces the initial list Qa ^ w ; P a which must be followed by an attack of P . Certainly, regulations circumventing these diÆculties may be formulated, but apparently only at the cost of a loss in intuitive appeal. Obernau/Neckar

BIBLIOGRAPHY [Barth and Krabbe, 1982] E. M. Barth and E. C. W. Krabbe. From Axiom to Dialogue, De Gruyter, Berlin, 1982. [Curry, 1963] H. B. Curry. Foundations of Mathematical Logic. McGraw-Hill, New York, 1963. [Ehrensberger and Zinn, 1997] J. Ehrensberger and C. Zinn. DiaLog | a system for dialogue logic. In CADE - 13, Conference on Automated Deduction, Townsville, North Queensland, Australia. Pp. 446{460. Lecture Notes in Arti cial Intelligence, Vol. 1249, Springer-Verlag, 1997. [Felscher, 1981] W. Felscher. Intuitionistic tableaux and dialogues. Prepared notes, distributed at the conference The Present State of the Problem of Foundation of Mathematics, Firenze, June 1981. [Felscher, 1985] W. Felscher. Dialogues, strategies and intuitionistic provability. Annals of Pure and Applied Logic, 28, 217{254, 1985. [Haas, 1980] G. Haas. Hypothesendialoge, konstruktiver Sequenzenkalkul und die Rechtfertigung von Dialograhmenregeln. In Theorie des wissenschaftlichen Argumentierens, C.F. Gethmann, ed. pp. 136{161. Suhrkamp, Frankfurt, 1980. [Hintikka, 1968] K. J. J. Hintikka. Language-games for quanti ers. In Studies in Logical Theory, pp. 46{72. American Philosophical Quarterly Monograph Series 2, Blackwell, Oxford, 1968.  [Kambartel, 1979] F. Kambartel. Uberlegungen zum pragmatischen und argumentativen Fundament der Logik. In Konstruktionen versus Positionen, K. Lorenz, ed. pp. 216{ 228. de Gruyter, Berlin, 1979.

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[Kamlah and Lorenzen, 1967] W. Kamlah and P. Lorenzen. Logische Propadeutik. Bibliograph.Institut, Mannheim, 1967. [Kindt, 1970] W. Kindt. Dialogspiele. Diplomarbeit, Math. Institut Universitat Freiburg, 1970. [Kindt, 1972] W. Kindt. Eine abstrakte Theorie von Dialogspielen. Dissertation, Universitat Freiburg, 1972. [Krabbe, 1982] E.C. W. Krabbe. Studies in dialogical logic. Dissertation, Rijksuniversiteit Groningen, 1982. [Lorenz, 1961] K. Lorenz. Arithmetik und Logik als Spiele. Dissertation, Universitat Kiel, 1961. Partially reprinted in [Lorenzen and Lorenz, 1978]. [Lorenz, 1968] K. Lorenz. Dialogspiele als semantische Grundlage von Logikkalkulen. Archiv Math. Logik Grundlagenforsch, 11, 32{55, 73{100, 1968. Reprinted in [Lorenzen and Lorenz, 1978]. [Lorenz, 1973] K. Lorenz. Die dialogische Rechtfertigung der eektiven Logik. In Zum normativenFundament der Wissenschaft, F. Kambartel and J. Mittelstra, eds, pp. 250{280. Athenaum, Frankfurt, 1973. Reprinted in [Lorenzen and Lorenz, 1978]. [Lorenzen, 1960] P. Lorenzen. Logik und Agon. In Atti Congr. Internat. de Filoso a, Vol. 4, Sansoni, Firenze, pp. 187{194. Reprinted in [Lorenzen and Lorenz, 1978]. [Lorenzen, 1961] P. Lorenzen Ein dialogisches Konstruktivitatskriterium. In In ntinistic Methods, Proceed. Symp. Foundations of Math, PWN, Warzawa, pp. 193{200, 1961. Reprinted in [Lorenzen and Lorenz, 1978]. [Lorenzen, 1962] P. Lorenzen. Metamathematik Bibliograph.Institut, Mannheim, 1962. [Lorenzen, 1967] P. Lorenzen. Formale Logik, 2nd ed., de Gruyter, Berlin, 1967. [Lorenzen and Lorenz, 1978] P. Lorenzen and K. Lorenz. Dialogische Logik, Wissenschaftl. Buchgesellschaft, Darmstadt, 1978. [Lorenzen and Schwemmer, 1973] P. Lorenzen and D. Schwemmer. Konstruktive Logik, Ethik und Wissenschaftstheorie, Bibliograph.Institut, Mannheim, 1973. [Mayer, 1981] G. Mayer. Die Logik im deutschen Konstruktivismus. Dissertation, Univesitat Munchen, 1981. [Saarinen, 1979] E. Saarinen, ed. Game-Theoretical Semantics, D. Reidel, Dordrecht, 1979. [Smullyan, 1968] R. M. Smullyan. First Order Logic. Springer-Verlag, Heidelberg, 1968. [Stegmuller, 1964] W. Stegmuller. Remarks on the completeness of logical systems relative to the validity concepts of P. Lorenzen and K. Lorenz, Notre Dame Journal of Formal Logic, 5, 81{112, 1964.

EDITOR'S NOTE The dialogue system of this chapter has recently been implmented as a theorem prover DiaLog [1] and Colosseum [2]. DiaLog, written in Lisp, oers a rule language for rede ning the rules of the game. It also supports automatic and interactive proving and has a user-friendly interface. EÆciency, however, has not been the major concern, and therefore, DiaLog may serve more as a tool for teaching or for experimenting with the dialogue rules. Colosseum is a no-frills re-implementation of Dialogue Games in Prolog. Its dialogue rules are hardwired for intuitionistic rst order predicate logic (as speci ed by Felscher above). Colosseum allows automatic proving only, but it is much faster than DiaLog and is web accessible.1 1 http://www8.informatik.uni-erlangen.de/IMMD8/staff/Zinn/Dialogue/Colosseum.html

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[1] J. Ehrensberger and C. Zinn. A system for dialogue logic. In 14th. Int'l Conf. on Automated Deduction (CADE-14), number 1249 in LNAI. Springer, 1997. [2] C. Zinn. Colosseum { An Automated Theorem Prover for Intuitionistic Predicate Logic based on Dialogue Games. In Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX): Position Papers. Institute for Programming & Logics, University at Albany - SUNY, TR 99-1, 1999.

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FREE LOGICS I:

Introduction

1 WHAT ARE FREE LOGICS? Some theorems of CQC =, such as those of the form (1)

9x(x =  )

and (2) '[=x] ! 9x'; are often accused of introducing into that theory|and thus into the very core of `our logic'|undesired `existential commitments'. However, the mere derivability of these sequences of symbols can hardly accomplish such a major feat by itself, and even when the theory is supplied with the usual `referential' semantics, metaphysics is still far from being determined one way or another. 1 and 2 certainly require|by way of this semantics| that every singular term of the language receive an interpretation in the domain of quanti cation, but so what? The formal instrument does not specify the metaphysical counterpart of the relation between a symbol and its interpretation, nor does it tell you which things can or cannot belong to a domain of quanti cation. The formal instrument is neutral with respect to all these questions, and thus by itself cannot introduce any metaphysical commitments, existential or otherwise. Things get more complicated when one takes into account the ideology most commonly associated with CQC= and its referential semantics. Then it becomes very `natural' to think of a singular term as denoting its interpretation, hence to read the semantical requirement evoked by 1 and 2 as the requirement that every singular term denote. Even more importantly, if one agrees with Quine that `to be is to be a value of a bound variable'1|that is, if one assigns `existential import' to quanti ers|the domain of quanti cation becomes the set of all and only those objects which exist in a given (possible) situation, and the above requirement is drastically strengthened, to the demand that every singular term denote an existing object. Now the ontological commitments are certainly apparent, and someone is bound to react to them in the name of logic's `purity'. 1 See for example [Quine, 1939]. In what follows, we will sometimes refer to this statement as Quine's dictum.

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Free logics2 result from this reaction. However, since what they are a reaction to is a very delicate combination of many factors|a certain philosophical understanding of a certain formal interpretation of a certain formal system|it is diÆcult to say exactly what they are and how far they extend. To say|as is often said|that they are `logics free of existence assumptions with respect to their singular terms' is too vague to be of much help, and also somewhat inaccurate from a historical point of view. For every formal system and every formal semantics can be free in this sense, given a suitable ideology, but this much tolerance was certainly not in the minds of the people who created free logics.3 They wanted to reform classical logic, and substitute for it a better instrument, they thought that both the usual formal systems and the usual formal semantics were faulty in important ways, and it is only fair to de ne free logics so as to make sense of the precise task that they set for themselves. On the other hand, it would not do to identify free logics with a certain class of theorems. For one thing, there is no one such class (as the expression `free logics' should make clear),4 and there is even some debate as to whether free logics result from restricting or rather extending classical logic.5 But more importantly, we suggested above that all these modi cations|whether restrictions or extensions|would make no sense (and in particular would not be legitimately referred to as free logics) if not in the context of certain interpretations of the formal systems, and of a certain understanding of these interpretations. And nally, it would be totally unsatisfactory to de ne free logics in terms of a given semantics, or even a given class of semantics. For not only is a formal semantics (as well as a formal system) not enough to characterise the present enterprise in the absence of some `intuitive reading' of it, but also the choice of a semantics is probably the most important question in this area, and we have to be careful not to prejudge such a fundamental issue by a biased de nition. Keeping all these reservations in mind will inevitably result in a less than straightforward characterisation of our subject, but the complications we will have to go through will prove instructive. For in this subject more than in others, logic, philosophy of logic and philosophy in general (especially metaphysics) are intertwined in a very delicate way, and it does not hurt if this delicate relation is emphasised right from the beginning. In conclusion, I propose the following de nition. A free logic is a formal system of quanti cation theory, with or without identity, which allows for some singular terms in some circumstances to be thought of as denoting no 2 This expression was rst used by Karel Lambert in 1960. 3 See for example [Leonard, 1956] and [Lambert, 1967]. 4 Thus `free logics' is the correct expression to refer to the

whole subject, but `free logic' is also very common. 5 In this regard, see van Fraassen's position sketched in Section 11.

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existing object, and in which quanti ers are invariably thought of as having existential import. A few comments and clari cations are in order. First of all, a terminological matter. The expression `thought of', which occurs twice in the de nition, must be regarded as inclusive of both the formal interpretation of the system and the intuitive (or philosophical) reading of this interpretation. When the formal semantics is missing (as was the case in free logics for several years), this `thinking of' reduces entirely to its intuitive component. Secondly, the de nition requires that there be in the language of a free logic expressions construed as singular terms. A language containing no individual constants or descriptions and allowing individual variables to occur only bound in well-formed formulas (and there are languages of this sort for CQC, for example some of Quine's) would hardly satisfy the present requirement. Thirdly, the de nition does not exclude the possibility that every singular term denotes in every circumstance, only that every singular term denotes an existing object in every circumstance. There are philosophers (Meinongians for example) who think that there are non-existing objects, and that singular terms may well denote them: the de nition is neutral with respect to such views. However, to avoid awkwardness, usually I will refer to singular terms not denoting an existent simply as non-denoting. Fourthly, the de nition is concerned not with whether there actually are non-denoting singular terms, but only with whether there may be. A free logic is after all a logic; hence all that it can reasonably care for is logical possibility. When a logic acknowledges the possibility of non-denoting singular terms, we will say that it allows for non-denoting singular terms. Fifthly, not every logic allowing for non-denoting singular terms is a free logic by our de nition. In particular, all attempts at saving the formal system (and the formal semantics) of classical logic by some substitutional or Meinongian reading of the quanti ers are ruled out. On the other hand, it is perfectly possible to add to a free logic substitutional or Meinongian quanti ers, thus extending its expressive power. Finally, even though referential semantics played a major role in the discussion above, the de nition does not mention this semantics. The reason is that the existential import of quanti ers, and even the distinction between denoting and non-denoting singular terms, can be eectively mimicked in some non-referential semantics (for example, in Leblanc's truth-value semantics),6 even if the best way to understand what is going on in these semantics is still to compare them with their referential analogues. Thus the three factors to whose combination a free logic is a reaction come to play dierent roles in its de nition: a free logic is the result of a modi cation of 6 On this and other alternatives to the standard referential approach, see the chapter by Leblanc in Volume 2 of the present 2nd edition of this Handbook.

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the formal system of CQC (or CQC=), motivated by a certain intuitive reading of it, which is best understood (at least so far) in the context of the usual referential interpretation of that system. 2 WHY FREE LOGICS? The most general answer to this question has already been suggested in the discussion preceding my de nition of a free logic. Though vigorously attacked from some quarters, the neopositivistic suspicion towards metaphysics is still highly in uential in contemporary logic. Whether they regard metaphysics as sheer `nonsense' or as a set of `synthetic' statements to be neatly distinguished from the `analytic' ones constituting their discipline, many logicians like logic to be metaphysically `pure', or not to carry any metaphysical `baggage'|as the many debates in the area of quanti ed modal logic show sometimes quite dramatically. To apply such a general motivation to the present case, it is enough to regard even the simplest existential statements as metaphysical in nature. However, this motivation by itself does not go very far towards motivating anything close to free logics. As we will see in the next section, classical logic has its own ways of dealing with these matters, and certainly many classical logicians would not accept without a ght the claim|presupposed by the alleged `justi cation' of free logics suggested above|that classical logic is in any sense existentially committed or metaphysically `impure'. To get closer to the justi cation we are looking for, we need to weaken that claim as follows. Classical logic (if ltered through the usual interpretation, and the usual reading of this interpretation) does not allow for non-denoting singular terms. To be sure, this logic can be used in such a way as to avoid any philosophical commitments or any problems resulting from the limitation in question, but this requires the adoption of convoluted and ad hoc procedures of translation from natural language into the formal language and back (in a word, of a number of epicycles). Free logics, on the other hand, represent a much more straightforward and direct approach to the same problems: they make the translations easier, they allow expressions of natural language to be taken more often at face value, and they require fewer ad hoc assumptions. This justi cation is certainly better than the rst one, but still, it does not entirely ful l its purpose. For it does not take into account the fact that the classical logician can shape his philosophy of language so as to make it t his logic perfectly (and make his logic the most `natural' thing in the world): Russell's position|to be mentioned brie y in the next section|is in this respect typical. And this makes it clear once and for all that the adoption of some speci c view in the philosophy of language is an essential step towards the justi cation of free (and perhaps all) logics.

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There is a whole spectrum of such views that would do the job nicely, ranging from an extremely `metaphysical' one to an extremely `pragmatic' one. For the sake of illustration, let me brie y discuss these two extremes. The `metaphysical' extreme states simply that in natural language there are non-denoting singular terms. A singular term is an expression that purports to denote a single object, and many a singular term fails to achieve this purpose. Nonetheless, they are still singular terms: `Pegasus' is as much a singular term as `Caesar' or `3', and `the winged horse' or `the round square' are as much singular terms as `the President of the USA in 2000'. Hence no formal system can give a faithful representation of the structure of natural language (and so be reasonably applied to it) if it does not allow for non-denoting singular terms. The `pragmatic' extreme, on the other hand, regards the real existence of non-denoting singular terms in natural language as totally irrelevant. Whether there are or there aren't any, there are contexts in which some people use expressions as singular terms without assuming that they denote anything, or maybe even in the process of wondering whether they denote or not. For example, an attempt by a person to prove that God exists| or that `God' denotes|might be conceived as a case in point. Whether these people are right or not, a logic allowing for non-denoting singular terms would also allow for a more direct and faithful representation (and evaluation) of their reasoning in those contexts. So this logic would be an instrument of wider and simpler applicability than classical logic, and would not prejudge important issues which it is inappropriate for logic to decide. Of course, the classical logician can be expected to have responses to these motivations. It is certainly not news that in philosophy, or anywhere else, you can't get something valuable for nothing. In the present case, this suggests that you need a position in between the two above extremes to transform the fear of metaphysical commitment so well entrenched in most contemporary logicians into a defence of free logics. 3 CLASSICAL LOGIC AND NON-DENOTING SINGULAR TERMS As suggested earlier, the classical logician is not forced to modify his formal instrument by the mere presence in natural language of expressions like `Pegasus' or `the round square'. He has at his disposal several techniques for dealing with alleged non-denoting singular terms within his own framework. Since all these techniques are treated extensively in other parts of the Handbook.7 I will limit myself here to little more than listing them. In the rst section, I pointed out that the problem free logicians see in classical logic (and try to solve with their logics) is the following: classical 7 In particular, in Hodges' chapter in Volume 1 and Salmon's chapter \Reference and Inforamtion Contents: Names and Descriptions" in a later volume.

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logic makes it impossible to combine the presence of non-denoting singular terms with an `existential' reading of quanti ers. A classical logician willing to avoid this problem, then, has two main options available: he can deny existential import to quanti ers, or exclude the possibility of non-denoting singular terms. If he wants to go the rst way, he will nd two basic suggestions in the literature. One is to drop the referential scheme of interpretation altogether, and go back to the old substitutional scheme, quite popular in the days before Tarski's systematisation of formal semantics. The other is to remain within the referential framework, but admitting non-existing objects (as well as existing ones) in the range of quanti ers.8 If he wants to go the second way, he will again have a choice between two alternatives: Russell's theory of descriptions and Frege{Carnap's chosen object theory. Within the rst alternative, he will rule out non-denoting singular terms by simply denying the status of singular terms to all those expressions of natural language (that is, de nite descriptions and `grammatically proper names') that can ever be non-denoting, and retaining it only for those other expressions (that is, demonstratives) that look absolutely `secure' from a denotational point of view. Within the second alternative, his strategy will be more subtle. For Frege never really denied (as Russell did|at least as far as logical form was concerned) that there are in natural language non-denoting singular terms, but claimed that their presence constitutes a defect, to be repaired in a `logically perfect' language (see [Frege, 1892]). Thus, whereas Russell's proposal extends very naturally to a complex philosophical position, which includes (at least) metaphysical and epistemological themes, Frege's quali es as an intrinsically pragmatic one, in whose favour nothing can be said better than Carnap's words in [Carnap, 1947]: `there is no theoretical issue of right or wrong between the various conceptions, but only the practical question of the comparative convenience of dierent methods' (p. 33). 4 INCLUSIVE LOGICS Chronologically, some of the rst instances of a revisionary attitude about the existential `commitments' of classical logic can be found in what Quine called inclusive logics, that is, logics allowing the domain of quanti cation to be empty. To dispel what seems to be a quite common misunderstanding, it needs to be pointed out once and for all that inclusive logics and free logics are two dierent subjects. A logic can be free without being inclusive, and can be inclusive without being free. However, it is also convenient to treat 8 The rst suggestion is usually associated with Le sniewski, the second one with Meinong. For more recent formulations, see in the rst case Lejewski [1954; 1958] and [Luschei, 1962], in the second Parsons [1980] and Routley [1980].

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the two subjects together. For, on the one hand, the problems they face are strictly connected, and on the other, as for example [Belnap, 1960] has pointed out, it is quite natural to require inclusiveness of a free logic and vice versa. The rst inclusive logic was developed (twenty- ve years before the rst free logics) by [Jaskowski, 1934]. Jaskowski's is a natural deduction system, which, in contrast with most other such systems, allows for two dierent kinds of assumptions (or `suppositions'). One can assume formulas (which one indicates by pre xing the formula with the metalinguistic symbol S ), and one can assume singular terms (which one indicates by pre xing the term with the metalinguistic symbol T ). The way the assumption of terms works is made clear by the quanti cational rules of the system, which are given below. 1. Supposition of a term: at any point in a deduction it is possible to introduce an assumption of the form T  , where  is a new term. 2. Universal Instantiation: '[=x] follows from 8x' and T  . 3. Universal Generalisation: if ' follows from T  then it is possible to deduce 8', and this conclusion does not depend on the assumption T  (which is thus `discharged').9 To explain how these rules allow the domain to be empty (by disallowing proofs of formulas which would exclude this possibility), it is best to use an example. Consider then (3)

8x' ! 9x';

a typical instance of an `exclusive' formula and a theorem of classical logic, and try to prove it in Jaskowski's system. A reasonable way to go about this is to assume the antecedent of 3 and the negation of its consequent, that is, to start out with (4) S 8x' (5) S 8x:': However, given the particular form of (2) above, nothing follows from 4 or 5 without also supposing a singular term. Let us do so, and continue with (6) T : Now from 4 and 6 we get 9 The fact that the Universal Generalisation can be given in this form depends on speci c features of Jaskowski's system: in particular, on the fact that his only terms are variables.

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(7) '[=x] and from 5 and 6 we get (8)

:'[=x]

which of course contradict each other. So the assumptions are not consistent, but the key point here is that there are not two but three assumptions, and in particular 4 and 5 can still be perfectly consistent if nothing like 6 is accepted (which is exactly what one would nd most natural in the case of the empty domain).10 Thus the attempted proof of 3 is blocked. Jaskowski considers this quanti cational system very brie y, almost as an appendix to a paper mostly devoted to propositional logic. Possibly for this reason, the system has a number of unnecessary limitations, and the consequences of removing them are not explored. If they had been explored, the system might have turned out to be the rst free logic as well as the rst inclusive logic. To understand what I mean, consider that in the system in question (i) open formulas are not provable, (ii) there are no individual constants, and (iii) the metalinguistic symbol T has no objectlanguage counterpart. If (iii) and either (i) or (ii) were dropped (and, say, T  were the object-language counterpart of T ), rules (1){(3) of p. 153 would immediately yield (in conjunction with the propositional rules) theorems like (9) (8x' ^ T   ) ! '[=x] (10) 8xT x; while at the same time blocking the proof of formulas like (11) 8x' ! '[=x]; and as we will see these are the key features of most free logics.11 When something like the above happens, and the solution of a problem can be found almost automatically by solving another problem, one naturally is led to suspect that there exists something more than a coincidence, that there is indeed a real connection between the two problems. In retrospect, it is not diÆcult to see what the connection is. Free logics are logics allowing for non-denoting singular terms, and of course if the domain is empty then all singular terms are non-denoting; hence if an inclusive logic 10 For in this domain there are no objects, hence nothing to talk about by using singular terms. 11 This of course when T  is read as a substitute of the more common E !. Furthermore, notice that removal of (iii) is not critical to generate a free logic: if (iii) is not dropped (but either (i) or (ii) is) what we obtain is a `pure' free logic, that is, a free logic without existence or identity. Indeed, making the existence symbol metalinguistic is one way of constructing a natural deduction or Gentzen formulation of such a pure free logic.

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allows for any singular terms at all, it must allow for non-denoting singular terms, and thus be free as well. In light of this consideration, it is easy to see that the only way Jaskowski's logic (or any inclusive logic for that matter) could avoid being free was by refusing to admit any singular terms (which is the philosophical meaning of limitations (i) and (ii) above). And one might expect that, simply by developing their instrument a little further, inclusive logicians would have nally `reached' free logics in a very natural way. However, this is not what happened, and the reason is interesting. As we will argue later at great length, the fundamental problem to be solved in the development of free logics is a semantical one: the problem of assigning reasonable truth-conditions to sentences containing nondenoting singular terms. Inclusive logicians went very close to hitting this problem when they considered dropping some of Jaskowski's limitations. Thus Mostowski [1951], when constructing an inclusive logic contravening (i) above, had to decide what to do with open formulas in the empty domain. In a language without individual constants (as his was), free variables are the only possible place-holders for singular terms, hence Mostowski's problem was at least in part a special case of the fundamental problem of free logics. But there was at the time no awareness of this, so he simply treated free variables in analogy with (universally) bound ones, and he made all open formulas true in the empty domain. The system resulting from this choice had a surprising anomaly: modus ponens was not truth- or validity-preserving in it, as the following example illustrates. (12) '(x) '(x) ! 9y'(y) 9y'(y) The presence of this anomaly could have worked as a stimulus towards more satisfactory solutions, if the general problem lingering in the background had been perceived. Since it was not, subsequent authors such as Hailperin [1953] and Quine [1954] regarded the anomaly as a mere nuisance, and preferred to avoid the question entirely by returning to Jaskowski's practice of excluding open theorems, thus contributing in a decisive way to sealing o what could otherwise have been a promising line of enquiry. II:

Proof-Theory

5 AXIOMATIC SYSTEMS We saw in the last section how inclusive logicians avoided the crucial (semantical) problem of free logics. It might be surprising to nd out that

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even most free logicians basically side-stepped this problem for ten years (at least in their published works), by limiting themselves to a purely prooftheoretical development of their logics. As a result, the history of free logics can be neatly divided into two (partly overlapping) periods: the rst one mostly devoted to proof-theory and the second one mostly devoted to semantics. It is natural, then, in accounting for the subject, to follow the same pattern, and here we will do just that. In the present part we will discuss the formal systems of free logic, as elaborated largely between 1956 and 1967,12 and in the next one the interpretations of these systems, whose development took o only beginning in 1966. Within each part, however, we will make no attempt at preserving any chronological order, but will be guided entirely by considerations of systematicity. Every axiomatic formulation of CQC contains as a primitive assumption either the so-called Law of Speci cation (13) 8x' ! '[=x] or some other principle or rule deductively equivalent to it. (For de niteness, we will refer from now on to a system containing 13 as a primitive assumption.)13 Furthermore, all the theorems of CQC = that free logicians nd questionable (including 1 and 2) are proved by making a substantial use of 13. It is natural to conclude, then, that the rst step in the construction of an axiom system for free logic is going to be dropping 13. When this is done, the remaining axioms permit the proof of the following weakened form of 13 (that we might call Restricted (Law of) Speci cation): (14) (8x' ^ 9x(x =  )) ! '[=x] Far from representing a problem for the free logician, however, this result is most welcome to him; for Restricted Speci cation (in contrast with Speci cation proper) is a law that makes perfectly good sense even in the presence of non-denoting singular terms (and existentially loaded quanti ers). To understand why this is so, consider that the supplementary condition required in 14 to instantiate the universal quanti cation 8x' with respect to the singular term  can be legitimately read as stating that  denotes a value of a bound variable, or more simply (via Quine's dictum) that  is denoting. Thus on the one hand 14 says nothing (and in particular nothing 12 For these systems, see Leonard [1956]; Leblanc and Hailperin [1959]; Hintikka [1959a] and Lambert [1963; 1967]. 13 Also, we will refer to a language without function symbols and with 8 as the only primitive quanti er. As a consequence of the latter, the counterpart of 13 in terms of 9 (that is, the Law of Particularisation 13 '[=x] ! 9x') will not occur among the primitive assumptions. And nally, let me notice once and for all that here we will try to give a uniform treatment of the various free logics, disregarding notational and stylistic dierences among their authors.

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questionable) about non-denoting singular terms, and on the other, though it cannot be used to justify the dubious inference from (15) Nothing (existent) is a winged horse to (16) Pegasus is not a winged horse; it can be used to justify the perfectly legitimate one from 15 and (17) Secretariat exists to (18) Secretariat is not a winged horse: Beginning with a seminal paper by Leonard [1956], that practically inaugurated the subject, free logicians have insisted that two of their most important tasks are (a) making explicit the existential assumptions that are tacit in classical logic (and that only can justify|in their opinion|the presence there of `laws' like 13), and (b) discriminating between the cases in which these assumptions are relevant and the cases in which they are not. 14 is a good example of how these two tasks can be successfully performed: on the one hand, the assumption that (the singular term)  be denoting|taken for granted by classical logic|is here expressed by (19) 9x(x =  ) and on the other the relevance of this assumption is signalled by its very presence, thus distinguishing the case of 14 from, say, that of (20) '( ) ! ::'( ); which is also a theorem of both classical and free logic and in which no supplementary existential condition is given (or needed). All of the above, however, is made possible by the fact that CQC = is a logic with identity, for the identity symbol plays a vital role in expressing existence in 19 and substitutivity of identicals a vital role in proving 14. What would happen if the starting point were an axiom system for CQC, that is, for classical logic without identity? We can approach this problem in stages. First of all, notice that if indeed 19 expresses an existential commitment to the denotational character of  , it seems legitimate to use it as de niens for a new existence symbol, say in the following way: (21) E ! =df 9x(x =  ), where x is alphabetically the rst variable dis-14 tinct from  .

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By using this abbreviation, 14 could be rephrased as (22) (8x' ^ E ! ) ! '[=x]; thus making the meaning of the extra assumption even more explicit. In the system resulting from CQC by dropping 13, neither 14 nor its de nitional abbreviation 22 can be proved; yet, on the other hand, something like 14 or 22 is certainly needed. For, as already noted, the procedures of classical logic (and in particular, Universal Instantiation), though based on tacit existential assumptions, are of course unquestionable when these assumptions are true. The simplest way of reintroducing the legitimate cases of instantiation after dropping 13 from CQC would be to add E ! to the set of primitive symbols, and 22 to the set of axiom-schemata. There is however a more ingenious way, which makes use of neither the existence nor the identity symbol, and is due to [Lambert, 1963]. To understand this alternative, it is enough to take a closer look at 14. What this `law' says is that if something is a value of a bound variable then it has all the properties (expressible in the language and) shared by all such values. This conditional statement, however, could be reformulated in universal terms: every value of a bound variable has all the properties (expressible in the language and) shared by all such values. And this reformulation in turn suggests (23) 8y(8x' ! '[y=x]) as a possible replacement for 14 or 22. I have now developed the core of a `pure' free logic FQC, of a free logic with existence FQCE!, of a free logic with identity FQC =, and of course of a free logic with existence and identity FQCE!=. Before presenting their nal formulations, however, two further problems must be mentioned. First of all, consider the system obtained from CQC= by substituting 23 for 13. In this system (24) 8x9y(y = x) is provable, which seems to be a perfectly reasonable result. For every value of a bound variable is certainly also a value of any other bound variable. However, as shown by Bencivenga [1978a; 1980a], this very natural result is not provable in the system obtained from CQC= by simply dropping 13, not is its counterpart in terms of the existence symbol (25) 8xE !x 14 This existence symbol was rst used by Russell, but only with descriptions. It was [Leonard, 1956] who generalised its application to all singular terms.

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provable in the system obtained from CQC= by substituting 22 for 13 (which again is not good news, given the evident connection between 25 and that `existential import' of quanti ers that we regarded as a de ning feature of free logics). Secondly, it long remained an open problem in pure free logic whether (26) 8x8y' ! 8y8x' is provable in the system obtained from CQC by substituting 23 for 13. Fine [1983] solved this problem in the negative, showing the independence of 26 from the system in question. In conclusion, then, let us agree on what follows. FQC is obtained from CQC by substituting 23 and 26 for 13. FQC= is obtained from CQC= by substituting 23 for 13. FQCE! and FQCE!= are obtained from CQC and CQC=, respectively, by substituting 22 and 25 for 13. Two nal remarks. First, all of the above are in a sense minimal systems of free logic: a few stronger systems will be considered in the part on semantics. Second, it will also become clearer in the part on semantics that all these systems are inclusive as well as free: once again, it is the strict connection between the two sets of problems that allows us to automatically solve the one while addressing the other. 6 NON-AXIOMATIC SYSTEMS Something must be said about natural deduction and Gentzen formulations of free logics. Indeed, the rst two formal systems for free logics|those by Leblanc and Hailperin [1959] and Hintikka [1959a]|were natural deduction systems, which however did not receive much currency in the literature. As to Gentzen systems for free logics, they can be found in [Routley, 1966; Trew, 1970; Bencivenga, 1980b]. Here in formulating both kinds of systems we will take for granted standard rules for connectives and identity (as well as, in the case of Gentzen systems, standard axioms), and we will make a substantial use of the existence symbol in the quanti cational rules. Systems for pure free logic (or free logic with identity but not existence) may be obtained by using the same rules but making `E!' into a metalinguistic symbol, and thus accepting as theorems only formulas not containing it.15 With all these quali cations, a natural deduction system for free logic can be characterised by the following four rules.

15 This is the strategy suggested in note 11. In the case of a free logic with identity but without existence, it would also be possible to have 9x(x =  ) do the job of E ! , but this would have the `unnatural' consequence of making quanti cation theory dependent on identity theory.

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(27) Introduction rule for 8 : fE !ag .. . '[a=x] 8x' where a is a new individual constant not occurring in '. (28) Elimination rule for 8 : 8x' E !a '[a=x] (29) Introduction rule for 9 : '[a=x] E !a 9x' (30) Elimination rule for 9 : f'[a=x]g fE !ag .. . 9x'

where a is a new individual constant not occurring in ' or . On the other hand, a Gentzen system for free logic can be characterised by the following four rules. (31) Introduction of 8 in the antecedent: ; '[a=x] `
0 `
0 ; E !a ; 0 ; 8x' `
;
0 (32) Introduction of 8 in the succedent : ; E !a `
; '[a=x] `
; 8x' where a does not occur in ;
or '. (33) Introduction of 9 in the antecedent : ; E !a; '[a=x] `
; 9x' `
where a does not occur in ;
or '.

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(34) Introduction of 9 in the succedent : `
; '[a=x] 0 `
0 ; E !a ; 0 `
;
0 ; 9x' III:

Semantics

7 THE PROBLEM Consider a simple subject-predicate sentence, say, (35) Socrates is a man: How is a truth-value to be assigned to 35 according to the usual referential semantics for classical logic (brie y, classical semantics)? Very simply put, the answer is as follows. First of all, we establish a domain of quanti cation (which, given our adoption here of Quine's dictum, can be identi ed with the set of existing things). Then we look for the denotation of the singular term `Socrates' and for the extension of the general term (or predicate) `being a man' in that domain. And nally, we pronounce 35 true if that denotation is a member of that extension, and false otherwise. There is more to this procedure than meets the eye. Indeed, it is impossible to set up (in a reasonable way) the conditions at which a given sentence is true without having some theory of truth, and the procedure in question is based on one such theory, that is, on what is usually called the correspondence theory of truth. 35 is (say) true|according to this theory|because it corresponds to reality, and it would be false if it did not. More generally, 35 is true in a given state of aairs (or `possible world') if it corresponds to reality there, and false otherwise. If we were doing propositional logic, this correspondence between (atomic) sentences and reality would be the bottom line, but at the level of analysis of quanti cation theory, that is, when sentences are analysed into (singular and general) terms, the correspondence in question is to be reduced to some more basic correspondences: the ones between singular terms and the objects constituting extensions. If in general the correspondence theory wants to establish the truth of a sentence in terms of a t between what the sentence says and the way the world is, then such basic correspondences represent at this level of analysis the points at which the t must be sought. To go back to our example once more, 35 is true just in case the object corresponding to `Socrates' is a member of the set corresponding to `being a man'. But then of course basic correspondences are the key to the whole matter. Once we have the basic correspondences relative to some sentence we can

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determine whether the sentence corresponds to reality or not, but not before. In the world in which we live, we know that 35 is true and (36) Plato is a table is false, but this is because we know who Socrates and Plato are, and which things are men and tables. Probably we would not know if we did not know who Socrates is, and certainly we would be in big trouble if there were no Socrates. This kind of trouble is exactly what awaits us when we introduce nondenoting singular terms into the picture. Non-denoting singular terms denote nothing existent. Of course, they could denote something else, and in what follows we will consider some such position, but this is one possibility among many, and we must also take into serious account the possibility that they denote nothing at all. And taking this possibility seriously means considering situations in which some of the basic correspondences required by classical semantics are simply not there. This is more than an epistemological problem. Consider for example (37) Secretariat is white and (38) Pegasus is white; and suppose that Secretariat be taken to a remote planet, where its colour could not be ascertained. Also, to simplify things, suppose that none of the ctional writings about Pegasus said anything about its colour. Still, there would be a fundamental dierence between 37 and 38. For the colour of Secretariat could not be ascertained in fact, due to the practical limitations of human beings, but could be ascertained in principle, by somebody able to overcome those practical limitations, whereas in the case of Pegasus the thing would be impossible in principle, too: since Pegasus is nowhere to be seen, no matter how our powers were to improve, they would not in uence our ability (or rather, inability) to verify its colour. So it is not a matter of what we know, but of what we think truth is. Under the circumstances imagined above, it looks like it's not the case that Pegasus is white. It is the case that it is not white? (Or|which is the same|it is false that it is white?) Maybe, but if it is so, it must be for (at least partly) dierent reasons than (say) in the case of 36, and we need our theory of truth to tell us exactly what the analogies and the dierences are between the two cases. The correspondence theory by itself cannot tell us this, because its verdicts are based on data|the basic correspondences| that here are not always available. Perhaps all we need is a small clause taking explicit care of such `exceptions', but still we need something, we need some way of deciding when sentences containing non-denoting singular

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terms are true, and why. This is the main question to be faced in the course of constructing a semantics for free logics|and in my opinion in free logics in general. It is an important question because any answer to it is inevitably going to provide an alternative to, or at least a generalisation of, the correspondence theory of truth. It is a delicate question because the correspondence theory is an old and venerable one, and challenging it represents a true act of `revolutionary science'. In the rest of the present part, I will give an account of this revolution. 8 OUTER DOMAINS Given the way in which we set up the problem of `free' semantics (that is, accommodating for the presence of gaps in the basic correspondences), the easiest way to `solve' this problem consists simply in avoiding any such gaps. This is substantially the way most classical logicians operate, either by assigning arbitrary denotations to (previously) non-denoting singular terms (a la Frege{Carnap) or by excluding (a la Russell) such (alleged) terms from the class of things in need of a direct semantical counterpart. However, there is a way of going in this direction without ending up in classical logic: all that we have to do is to acknowledge that `Pegasus' or `the present King of France' have a semantical counterpart (or a denotation) just as much as `Bill Clinton' or `the present President of France' do, only that such counterparts (or denotations) are not members of the domain of quanti cation, or, to put it more bluntly, do not exist. Even if some suggestions of this kind are much older,16 the rst such proposal that appeared in print was contained in the review by Church [1965] of Lambert [1963]. The purpose of the review was a critical one: Church indeed meant to show that the whole enterprise of free logic was of very little philosophical signi cance. Actually however (and a little ironically), the main result it achieved was that of sketching one of the very rst semantical treatments of the subject, and one that was going to have a lot of success in the next few years. Brie y, the substance of Church's contribution was as follows. Let S be any set, and let a classical interpretation of individual and predicate constants be de ned on S . Let P be any monadic predicate, and let two new quanti ers be de ned, to be read `for every x, if x is P then . . . ' and `there is an x such that x is P and . . . '. Church suggested (without actually proving it, but the claim was indeed true, and was proved later)17 that the set of theorems of Lambert's axiomatic system would coincide with the set 16 For example [Leblanc and Thomason, 1968] mention a suggestion of outer domains made (to Leblanc) by Joseph Ullian in 1962, and apparently both Belnap and Lambert had outer domain semantics very early (but never published their results). 17 A sketch of the proof is given in Section 11.

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of (classically) logical truths containing only the new quanti ers. As Meyer and Lambert [1968] put it, free logic was then just a `simple exercise in a theory of restricted quanti cation'. Shortly after Church's `proposal', at least three major attempts were under way at constructing free semantics along the lines implicitly (and unwillingly) suggested by it. None of them made explicit reference to Church, and quite possibly they were all totally independent of his review, but there is a factual, objective sense in which they were all developing the suggestions contained in it, and emphasising dierent aspects of them. The system which went the closest to reproducing Church's intuitions was the `logic of possible and actual objects' proposed by Cocchiarella [1966]. Semantically, the basic unit of this logic (a Cocchiarella structure) can be conceived of as an ordered triple hA; A0 ; I i, where A is as usual a nonempty set and I is a (total) function interpreting individual and predicate constants on A. The new character in this story is A0 , which is just any (possibly empty) subset of A. A is the range of quanti ers, but not of quanti ers having existential import: rather, its members are to be construed intuitively as `possible objects'. A0 , on the other hand, is the range of another pair of quanti ers, which do have existential import. If we adopt the usual the `existentially committed' quanti ers and for example V andsymbols W for theformore general ones, it is easy to see that 8x' can be true in a Cocchiarella structure while '[=x] is not (indeed, even while 9x' is not, if A0 is empty), hence that Speci cation fails for the restricted quanti ers. On the other hand, this principle does hold for the unrestricted quanti ers, which suggests that a formal system for theVlogic inW question can be obtained simply by pairing a classical logic for and with a free logic for 8 and 9 and adding the schema V (39) x' ! 8x'; which supplies the connection between the two sets of quanti ers. Due to the presence of two sets of quanti ers and of principles like 39, Cocchiarella's logic of possible and actual objects is in fact more than a minimal free logic in the sense of Part II, but by dropping the unrestricted quanti ers from the language and all the theorems containing them from the formal system, we would obtain exactly a minimal free logic in that sense. On the other hand, if we were to do this then the larger set A would not be the range of any quanti ers but would only be providing denotations for the individual constants not interpreted in A0 . It might be natural then to represent the situation in a slightly dierent way and, instead of insisting on the set of existents being a subset of a larger set of possibles, focus on the distinction between existents and non-existents (that is, in terms of a Cocchiarella structure, between A0 and A A0 ). And this in turn would bring us immediately to the variant of the present approach proposed by Leblanc and Thomason [1968]. Since this variant is probably the most

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popular, we will give here a slightly more detailed account of it than we did of Cocchiarella's (or than we will do of Scott's). The reader can easily accommodate our remarks to the other variants. A Leblanc{Thomason (or, more simply LT) structure is again an ordered triple hA; A0 ; I i, where however A and A0 are two disjoint sets (called the inner domain and the outer domain, respectively) such that their union is non-empty (this union of course corresponds to the set of possibles in a Cocchiarella structure, and it still plays an important role in the present context, though in a way it shifted to the background). I is a (total) function interpreting individual and predicate constants on A [ A0 . An LT structure is null if its inner domain is empty, and non-null otherwise. In a non-null LT structure, an assignment is a (total) function from the set of variables to the inner domain. Satisfaction is then de ned as usual, but the fact that variables can only get values in the inner domain makes of this domain the range of quanti ers. Leblanc and Thomason's semantics is inclusive as well as free, as is shown by the presence of null LT structures. Thus the problem arises once again of what to do in those structures with open formulas. However, this is not a problem that we need consider. Simplifying on Leblanc{Thomason's own treatment, we can agree to adopt the Jaskowski{Hailperin{Quine suggestion of accepting only closed theorems, and thus leave open formulas simply uninterpreted in null LT structures.18 In contrast with the above authors, this won't produce any limitation in our expressive powers, because we already have individual constants as place-holders for singular terms, and individual constants behave in null LT structures just as they behave (when they are non-denoting) in the non-null ones.19 Besides, we need not deal with open formulas as a preliminary step for evaluating quanti ed sentences in null LT structures, since we can agree once and for all that for all such structures A and all sentences 8x' (40) A  8x':

An analogous attitude will be adopted (without further mention) with respect to all the alternative semantics to be presented here. We have thus considered two variants of what we will call in general the outer domain approach to free semantics. As suggested above, they emphasise dierent aspects of this approach (and of the original suggestions by Church). Cocchiarella makes the most of the notion of restricted quanti cation, whereas Leblanc and Thomason make the most of the presence of two kinds of denotata. A third aspect of this approach is its similarity to Frege{Carnap's classical device; in both cases indeed the problem of nondenoting singular terms is solved by making them denoting (in a sense). It 18 As to Leblanc and Thomason themselves, they preferred to follow Mostowski in weakening modus ponens. 19 That is, in both cases they denote members of the outer domain.

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is not surprising then that there be a further variant of the approach in question which makes the most of this similarity. Such a variant is due to Scott [1967]. Scott's theory (as on the other hand both Frege's and Carnap's) is actually a theory of (de nite) descriptions, but the general semantical strategy it embodies makes perfectly good sense at the level of unanalysed singular terms, too. Very simply put, this strategy is as follows. Associate with each domain of quanti cation an entity not belonging to it, say the entity . Since  is outside the range of quanti ers, by Quine's dictum it does not exist, but still it can be assigned as a semantical value to singular terms; hence fg works practically as an outer domain. At the same time however, since this outer domain is a singleton,  works also like a Carnapian chosen object, in that all the (originally) non-denoting singular terms have it as their common semantical counterpart (or `denotation'). Because of this last feature of Scott's semantics, (41) (:E ! ^ :E ! 0 ) !  =  0 is logically true in it. Since of course 41 is not provable in the minimal free logics of Part II, they should be strengthened somewhat to generate a formal system adequate to the semantics in question. The simplest way to do this consists in adding 41 itself as a further axiom-schema. A more elaborate alternative would require the addition of a new symbol (for example, `') to the language, of a clause xing its interpretation on the `non-existent object' to the semantics, and of the schema (42) :E !

! =

to the deductive apparatus. Actually, if we were dealing with descriptions, this more elaborate alternative might turn out to the be the simpler of the two, because in description theory `' could be introduced by de nition, say by

 =df x(x 6= x): 

(43)

Our presentation of the outer domain approach to free semantics ends here, and we can conclude the present section with a brief appraisal of this approach. Such remarks will be of a general nature, and will leave aside the special developments recommended by Scott, which will be the subject of further discussion in the section on descriptions. First of all, then, the positive side. Outer domain semantics is simple, and we know why: lling all the gaps left by non-denoting singular terms in the basic correspondences allows one to stick to the standard evaluation procedures, thus generating a feeling of familiarity for the whole enterprise. The problem of non-denoting singular terms is not so much solved as it is dissolved. The reason why `Pegasus is white' is (say) true is not at all

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dierent from the reason why `Secretariat is white' would be: `Pegasus is white' is true because the (non-existent) object Pegasus falls within the (non-existent part of the) extension of `white'. Furthermore, outer domain semantics is formally very convenient. Semantical completeness is provable here in its stronger form: not only the set of logically true sentences but also the set of valid arguments is recursively enumerable, and thus the whole logic is under complete (proof-theoretical) control. Once again, the reason is not hard to nd, even though probably it will be fully appreciated only later, when this semantics is contrasted with some of its alternatives.20 The fact is that the semantics in question is bivalent: every sentence, whether or not it contains non-denoting singular terms, is either true or false (in any structure). Whereas all the positive comments on this semantics have to do with practical or technical matters, all the negative ones have to do with philosophical matters. The rst (and most common) of these comments is probably best put in the form of a question: what exactly is the status of the members of the outer domain? The most natural answer to this question is `non-existent objects', but such an answer generates trouble. It is not that non-existent objects are not philosophically `respectable'. On the contrary, they are quite popular in philosophy today, probably more than they ever were after Russell's alleged `refutation' of Meinong. Scholars of Meinongian inclination, for example Parsons [1980], have questioned the validity of that refutation, and constructed ingenious philosophical theories of non-existent objects. To be sure, such objects are diÆcult to deal with, mostly because|as noted by [Quine, 1948]|their identity conditions are far from clear, but to say that they are diÆcult is not to say|as Quine concluded a bit too hastily|that they are `well-nigh incorrigible'. After all, if we are not ready yet to give a satisfactory account of non-existent objects|and chances are that soon this will no longer be true, if indeed it is true now|the problem might be with us and our philosophy, rather than with the objects themselves. All of this is very good, but unfortunately it does not even get close to removing the trouble we mentioned above. For the diÆculty with using nonexistent objects to construct a semantics for free logics is not that we are not ready to accept them or to account for them, but that accepting them and accounting for them should have little to do with one's logic, and should depend instead on one's metaphysical position|at least according to a quite common conception of logic and metaphysics.21 And if this conception is 20 See in 21 Given

this connection the end of Section 10. that metaphysics is often de ned as the study of what is (insofar as it is), it may sound awkward to say that this discipline should also be concerned with what is not. The awkwardness however is reduced when we consider that most supporters of nonexistent objects ascribe to them some sort of (watered-down) `being' (often discriminating between `to be' and the stronger `to exist').

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correct, then outer domain semantics is in con ict with what we regarded as the most basic motivation for having a free logic in the rst place! Another criticism of outer domain semantics has to do less with nonexistent objects in themselves than with the particular use the semantics in question makes of them. For many supporters of such objects (including Meinong)22 have held that at least some of them are `incomplete', that is, such that for some property P , they have neither P nor not-P , and indeed, there seems to be something to the claim that, if for example none of the stories about Pegasus says anything about its length, then (say) the sentence (44) Pegasus is six feet long is neither true nor false, but simply indeterminate. The present semantics, however, allows for no such `truth-value gaps', and we must be careful not to introduce them too hastily into the picture. For if we decided to simply leave some members of the outer domain `unde ned' with respect to some predicate constant P , this would determine the immediate collapse of such logical laws as (45) ' _ :'; and with them of most of CPC. Though certainly quite serious, this criticism of outer domain semantics is not as damaging as the rst one was. Truth-value gaps cannot be introduced too hastily in this semantics, but can be introduced after all. In a later section, we will mention a compromise between the outer domain and the supervaluational approach which saves much of the spirit of both while allowing (as supervaluations do) for truth-value gaps and (in a sense) `incomplete' objects. A third negative comment on outer domain semantics is even more dependent than the previous one on the particular ways this semantics has been formulated so far, and furthermore is itself grounded on a debatable philosophical position. It is just that some people nd it objectionable that there be `genuine' relations between existent and non-existent objects, and the semantics in question (in its usual formulations) seems to allow for such relations. What I mean by `genuine' deserves some words of explanation. If we admit non-existent objects, there are inevitably going to be some relations between them and the existent ones, because it is simply true that, say, (46) I am thinking of Pegasus: Relations such as the one expressed by 46, however, are of a very special kind; without entering into any detail, we can qualify them as `intentional' or in some sense `modal', and contrast them with such `purely descriptive' (or `genuine') relations as the one expressed by 22 See

[Findlay, 1963, p. 57].

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(47) Peter is taller than Mary: Now according to some philosophers of logic the predicate constants P; Q; R; : : : of quanti cation theory stand for genuine (nonintentional) relations, hence if you think that no such relations hold between existents and nonexistents (and share this position in the philosophy of logic), you might be embarrassed by the fact that in the usual formulations of outer domain semantics an ordered pair ho1 ; o2 i can fall into the extension of (say) P even when o1 is a member of the inner domain and o2 a member of the outer domain (or vice versa). It would be possible to reformulate outer domain semantics so as to avoid this (for some people) unwelcome feature of it. However, this has not been done yet, and the present context is certainly not the right place to do it. Let me just notice in closing that at some point the problems raised by these reformulations will become interwoven with the problems raised by our second criticism of outer domain semantics. For there are several ways to go after claiming that the existent object a cannot hold the genuine relation P to the non-existent object b, and one of them is to say that (48) P ab is an indeterminate sentence. 9 CONVENTIONS The positions that we will consider in the present section are quite disparate. What holds them together (in my opinion at least) is the fact that they determine truth-values for sentences containing non-denoting singular terms pretty much by at, and that whatever discussions or justi cations they oer of their choices are more or less of an `external' nature, that is, have mostly to do with the practical consequences of these choices or with how much they ` t' with other (already accepted) linguistic theories.23 For this reason, I found it suggestive to group them around the word `convention'. The most typical `conventional' positions can be described very easily. Their basic semantical unit is a `partial' structure hA; I i, where A is the usual domain of quanti cation and I interprets (on A) all the predicate constants and some (possibly all, possibly even none) of the individual constants. Truth-values for atomic formulas not containing non-interpreted constants are determined as usual, whereas all the atomic formulas containing such constants have the same truth-value, true or false as the case 23 For example, [Burge, 1974] takes it as a crucial argument in favour of his approach that it allows him to save a Tarskian theory of truth. Of course, even authors going in dierent directions do sometimes oer `external' justi cations, but nowhere seem such justi cations as crucial as in the semantics discussed in the present section.

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may be. Borrowing (and adapting) some terminology by Lambert [1981], we can thus distinguish (on obvious grounds) between positive and negative conventional semantics. The truth-values of complex formulas are also determined as usual. In particular, assignments are de ned as (total) functions from the set of variables to the domain, and the satisfaction-condition for quanti ed formulas is the standard (49) A  8x'[f ] if and only if for every member  of the domain of A; A  '[f; =x]: Completeness and the other usual metatheoretical results are not diÆcult to establish for conventional semantics: once more, bivalence makes things relatively easy. Indeed, there is in general not much to be said about the technicalities of these semantics; hence we might turn right away to some considerations for and against accepting them. Given the present state of the literature, positive conventional semantics are little more than a theoretical possibility. Of course, it is a possibility of which most scholars in the eld are aware, but nonetheless it has not become yet the core of a full- edged semantical approach. In such a situation, we cannot expect to nd a great deal of (published) discussion on the semantics in question; hence most of this discussion we will have to supply on our own. An argument which could be given in favour of positive conventional semantics is that they make it very easy to validate the schema (50)  = ; which is usually regarded as expressing a logical law. However, this does not mean that such logics allow for a standard treatment of identity: they create problems with respect to the substitutivity of identicals. For consider a structure hA; I i such that I (a) is not de ned, I (b) is de ned and I (b) 62 I (P ). In this structure, (51) a = b and (52) P a are both true, but (53) P b is false. On the other hand, the situation is not so desperate as it might seem. As we will see, many free semantics are forced to treat identity in some special way, and in particular to require explicitly that a sentence of the form 51 be false when exactly one of a and b is denoting. A similar special provision is all that we would need here.

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Another criticism of positive conventional semantics could be that they, too, allow for `genuine' relations between existents and non-existents. Of course, this criticism should be slightly reformulated, since the semantics in question do not literally allow for any non-existents at all, but only for the truth of (atomic) sentences containing both denoting and non-denoting singular terms. And of course, once reformulated, the criticism might be easily answered in a number of dierent ways, by adopting some more `special provisions'. Turning now to negative conventional semantics, we must notice rst of all that they have had much more success than their positive counterparts. Indeed, the rst semantical account of a free logic ever published, the one by Schock [1964; 1968], was a negative conventional semantics, and so was one of the latest ones, by [Burge, 1974]. If we add that Russell's classical description theory has a lot in common|in the results if not in the methods or the motivations|with these semantics, and that even authors going in a dierent direction|such as Scott [1967]|tend to agree with them when it comes to determining the truth-values of sentences,24 we will have an idea of the persistent attraction of the approach in question. What are the reasons for this attraction? Schock [1968] expresses his motivation as follows: `The application of a predicate to various terms holds just when the denotations of the terms stand in the relation denoted by the predicate; if not all of the terms denote, then their denotations cannot stand in the relation and the application does not hold' (p. 21). In other words, since there is no denotation of `Pegasus', the denotation of `Pegasus' cannot stand in any relation with (say) the denotation of `Bellerophon'; hence (54) Pegasus is loved by Bellerophon is false. As it is, this is not much of an argument. For it basically reduces to the circular claim that 54 is false because (55) The denotation of `Pegasus' stands in the relation of being loved by with the denotation of `Bellerophon' is|where the falsity of 55 is as much in need of a justi cation as that of 54. One might try to shore it up by pointing out that|in the usual set-theoretical terms of classical semantics|the set of ordered pairs corresponding to the relation of being loved by is not going to contain any member corresponding to Pegasus and Bellerophon. But one might answer that classical set theory|just as classical semantics|is not prepared to deal with non-denoting singular terms, and that things could be dierent 24 See the schema (I3) on p. 188 of [Scott, 1967], which Scott considers very reasonable when giving axioms for a theory (not however for pure logic).

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in a `free set theory' devised for this purpose.25 The negative conventional semanticist might then reply that his approach does not require any such new technical instrument, that it allows one to preserve most of the classical framework, and that these `conservative' features are very important from a pragmatic point of view. With this appeal to conservatism we have struck a key note. For what exactly is so great about preserving the classical framework? And why should pragmatic arguments be so crucial in choosing a free semantics? Of course, one may think that pragmatic arguments are always crucial, or even that they are the only arguments one can give in favour of any theory, and that the traditional framework is always to be preserved whenever it is possible. This is a general position in the philosophy of logic (and of science), and I have nothing much to say about it|except that it does not seem to be shared by many free logicians. What I think deserves some comment is the opinion somebody might have that in this particular case, because of the particular nature of the problem, pragmatic arguments are more important than usual. More precisely, I refer to the opinion that, since non-denoting singular terms are basically `don't cares', the only criteria to use in assessing an attitude towards them are whether the attitude is simple, eÆcient, and does not require a vast revision of our conceptual framework. I think that this opinion may be very dangerous for free logic as a whole.26 For after all, what is simpler and more conservative in this case than just sticking to classical logic, supplemented by some of the policies mentioned in Section 3? Thus, omitting any further comment on the speci cs of negative conventional semantics,27 I will conclude the present section with some remarks on why free logicians may think that non-denoting singular terms are not `don't cares', and why they might want something more than an eÆcient way to accommodate them in the classical framework. It is not that free logicians are interested in non-denoting singular terms in themselves; it is not that they have some kind of perverse attraction for what does not exist. However, they are not bound to the realm of existents either; they do not share that `prejudice in favour of the actual'28 which is so common (and possibly healthy) in other branches of knowledge. A scienti c truth is true (at least in part) because of the way the world is, and given a suÆciently wide conception of the `world' the same might be said of many philosophical truths, but a logical truth should be independent of any such factual matters, and in particular of what exists and what does not exist. 25 Such free set theories have been developed in [Scott, 1967] and [Bencivenga, 1976]. To my knowledge, however, they have never been used in formulating semantics. 26 Of course, so may be the more general position mentioned above. but that is also too general to be discussed here. 27 But let me stop a minute to notice that|as far as identity is concerned|these semantics are in a sense in a dual position with respect to their positive counterparts: they easily validate substitutivity of identicals but invalidate self-identity. 28 The expression is Meinong's. See [Meinong, 1904].

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A logical truth should depend only on (logical) form, and there seems to be no plausible (non ad hoc) ground for distinguishing between the form of `Pegasus is white' and that of `Secretariat is white'. Of course, we know that somewhere there is a dierence, because Secretariat exists and Pegasus does not, but invoking that piece of information, and discriminating on that score between the procedures involved in evaluating the two sentences, would be contaminating logic with mere contingencies. In conclusion, non-denoting singular terms represent an important challenge for logic in general. For this reason, even though the free logician is not going to forget considerations of simplicity and theoretical conservatism, he may think that it is more crucial to rethink the whole subject, no matter how complicated and revisionary this process is going to be. 10 SUPERVALUATIONS AND BEYOND The most organic attempt to date at rethinking the whole subject of truth theory in view of the presence of non-denoting singular terms was initiated in 1966 by two seminal papers by van Fraassen [1966a; 1966b], and pursued by van Fraassen himself and several other authors, including Skyrms [1968], Meyer and Lambert [1968], Woodru [1971] and Bencivenga [1980b; 1981]. In the present section we will study this approach, which from its most characteristic technical instrument may be called the supervaluational approach. The starting point of our analysis is once more conventions. According to van Fraassen [1966a], the truth-value of a sentence like (56) Pegasus has a white hind leg; or even the fact that this sentence has a truth-value, is ultimately to be established on the ground of some convention. This convention, however, belongs to the philosophy of language, and should receive there whatever justi cation it is going to receive. Logic, on the other hand, has nothing to do with any such conventions and justi cations: the set of logical truths should be absolutely independent of the philosophy of language we decide to adopt. In particular, there will be conventions assigning True to 56 and conventions assigning False to it, but logic should not be committed to any of them. At the very most, we can think of logic as committed to the logical product of all possible conventions, to what all these conventions have in common, to what is going to be true (or false) no matter what convention we adopt. This notion of the logical product of all possible conventions leads very naturally to the idea of a supervaluation, in the following way. Let a partial structure A = hA; I i be given, and suppose that I (a) is not de ned, I (b) is de ned and I (b) 2 I (P ). Application of the standard evaluation procedures establishes the truth of sentences like

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(57) P b (58) P b _ :P b (59) 9xP x as well as the falsity of (60) :P b (61) P b ^ :P b (62) 8x:P x; but determines no truth-value at all for (63) P a (64) :P a (65) P a _ P b (66) P a ^ :P b (67) P a _ :P a (68) P a ^ :P a: Of course, 63{68 might receive any combination of truth-values on the ground of some convention or other, but it seems reasonable to restrict our attention to those conventions that are classical in the following sense: they assign truth-values to atomic formulas containing non-denoting singular terms in some way that it is not our present concern to examine (indeed, that we could for our present purposes regard as totally arbitrary), but then they proceed to evaluate complex formulas in the standard way. The combination of any such classical convention and the information supplied by the partial structure will determine a valuation of all the sentences of the language. Let us agree to call any such valuation a classical valuation (on A). Of course, all classical valuations will agree on all the sentences (like 57{62) that contain no non-denoting singular terms, but the interesting thing is that they will also agree on many sentences which do contain non-denoting singular terms. Thus for example 63 will receive the value True in some classical valuations and the value False in some others, but every classical valuation will verify 65 and 67 and falsify 66 and 68. In other words, there will be cases|and many of them|in which the logical product of all classical valuations will be non-empty, and as such informative, beyond what is determined by the partial structure. As the fate of

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67 and 68 suggests, this supplementary information is enough to extend to non-denoting singular terms all of CPC. The essence of van Fraassen's approach consists simply in using this supplementary information. More precisely, the supervaluation WA for a partial structure A is characterised by him as the (partial) valuation which assigns True to the sentences that are true in all classical valuations on A, False to the sentences that are false in all classical valuations on A, and no truth-value at all to the remaining sentences. Then, in at least one of the alternatives he contemplates,29 supervaluations are to constitute the basic (or admissible) valuations of free semantics, and all the other semantical notions are de ned in their terms. The fact that supervaluations preserve all of CPC without espousing any speci c convention or admitting non-existent objects is certainly remarkable, but it is also important to point out that when we move beyond propositional logic supervaluations create serious problems. The most apparent of these problems concern identity. When a is nondenoting, nothing so far prevents a classical valuation from falsifying (69) a = a; and when both a and b are non-denoting, nothing so far prevents a classical valuation from verifying 51 and 52 and falsifying 53 on p. 170, thus invalidating substitutivity of identicals. Further (and more subtle) problems concern quanti cation. To understand them, we must rst of all ask ourselves how the present approach can be extended to deal with variables and open formulas. A natural way would seem to be the following. De ne a convention for a partial structure A as a binary function from the set of atomic formulas and the set of assignments for A to fT; F g (that is, as a function assigning (arbitrary) truth-values to atomic formulas relative to assignments). Then let A, any convention and any assignment determine an auxiliary classical valuation, by using standard evaluation techniques for atomic formulas not containing non-denoting singular terms and for complex formulas, and relying on the convention for atomic formulas containing non-denoting singular terms. Point out that in the case of sentences assignments make no dierence, and de ne on this ground the notion of a classical valuation on A, as determined only by A and a convention. Finally, de ne the supervaluation for A as the logical product of all classical valuations on A. All of this sounds very good, but unfortunately it does not work: for it may well be that in the case of sentences assignments do make a dierence. Indeed, nothing so far prevents a convention from assigning True to some atomic sentence 29 He also considers an alternative in which classical valuations themselves are the admissible valuations, but such an alternative is far less interesting or philosophically defensible; hence we will totally disregard it here.

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(containing a non-denoting singular term) relative to some assignment and False to the same sentence relative to a dierent assignment. Hence the `de nition' suggested above of a classical valuation is not legitimate. Van Fraassen's solution of these problems is disappointing. Very simply put, he adds to the de nition of a convention a number of ad hoc clauses which rule out|by at|all the possibilities contemplated above. More precisely, a convention K is to be de ned in such a way that 1. K ( = ; f ) = T ; 2. K ('[=x]; f ) = K ('[ 0 =x]; f ) if either  [f ] =  0 [f ] or K ( =  0 ; f ) = T; 3. K ('; f ) = K ('; f 0) if f (x) = f 0 (x) for every variable x occurring (free) in '. For analogous reasons, it is also required that 4. K ( =  0 ; f ) = F if exactly one of  [f ] and  0 [f ] is de ned.30 These additional clauses simplify the technical developments, and make some of the desired metatheoretical results easily available, but certainly don't go in the direction of providing a satisfactory philosophical motivation for the resulting semantics. Indeed, they rather weaken whatever motivation there was after our rst introduction of the supervaluational approach. For remember, the crucial point there was the neat separation promised by supervaluations between logic and philosophy of language, and the fact that they were supposed to be independent of speci c conventions, and committed only to the logical product of all conventions. Now it would be hard to hold this point of view|in presence of so many restrictions on what counts as a convention. Even the fact that we should limit ourselves to classical conventions (or valuations) might begin to look suspicious, and the whole enterprise appear dangerously close to a gigantic circle. To put it bluntly, it seems that van Fraassen can assign truth-values to sentences containing non-denoting singular terms only to the extent to which he is not independent of a conventional attitude. In my opinion, these shortcomings of supervaluational semantics are due less to the general idea of a supervaluation than to a failure on van Fraassen's part to get deeper into its analysis. Indeed, I think that supervaluations come very close to providing that generalisation of the correspondence theory of truth that we judged necessary for a reasonable treatment of nondenoting singular terms. To justify this claim, it will be convenient to have a fresh look at the whole thing. 30 Van Fraassen's language does not contain E !. If it did, it would probably be necessary to add one more clause: (e) k(E !; f ) = T if and only if  [f ] is de ned.

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Let me begin by asking a direct question. Why is a sentence like 67 always true in supervaluational semantics, even when its only atomic component 63 is truth-valueless? One way to answer this question could be the following: even though 63 has no truth-value, if it did have a truth-value, any truthvalue, 67 would be true. But what is required for 63 to have a truth-value? The semantics itself gives the answer: 63 has a truth-value if and only if a is denoting. Hence the answer to our original question may be rewritten as follows: 67 is true even when a is non-denoting (and 63 truth-valueless) because if a were denoting then it would be true. This answer constitutes the core of a new theory of truth, which for the sake of a label we might call the counterfactual theory of truth. This theory substantially agrees with the correspondence theory on all sentences not containing non-denoting singular terms, but develops in an original way beyond that scope. Its most basic principle may be formulated as follows: a sentence containing non-denoting singular terms is true (false) if and only if it would be true (false) in case these terms were denoting, no matter what their denotations were. According to this principle, not only is 67 always (hence logically) true and 68 logically false, but also 69 is logically true and substitutivity of identicals is truth-preserving, and all of this as a consequence not of the adoption of ad hoc clauses but of the use of normal evaluation procedures. Also, it will be useful to point out right away that accepting the principle in question does not commit one in any way to outer domains or non-existent objects. For in outer domain semantics non-denoting singular terms simply `denote' non-existents, whereas in the present approach these terms denote nothing, and we only take the liberty of considering alternative situations (or `possible worlds') in which they denote, and of making their behaviour there relevant for the evaluation of sentences containing them in the situations (or worlds) in which they do not denote. We will see that compromises are possible between the counterfactual theory of truth and outer domain semantics, but such compromises are not inevitable. Supervaluational semantics|as developed by van Fraassen|suggests the counterfactual theory, but does not explicitly espouse it. More precisely, this semantics does not get to the point of assigning a truth-value (or no truthvalue) to a sentence containing non-denoting singular terms by considering situations in which these terms are denoting. Rather, it considers situations in which the atomic formulas containing these terms receive truth-values. In a way, it is as though supervaluational semantics were developing the suggestions leading to the counterfactual theory only at a propositional level of logical analysis. From this limitation springs in my opinion all the talk about conventions, for it seems that only on a conventional basis we can assign (what look like) arbitrary truth-values to unanalysed atomic formulas. And from the same source springs also the necessity of adding ad hoc clauses to the de nition of a convention; for from a purely propositional

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point of view there is simply no reason why a sentence like (69) should not be false, or more generally why the assignment of truth-values should t the non-propositional logical structure of sentences. The message sent by these considerations is quite clear: what we have to do to remove all that sounds ad hoc and circular in the semantics of supervaluations is carry the approach expressed by this semantics to a more speci cally quanti cational level. Several authors have received this message, and several semantics have been developed along these lines, but all of them had to face, and solve one way or another, a serious problem, whose realisation might well have been the main reason for van Fraassen's adoption of a `propositional' treatment of non-denoting singular terms. The problem is as follows. Suppose that supervaluational semantics be developed at a quanti cational level in what looks like the most natural way. Given a partial structure A and a sentence ' containing singular terms that are non-denoting in A, one considers all extensions of A which make those terms denoting, and pronounces ' true (in A) if it is true in all such extensions, false if it is false in all such extensions, and truth-valueless otherwise. Now consider the sentence (70) 8xP x ! P a; and suppose that a be non-denoting (in some structure A). 70 is an instance of Speci cation, and we know that rejecting Speci cation is the most distinctive feature of a free logic from a proof-theoretical point of view. In particular, 70 is not provable in any free logic unless a special clause is added to it which makes sure that a is denoting; hence we would expect that when a is non-denoting a free semantics had a way of invalidating 70. But this is simply not the case in the semantics sketched above. For 70 is certainly true in all extensions of A in which a is denoting, and so it is true in A itself. This argument can be easily generalised to any other instance of Speci cation, and the conclusion is startling: the most natural `quanti cational' development of supervaluational semantics leads not to free but to classical logic! There are in the literature at least four dierent ways of addressing this problem.31 The simplest one is advocated by Woodru [1971], and consists substantially in mixing the supervaluational approach with the outer domain approach. To get the compromise in question we need to qualify the counterfactual theory of truth in the following way: a sentence containing non-denoting singular terms is true (false) if and only if it would be true (false) in case these terms were denoting, no matter what their denotations were but provided that they were non-existent objects. Thus all 31 Except for the last one, the semantics to be discussed below do not present themselves explicitly as `ways of addressing this problem'. But this is a good way of perceiving the substance of their contribution.

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extensions of a partial structure A are to be conceived as all possible ways of adding an outer domain to it, and 70 is easily invalidated. The resulting semantics has a few advantages over the straight outer domain approach, especially because it does not assign a truth-value to every sentence containing non-denoting singular terms, and thus in a sense can accommodate for `incomplete' objects, but it shares the most substantial problem of that approach, that is, the metaphysical commitment to non-existents. A more sophisticated variant of this strategy was proposed (earlier) by Meyer and Lambert [1968]. It still consists substantially in allowing for outer domains, but these domains are thought of as constituted by words, not by objects. More precisely, non-denoting singular terms are thought of as themselves contained in what Meyer and Lambert call the semantical| not outer|domain of a nominal interpretation, and then predicates are distributed in all possible ways over these new `entities' to form the logical points over the nominal interpretation. A sentence is true (false) in a nominal interpretation if and only if it is true (false) in all logical points over it, and true (false) in the underlying real interpretation (which corresponds to a partial structure) if and only if it is true (false) in all the nominal interpretations which `complete' it. This approach is certainly suggestive, but insuÆciently motivated, and it needs further elaboration before becoming really practicable. The main problem with non-denoting singular terms is that of explaining why a sentence like (71) Pegasus is a horse has whatever truth-value it has (if any), but just on this question the authors become elusive. 71 is true in a logical point|they say|not because the object Pegasus is a horse there, but because there the word `Pegasus' is a horse-word. Again, this is suggestive, but what exactly is implied by being a horse-word? And how are horse-words to be identi ed if not in terms of the truth of sentences of the form (72)  is a horse? Unless we answer these questions (and the authors don't), the whole strategy might look circular, and haunted by the ghost of an ultimately `conventional' attitude. The third attempt at developing the suggestions contained in supervaluational semantics at a `deeper' level of analysis is due to [Skyrms, 1968], and can be described as resulting from two distinct applications of those suggestions. Straight supervaluational technique (that is, assignment of arbitrary truth-values to atomic sentences containing non-denoting singular terms, and subsequent construction of the logical product of all the valuations so obtained) is used with truth-functional compounds, whereas with

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atomic sentences (and in particular identities) containing non-denoting singular terms Skyrms substantially adopts the counterfactual theory of truth, by constructing the logical product of (the truth-values the sentences in question have in) all the extensions of the original structure that assign denotations to the (originally) non-denoting singular terms. Quanti ed sentences are treated in yet a third way, that is, in the standard way: 8x' is true (or, in Skyrms' terminology, holds) in a structure A just in case ' holds in A for every assignment. Skyrms' motivations are expressed very clearly. Frege seems to have thought|he says|that all sentences containing non-denoting singular terms should be truth-valueless, but supervaluations add `an Aristotelian notion of Redemption to the Fregean notion of Sin', in that `if the logical structure is such that every way of lling up the \holes" makes it true (false), then the sentence is true (false) regardless of the holes' (his italics). `Van Fraassen', he continues quite correctly, `applies this idea only to the extent to which logical structure is determined by the sentential connectives', but `identity is also a logical constant, and I suggest that we apply this idea to identity statements' (p. 479). Unfortunately, Skyrms stops short of noticing that quanti ers, too, are logical constants, and thus should also contribute to `determining the logical structure'. As a result, the supervaluational idea is not applied to quanti cation, and very little of quanti ed logic is `redeemed'. In particular, when a and b are non-denoting, not only 70 but also

(73) 8xRxa ! 8yRya (74) 8x(Qxa ! Rxa) ! (8xQxa ! 8xRxa) (75) P a ! 8xP a (76) a = b ! (8xRxa ! 8xRxb) (77) (8xP x ^ 9x(x = a)) ! P a are truth-valueless. Skyrms does not propose a formal system adequate to his semantics, nor did anybody else, and in fact David Kaplan has apparently proved that the semantics in question is not recursively axiomatisable. On the other hand, the approach advocated by Bencivenga [1980b; 1981] falls well within the mainstream of the `standard' free logics we presented in Part II. To get immediately to the core of Bencivenga's semantics, let us concentrate on the solution he oers for the problem connected with 70. Once more, consider a structure A in which a is non-denoting. Suppose that the antecedent of 70 be true in A, and consider an extension A0 of A which assigns a denotation to a. Of course, 70 is true in A0 : if for example we

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assume that its consequent be false there, its antecedent will be false, too. The situation can be depicted as follows:

A A0

8xP x P a T F

F

Now Bencivenga points out that, since we are trying to evaluate 70 in A, the truth-values this sentence has in other structures (such as A0) are really of no independent interest. The only reason why we refer to A0 and other extensions is that A gives no information about the consequent of 70, and we hope that this lack of information can be remedied by extending A. Thus in the case of 69, in which, too, A gives no direct information, we are able by extending it to determine the value True (and save the `logical law' of self-identity). However, we must not forget the purely instrumental character of the extensions in question, and in particular must not let them prevail over the information A already gives. What this means|in terms of the above diagram|is that it is perfectly legitimate to take into account the truth-value assigned by A0 to P a, since A assigns no truth-value to it, but this truth-value should be combined|to the extent to which our evaluation procedure is relative to A|with the truth-value A| not A0 |assigns to 8xP x, since in this case A already gives a de nite response, and one that A0 does not `complete', but simply contradicts. And of course if truth-values are combined in this way, 70 turns out to be false. In general, then, it is all right to extend A in all possible ways and to construct the logical product of all (the valuations relative to) such extensions, but in de ning the valuations in question whatever information is provided by A must always weigh more than the information provided by the other (auxiliary) sources. This discussion leads very naturally to the de nition of a new technical instrument: the valuation VA0 (A) for an extension (or more precisely, a `completion')32 A0 of A from the point of view of A. Without entering into the details of this de nition, we can say that VA0 (A) is determined by A wherever A assigns de nite truth-values, and is determined by A0 elsewhere. The supervaluational instrument is then applied to all these VA0 (A) (where A0 is a completion of A), and gives the nal truth-values (or lack of truth-values) relative to A. Bencivenga's semantics does not show any of the asymmetries or oddities of Skyrms'. There are not three dierent ways of evaluating sentences, and the formal systems introduced in Part II are provably adequate to (suitable versions of) it. Furthermore, this semantics is not committed in any way to outer domains, for exactly the same reasons for which the counterfactual theory of truth in general is not. In this semantics, the truth-value of a

32 A completion of A is an extension of A which assigns a denotation to all singular terms.

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sentence in a given structure often depends on the truth-values some parts of the sentences have in other structures (in which the non-denoting singular terms occurring in the sentence are denoting), but this does not add to any structure any new category of objects, even less non-existent objects. There are dierent structures, and dierent sets of objects exist in them: this much seems pretty safe to say, and this is all that the semantics in question needs. Of course, the price must be paid somewhere. Here the price of this metaphysical simpli cation is paid in terms of a number of logical complications, rst and foremost the de nition of VA0 (A) . Some people (for example, [Posy, 1982]) have objected to this de nition, mostly because the valuation in question does not correspond to any (single) structure: what is true (false) in it is not just what is true (false) in A, nor just what is true (false) in A0 , but some combination of the two. To this objection one might answer that it seems to be a tendency of contemporary philosophical logic to regard structures as themselves constituting a structure, rather than just a set, that is, as bearing to one another relations that are semantically signi cant. Kripke's semantics for modal logic is a sign of this tendency, and Bencivenga's doubly determined valuations may be another (perhaps more radical) sign of it. Before concluding the present section, something must be said about a few formal properties of supervaluations. Such properties have been proved within the context of van Fraassen's original semantics, but the proofs could be easily adapted to most of the variants we presented here. Let us begin by considering the simple sentence (78) P a: We have already mentioned (and used) the fact that in supervaluational semantics 78 is true in a structure A only if (79) E !a is also true there. And we also noticed that 78 cannot even be false in A unless 79 is true, or, to put it otherwise, that 79 is a semantical consequence not only of 78 but also of (80) :P a: There are important historical connections here. Frege [1892] and Strawson [1950; 1952] emphasised the role that relations of presupposition play in natural language. According to their characterisation, a sentence ' presupposes a sentence just in case the truth of is a necessary condition for ' to have any truth-value at all. Thus for example

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(81) John stopped beating his wife presupposes both (82) John has a wife and (83) John used to beat his wife. For, if either 82 or 83 were not true, 81 would be neither true nor false: it would simply represent a `spurious' use of language. Particularly important are the relations of existential presupposition. According to Frege and Strawson, a sentence like (84) The present King of France is wise does not imply (85) The present King of France exists (as Russell claimed), but rather presupposes it, and in general any simple sentence containing singular terms presupposes the existence of denotations for those terms. Classical semantics cannot express any non-trivial relation of presupposition (and in particular existential presupposition). For in classical semantics every sentence (in every situation) has a truth-value, and thus the only sentences that can be presupposed are the logically true ones. Strawson used this fact as evidence that formal logic is in general inadequate to deal with natural language. On the other hand|as is illustrated by the relations between 78 and 79|supervaluational semantics does allow for non-trivial relations of existential presupposition (since 79 is not logically true in it); hence this semantics constitutes an implicit answer to Strawson's challenge.33 It is crucial to the above argument that supervaluational semantics is nonbivalent. There are less positive sides to this failure of bivalence. For example, van Fraassen proved that a suitable formal system of free logic is weakly complete with respect to his semantics, but proved also that this system is not strongly complete with respect to the same semantics.34 The essence of the proof is as follows. Suppose the system were strongly complete. Then, since (86) P a  E !a;

33 For some developments along these lines, see van Fraassen [1968; 1969]. Apparently, however, the discovery that supervaluations allow for non-trivial presuppositional relations is due to Lambert (see [van Fraassen, 1968, p. 151]). 34 For the rst result, see [van Fraassen, 1966a], for the second one see [van Fraassen, 1966b].

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we would have (87) P a ` E !a; and since the Deduction Theorem is provable for the system in question, we could also conclude that (88) P a ! E !a is a theorem in it. But this is impossible, because 88 is not logically true and the system is provably sound. Once again, it is the failure of bivalence that allows for this result. For when 78 is true, we know from 86 that 79|hence also 88|is true, and when 78 is false 88 is true on purely propositional grounds. But 78 can also be neither true nor false, and in that case 88 has no truth-value either|which explains why it is not logically true. The result in question leaves two interesting problems open. On the one hand, there is the obvious problem of whether or not a dierent formal system could be strongly complete for supervaluational semantics|that is, whether or not the set of supervaluationally valid arguments is recursively enumerable. On the other hand, we know that in bivalent semantics weak completeness (for a given formal system) plus compactness gives strong completeness (for the same system), but there is no reason to think that this implication should hold when bivalence fails. In particular, the above argument against strong completeness makes no reference to in nite sets of sentences; hence it still leaves the possibility open that the semantics be compact. Whether or not it is, is our second problem. Woodru [1984] answered both questions in the negative. The set of supervaluationally valid arguments is not recursively enumerable, and supervaluational semantics is not compact. On the other hand, [Bencivenga, 1983] has shown that the quanti er-free fragment of the semantics is compact. This results is interesting because our argument against strong completeness does not depend on quanti ers either; hence in quanti er-free supervaluational semantics it is indeed the case that weak completeness plus compactness does not give strong completeness. IV:

Extensions and Connections

11 FREE LOGIC AND CLASSICAL LOGIC We already know that it is possible to deal with free logic as restricted quanti cation theory. We saw the semantical side of this when introducing the outer domain approach. A syntactical result along the same line was

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proved by [Meyer and Lambert, 1968]. We will give now a brief sketch of their proof. Let L be a rst-order language with the existence but without the identity symbol. Let : and ! be the only primitive connectives of L, and 8 its only primitive quanti er. Let L0 be the result of adding a new monadic predicate constant Q to L, and let a translation  of L into L0 be de ned as follows: 1. (P 1 : : : n ) = P 1 : : : n ; 2. (E ! ) = Q ; 3. (:') = :(' ); 4. (' ! ) = ' !  ; 5. (8x') = 8x(Qx ! ' ). What Meyer and Lambert showed (in eect) is that a sentence ' of L is a theorem of FQCE! if and only if ' is a theorem of classical logic (brie y, a classical theorem). The `only if' part of this biconditional is straightforward. One need only show that the translation of every axiom of FQCE! is a classical theorem, and that if ' and (' ! ) are classical theorems, so is  . The `if' part is more complicated. The reason is obvious: classical logic is more powerful than free logic, hence it is not at all trivial that classical logic does not allow one to prove translations more than free logic allows to prove theorems. What we need here is a conservative extension result, showing that the more powerful deductive tools available in classical logic (in particular Speci cation) do not extend the class of provable sentences of a certain form. First of all, then, we must give a clear formulation of the result we need. For this purpose, Meyer and Lambert construct, for every sentence ' of 0 1 L , the sentence ' , by substituting E ! for Q. Given that (89) 8x(E !x ! ) $ 8x

is provable in FQCE!, '1 is provably equivalent to ' in FQCE!. Furthermore, it is obvious that '1 is a classical theorem just in case ' is. In conclusion, our problem reduces to showing that '1 is a classical theorem only if it is provable in FQCE!|and this is the conservative extension result we need. Of course, if '1 is a classical theorem, its proof (in classical logic) may well contain instances of Speci cation. However, Meyer and Lambert want to show that, given the particular form of '1 , all the instances of Speci cation that might be needed to prove it are also instances of the weaker schema

186

(90) 8x(E !x ! ) ! (E !

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! [=x]);

which is provable in FQCE!. And this would follow if it were possible to show that, every time a universal quanti cation 8x occurs in a proof of '1 (in classical logic),  is of the form E !x ! . That '1 itself has the above property is trivial, but in an axiomatic system this shows nothing about the structure of the sentences that can be used to prove '1 . However, a solution is readily at hand. It is enough to reformulate the problem within a Gentzen system for classical logic.35 Since this system has the subformula property, we may be sure that if ` '1 is provable in it then universal quanti ers occur in the appropriate contexts in the whole proof, hence that the following Gentzen-variant of 90 is all that is ever applied in the proof:

; E ! ! [=x] `
: ; 8x(E !x ! ) `
This concludes Meyer and Lambert's argument. An analogous (and simpler) result is available in the opposite direction. Let L be as before, except that it contains the identity but not the existence symbol. Consider the exclusive free logic EFQC= obtained by adding to FQC= the axiom-schema (91) 8x' ! 9x';

and the translation + of L into L de ned as follows:

1. '+ = (9x(x = a1 ) ^ : : : ^ 9x(x = an )) ! ', where a1 ; : : : ; an are all the individual constants occurring in '.

It is easy to show that a sentence ' is a classical theorem if and only if '+ is a theorem of EFQC=. For more formal connections between free logics and classical logic, the reader may consult [Trew, 1970]. We prefer to close the present section by discussing an opinion that challenges the most common view of the relations between these two (kinds of) logics, a view that we have endorsed here. It is quite natural to think of free logics as alternatives to classical logic.36 After all, the people who created the subject were reacting against principles of classical logic (such as Speci cation) that they considered wrong. Van Fraassen [1969], however, would rather think of a free logic as an extension of a classical logic, obtained by adding to it a theory of singular terms that was simply not available in the classical framework. 35 To be precise, Meyer and Lambert do not refer to a Gentzen system but to a variant of such a system proposed by Anderson and Belnap. But the essence of their argument is the same as given here. 36 In the case of minimal free logics without E !, these alternatives qualify as fragments of classical logic.

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The rationale of van Fraassen's position is as follows. We have already mentioned the fact that classical logic may be formalised so as to exclude both open theorems and individual constants. Quine [1940], for example, proceeds in this way. And Quine's system|which basically has no place for singular terms|is a subsystem of some free logics, for example of the pure exclusive free logic EFQC, which bears to FQC the same relation 37 EFQCE! bears to FQCE!. Of course, there are formalisations of classical logic that do account for singular terms, for example by admitting individual constants, and they are subsystems of no free logic whatsoever, but in van Fraassen's opinion these formalisations were adopted faute de mieux. In absence of an adequate theory of singular terms, classical logicians extended to these terms the principles of their logic of bound variables (or `bound' logic). The extension was faulty, but this fault did not touch the substantial validity of classical logic as a bound logic. Free logics on the other hand set things right, restricting classical logic to its proper scope and supplying a speci c treatment of singular terms (indeed, several such treatments). In assessing this argument, it is of fundamental importance to notice that it operates at three dierent levels. At bottom, there is the simple fact that free logics handle quanti ers and bound variables in the standard (referential) way. As we said a number of times, free logics confer existential import to quanti ers, and accept Quine's dictum that to be is to be a value of a bound variable. Next, there is the fact that it is possible to construe classical logic as a bound logic, and thus make it a subsystem of some free logic. But nally, there is also the suggestion that it is better to construe classical logic in this way, that such a construal is more likely to `capture the spirit' of both classical and free logics, and that any other position on the matter would be adopted faute de mieux. This last is basically a value judgement, and as such more prescriptive than descriptive in nature. Its supporters might claim that its adoption would allow one to maintain a conservative attitude with respect to logic, and possibly remove some psychological obstacles to accepting free logics. I would rather insist that such a claim of conservatism does very little historical justice to both classical and free logicians. For classical logicians| pace van Fraassen|did have their own views about singular terms, views that they at least considered adequate and that free logicians did very little to preserve.

37 Of course, Quine's system is also a subsystem of EFQCE!, but its relation to EFQC is more interesting, because they have the same language.

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12 DESCRIPTIONS From the very beginning, free logicians were concerned with de nite descriptions. There are at least two reasons for this interest. The rst one is historical: Russell's description theory was one of the most important instruments in the hands of classical logicians to deal with (alleged) nondenoting singular terms, hence an important test for free logics was whether or not they were able to handle the same subject in a more satisfactory way. The second reason is theoretical: the necessity of a free logics is more apparent the more inevitable the presence of non-denoting singular terms seems to be, and certainly descriptions (if they are considered singular terms) make it very diÆcult to deny that there are non-denoting singular terms. You may think that `Pegasus' does not denote, but no major problem would follow (apart of course from a con ict with your intuitions) if you were to decide instead that it does. On the other hand, if you decide that `the winged horse' is denoting, this will appear to contradict the truth of (92) No (existing) horse is winged; and even worse consequences will follow if you decide that `the round square' or `the entity dierent from itself' are denoting. The basic principles of Russell's description theory|as given for example in Whitehead and Russell [1910]|were the two de nitions38 (93) E ! x' =df 9y(8x(' $ x = y))



[ x'=y] =df 9y(8x(' $ x = y) ^ ):



(94)

However, free logicians usually regard de nite descriptions as genuine singular terms; hence they are not interested in the elimination procedures connected with de nitions like 93{94. Rather, they are interested in the acceptability of the corresponding biconditionals (95) E ! x' $ 9y(8x(' $ x = y))



[ x'=y] $ 9y(8x(' $ x = y) ^ ):



(96)

Now free logicians never questioned 95; they usually regarded its right-hand member as giving both a necessary and a suÆcient condition for the existence of a denotation of x'. Similarly, one half of 96, that is,



(97) 9y(8x(' $ x = y) ^ ) ! [ x'=y]



38 For precision's sake, it must be noted that the two Russellian de nitions contained scope operators. But these operators play practically no role in free description theories (the only exception I know of is [Scales, 1969]): hence to simplify things we will disregard them here. Also, in this paragraph we will always assume that y is free in .

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is universally accepted in free logic: denoting de nite descriptions appear to everybody to conform to Russell's analysis. What is in question is the other half of 96, that is, (98) [ x'=y] ! 9y(8x(' $ x = y) ^ );



and the main reason why it is in question is that it implies (99) [ x'=y] ! E ! x':





For 99 forces one to consider false most sentences containing non-denoting descriptions,39 including such sentences as (100) x' = x';





which most free logicians regard as logically true. The rst free description theory was proposed by [Leonard, 1956], but in a second-order modal language|which explains why it did not generate much response in the literature. A more accessible suggestion came from [Hintikka, 1959b]. Hintikka's theory is based on a single principle, the biconditional (101)  = x' $ ('[=x] ^ 8x(' ! x =  )):



101 implies both 95 and 97, but it also has a number of unwelcome consequences. In particular, [Lambert, 1962] showed that (102) '[ x'=x]



follows from 101 and 100, and some instances of 102, such as (103) P ( x(P x ^ :P x)) ^ :P ( x(P x ^ :P x));





are contradictory sentences! Lambert's own solution of this problem consists in weakening Hintikka's theory, by substituting (104) 8y(y = x' $ ('[y=x] ^ 8x(' ! x = y)))



for 101. Now in a free logic assuming 104 as an axiom-schema is equivalent to assuming (105) E ! x' ! ( = x' $ ('[=x] ^ 8x(' ! x =  )));





39 Again, we must notice that unless we adopt scope operators or deny to descriptions the status of singular terms (and Russell did both) 99 leads to downright inconsistency. But the main point of the argument is independent of this, since it can be made in connection with such simple sentences as 100. So once more we need not enter into unnecessary complications.

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which shows that Lambert's theory|to be called FD|has something speci c to say about descriptions40 only to the extent to which they are denoting. For this reason, several authors (including Lambert, van Fraassen and Scott)41 have considered FD a minimal free description theory, the common core as it were of all such theories. The intuitive idea behind this characterisation is that, again, everybody agrees on how to treat denoting descriptions, and FD says nothing (speci c) beyond that. Disagreements will arise among free description theorists only with respect to non-denoting descriptions and in this area a large number of alternatives are possible, which in general require the addition of further schemata to FD. Lambert [1962] mentions one of these alternatives, that is, the theory (to be called FD1 ) which results from adding to FD the schema (106)  = x(x =  );



and [Lambert, 1964] a dierent one, obtained by replacing 104 with (107) x' = 

$ 8y( = y $ ('[y=x] ^ 8x(' ! x = y)));



from which however 104 is derivable. This last theory|to be called FD02 |is an interesting one. For it turns out that it is equivalent to the theory FD2 which is obtained by adding to FQCE!= 104 and the principle 41 on p. 166|that is, by combining minimal description theory with Scott's free logic. Van Fraassen and Lambert [1967] make some interesting remarks about the philosophical signi cance of the dierences between all these description theories (which remarks|in view of the above equivalence result|apply mutatis mutandis to Scott's free logic). FD2 (or FD02 )|they say|may be the right theory for some speci c (and limited) purposes. For example, in the course of reconstructing mathematics non-denoting descriptions may well be regarded as `don't cares', and a compromise between free logic and the chosen object theory may be the most eÆcient way to handle them. On the other hand, if we are interested in natural language, FD2 is going to be too strong. To give just one example, such a theory would allow us to derive (108) John avoided the explosion of the White House in 1965; from (109) John avoided the accident at the corner of High Street and Pleasant Street, 40 That is, something that does not follow simply from treating descriptions as genuine singular terms, and thus extending to them the laws of (free) quanti cation and identity theory. 41 See [van Fraassen and Lambert, 1967; Scott, 1970; Lambert, 1972].

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and the fact that there was no accident at that corner or any explosion of the White House in 1965. For these other, more `philosophical' purposes, a weaker theory like FD or FD1 might be preferred. The `liberality' of this position is certainly attractive; however, there are problems with it. On the one hand, if FD2 or FD'2 are recommended on pragmatic grounds, to people who don't care much about non-denoting descriptions, how can they be preferred to the original chosen object theory| which is certainly simpler and thus even more recommendable from a pragmatic point of view? On the other hand, if we reject FD2 and move to weaker theories, how are we going to choose among them? The intuitive acceptability of `laws' like 106 by itself won't do, for we need a way of checking our intuitions on the matter, and even more importantly we need some kind of evidence that we have found all the relevant laws. The problem with van Fraassen and Lambert's approach is that they do not give a semantical analysis of their theories, except for the minimal FD. They do present semantics for all these theories, and completeness theorems for them, but such `semantics' do little more than duplicating the theories, and the arbitrary selections that seem to be at their foundations. Thus for example the fundamental unit of the `semantics' for FD1 |the FD1 -structure|is de ned essentially as an FD-structure that veri es all instances of 106, and such an approach certainly says very little about why 106 is a logical law, and which other laws (if any) should be accepted. This leaves us with the semantics for FD, but FD is a very weak theory, too weak even for its author Lambert.42 The above is substantially the same criticism already raised against van Fraassen's semantics for (free) quanti cation and identity theory. Just as in that case (say) self-identity was validated by at, so it happens now for 106. Thus we may expect to nd here the same kinds of developments of van Fraassen's approach that we found there. And indeed, at least one such development is available, by Bencivenga [1978b; 1980c]. Once again, Bencivenga's starting point is the counterfactual theory of truth: a sentence containing non-denoting singular terms is true (false) if and only if it would be true (false) in case these terms were denoting. However, a major complication arises in applying this theory to descriptions, in that it is not always possible for a description to denote, or for a set of descriptions to denote simultaneously. Thus (110) x(P x ^ :P x)



will never have a denotation (if not in some variant of the chosen object theory, which Bencivenga is not willing to accept), and (111) xP x



42 In

this regard, see the conclusion of [Lambert, 1962].

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(112) x:P x



(113) x(x = x);



though being all `consistent' descriptions (and having denotations somewhere) will never have denotations together. With respect to sentences containing either 110 or all of 111{113, Bencivenga faces a choice: he can either make them all vacuously true (or perhaps false) or modify his approach, for example by requiring as an additional condition for the truth (or falsity) of sentences containing non-denoting descriptions that there be at least one structure in which all such descriptions are denoting. He chooses the second route, and this choice gives rise to further complications. For now every sentence containing (say) 110, even such a sentence as (114) x(P x ^ :P x) = x(P x ^ :P x);





whose logical truth seems not to depend on a logical analysis of descriptions, becomes `essentially truth-valueless'|that is, does never receive a truthvalue. Bencivenga's assessment of the situation is that free quanti cation and identity theories, though successful in removing the existential assumptions of classical logic, still carry with them some weaker assumptions, of possibility of existence. Such assumptions, however, are contradicted by (some) descriptions, hence our quanti cational logic should be modi ed if we want to allow for a natural extension of it to descriptions. Whether we will actually make the modi cation in question or instead worry about assumptions of possibility where they really matter (that is, in languages with descriptions) will ultimately be decided|Bencivenga thinks|on practical grounds, and certainly relevant to these practical considerations is his proof that the set of logically true sentences of his (possibility-free) semantics for descriptions is not recursively enumerable. So much for the extensions to descriptions of the supervaluational approach. Analogous extensions of the outer domain approach and of the `conventional' approach were proposed by [Grandy, 1972] and by [Burge, 1974], respectively. Since Grandy's development is less immediate than Burge's, and contains at least one new theoretical notion, we will conclude the present section by brie y describing it. The novelty of Grandy's semantics is a function , de ned on all subsets of the union A [ A0 of the inner and the outer domain of an LT-structure and with values in A [ A0 . By de nition, (S ) 2 A if and only if S \ A = (S ), that is, the value of  for a given subset S of A [ A0 `exists' just in case it is the only existing member of S . In a Grandy structure, de ned as an ordered 4-tuple hA; A0 ; I; i, the denotation of a description x' is the value  has for the subset of A [ A0 constituted by all objects satisfying '.



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Grandy's semantics does not force one to identify all non-existents, as for example Scott's does. Thus, in intuitive terms, if  assigns dierent values to the set of winged horses and the set of golden mountains, the sentence (115) The winged horse = the golden mountain turns out false. On the other hand, however, if (116) '[=x] $ [=x] is logically true, then in every Grandy structure ' and the same objects; hence

are satis ed by

(117) x' = x





is logically true, too. This is certainly an asset of Grandy's approach: in it one can validate in a natural way such schemata as (118) x' = x(' ^ ');





which are certainly as `intuitive' as (say) 106 was and which in van Fraassen{ Lambert's framework would require the addition of further ad hoc clauses. From a proof-theoretical point of view, the approach in question is characterised by the rule `  ! ('[=x] $ [=x]) (119) ; if  does not occur in ; `  ! x' = x which allows for a simple proof of 118 and the like.





ACKNOWLEDGEMENTS Work for the completion of this paper was partly supported by a Faculty Fellowship of the School of Humanities, University of California at Irvine. Thanks are due to Nuel Belnap, Gerald Charlwood, Wilfrid Hodges, Karel Lambert and Brian Skyrms for comments on earlier drafts of the paper. University of California at Irvine

BIBLIOGRAPHY [Barba,, 1989] J. L. Barba. A modal version of free logic, Topoi, 9, 131{135, 1989. [Belnap, 1960] N. D. Belnap, Jr. Review of [Hintikka, 1959a], Journal of Symbolic Logic, 25, 88, 1960. [Bencivenga, 1976] E. Bencivenga. Set theory and free logic. Journal of Philosophical Logic, 5, 1{15, 1976. [Bencivenga, 1978a] E. Bencivenga. A semantics for a weak free logic. Notre Dame Journal of Formal Logic, 19, 646{652, 1978.

194

ERMANNO BENCIVENGA

[Bencivenga, 1978b] E. Bencivenga. Free semantics for inde nite descriptions. Journal of Philosophical Logic, 7, 389{405, 1978. [Bencivenga, 1980a] E. Bencivenga. A weak free logic with the existence sign. Notre Dame Journal of Formal Logic, 231, 572{576, 1980. [Bencivenga, 1980b] E. Bencivenga. Una logica dei termini singolari, Boringhieri, Torino, 1980. [Bencivenga, 1980c] E. Bencivenga. Free semantics for de nite descriptions. Logique et Analyse, 23, 393{405, 1980. [Bencivenga, 1981] E. Bencivenga. Free semantics. Boston Studies in the Philosophy of Science, 47, 31{48, 1981. [Bencivenga, 1983] E. Bencivenga. Compactness of a supervaluational language. Journal of Symbolic Logic, 48, 384{386, 1983. [Bencivenga, 1990] E. Bencivenga. Free from what? Erkenntnis, 33, 9{21, 1990. [Bencivenga, Lambert and van Fraassen, 1991] E. Bencivenga, K. Lambert and B. C. van Fraassen. Logic, Bivalence and Denotation, 2nd edition, Ridgeview, Atascadero, (Califormia), 1991. [Burge, 1974] T. Burge. Truth and singular terms. Nous, 8, 309{325, 1974. [Carnap, 1947] R. Carnap. Meaning and Necessity. University of Chicago Press, Chicago, 1947. [Church, 1965] A. Church. Review of [Lambert, 1963]. Journal of Symbolic Logic, 30, 103{104, 1965. [Cocchiarella, 1966] N. Cocchiarella. A logic of possible and actual objects. Journal of Symbolic Logic, 31, 688, 1966. [Findlay, 1963] A. N. Findlay. Meinong's Theory of Objects and Values, Clarendon Press, Oxford, 1963. [Fine, 1983] K. Fine. The permutation principle in quanti cational logic. Journal of Philosophical Logic, 12, 33{37, 1983.  [Frege, 1892] G. Frege. Uber Sinn und Bedeutung. Zeitschrift fur Philosophie und philosophische Kritik, 100, 25{50, 1892. [Grandy, 1972] R. Grandy. A de nition of truth for theories with intensional de nite description operators. Journal of Philosophical Logic, 1, 137{155, 1972. [Hailperin, 1953] T. Hailperin. Quanti cation theory and empty individual-domains. Journal of Symbolic Logic, 18, 197{200, 1953. [Hintikka, 1959a] J. Hintikka. Existential presuppositions and existential commitments. Journal of Philosophy, 56, 125{137, 1959. [Hintikka, 1959b] J. Hintikka. Towards a theory of de nite descriptions. Analysis, 19, 79{85, 1959. [Jaskowski, 1934] S. Jaskowski. On the rules of supposition in formal logic. Studia Logica, 1, 5{32, 1934. [Lambert, 1962] K. Lambert. Notes on E ! III: A theory of descriptions. Philosophical Studies, 13, 51{59, 1962. [Lambert, 1963] K. Lambert. Existential import revisited. Notre Dame Journal of Formal Logic, 4, 288{292, 1963. [Lambert, 1964] K. Lambert. Notes on E ! IV: A reduction in free quanti cation theory with identity and descriptions. Philosophical Studies, 15, 85{88, 1964. [Lambert, 1967] K. Lambert. Free logic and the concept of existence. Note Dame Journal of Formal Logic, 8, 133{144, 1967. [Lambert, 1972] K. Lambert. Notes on free description theories: some philosophical issues and consequences. Journal of Philosophical Logic, 1, 184{191, 1972. [Lambert, 1981] K. Lambert. On the philosophical foundations of free logic. Inquiry, 24, 147{203, 1981. [Lambert, 1991] K. Lambert. Philosophical Applications of Free Logic, Oxford University Press, New York, 1991. [Leblanc and Hailperin, 1959] H. Leblanc and T. Hailperin. Nondesignating singular terms. Philosophical Review, 68, 239{243, 1959. [Leblanc and Thomason, 1968] H. Leblanc and R. H. Thomason. Completeness theorems for some presupposition-free logics. Fundamenta Math., 62, 125{26, 1968.

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[Lejewski, 1954] C. Lejewski. Logic and existence. British J. Philosophy of Science, 5, 104{119, 1954. [Lejewski, 1958] C. Lejewski. On Lesniewski's ontology. Ratio, 1, 150{176, 1958. [Leonard, 1956] H. S. Leonard. The logic of existence. Philosophical Studies, 7, 49{64, 1956. [Luschei, 1962] E. C. Luschei. The Logical Systems of Lesniewski. North-Holland, Amsterdam, 1962.  [Meinong, 1904] A. Meinong. Uber Gegenstandstheorie. In Untersuchungen zur Gegenstandstheorie und Psychologie, Barth, Leipzig, 1904. [Meyer and Lambert, 1968] R. K. Meyer and K. Lambert. Universally free logic and standard quanti cation theory. Journal of Symbolic Logic, 33, 8{26, 1968. [Mostowski, 1951] A. Mostowski. On the rules of proof in the pure functional calculus of the rst order. Journal of Symbolic Logic, 16, 107{111, 1951. [Parsons, 1980] T. Parsons. Nonexistent Objects, Yale University Press, New Haven, 1980. [Posy, 1982] C. Posy. A free IPC is a natural logic. Topoi, 1, 30{43, 1982. [Quine, 1939] W. V. O. Quine. Designation and existence. Journal of Philosophy, 36, 701{709, 1939. [Quine, 1940] W. V. O. Quine. Mathematical Logic, Harvard University Press, Cambridge, 1940. [Quine, 1948] W. V. O. Quine. On what there is. Review of Metaphysics, 2, 21{38, 1948. [Quine, 1954] W. V. O. Quine. Quanti cation and the empty domain. Journal of Symbolic Logic, 19, 177{179, 1954. [Routley, 1966] R. Routley. Some things do not exist. Notre Dame Journal of Formal Logic, 7, 251{276, 1966. [Routley, 1980] R. Routley. Exploring Meinong's Jungle and Beyond. Australian National University, Canberra, 1980. [Scales, 1969] R. Scales. Attribution and Reference. PhD Thesis, University of California at Irvine, 1969. [Schock, 1964] R. Schock. Contributions to syntax, semantics, and the philosophy of science. Notre Dame Journal of Formal Logic, 5, 241{289, 1964. [Schock, 1968] R. Schock. Logics without Existence Assumptions. Almqvist and Wiksell, Stockholm, 1968. [Scott, 1967] D. Scott. Existence and description in formal logic. In R.Schoenman, ed. Bertrand Russell, Philosopher of the Century, pp. 181{200. Allen and Unwin, London, 1967. [Scott, 1970] D. Scott. Advice in modal logic. In K. Lambert, ed. Philosophical Problems in Logic, pp. 143{173, D. Reidel, Dordrecht, 1970. [Skyrms, 1968] B . Skyrms. Supervaluations: identity, existence, and individual concepts. Journal of Philosophy, 69, 477{482, 1969. [Strawson, 1950] P. F. Strawson. On referring. Mind, 59, 320{344, 1950. [Strawson, 1952] P. F. Strawson. Introduction to Logical Theory. Methuen, London, 1952. [Trew, 1970] A. Trew. Nonstandard theories of quanti cation and identity. Journal of Symbolic Logic, 35, 267{294, 1970. [van Fraassen, 1966a] B. C. van Fraassen. The completeness of free logic. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik, 12, 219{234, 1966. [van Fraassen, 1966b] B. C. van Fraassen. Singular terms, truth-value gaps, and free logic. Journal of Philosophy, 67, 481{495, 1966. [van Fraassen, 1968] B. C. van Fraassen. Presupposition, implication, and self-reference. Journal of Philosophy, 69, 136{152, 1968. [van Fraassen, 1969] B. C. van Fraassen. Presuppositions, supervaluations, and free logic. In K. Lambert, ed. The Logical Way of Doing Things, pp. 67{91. Yale University Press, New Haven, 1969. [van Fraassen and Lambert, 1967] B. C. van Fraassen and K. Lambert. On free description theory. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik, 13, 225{240, 1967.

196

ERMANNO BENCIVENGA

[Whitehead and Russell, 1910] A. N. Whitehead and B. Russell. Principia Mathematica, Vol. 1, Cambridge University Press, Cambridge, 1910. [Woodru, 1971] P. W. Woodru. Free logic, modality and truth. Unpublished manuscript, 1971. [Woodru, 1984] P. W. Woodru. On supervaluations in free logic. Journal of Symbolic Logic, 49, 943{950, 1984.

SCOTT LEHMANN

MORE FREE LOGIC

By a free logic is generally meant a variant of classical rst-order logic in which constant terms may, under interpretation, fail to refer to individuals in the domain D over which the bound variables range, either because they do not refer at all or because they refer to individuals outside D. If D is identi ed with what is assumed by the given interpretation to exist, in accord with Quine's dictum that \to be is to be the value of a [bound] variable,"1 then a free variation on classical semantics does not require that all constant terms refer to existents, and in this sense such terms lack existential import. Classical semantics treats free variables like constants, at least in the quanti er clause of the valuation rules. When we stipulate that 9xA is true i A is true for some assignment of a value (x) in D to x, we are treating x at its free occurrences in A as a constant that refers to (x). In free semantics, free variables are also generally treated like constants, which means that they need not be assigned values in D; thus free variables and variable terms (such as x + y or 1=x) constructed from them also lack existential import. However, when reckoning the truth of 9xA in terms of the truth of A for assignments of values to x, we consider only assignments  for which (x) 2 D. Thus, although neither constant nor variable terms need refer to individuals in D, free semantics honors Quine's dictum.2 In classical semantics, free variables have existential import because D is non-empty: there is always something in D for x to be assigned by . Variants of classical semantics in which this requirement is relaxed so that D may be empty are said to be inclusive. A semantics that is free and inclusive is said to be universally free: the range of the bound variables may be empty, and even if it is not, neither constant nor variable terms have existential import. This survey of free logic will begin by considering its motivation, then move to reviewing various kinds of free semantics and the syntactic proof systems designed to capture the forthcoming notions of logical truth or logical consequence, and conclude by describing some applications of free logics, notably free description theory. As this summary may suggest, my emphasis throughout will be on semantics. The account is self-contained 1 Quine [1948, p. 15]. That Quine means bound variables here is clear from his earlier statement [p. 13] that \a theory is committed to those and only those entitites to which the bound variables of the theory must be capable of referring in order for the aÆrmations made in the theory to be true." 2 Compare Bencivenga's [1986, p. 375] characterization: \A free logic is a formal system of quanti cation theory, with or without identity, which allows for some singular terms in some circumstances to be thought of as denoting no existing object, and in which quanti ers are invariably thought of as having existential import."

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and does not presuppose familiarity with [Bencivenga, 1986], reproduced in this volume. There is new material on motivation, applications, and neutral free semantics, while areas of overlap dier in detail and emphasis. Semantic options are laid out in greater detail, as are free description theories built upon them, but I pay less attention to the history of ideas. 1 QUICK REVIEW OF CLASSICAL FIRST-ORDER LOGIC Partly to settle notation and terminology, and partly because free logics are variants of it, let us rst quickly review classical rst-order logic with identity, which we may take to be framed in formal rst-order languages L: The logical vocabulary of L includes the identity operator =, plus an adequate set of quanti ers and truth-functional operators. Let us assume they are the universal quanti er 8, negation :, and material conditional !. The non-logical vocabulary of L includes (individual) variables, plus perhaps constants, (k place) function-names , and (k place) predicates. We need not specify these symbols, which will vary with L; its sentences are to represent the logical forms of certain sentences of natural language, and its non-logical vocabulary will be chosen accordingly. Many formulations of free logic employ a 1-place existence predicate E or E !, but such a predicate can generally be de ned in terms of identity,3 so we need not include it in the non-logical vocabulary. The proof method of L will require an unbounded list of variables or special constants. After de ning terms as (1) variables, (2) names, and (3) complex terms ft1 : : : tk , where the ti are terms and f is a k place function-name, the formation rules of L identify formulae as (4) subject-predicate formulae P t1 : : : tk , where the ti are terms and P is a k place predicate, (5) identities s = t, where s and t are terms, (6) negations :A, where A is a formula, (7) conditionals (A ! B ), where A and B are formulae, and (8) universals 8xA, where x is a variable and A is a formula.4 Identities and subject-predicate formulae are atomic; atomic formulae and their negations are elementary. Subsequently, I shall use the following syntactical variables, with or without subscript: for variables: x, y, and z ; for constants: a, b, and c; for function-names: f ; for predicates: P ; for terms: s and t; for formulae: A, B , and C ; for sets of formulae: X . Conjunctions (A&B ), disjunctions (A _ B ), biconditionals (A $ B ), and existentials 9xA may be de ned as usual in terms of :, !, and 8. s 6= t abbreviates :s = t. The outermost parentheses in conditionals, conjunctions, disjunctions, and biconditionals standing alone will be omitted. ft1 : : : tk and P t1 : : : tk will be used with 3 For exceptions, see [Garson, 1991], discussed below in Section 5.3, and [Gumb, 1998]. 4 To avoid the notational clutter that attends the use of single- and quasi-quotation,

I shall generally follow Church [1956] in using symbols of L as names for themselves and juxtaposition for juxtaposition.

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the assumption that f and P are k place; where necessary, commas and parentheses will disambiguate expressions, as in P f (x; y), and may also be inserted to enhance readability, as in 9x(x = fx): An occurrence of a term t in a formula A is bound in A provided it is an occurrence in a part 8xB of A, where x occurs in t; an occurrence of t in A is free if it is not bound. The bound (free) variables of A are those with a bound (free) occurrence in A. A sentence is a formula without free variables. A(x1 ; : : : ; xk =t1 ; : : : ; tk ) is the result of simultaneously replacing the xi at each free occurrence in A by ti , having (if necessary) rst made such occurrences free for ti in A: if a free occurrence of xi in A is in a part 8yB , where y occurs in ti , replace each occurrence of y in 8yB by the rst variable that occurs in neither A nor any of the tj ; relabel the result `A' and repeat until there are no such occurrences. I shall write A(x1 ; : : : ; xk ) for A and A(t1 ; : : : ; tk ) for A(x1 ; : : : ; xk =t1 ; : : : ; tk ). 9!xA or 9!xA(x) abbreviates 9x8y(A(x=y) $ y = x), where y is not x. In writing 9x(x = t), I assume that x does not occur in t. The universal closure 8A of A is 8x1 : : : 8xk A, where the free variables of A are x1 ; : : : ; xk : An interpretation I of L is a pair hD; di; where D is a set and d is a denotation function de ned on the constants, function-names, and predicates of L, such that: i1.

D is non-empty;

i2.

d(a) 2 D;

i3.

If f is k place, d(f ) is a total k

i4.

If P is k place, d(P ) is a k

ary function D ! D:

ary relation in D.

An assignment  is a function that assigns individuals (x) in D to the variables. An x-variant of  is an assignment that diers from  at most at x: Under I and , terms refer to individuals of D according to the reference rules: r1. x refers to (x): r2. a refers to d(a) r3. ft1 : : : tk refers to d(f )(1 ; : : : ; k ), if ti refers to i . Under I and , formulae are true or false (and false if not true) according to the valuation rules: v1. P t1 : : : tk is true i h1 ; : : : ; k i 2 I (P ), if ti refers to i . v2. If s refers to  and t to , then s = t is true i  is :

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:A is true i A is false. A ! B is false i A is true and B is false. 8xA is false i A is false for some x variant of .5

Since the referents of terms without variables and the truth-values of sentences are independent of , I shall speak of referents and truth-values under I in such cases. Logical relations and properties are de ned as usual in terms of the totality of interpretations: A is a logical consequence of X (X j= A) i there is no interpretation and assignment under which all the X -formulae are true and A is false; X is satis able i there is some interpretation and assignment under which all the X -formulae are true; A is logically true (false) i A is true (false) under each interpretation and assignment; A and B are logically equivalent i, under each interpretation and assignment, A is true i B is true. A1 ; : : : ; Ak j= B means: fA1 ; : : : ; Ak g j= B . X; A j= B means: X [ fAg j= B . X 6j= A means: not X j= A: These de nitions embody what Kleene [1967, p. 103] terms the conditional reading of free variables: free variables are treated by r1 as names of D-individuals. By contrast, the generality reading treats free variables as if they were universally quanti ed. It may be captured by stipulating that A is true (false) under I i A is true (false) under I and  for each . We can then drop \and assignment" from the above de nitions. However, we end up with weaker notions of logical consequence and logical equivalence (and a stronger notion of satis ability). For the logical consequence relation j=g , we have X j=g A i 8X j= 8A, where 8X = f8B : B 2 X g, so that X j=g A if X j= A but not conversely (e.g., P x j=g 8xP x, but P x 6j= 8xP x). If X is a set of sentences, the two consequence relations coincide, since X j= A i X j= 8A and here we have 8X = X: From the semantic perspective assumed here, the aim of proof theory is to provide syntactic characterizations of logical properties and relations, which are de ned in semantic terms. In particular, we want a syntactic notion of proof from hypotheses that captures the logical consequence relation: A is provable from hypotheses in X (X ` A) i A is a logical consequence of X (X j= A), at least if X is a set of sentences. A proof system with this property is said to be strongly complete. A proof system in which A is 5 Most presentations of free logic give a substitutional account of quanti cation, on which v5 would read instead: 8xA(x) is false i A(a) is false for some constant a. If 8x is to have the force of `for all individuals x', every individual in D must be named by some constant or other. If D is uncountable, the terms and formulae of L will then be undecidable. This awkward result may be avoided by proving, via the Lowenheim-Skolem theorem, that interpretations may be restricted to countable universes without altering logical consequence relations, so that no more than a countable in nity of constants need be assumed. By contrast, the objectual account of quanti cation given in v5 does not require an elaborate justi cation.

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provable (from no hypotheses) i A is logically true (` A i j= A) is weakly complete. If detachment or modus ponens MP

A; A ! B ` B

holds, and the deduction theorem or conditional proof CP

X ` A ! B provided X; A ` B

holds for sentences A, then weak completeness is equivalent to: X ` A i X j= A for nite sets X of sentences. Many strongly complete systems in a variety of styles are known for classical rst-order semantics. It will be useful to give one that can be modi ed in simple ways to capture logical consequence for at least some free variations on classical semantics. The simplest proof systems to describe are Hilbert-style systems, which specify logical axioms and rules of inference, and de ne a proof of A from hypotheses X as a nite sequence hA1 ; : : : ; Ak i such that Ak = A and each Ai is either a member of X , or a logical axiom, or is derived from previous formulae in the sequence by a rule of inference. Unlike natural deduction systems, in which some inference rules (such as CP) are conditional, those of a Hilbert-style system are (like MP) categorical. Since the propositional part of the system does not matter here, we may adopt the simple inference rule T

A1 ; : : : ; Ak ` B

if B is a tautological consequence of fA1 ; : : : ; Ak g, that is, there is no assignment of truth-values to universals and atomic formulae for which each Ai is true and B is false in virtue of rules v3 and v4. The quanti er rule and axioms are as in [Church, 1956, p. 172]; the rule is generalization: UG

A ` 8xA

and the axiom schemas are distribution and speci cation: A1

8x(A ! B ) ! (A ! 8xB ); if x is not free in A

A2

8xA(x) ! A(t)

Finally, we have the identity axiom schemas: A3

x=x

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A4 x = y ! (A(x) ! A(y)); where A is atomic. Let us agree that A(x) is A(z=x) and A(y) is A(z=y), so that A(y) results from A(x) by replacing x at some occurrences by y. Let us call this system CL. Like most Hilbert-style systems, CL is sound with respect to j=g but not j=: if X ` A, then X j=g A but not necessarily X j= A (consider UG). The limitation is immaterial if X consists of sentences, as it will if L is used to represent arguments in some natural language. 2 MOTIVATIONS FOR FREE LOGIC The motivations for free logic may be grouped under four headings. (1) Classical rst-order semantics embodies existence assumptions that can produce weird results and constrain what can be done to avoid them. Accordingly, (2) certain philosophical doctrines can be expressed in rst-order languages, classically conceived, only with diÆculty and in ways that will seem arti cial. Then there are general considerations of logical form: (3) if logical form captures truth-conditions in the sense of determining correct truthvalues in all possible situations, then logical semantics must be universally free. Finally, for those who remain unconvinced, there is a pragmatic argument: (4) the representation of logical moves in a classical system can often be considerably simpli ed if we pretend that certain expressions are terms that need not refer to any existent, either because they do not refer at all or because they refer to pretend objects.

2.1 Classical Existence Assumptions The existence assumptions built into classical rst-order semantics are implicit in i1{i3: D is non-empty, constants refer to individuals of D, and function-names refer to total functions D ! D. These assumptions constrain the meaning of logical forms. As implicit premises, they permit some surprising inferences. While such unwanted conclusions can be avoided, the ways of doing so, constrained as they are by these assumptions, may seem arti cial and too complex.6 Let us consider the existence assumptions in turn. 6 Various free logics can be represented as classical rst-order theories. Let L be the f rst-order language of a free system FL in which free logical truth has been characterized in terms of Hilbert-style provability from logical axioms by logical rules of inference: j= A i `FL A; and let Lc result from Lf by adding a 1-place predicate E . Trew [1970] shows the dedicated reader how to (1) translate sentences A of Lf into sentences tr(A) of Lc and, for each of a variety of systems FL, how to (2) write axioms Ax(FL) that classically constrain the interpretation of E , so that `FL A i Ax(FL) `CL tr(A), i.e., i tr(A) is classically provable from Ax(FL).

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(a) Existential conclusions may be validly drawn from premises that are not overtly existential, because the assumption of a non-empty universe operates as an implicit existential premise. For example, `A person is good i she loves everyone, so some person loves all good persons' is valid when assigned the simple form

8x(Gx $ 8yLxy) 9x8y(Gy ! Lxy) For if there is a good person x, then x loves everyone, hence all good persons. If there is no good person, let x be anybody: x loves all good persons (at least on the standard no-counterexample interpretation of `all'), so someone loves all good persons. The crucial step is the passage from `8y(Gy ! Lxy) is true of any x' to `8y(Gy ! Lxy) is true of some x', a move licensed (albeit sotto voce) by the assumption that the range of x is non-empty.7 These awkward results can be sidestepped by complicating logical form: take the variables to range, not over persons but over some wider class, and relativize the quanti ers to the subclass of persons by introducing an appropriate 1-place predicate. The resulting argument

8x(P x ! (Gx $ 8y(P y ! Lxy))) 9x(P x & 8y(P y ! (Gy ! Lxy))) is invalid, since the premise is true and the conclusion false when P is assigned the empty extension, which of course is permitted in classical semantics. Constants and function-names complicate the transformation. To preserve the validity of `Pope John-Paul II is nobody's spouse, so he isn't his own spouse', we will need to add a premise Pj to do the work of i2 and a premise 8x(P x ! P Sx) to do the work of i3:

:9x(P x & j = Sx) 8x(P x ! P Sx) Pj j 6= Sj

7 A more mathematical example is the generation of an empty set, apparently ex nihilo, in some ZF-formulations of pure set theory, e:g. [Shoen eld, 1967]: 9y8z (z 2= y) follows from the subset axiom in the form 8x9y8z (z 2 y $ (z 2 x&A(z ))). For let x be a set; by the subset axiom, there is a set y whose members are the sets z such that z 2 x and z 6= z ; since no set z is such that z 6= z , there are no such sets and y has no members; so there is a set with no members. What is disturbing here is that the subset axiom has an essentially conditional form: if x exists, so does any describable subset of x. How can it yield a categorical existence claim? The answer is that classical semantics guarantees that the range of x is non-empty, so some set will exist.

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If P is assigned the empty extension, then Pj will be false, but for a reason that will strike some people as incorrect: j refers to something not in the extension of P . If there were no people, `Pope John-Paul II is a person' might be false, but not because `Pope John-Paul II' refers to something (zero, say) which is not in the now empty extension of `person'; `Pope John-Paul II' doesn't refer at all in this situation. At best, we have a semantic proxy for failure of reference that delivers the right truth-values | at best because one might want to hold that `Pope John-Paul II is a person' is neither true nor false when `Pope John Paul II' does not refer. None of these manoeuvres will alter the validity of such arguments as `Everything is self-identical, so something is'

8x(x = x) 9x(x = x) since the range of x is non-empty. And it might be objected that the account of logical possibility given by classical semantics is defective, for surely the range of the variables could be empty. (b) i2 requires that constants refer to something in the range of the variables, which permits quick proofs of the existence of God (or Grendel, or anything you like), since 9x(x = g) is logically true if g refers to something in the range of x. If you feel bad about doing so little work for such large results, you can give a short CL-proof: 1. 2. 3. 4. 5.

x=x 8x(x = x) 8x(x = x) ! g = g 8x(x 6= g) ! g 6= g 9x(x = g)

A3 UG(1) A2 A2 T(2,3,4)

The instances of A2 are logically true because g must refer to something in the range of x: The standard Russellian x for these problems is to replace constants g that may not refer by predicate constructions: nd a singular predicate G true of at most one thing, which you'd be willing to label g if there were such a thing (in the present case, perhaps Anselm's `nothing greater than x can be or be conceived' will do) and replace atomic parts A(g) by 9y(8z (Gz $ y = z ) & A(y)), where y is not free in A. Then the existence claim 9x(x = g) becomes 9x9y(8z (Gz $ y = z ) & x = y), which is obviously not logically true.

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Alternatively, add 8x8y((Gx & Gy) ! x = y) as a premise (an axiom) to constrain interpretations of G, and replace atomic parts A(g) by 9y (Gy & A(y)). This will transform the existence claim 9x(x = g) into 9x9y(Gy&x = y), and the issue of its logical truth into that of the validity of

8x8y((Gx & Gy) ! x = y) 9x9y(Gy & x = y) which is obviously invalid. This procedure will make atomic sentences A(g) with non-referring terms g false, since the predicate G will be true of nothing. If you think that some subject-predicate sentences with non-referring subjects | such as `Grendel was slain' | are true, you can use the replacement 8y(Gy ! A(y)) for them, as Mendelson [1989, p. 613] observes. But something seems to be missing here. The reason for the truth of `Grendel was slain' and the falsity of `Grendel was pink' is really the same: whatever singular predicate we nd for Grendel is true of nothing. Moreover, if you think that some subjectpredicate sentences with non-referring subjects, such as van Fraassen's [1966, p. 82] `Pegasus has a white hind leg', are neither true nor false, you will not be happy with Russell's way of dealing with them, since classical semantics is bivalent: any sentence is true or false. (c) i3 requires that functions be total, which validates such arguments as `Every spouse loves his or her spouse (and, of course, the spouse of one's spouse is oneself), so nobody is unloved', if it is given the simple form

8x8y(x = Sy ! (y = Sx & LxSx)) :9x:9yLyx For any person x is such that x's spouse loves the spouse of x's spouse, who of course is x, and everyone has a spouse, since S is total. In real life, not everyone has a spouse, but the straightforward way of saying this, :8x9y(y = Sx), is logically false. In mathematical applications of logic, it would be convenient to introduce notations for partial functions, e:g:, to de ne predecessor P in terms of successor S in the natural numbers by 8x8y(P x = y $ Sy = x) or division = in terms of multiplication  in the reals by 8x8y8z (x=y = z $ z  y = x). However, such de nitions will not do: given :9x(Sx = 0), the rst entails :9x(P 0 = x), which is logically false; given 8x(x  0 = 0), the second entails S 0 = 0, which contradicts :9x(Sx = 0). Nor can we simply exclude the troublesome arguments. 8x(x 6= 0 ! 8y(P x = y $ Sy = x)) leaves P unde ned at 0, and

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8x8y(y 6= 0 ! 8z (x=y = z $ z  y = x)) leaves x=y unde ned when

the value of y is 0, whereas classical semantics requires that functions be everywhere de ned. These diÆculties may be avoided by replacing function names with relational predicates for their graphs.

8x8y(Syx ! (Sxy & LxSx)) :9x:9yLyx is invalid, and de nitions like 8x8y(P xy $ Sy = x) or 8x8y8z (Dxyz $ z  y = x) are ne. For some applications, however, we will need to add a premise (an axiom) to the eect that the de ned predicate is functional, e:g:, 8x8y8z ((Sxy&Sxz ) ! y = z ). And we lose the considerable advantages of functional notation. Alternatively, we can represent a partial function f by a total function F that coincides with f where f is de ned and is given some arbitrary value elsewhere. If the arbitrary value for the predecessor and division functions is 0, then their de nitions may be given as (8x(x 6= 0 ! 8y(P x = y $ Sy = x))&P 0 = 0) and 8x(8y(y 6= 0 ! 8z (x=y = z $ z  y = x)) & x=0 = 0). In the case of the spouse function, we might pick an unmarried person (the Pope, say) and extend the spouse function to unmarried people x by stipulating that the spouse of x is the Pope. If S now represents this function and p names the Pope, then Sx 6= p will tell us that x has a spouse (that the partial spouse function is de ned at x), and we can recast the premise of the argument as 8x(Sx 6= p ! 8y(x = Sy ! (y = Sx & LxSx))): The price is a violation of ordinary usage: `The Pope's spouse is the Pope' is not true. Since the Pope is unmarried, `The Pope's spouse' doesn't refer to the Pope or to anyone else; it doesn't refer to anything. Where the partial function f is onto, as is the predecessor function, we cannot identify the arguments at which it is de ned in this way. No matter what individual i is picked for the arbitrary value, F x 6= i will be false for some x at which f is de ned.

2.2 Logical Habitats for Philosophical Doctrines If free semantics permits us to avoid strange results, it also permits us to state strange doctrines. Free semantics provides a more neutral logical setting than classical semantics for certain philosophical views. It permits distinctions upon which they depend to be made in a straightforward way and does not prejudice the case against them by rendering important claims logically false. Let us consider an assortment.

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a. Meinong claimed, notoriously, that there are non-existent objects, that things like the golden mountain may have being while lacking existence. The straightforward form of `a does not exist' is :9x(a = x), but this is logically false in classical semantics, since a must refer to something in the range of x. A device like Russell's will avoid this, but at the expense of a complex form :9x9y(8z (Az $ z = y) & y = x), where Az is uniquely true of a. We could instead take the variables to range over beings and introduce a 1-place existence predicate E ; quanti cation over existents would be represented by relativizing quanti ers to E , 8x(Ex ! A(x)) indicating that A(x) is true of all existents, 9x(Ex&A(x)) that it is true of some existent. Then 9x:Ex will represent `there are non-existent objects.' We can even let Ex abbreviate 9y(y = x), provided we follow Lejeweski and give identity a non-standard meaning: an interpretation is hD; d; oi, where hD; di is classical and o 2 D, and s = t is true under I and  i s and t refer under I and  to the same individual of D and this individual is not o. Relative to interpretations and valuations of this sort, 9x:9y(y = x) is logically true. These classical or quasi-classical approaches do not really honor Quine's dictum. Here to be is to be the value of a variable, all right, but to exist is not. By contrast, a free semantics that permits constants to refer to individuals outside the range of x allows us to say simply that a does not exist without immediately contradicting ourselves and without abandoning Quine's useful connection between quanti cation and existence. If we wish to make the more general Meinongian claim that there are non-existent objects | if we wish to quantify over them | then it may be argued that we are really committed to objects with being and should simply treat them classically, delimiting the subclass of existents with E . Alternatively, we could add a special quanti er 9b y, meaning `there is a being y such that' and write 9b y:9x(y = x) : to be is to be the value of a variable bound by 9b , to exist is to be the value of a variable bound by 9. For free semantics of this sort | and an argument that the semantics of any modal or tense logic can be built up from it | see [Cocchiarella, 1991]. More modestly, we could limit what does not exist to a single Lejewskian object, named by o, and then de ne 9b xA(x) as A(o) _ 9xA(x). For this approach | and a proof that it is deductively equivalent to Lejewski's | see [Lambert and Scharle, 1967]. b. The truth of `Fred exists, but might not have' is typically explained by gesturing toward a world that is possible relative to ours at which `Fred does not exist', i.e., :9x(x = f ), is true. Accordingly, at such a

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world, f cannot refer to something in the range of x: Fred is not among the existents of that world. Standard Kripke-style modal semantics is free. c. Intuitionists and constructivists in uenced by them reject nonconstructive proofs in mathematics. A proof of 9xA(x) is, for them, nothing more nor less than a proof of A(t) for some t, and such a proof is not secured merely be showing that a contradiction can be obtained from the assumption that 8x:A(x). Now if we have not established that f is de ned at s, we can hardly claim to have a proof of fs = fs and therefore a proof of 9x(x = fs). Yet standard formulations of intuitionist logic follow classical logic in regarding t = t as a logical axiom and licensing the inference to 9x(x = t). Accordingly, we must either banish partial functions from intuitionist logic, thereby limiting its reach in mathematics and ignoring the views of patriarchs like Brouwer, or further modify the logic, this time in the direction of free logic. For free variations on Kripke-style semantics for intuitionistic logic, see [Posy, 1982]. d. Evans [1979] has noted that the standard examples of contingent a priori truths presuppose a free semantics. If we stipulate that `Julius' refers to whoever (uniquely) invented the zipper, then `if someone (uniquely) invented the zipper, Julius did' appears to be (1) true, (2) a priori, but (3) contingent. It is true because if someone (uniquely) invented the zipper, that person is Julius because `Julius' refers to whoever (uniquely) invented the zipper. It is a priori because we need only understand the stipulation to see that it is true. It is contingent because there is a possible world in which it is false, given that someone actually did (uniquely) invent the zipper: in the actual world Julius (uniquely) invented the zipper, but he might not have: at a possible world in which someone else (uniquely) invented the zipper, `Julius did' is false. A free semantics is presupposed because in classical semantics we cannot introduce a constant like j with a de ning axiom 8x(x = j $ A(x)) without rst establishing that 9!xA(x). Otherwise, we could stipulate that 8x(x = j $ x 6= x) and end up with the logically false j 6= j . In this case, A(x) is 8y(Zy $ y = x), representing `x (uniquely) invented the zipper'. Thus, the candidate sentence (9!xZx ! 8y(Zy $ y = j ))) | if someone (uniquely) invented the zipper, Julius did | is not well-formed unless 9!xZx is true, so it cannot be known a priori to be true. If, however, we allow non-referring names with the understanding that atomic constructions involving them are false, 8x(x = j $ 8y(Zy $ y = x))) will be true whether or not 9!xZx is true; and if 9!xZx is true, j must refer and 8y(Zy $ y = j ) will also be true.

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e. Aristotle held that true predication requires a subject: the truth of `Socrates is well' presupposes the existence of Socrates: `Socrates' must refer. However, the falsity of `Socrates is well' does not: `Socrates is well' is false if Socrates exists but illness is attributable to him; it is also false if Socrates does not exist. `Socrates is ill' and `Socrates is not well' do not give the same information: the former attributes illness to Socrates, the latter merely denies that wellness is attributable to him: he may be ill, or he may not exist at all. While classical semantics can handle contrary predications like `John is rich' and `John is poor' simply by not identifying the extension of `rich' with the complement of the extension of `poor', it does not allow `Socrates is well' and `Socrates is ill' to be false if `Socrates' does not refer: all names refer. A free semantics in which non-referring subject-terms render subject-predicate sentences false embodies Aristotle's view in a natural way. Scales [1969] has extended it by allowing complex predicates xA(x) to be formed from open sentences A(x) : xA(x)t is true i t refers and A(t) is true. Since A(t) may be true when t does not refer, xA(x)t and A(t) may dier in truth-value. For example, :W p representing `Pegasus is not winged' is true, while (x:W x)p, representing `Pegasus is wingless' is false, if `Pegasus' does not refer. f. Russell and Meinong go further than Aristotle, holding that ascribing to a sentence a subject-predicate form requires that the subject-term refer, whether the sentence is true or false, a doctrine now embodied in classical semantics. As Lambert [1986, p. 276] notes, Meinong held that `the golden mountain is golden' is a predication, so there must be a golden mountain; Russell couldn't swallow the conclusion, and so denied the sentence a subject-predicate form. If the Russell{Meinong view of predication is accepted, `exists' cannot be a predicate. One argument, extracted from Mendelson [1989, p. 609], is this: (1) If `exists' is a predicate, then singular existentials of the form `s exists' are subject-predicate sentences with subject s. (2) In a subject-predicate sentence, \the subject stands for something and the predicate says something about that for which the subject stands." So (3) if `exists' is a predicate, `s exists' is trivially true. But (4) some singular existentials (such as `Neptune exists') are not trivially true, and others (such as `Vulcan exists') are simply false. So (5) `exists' is not a predicate. Since (2) is enshrined in classical semantics, it will be diÆcult to treat `exists' as a predicate E in the classical setting | unless we abandon (1) or Quine's dictum.8 For the 8 Mendelson,

under the spell of (2), does abandon (1): he construes \atomic-looking"

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extension of E will be the range of the bound variables and any term s must refer to something in this range, so Es will indeed be trivially true. g. An alternative to both Aristotle and Russell-Meinong holds that `The present king of France is bald' is a subject-predicate sentence whose truth or falsity requires the existence of the present king of France, or the truth of `The present king of France exists.' `The present king of France is bald' then presupposes `The present king of France exists' in Strawson's [1952, p. 175] sense: a statement S presupposes a statement S 0 i \the truth of S 0 is a precondition for the truth-or-falsity of S ." Presupposition is an interesting relation only if S can be neither true nor false, since otherwise S presupposes S 0 i S 0 is necessarily true. So a formal treatment will require giving up bivalence. If classical semantics is assumed, then Bk does indeed presuppose 9x(x = k), for the latter is logically true. However, as just observed in Section 2.2f, 9x(x = k) cannot be an adequate representation of the contingent truth `The present king of France exists'. The natural semantic setting for presupposition is a free semantics in which (1) subject-predicate sentences with non-referring subjects are neither true nor false, (2) existence claims of the form 9x(x = t) are false if t does not refer, (3) :A is neither true nor false i A is neither true nor false, and (4) X j= A i A is true whenever all the X sentences are true. Presupposition may then be characterized as A presupposes B i A j= B and :A j= B

and Bk presupposes 9x(x = k), because both Bk j= 9x(x = k) and :Bk j= 9x(x = k), although 6j= 9x(x = k). Free semantics of this \neutral" kind are discussed in Section 3.6. For a dierent supervaluational treatment of presupposition, see [van Fraassen, 1968]. h. Mereology conceives of individuals as wholes with parts. For the mereologist, `The rivers of Canada are numerous' does not claim that a particular set | the set of Canadian rivers | has many members, but that a particular whole | the mereological sum of the Canadian rivers | has many parts. Wholes whose parts are spatio-temporal individuals are also spatio-temporal, though perhaps spatio-temporally discontinuous; by contrast, sets of spatio-temporal individuals are abstract objects. Since every whole is a part of itself, there cannot be sentences Ps as either 9x(Sx&P x) or 8x(Sx ! P x), where S is a singular predicate. However, his non-standard semantics generates the truth-values that Ps would have if it were a subject-predicate sentence whose subject may refer to individuals outside the range of the bound variables.

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a whole with no parts (though there may be wholes | true atoms | with no proper parts). While set theory can embrace the empty set, mereology does not recognize an empty whole. In thinking rigorously and systematically about parts and wholes, it is very convenient to employ operators such as binary sum (x + y is the whole whose parts z are such that z is part of x or z is part of y), generalized sum (xA(x) is the whole whose parts z are such that there is a y such that A(y) of which z is part), binary product (x  y is the whole whose parts z are such that z is part of x and z is part of y), etc. But such operators will not be everywhere de ned, if there is no empty whole: both x(x 6= x) and x  y, where x and y have no common part, would be the empty whole, if there were such a thing, but there is not, so both x(x 6= x) and x  y, where x and y have no common part, are unde ned. If mereological theories are to be framed as rst-order theories, classical semantics is an unwelcome constraint: the mereologist must either do without these operators and conduct all logical business in terms of cumbersome predicates, or he must hold his nose and introduce a constant for something he denies exists, viz. the empty whole. A free semantics that permits partial operators is much more congenial. For a discussion of free mereological theories, see [Simons, 1991].

2.3 Logical Form I take the present view of logical form to have these elements: a. Sentences-in-context have a semantic structure or logical form: their truth-values (truth, falsity, or lack thereof) reduce, via recursive semantic rules, to the semantic values of their unstructured parts or elements (e:g:, a subject-predicate sentence is true i the referent of the subject term belongs to the extension of the predicate). b. An interpretation assigns appropriate semantic values to such elements (e:g:, extensions to predicates). c. Any logically possible situation is represented by some interpretation; in particular, the actual situation is represented by an interpretation, so that the actual truth or falsity of a sentence reduces to the actual semantic values of its elements. d. The logical properties and relations of sentences-in-context are determined by their semantic structures and the totality of interpretations via semantic de nitions of these properties and relations (e:g:, logical truth is truth under every interpretation); such de nitions provide the

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basis for judging the soundness and adequacy of proof systems that give purely syntactic accounts of proof and proof from hypotheses. Granting this, it is easy to argue for inclusive semantics: bound variables may range over the empty domain, for surely it is logically possible that there be nothing at all. To be sure, once we have investigated this case, we might want to heed Quine's advice [1953, p. 161] and put it aside for pragmatic reasons if, by including it, we would \cut ourselves o from laws applicable to all other cases". Or we might include it by using those laws but performing an extra check on results: existentials will be false and universals true in the empty domain, and the other sentences will be truth-functional compounds of these if we have followed Quine and purged the language of all singular terms but variables. Still, what justi es the truth-value assignments is a look at this case and thinking through the application of general semantic rules to it. Note that the no-counterexample interpretation is required if universals are to be true and that vacuous quanti ers are not idle in the empty domain (9y8xP x is false while 8xP x is true).9 We may also argue for free semantics as follows: (1) In accord with Quine's dictum, the range of bound variables in a given interpretation is restricted to individuals that exist in the possible situation represented by the interpretation. (2) There are sentences in which expressions that look for all the world like names do not, in actual use, refer to actual individuals. So (3) if these expressions are treated as names, no interpretation that represents the actual situation can assign them referents in the range of the bound variables. (4) What is permitted in interpretations that represent the actual situation must also be permitted in interpretations that represent possible situations. So (5) if these expressions are treated as names, interpretations in general need not assign them referents in the range of the bound variables. (1) is true of classical semantics, whether we identify an interpretation with (i) a meaning function that associates with each element an appropriate semantic value in the actual world, or (ii) a possible world, at which the meanings of elements in the actual world determine their semantic values, or (iii) a meaning function at a possible world.10 Sentences of type (2) include: 9 For discussion of systems that treat vacuous quanti ers in the empty domain dierently, see [Lin, 1983]. 10 The main defect of (i), aside from having to represent reasoning about hypothetical situations indirectly, is that the valid arguments depend upon how many individuals there are in the actual world. If the number is nite | k, say | any argument with a premise stating that there are more than k individuals will be valid, since the premises are not satis able. The main defect of (ii), aside from issues of epistemic access to possible worlds, is that arguments like `Some even number has an irrational square root, 2 is an even number, so 2 has an irrational square root' will be valid, since the conclusion is presumably true at any possible world. Hence, (iii).

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s1. If Vulcan exists and its orbit lies within that of Mercury, it's going to be very hard to observe. s2. Vulcan doesn't exist. s3. Zeus is not Allah. s4. The ancient Greeks worshipped Zeus. s5. Pegasus is a winged horse. s6. Bosch painted a picture of hell. s7. Mary admires Eustacia Vye. The simple thought behind (4) is that the actual is a special case of the possible. A name like `Hillary Clinton' happens to refer to a particular individual of the actual world; but the name could have been attached to some other individual of this world or to an individual of some other possible world. If we are prepared to regard an expression like `Eustacia Vye' as a name that does not in fact refer to an individual of this world (but to the heroine of Thomas Hardy's novel, The Return of the Native), then we should allow that in a possible situation, however conceived, it need not refer to an individual that exists in that situation. (5), the conclusion of the argument, is conditional, and we may avoid adopting free semantics by refusing to admit that such expressions are, despite appearances, singular terms. The cost of such denial is dealing with them in some other way, and those we have seen above are awkward. What basis is there for holding that `Mary admires Hillary Clinton' is a relational subject-predicate construction, while `Mary admires Eustacia Vye' is not? The obvious dierence is that `Hillary Clinton' refers to a real person and `Eustacia Vye' does not. But why should logical form depend upon that? Lambert [1998, p. 157] and Kroon [1991, p. 21] suggest that logical form is independent of empirical fact, and hence independent of whether terms actually refer. The premise, however, seems too strong. We may agree that the logical form of a sentence cannot depend upon its truth or falsity, since its truth or falsity is determined by its logical form and the actual semantic values its elements, including the referents of its terms. But it does not follow that form cannot depend on whether expressions that we are tempted to classify as terms refer, particularly if we cannot gure out how to get semantic rules to deliver truth-values smoothly in such cases. Indeed, it may be argued that logical form does depend to an extent on empirical fact, since it is sentences-in-use that have such forms. When Alice says, `I'm hungry', and Bill adds, `But I'm not', there is no contradiction: the form of what Alice says is represented by Ha and the form of what Bill says by :Hb. If logical forms re ect the dierent uses of indexicals

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like `I', as they must in one way or another, then why may they not re ect such empirical facts as whether \terms" refer? We might want to say that the logical form of `I'm hungry' is the same in both cases, namely, the simple subject-predicate form Ps, and that Ha represents, in this form, the propositional content of what Alice says. But then it is representations of propositional content that will do the work that (a){(d) require of logical form. I prefer a more pragmatic argument: logical form should be reasonably accessible, and the closer it is to surface form the better, other things equal. We do not want to misconstrue logical form and have to reformulate reasoning when we discover our error. If logical forms are contingent upon whether certain expressions refer | a matter that may be very diÆcult to settle | then logic may not be very useful. We don't want to have to revise reasoning about the unknown solution to some equation if we nd out, as a result of that very reasoning (how else?), that there is no solution (or no unique solution). Moreover, we'd like to be able to conduct such reasoning using a term t for the solution, rather than (say) in Russell's indirect and clumsy fashion.11 But perhaps we need not abandon the classical perspective, even if we admit that the italicized terms in s1-s7 are singular terms. Let me sketch two objections of this kind: OBJECTION 1. Suppose we insist that sentences with terms that do not refer, or that seem to refer to things outside the range of the variables, are neither true nor false. Then applying classical laws will lead from truths to truths: we may apply existential generalization to `Mary admires Eustacia Vye' to get `Mary admires someone', but the premise is untrue, so the falsity of the conclusion (if it is false) does not invalidate the rule. The reason for including the actual situation (in which such terms as `Eustacia Vye' do not refer to anyone) among the possible situations is to mesh logic with truth: the conclusion of a sound argument should be true. But if we insist that sentences with terms like `Eustacia Vye' are truth-valueless, we need not worry that classical logic will lead us from true premises to untrue conclusions. Accordingly, such sentences can be set aside as `don't cares'.12 There are three problems with this proposal. First, it is diÆcult to maintain that all such sentences are truth-valueless; indeed, anyone not bewitched by some theory will take most of s1{s7 to be true. Second, familiar rules of inference such as addition 11 Note, however, that a free semantics that does not support extensionality in the sense that 8x(A(x) $ B (x)) j= A(t) $ B (t) is probably not going to support this reasoning either, since it will typically involve moving from A(t) and 8x(A(x) $ B (x)) to B (t). We shall probably have to conduct it under the additional assumption that t exists: 9x(x = t). 12 This objection is suggested by van Fraassen [1966, Section 3].

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A `A_B

will then not preserve truth, contrary to what is alleged: `Washington is the capitol of the USA so either Washington or Atlantis is the capitol of the USA' has a true premise and a truth-valueless conclusion. Third, no justi cation has yet been given for the claim that terms which do not refer to things in the range of the bound variables render sentences truth-valueless; such a justi cation requires an extension of the ordinary semantic rules to this case. OBJECTION 2. Let us concede that sentences with terms that do not refer to existents can be true or false; still, it does not follow that we must reject classical semantics for free semantics: a. Perhaps, as Stenlund [1973], Burge [1974], and Kroon [1991] suggest, some sentences of this kind can be handled classically, albeit by shifting from possible to imaginary or hypothetical worlds. In actual use, a sentence like `Pegasus is a winged horse' invokes an implicit `in myth' operator that, in eect, shifts attention from the actual situation to an imaginary one in which Pegasus exists (and indeed turns out to be a winged horse). That is, in actual use or context, the sentence is not about this world, but about another one, at which the normal reference conditions of classical semantics are ful lled. Thus, classical semantics is all we need to understand why the reasoning of Sherlock Holmes in \Silver Blaze" about the missing horse is sound in the world of Doyle's story: \. . . he must have gone to King's Pyland or to Mapleton; he is not at King's Pyland. Therefore he is at Mapleton."13 However, this manoeuvre works only for sentences like s5. Consider s3: Zeus is not Allah, but where? Insofar as we can understand the truth of most of the sentences s1-s7 in terms of reference to imaginary individuals, it is reference across, not within, worlds, and that is not going to be accommodated by classical interpretations. All we need do to create problems for the recommended treatment of `Pegasus is a winged horse' is to add `though such things do not exist'. If `Pegasus is a winged horse' is true, so is `Pegasus is a winged horse, though such things do not exist', but moving to a world of myth will render it self-contradictory. We can x this by staying here in the actual world and understanding `Pegasus' to refer to something in a world of myth, but that abandons classical semantics for some free variant of it. b. Perhaps what is problematic about sentences like s4, s6, and s7 is not failure of reference, but referential opacity, which we are not going to 13 A. C. Doyle, \Silver Blaze", in The Complete Sherlock Holmes (Garden City: Garden City Books, 1930), pp. 383{401 at p. 393.

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be able to handle with extensional logical forms anyway. Contexts like `worshipped . . . ', `picture of . . . ' and `admires . . . ', according to this view, are not referentially transparent: truth-value need not be preserved by replacing a referential term by a co-referring term. The suggestion is that if John worships God, it does not follow that John worships Satan, even if God and Satan turn out to be identical (or as identical as the members of the Trinity). And similarly for the others. This objection does not help in the other cases. Nor does it seem very convincing. There is certainly a sense in which John does worship Satan if he worships God and God is Satan. And similar things may be said about the other contexts. A picture of Fred is a picture of the Grand Dragon of the Ku Klux Klan, if that's who Fred is. `Admires' is no more intensional than `loves', the standard logic-text example of a relational predicate.

2.4 Derived Rules and Axioms Re ection on the cases discussed in Section 2.1 will suggest that we can work around the constraints of classical semantics in various ways without giving up its familiar simplicity. We can relativize the quanti ers to a predicate E that we interpret as true of existents; irreferential terms are those that refer to individuals of D that are not in d(E ), function-names f such that d(f )() 2= d(E ) for some  2 d(E ) represent functions d(E ) ! d(E ) that are not total. Or we can eliminate non-referring terms in Russell's way. While such representations may not be perfect, they may be good enough for most purposes. However, classical representations of irreferential names and partial functions can be cumbersome, and at some point those who work with them will want to develop some derived rules to facilitate reasoning and its formal or informal representation. Such rules will be those of a corresponding free logic. Development of various free logics can therefore provide a number of `o the shelf' systems that can be applied in such cases. Let us consider two examples. a. Suppose that you believe Russell was correct in holding that descriptions like `the present King of France' are not genuine singular terms and that what appear to be subject-predicate constructions like `the present King of France is bald' are actually complex quanti er constructions. Still, working with these complex constructions is about as inviting as programming in machine language; you will want to develop some macros. So let's pretend that `the present King of France' is a term and `the present King of France is bald' is a subject-predicate sentence whose truth-value is given by Russell's quanti er construction, and attempt to develop a system of derived axioms and rules of

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inference that enables you to treat descriptions as if they were terms. The obvious minimal constraint on such a system is that A should be provable i the Russellian expansion of A is logically true. If we can pretend that descriptions are terms, we can also pretend that they can refer | or fail to refer, as the case may be | and ask what sort of logical semantics we may pretend underlies the system of derived axioms and rules. As Scales [1969] and Burge [1974] show, the result of this process is a free logic, whose underlying semantics may be suÆciently compelling to loosen your allegiance to Russell and to classical logic. b. Quine's pure set theory NF [1969] is framed in a rst-order language without identity, whose non-logical symbols are variables and the 2place predicate 2 : Identity (in the weak sense of indiscernibility) is introduced by de nition: x = y abbreviates 8z ((z 2 x $ z 2 y) & (x 2 z $ y 2 z )):14 The only axioms are extensionality

8z (z 2 x $ z 2 y) ! x = y and restricted comprehension

9x8y(y 2 x $ A(y)); where A is strati ed: it is possible to replace the variables by numerals so that subject-predicate constructions x 2 y become n 2 n + 1. The restriction is designed to secure the safety of type theory without the pain. Quine's view [1969, p. 16] is that \much. . . of what is commonly said of classes with the help of `2' can be accounted for as a mere manner of speaking, involving no real reference to classes or any irreducible use of `2'." Singular terms fx : A(x)g for classes | terms Quine calls class abstracts | are introduced by a pair of contextual de nitions: y 2 fx : A(x)g abbreviates A(y), and fx : A(x)g 2 , where  is a class abstract, abbreviates 9y(y = fx : A(x)g & y 2 ). If class abstracts are regarded as referring to pretend or virtual classes, then virtual classes which belong to virtual classes are real, though the virtual classes to which they belong need not be; sets are members of real classes. Identity may be extended to class abstracts by taking  = to abbreviate 8x(x 2  $ x 2 ), so that  = fx : A(x)g $ 8x(x 2 $ A(x)) and  = fx : x 2 g: Existential generalization fails for class abstracts: A() 6j= 9xA(x) for some A(). In particular, let A(x) be x = , where  is fx : B (x)g. Then the premise A() is  = , i:e:, fx : B (x)g = fx : B (x)g, i:e:, 14 Of course, this will not force `=' to be interpreted as numerical identity; no set of axioms constraining the interpretation of `=' can do that, which is why identity is commonly regarded not as a predicate but as a logical operator.

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8y(y 2 fx : B (x)g $ y 2 fx : B (x)g), i:e:, 8y(B (y) $ B (y)), which is logically true. The conclusion 9xA(x) is 9x(x = ), i:e:, 9x8y(y 2 x $ y 2 fx : B (x)g), i:e:, 9x8y(y 2 x $ B (y)), which amounts

to unrestricted comprehension, since B (y) can be any formula and the premise will be true. If B (y) is y 2= y, we obtain the logical falsehood 9x8y(y 2 x $ y 2= y) of Russell's paradox. However, the inference will be valid if 9x(x = ) is added as an extra premise: A(), 9x(x = ) j= 9xA(x). For then we can move from A() to A(x) (in virtue of j=  = ! (A() $ A( )) for variables or set abstracts ; ), from which of course 9xA(x) follows. This is not a problem here because the added premise is just the conclusion when B (y) is y 2= y. This is typical of free logics: A(t), 9x(x = t) j= 9xA(x), but not necessarily A(t) j= 9xA(x). A natural question now is whether we can capture the logical moves involving class abstracts  in a set of derived axioms and rules which treat them as genuine terms t and, if so, whether there is a natural free semantics that we might pretend underlies their use. A suÆciently natural semantics might blur the distinction between pretense and reality, especially in pure mathematics, where there does not seem to be much dierence between pretending that mathematical objects exist and asserting that they do. For a restructuing of Quine's NF along these lines, see [Scott, 1967]. 3 FREE SEMANTICS Free departures from classical semantics are usually | though not always, as in [Farmer, 1995], [Feferman, 1995], and [Woodru, 1984] | universally free. If we are going to permit terms that do not refer to individuals in the range of the bound variables, why not include the case where no term can refer to such an individual, simply because there are none? This is easy enough to do semantically, though the required adjustments to proof systems are a bit more trouble. There are two ways in which terms may fail to refer to individuals in the range of the bound variables: either they refer to individuals outside this range, or they do not refer at all. The rst way leads to outer domain semantics, a straightforward bivalent modi cation of classical semantics in which a classical domain D is divided into a possibly empty inner domain Di , over which the bound variables range, and an outer domain Do . With the exception of the quanti er valuation clause v5, in which x variants must now be understood to assign to x an individual of the inner domain, the interpretation and valuation clauses of classical semantics can be adopted without change. The second way involves partial interpretations I = hD; di, where D may be empty and the denotation function d is partial on constants and assigns

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partial functions D ! D to function-names. Assignment functions  are also typically partial on variables. Here we face the problem of \assigning reasonable truth conditions to sentences containing non-denoting singular terms," as Bencivenga [1986, p. 382] observes. This requires giving reasoned responses to the following questions: q1. If t does not refer, must ft also be non-referring? Does `Jack Aubrey's sovereign' refer to England's George III, although `Jack Aubrey' does not refer to a real person but to the ctitious captain of Patrick O'Brian's sea novels? q2. Should d treat predicates as names of partial truth-valued functions on D? Is `2 is green' false (because `2' refers to something that is not in the extension of `green') or is it truth-valueless (because `2' does not refer to something that is colored)? q3. If t does not refer, may Pt be true? If not, is it false or is it truthvalueless? q4. If t does not refer, is t = t true, false, or truth-valueless? If s refers and t does not, is s = t false or truth-valueless? q5. If formulae may lack truth-value, how are the classical truth-tables for the connectives to be extended? Shall we count A ! B true or truth-valueless if A is false and B is truth-valueless, or B is true and A is truth-valueless? q6. If formulae may lack truth-value, how are the quanti er clauses to be modi ed? Should v5 read `8xA is false if A is false for some x variant of , and 8xA is true otherwise' or `8xA is false if A is false for some x variant of , and 8xA is true if A is true for each x variant of ' or `8xA is true if A is true for each x variant of , and false otherwise'? q7. If formulae may lack truth-value, how are the de nitions of logical properties and relations to be modi ed? Should logical consequence preserve truth? non-falsehood? both? Applications may decide some of these questions. For example: 1. If we wish to allow for the non-strict functions and relations of computer science, then we will answer `No' to q1 and `Yes' to the rst part of q3. Following Gumb and Lambert [1997], we might then implement such permissions by adding a virtual entity u (for `unde ned') to D: d will assign to a k place function-name f a total k ary function d(f ) : D [ fug ! D [ fug and to a k place predicate P a k ary relation d(P ) in D [ fug as its extension. If we want ft to be unde ned, though t refers to , we will set d(f )() = u; if we want ft

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to be de ned, though t does not refer, we will require d(f )(u) 2 D; if we want Pt to be true when t does not refer, we will put u in the extension d(P ) of P . u is not to be regarded as a strange entity, but as a notational device for simplifying the statement of semantic rules, as in [Kroon, 1991]. 2. If we think that `2 is green' presupposes `2 is colored' in the sense of Section 2.2g, we will answer `Yes' to q2. We can then follow Smiley [1960] and Ebbinghaus [1969] and have I assign P both a domain of application D(P ) in D and, within that domain, an extension d(P ). If we want Px to be truth-valueless when x is assigned , then we put  outside D(P ); if we want Px to be false when x is assigned , then we put  in D(P ) d(P ): The large decision is whether to answer q3 and q4 in a way that permits truth-valueless atomic formulae | and forces us to answer q5-q7. We can prune the choice tree considerably by opting for bivalence at the atomic level. However, such a decision needs to be rationalized. It will not do simply to argue that atomic formulae with non-referring terms should be false because (a) any atomic formula A(t) is a predication which is true just in case the A(x) is true of the referent of t, so that (b) where t fails to refer, A(t) is not true, so (c) where t fails to refer, A(t) must be false. For what underwrites the move from (b) to (c) is bivalence. However, there may be applications which call for such a ruling. For example, Farmer [1995, p. 281] claims that in the \traditional approach to partial functions" in mathematics, variables and constants always refer, functions may be partial and ft does not refer if t does not refer or d(f ) is not de ned at d(t), while Pt is false if t does not refer. The usual route to true atomic formulae with non-referring terms is story semantics, a non-referential variant of outer domain semantics: treat nonreferring terms as if they referred to individuals in an outer domain, taking the formulae that are true under such a pretense to constitute a story S which supplements a partial referential interpretation I and assignment . Story semantics is equivalent to outer domain semantics in which the individuals of the outer domain Do are treated as virtual or pretend objects. Story (or virtual outer domain) semantics seems to provide a natural way of dealing with sentences that are about ctional or mythical entities, at least if we do not want to follow Meinong in reifying them. Note, however, that we don't yet have a justi cation for the bivalence that is built into this type of semantics, because actual stories or myths, unlike the stories of story semantics, are not complete. Nothing Doyle wrote decides the ctional truth-value of `Sherlock Holmes died before 1920,' yet if Bh represents this sentence and h does not refer, any story S will include either Bh or :Bh: If atomic formulae with non-referring terms are all to be false, then an excursion into stories is unnecessary: we can simply modify the valuation

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rules v1 and v2 for atomic formulae to require as much. Note, however, that t = t will then be false, if t does not refer. Even stranger results await those who follow Frege [1892] and deny truth-values to atomic formulae with non-referring terms. For if P t lacks truth-value, then :P t and P t _ :P t also appear to lack truth-value. Accordingly, some instances of A _ :A are not logically true, if logical truth is truth under each partial interpretation and assignment. This is suÆciently disturbing to motivate a search for an respectable alternative that permits P t to be truth-valueless while insuring that t = t and A _:A are logically true. The usual proposal is a supervaluational semantics, in which truth under a partial interpretation is understood as truth under all completions of it. A free semantics in which some atomic formulae with terms that do not refer to individuals in the range of the bound variables are true is said to be positive.15 If all such atomic formulae are false (truth-valueless), the semantics is said to be negative (neutral). Both outer domain and story semantics are positive in this sense, as is supervaluational semantics. However, the more signi cant divide is between bivalent and non-bivalent accounts. I shall rst discuss bivalent free semantics, both positive and negative, then non-bivalent free semantics, including supervaluations.

3.1 Positive Bivalent Semantics: Outer Domains Outer domain free semantics involves minimal change in classical semantics. An outer domain interpretation I = hD; di is classical, except that D is partitioned into an inner domain Di and an outer domain Do. Thus, i1 is altered to: io 1: D is non-empty, and D = Di + Do No change is needed in the classical notion of an assignment. However, bound variables are to range over Di , so the classical notion of an x variant must be modi ed to require that x is assigned a value in Di : an x-variant of  is an assignment that diers from  at most at x and assigns x a value in Di . Note that  now need not be an x-variant of . If Di is empty,  has no x-variants, so in this case universals are true and existentials are false. Evidently: 6j= 9x(x = x) P t 6j= 9xP x; but A(t), 9x(x = t) j= 9xA(x) 8xP x 6j= P t; but 8xA(x), 9x(x = t) j= A(t) P t 6j= 9x(x = t) j= t = t 15 Here I follow the recent usage of Lambert [1997, p. 62]. Other meanings of `positive' can be found in the free logic literature. Bencivenga [1986, p. 397] terms (conventional) semantics positive if each atomic formula containing a non-referring term is true, while Lambert [1991b, p. 344] uses `positive' merely as a synonym for `non-negative'.

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If desired, an existence predicate E ! may be introduced by taking E !t to abbreviate 9x(x = t). The extension d(E !) of E ! under I will be Di : A Hilbert-style system PFL (for `positive free logic') of axioms and rules16 that is strongly complete relative to outer domain free semantics may be obtained from CL by replacing A1 with

8x(A ! B ) ! (8xA ! 8xB ) A ! 8xA, if x is not free in A modifying A2 to (8xA(x) & 9x(x = t)) ! A(t) modifying A3 to

t=t modifying A4 to

s = t ! (A(s) ! A(t)), if A is atomic and adding

8x9y(y = x): Lambert [1991a, p. 9] characterizes outer domain semantics as embodying a \Meinongian world picture": the inner domain consists of existents, while beings that lack existence are relegated to the outer domain. This identi cation is a bit misleading, since Meinong held that non-existent beings are indeterminate with respect to certain properties | the golden mountain has no speci c height | whereas the objects of an outer domain are determinate in virtue of i4 and v1. A true Meinongian outer domain semantics would not be bivalent. Moreover, outer domains are sometimes taken to consist of pretend or virtual objects: that is, objects that we pretend exist so as to provide a referential semantics for terms that do not refer to existents. But Meinong did not regard having being as a matter of pretense.

3.2 Positive Bivalent Semantics: Stories Story semantics can be regarded as a non-referential version of outer domain semantics. A story interpretation hI; S i consists of a partial interpretation I that permits non-referring terms and a story (or convention) S that assigns truth-values to atomic formulae containing such terms. 16 This formulation is based on [Meyer and Lambert, 1968]; see also [Lambert, 1997, p. 39]. Leblanc [1968] has shown how to derive (8xA & E !t) ! A(t) from the other axioms by the rules of inference. For discussion of related systems, see [Bencivenga, 1986, Sections 5 and 6].

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A partial interpretation I = hD; di is classical except that D may be empty, d is partial on constants, and d assigns function-names partial functions D ! D. In particular: ip 1: D is a possibly empty set.

ip 2: If d is de ned at a, d(a) 2 D:

ip 3: If f is k place, d(f ) is a partial k-ary function D ! D: i4 remains unchanged. Assignments  may be partial, but as in ip 2 if  is de ned at x, (x) 2 D. The rules for reference under I and  must be reformulated to take account of non-referring terms: rp1: If  is de ned at x, then x refers to (x); otherwise, x does not refer. rp2: If d is de ned at a, then a refers to d(a); otherwise, a does not refer.

rp3: If each ti refers and d(f ) is de ned at h1 ; : : : ; k i, where ti refers to i , then ft1 : : : tk refers to d(f )(1 ; : : : ; k ); otherwise, ft1 : : : tk does not refer. However, irreferential terms do not lead to truth-valueless formulae, since the story S supplies the missing truth-values for atomic formulae. A substitutional account of quanti cation would permit us simply to identify S with a set of atomic sentences satisfying certain conditions.17 Objectual quanti cation requires a somewhat more complicated account, derived from Woodru [1984]. Here a story is a function S from assignments  to sets S () of atomic formulae with non-referring terms satisfying the following conditions: s1. If t does not refer, then t = t 2 S ():

s2. If just one of s and t refers, then s = t 2= S (): s3. If neither s nor t refers and s = t A(t) 2 S ():

2 S (),

then A(s)

2 S ()

i

s4. If both s and t refer to the same individual of D, then A(s) 2 S () i A(t) 2 S (): s5. If  and agree on the free variables of A, then A 2 S () i A 2 S ( ):

Truth-values for atomic formulae under hI; S i and  are xed by I and  if all terms refer, and by S and  otherwise: vs 1: If each ti refers, then P t1 : : : tk is true i h1 ; : : : ; k i 2 d(P ), where ti refers to i ; otherwise, P t1 : : : tk is true i P t1 : : : tk 2 S (): 17 The

conditions are s1{s4 with `S ()' replaced by `S '.

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vs 2: If both s and t refer, then s = t is true i  is , where s refers to  and t refers to ; otherwise, s = t is true i s = t 2 S (): The classical valuation rules v3{v5 are unchanged, except that truth and falsity are now relative to hI; S i and . Note that if t does not refer under I and , then 9x(x = t) is false under hI; S i and . If t does not refer because D is empty, there are no x-variants of , hence no x-variants of  for which x = t is true under hI; S i and ; so 9x(x = t) is false under hI; S i and . If D is not empty, let be any x-variant of . Since x refers under I and and t does not, x = t 2= S ( ) and x = t is false under hI; S i and ; therefore, 9x(x = t) is false under hI; S i and . It can be shown that to any outer domain interpretation hDi + Do ; di and assignment  there corresponds a story interpretation hDi ; d0 ; S 0 i and assignment 0 that preserves truth-values, and conversely. Thus, adopting story semantics does not require any change in PFL. Story semantics is now somewhat unfashionable. Lambert [2001] regards his creation as an unsuccessful attempt to develop \a philosophically palatable semantics for positive free logic whose domain consists of a single set of (intuitively) existing objects, whose denotation function is partial, and whose truth de nition makes no appeal to other worlds or `extensions' of the domain." It is unsuccessful, in his view, because a story is \simply a list of sentences governed by some logical laws, hence a story in a very Pickwickian sense indeed." However, it seems no more Pickwickian than identifying properties of D-individuals with the subsets of D, as is standard in classical semantics. Another objection is Bencivenga's: without a semantic rationale, conditions s1{s5 are ad hoc, and the \logical laws" that they build into a story S are without foundation.18 For example, if we are going to require that a = a 2 S () when a does not refer under I, why shouldn't we also require that P a 2 S () when 8xP x is true, but a does not refer, under I? Why can't 8xA(x) ! A(t) also claim the status of a logical law, contrary to the desires of free logicians? Perhaps this objection can be partially met by arguing that conditions s1{s5 capture the rules of language games about ctional or pretend entities.19 However, as noted above, this justi cation will be incomplete unless we can argue that such language games commit 18 Bencivenga [1986, p. 403], and [this volume, p. 176]. Woodru [1984, p. 944] characterizes conditions like s1{s5 as \constraints designed to ensure that we get the right results (for instance, that the laws of identity continue to hold)." 19 I confess that I do not nd Walton's use of this idea very illuminating. According to Walton [1990, p. 400], when Sally claims that Tom Sawyer attended his own funeral, she is claiming that The Adventures of Tom Sawyer is such that \to behave in a certain way, to engage in an act of pretense of a certain kind while participating in a game authorized for it, is ctionally to speak the truth." Unless Sally is a very unusual person, this is false. As an account of when Sally's assertions of `Tom Sawyer attended his own funeral' are true, it is more promising, but obscure.

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participants to bivalence, i:e:, to accepting A or :A, for any atomic sentence A with a non-referring term.

3.3 Negative Bivalent Semantics Negative free semantics is story semantics without the story: an interpretation I is precisely as it is in story semantics, that is, it is a partial interpretation de ned by ip 1{ip 3 and i4. The rules of reference rp 1{rp3 are unchanged, but vs 1 and vs 2 are altered to declare that atomic formulae with non-referring terms are false: vr 1: If each ti refers, then P t1 : : : tk is true if h1 ; : : : ; k i 2 d(P ), where ti refers to i ; otherwise, P t1 : : : tk is false. vr 2: If s and t refer, then s = t is true if  is , where s refers to  and t refers to ; otherwise, s = t is false. The subscript `r' is for `Russell'. In contrast to outer domain semantics, we have: P t j= 9x(x = t) 6j= t = t :9x(x = t) j= t 6= t: A Hilbert-style system NFL (for `negative free logic') of logical axioms and rules20 that is strongly complete with respect to this semantics may be obtained from PFL by altering the identity axiom schema t = t to

8x(x = x) and adding

A(t) ! 9x(x = t), if A is atomic. A somewhat less free version of negative free semantics is employed by Farmer [1995] and Feferman [1995] to formalize reasoning about partial functions in mathematics. Since mathematical domains | natural numbers, sets, etc. | are assumed to be non-empty, ip 1 is replaced by the classical i1. Since in mathematical practice variables and constants are assumed to refer, assignments are total and ip 2 is replaced by the classical i2; the classical reference rules r1 and r2 (resp.) replace rp 1 and rp 2 (resp.). For a Hilbert-style axiomatization LPT of this semantics, with extensions to a partial combinatory logic CLp and a partial  calculus p , see [Feferman, 1995]. For a type-theoretical extension LUTINS of this semantics that has been axiomatized to provide a basis for automated theorem proving, see 20 See

[Burge, 1974, p. 191] and [Lambert, 1997, p. 83].

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[Farmer, 1995]. Following Beeson [1985, p. 98], Farmer and Feferman replace the existence predicate E ! with # (read `is de ned'); t # is equivalent to 9x(x = t). Lambert [1991a, p. 9] characterizes negative free semantics as Russellian: there are no entities, Meinongian or virtual, beyond the existents, and nonreferring terms render atomic formulae false just as they do for Russell. However, Russell's truth-values result from treating non-referring terms as descriptive and atomic formulae containing them as abbreviations for more complex formulae that turn out to be false if the descriptions are empty. So his truth-values are rationalized by some analysis of constructions containing non-referring terms. That is not yet the case here. If we hold, with Burge [1974, p. 193] following Aristotle, that \true predications at the most basic level express comments on topics, or attributions of properties or relations to objects," then we will agree that \lacking a topic or object, basic predications cannot be true." But this will not get us all the way to negative free semantics unless we buy bivalence, for which Burge does not argue. The rest of the justi cation will probably have to be provided by particular applications, as when Farmer [1995, p. 282] notes that the \traditional approach to partial functions" in mathematics holds that \formulas are always true or false" and that \application of a predicate is false if any argument is unde ned."

3.4 Intermission: Axiomatizing Equivalence and Implication Before turning to non-bivalent free semantics, let us take note of Lin's [1983] study of equivalence and implication for various bivalent free semantics. Some elementary logic texts, such as [Tidman and Kahane, 1999], present natural deduction systems that include replacement rules A[B ] ` A[C ], where A[C ] results from A[B ] by replacing a part B by C . For each such rule, there is a decidable syntactic relation R such that (i) B R C or C R B and (ii) R related formulae are logically equivalent. Examples are double negation, where ::B R B , and DeMorgan's laws, where :(B _C ) R :B &:C and :(B &C ) R :B _ :C . Any rule of this kind is closed under ordinary replacement: if B ` C is an instance, so is A[B ] ` A[C ]. Let us call such rules replacement closed. Classical semantics supports ordinary replacement in the sense that if B is logically equivalent to C , then A[B ] is logically equivalent to A[C ]. So if B is provable from A by replacement closed rules, B is logically equivalent to A. For classical semantics and free variants that support ordinary replacement, it is natural to ask if the converse holds: is there a system S of replacement closed rules such that B is S -provable from A if B is logically equivalent to A? Positive results are summarized in Chart B of [Lin, 1983, p. 86]. Textbook authors warn students not to apply implicational rules like ADD to parts of formulae, since :B ` :(B _ C ) is unsound. However,

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applications of such rules to certain parts of formulae are sound. If a positive (negative) part B of a formula A[B + ](A[B ]) is characterized as in [Schutte, 1960, p. 11], then A[B + ] ` A[B _ C + ] is sound because the truth (falsity) of a positive (negative) part renders a formula true. Moreover, A[B _ C ] ` A[B ] is sound as well. So we may generalize addition to a pair of rules: A[B + ] ` A[B _ C + ] and A[B _ C ] ` A[B ]: A natural question is whether implication can be characterized by a system of such paired rules R1 and R2, where R1 is A[B + ] ` A[C + ], R2 is A[C ] ` A[B ], and B bears some decidable syntactic relation R to C . Given Schutte's account of positive and negative parts, classical semantics supports polar replacement: if B j= C , then A[B + ] j= A[C + ] and A0 [C ] j= A0 [B ]. This suggests that the paired rules R1 and R2 should be closed under polar replacement: if B ` C is an instance of R1 (R2), then A[B + ] ` A[C + ] is an instance of R1 (R2) and A0 [C ] ` A0 [B ] is an instance of R2 (R1). For then, from a proof hB = B1 ; : : : ; Bk = C i of C from B , we could obtain (1) a proof of A[C + ] from A[B + ] by Bi ! A[Bi + ], and (2) a proof of A0 [B ] from A0 [C ] by Bi ! A0 [Bi ] and reversing the resulting sequence of formulae. Schutte's notion of positive and negative part does not license closure under polar replacement, since a positive (negative) part of a negative part of A need not be a negative (positive) part of A. In the case of addition, for example, B _ A ` (B _ C ) _ A is an instance of R1, but :((B _ C ) _ A) ` :(B _ A) is not an instance of R2. However, there is another notion of positive and negative part that does support polar replacement and for which implicational replacement holds for classical semantics: B is a positive (negative) part of A i B occurs within the scope of an even (odd) number of negations in A, where the quanti er is 8 and the connectives are :, &, and _; and 9 and ! are de ned as usual in terms of them. It is this notion that Lin uses in de ning closure under polar replacement. He then characterizes implication in classical semantics and various free (bivalent) variations by systems of implicational rules closed under polar replacement; results are summarized in Chart A of [Lin, 1983, p. 47].

3.5 Non-bivalent Semantics: Supervaluations Frege [1892, p. 70] claims that \anyone who seriously took [`Odysseus was set ashore at Ithaca while sound asleep'] to be true or false would ascribe to the name `Odysseus' a reference." His view is that subject-predicate sentences with non-referring terms are truth-valueless. If interpretations and assignments are partial, then Frege's view dictates that vr 1 be replaced by vf 1: If each ti refers, then P t1 : : : tk is true if h1 ; : : : ; k i 2 d(P ) and P t1 : : : tk is false if h1 ; : : : ; k i 2= d(P ), where ti refers to i ; otherwise,

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P t1 : : : tk lacks truth-value. If `=' is regarded as a predicate | or as a binary truth-valued operator that has no output without a pair of inputs | then vr 2 must be replaced by vf 2: If s and t refer, then s = t is true if  is and s = t is false if  is not , where s refers to  and t refers to ; otherwise, s = t lacks truth-value. Once bivalence is lost at the atomic level, we must face questions q5{q7. There are two plausible answers to q5: either the weak or the strong truthtables, as Kleene [1950, p. 334] calls them. The weak tables assign precisely the same truth-values as do the classical tables, leaving the compound truthvalueless in all other cases. Thus :A is truth-valueless when A is truthvalueless, and A ! B is truth-valueless when A or B (or both) is truthvalueless. The strong tables treat negation in the same way, but preserve certain features of the classical tables for other compounds: A ! B is true when A is false or B is true, regardless of whether the other constituent has a truth-value. Thus, treating A _ B as :A ! B and A&B as :(A ! :B ), disjunctions are true if at least one disjunct is true, and conjunctions are false if at least one conjunct is false. Neither of these answers to q5 will prevent such classical logical truths as P t _ :P t and s = t ! (P s ! P t) from ending up with no truth-value when t and s do not refer. Note also that t = t will have no truth-value if t does not refer. Supervaluational semantics is an attempt to avoid such alien results, while permitting some atomic formulae to lack truth-values. The basic idea is to consider completions of a partial interpretation I and to revise the valuation rules so that A is supertrue (superfalse) under I and  if, for each completion I 0 of I and 0 of , A is true (false) under I 0 and 0 , and A is supervalueless under I and  otherwise. In van Fraassen's [1966] original development of the idea, completions of I are achieved by adding stories S , which he terms classical valuations over I : if I = hD; di, then I 0 = hI; S i, where S is de ned as in story semantics. Since P t _ :P t, s = t ! (P s ! P t), and t = t are true under any story interpretation hI; S i, they are supertrue under any partial interpretation I , and hence logically true with respect to supervaluational semantics. More generally, A is logically true with respect to supervaluational semantics (in the sense of being supertrue under every partial interpretation I ) i A is logically true with respect to story semantics. Accordingly, the Hilbert-style axiomatization PFL of story semantics is weakly complete with respect to supervaluational semantics. However, it is not strongly complete: P a j=s 9x(x = a) but P a 6` 9x(x = a), where X j=s A i A is supertrue under each partial interpretation I and assignment  for which each B 2 X is supertrue. Note rst that if a does not refer under I , P a has no supervalue, since there are stories S and S 0

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for which P a 2 S but P a 2= S 0 :21 Thus, if P a is supertrue under I , then a must refer under I . So 9x(x = a) is true under any I 0 , where I 0 is a completion of I , and therefore 9x(x = a) is supertrue under I . Accordingly, P a j=s 9x(x = a). However, 6j=s P a ! 9x(x = a), since if a does not refer under I , then, as noted in Section 3.2, 9x(x = a) is false under any hI; S i. However, for some completions hI; S i of I , P a is true under hI; S i and for others, it is false; thus, P a ! 9x(x = a) is not supertrue under every interpretation I . Accordingly, by weak completeness, 6` P a ! 9x(x = a). But the deduction theorem holds for PFL, so P a 6` 9x(x = a): Since PFL is strongly complete with respect to story semantics, we have P a j=s 9x(x = a) but P a 6j= 9x(x = a). This is analogous to the situation in classical semantics, where P x j=g Pa but P x 6j= P a. This suggests that, just as the generality interpretations of classical logic treat free variables as if they were universally quanti ed, supervaluations may also involve implicit quanti cation. Woodru [1984] has shown that they do indeed, and that it is second-order.22 Consider a subject-predicate formula P xa, a partial interpretation I , where d(a) is unde ned, and an assignment , for which (x) 2 D. The story S in a completion I 0 = hI; S i of I may be regarded as assigning an extension d(Pa ) to a predicate Pa de ned by Pa x $ P ax: (x) 2 d(Pa ) i P ax 2 S ( ), where is an x-variant of . Thus, P xa is supertrue under I and  i for each S , P xa is true under hI; S i and  i for each extension d(Pa ), Pa x is true under I and  i 8PaPa x is true under I and . This sketch of the argument assumes that a does not refer. Woodru shows how to conditionalize such assumptions to obtain, for any A, a second-order normal form tr(A), such that A is supertrue under I and  i tr(A) is supertrue under I and . Moreover, tr(A) is such that if A is supertrue under I and , a part of tr(A) of the form 8P1 : : : 8Pk B , where B contains no constant that does not refer under I and , is true under I and . 23 Woodru goes on to establish that supervaluational semantics inherits the pathologies of classical second-order semantics. Compactness (X is satis able if every nite subset of X is satis able), the upward Lowenheim{ Skolem theorem (X is satis able in ! if there is some k such that X is satis able in f0; : : : ; k + j g for each j ), and the downward LowenheimSkolem theorem (X is satis able in ! if X is satis able in some larger set) all fail; and nite logical consequence is not recursively axiomatizable (there is no recursive set of axioms such that A1 ; : : : ; Ak j=s B i A1 ; : : : ; Ak ` B ): 21 Since we are dealing with sentences here, I suppress mention of assignments. 22 Note that supervaluations are also like generality interpretations in not treating

connectives as (strict) truth-functions. P x _:P x can be true (supertrue) without either P x or :P x being true (supertrue). 23 Woodru's construction is carried out for languages without function-names. In addition, partial interpretations are partial only with respect to constants: domains are non-empty and assignment functions are total.

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The question naturally arises whether there are constraints on stories that restore a rst-order regime. That is, are there constraints C such that if supertruth under I is understood as truth under each completion hI; S i of I such that S satis es C , then the desirable properties of classical rstorder semantics (compactness, the Lowenheim{Skolem theorems, and the recursive axiomatizability of logical consequence) are assured? Woodru [1991] shows that what he terms `actualist' constraints will do the trick. Constraints C of this type, which require too much development to describe here, have the eect of making each formula A superequivalent to an actualist formula A0 , in which every occurrence of a constant a is in a part of of the form 9x(x = a), 9x(x = a)&B , or 9x(x = a) ! B . Superequivalence here means that A is supertrue under I i A0 is supertrue under I , where supertruth under I is now truth under hI; S i for each S that satis es C . What Woodru [1991, p. 227] terms \the rst-order character of actualist semantics" then follows from the fact that actualist formulae are stable: A is true under hI; S i i A is true under hI; S 0 i, for any stories S and S 0 :24 Woodru [1991, p. 225] suggests that \the text of some story, theory or myth" could function as an actualist constraint. Details, however, are left to the reader's imagination; as he notes at the outset, his treatment is quite abstract. The equivalence of story semantics and outer domain semantics will suggest another way to complete a partial interpretation I = hD; di: embed it in an outer domain interpretation I 0 = hD + Do; d0 i, where: ie 2: d0 (a) = d(a) if d is de ned at a, and d0 (a) 2 Do otherwise. ie 3: d0 (f )(1 ; : : : ; k ) = d(f )(1 ; : : : ; k ) if d(f ) is de ned at h1 ; : : : ; k i, and d0 (f )(1 ; : : : ; k ) 2 Do otherwise. ie 4: d(P ) is the restriction of d0 (P ) to Dk , if P is k place.

Partial assignments  are similarly completed by requiring that 0 (x) = (x) if  is de ned at x and 0 (x) 2 Do otherwise. We can then stipulate that A is supertrue (superfalse) under I and  i, for each completion I 0 of I and completion 0 of , A is true (false) under I 0 and 0 : Bencivenga [1980] develops an equivalent semantics that embeds partial interpretations in classical interpretations with non-standard valuation 24 Instead of stories S over I , Woodru speaks of conventions C over I , which he characterizes as consisting of (1) an equivalence relation  on the constants that do not refer under I and (2) an extension in D for each atomic formula whose terms are nonreferring constants and, for some k  0, the rst k variables. (2) treats atomic formulae as predicates (sentences as 0-place predicates) and is subject to the constraints that (a) a = x and x = a are true of nothing in D, (b) a = b is true if a  b, and (c) if A is obtained from B by replacing constants by equivalent constants, then A and B are true of the same tuples of individuals of D. Evidently, to each story S corresponds a convention C that gives us the same information about the truth values of atomic formulae with non-referring terms, assuming that D is non-empty and (x) 2 D, and conversely.

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rules. A classical interpretation I 0 = hD0 ; d0 i over a partial interpretation I = hD; di satis es the following conditions: ic1: D0 is non-empty and D  D0 : ic2: If d is de ned at a, then d0 (a) = d(a):

ic3: If d(f ) is de ned at h1 ; : : : ; k i, then d0 (f )(1 ; : : : ; k ) = d(f )(1 ; : : : ; k ):25 ic4: d(P )  d0 (P ):

Partial assignments  are completed by stipulating that 0 (x) = (x) if  is de ned at x. Note that, save for ic 1; these conditions are weaker than those on outer domain completions. If supertruth is reckoned in terms of classical completions, then P t _:P t, s = t ! (P s ! P t), and t = t will be logically supertrue, but so will 9x(x = a) and P t ! 9xP x | an unwelcome result in free logic. Bencivenga's technical solution to this problem is essentially to modify the valuation rules v1, v2, and v5 for classical interpretations over partial interpretations.26 The notion of an x-variant in v5 must be understood as in outer domain semantics: x must be assigned something in D. v1 and v2 become: vb 1: If each ti refers under I and , then P t1 : : : tk is true under I 0 and 0 if h1 ; : : : ; k i 2 d(P ) and P t1 : : : tk is false under I 0 and 0 if h1 ; : : : ; k i 2= d(P ), where the referent of ti under I and  is i ; otherwise, P t1 : : : tk is true under I 0 and 0 if h 1 ; : : : ; k i 2 d0 (P ) and P t1 : : : tk is false under I 0 and 0 if h 1 ; : : : ; k i 2= d0 (P ), where i is the referent of ti under I 0 and 0 : vb 2: If just one of s and t refers under I and , then s = t is false under I 0 and 0 ; otherwise, s = t is true under I 0 and 0 if s and t refer under I 0 and 0 to the same individual, and s = t is false under I 0 and 0 if s and t refer under I 0 and 0 to dierent individuals.27 25 Bencivenga's formal language does not contain function-names, but presumably they would be handled in this way. 26 The non-standard valuation rules capture valuation under I 0 \from the point of view of" I , as Bencivenga [1986, p. 409] and [this volume, p. 181], puts it. For his own somewhat dierent presentation of the rules, see [Bencivenga, 1980, pp. 101{103]. 27 The long-winded form of these rules permits their use in Bencivenga's [1980b] free description theory, where such atomic sentences as P (x(x 6= x)) lack truth-value under I 0 . See Section 4.4 below. vb 2; like s2 or ie 2; implies that s = t is superfalse if s refers and t does not. On Frege's view, re ected in vf 2; s = t should lack truth-value in this case. For a variant of supervaluational semantics that aims to honor Frege's position, see [Skyrms, 1968]. Skyrms account is not quite correct | subscripts on `G' must be reversed in (ii), lest s = t be supervalueless when s and t refer to dierent individuals (or, where s is not t, to the same individual) | and it has the strange consequence that 9x(x = a) is supervalueless, not superfalse, when a does not refer.

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These adjustments preserve the logical supertruth of P t _ :P t, t = t, and s = t ! (P s ! P t). However, if a does not refer under I , 9x(x = a) is superfalse under I and . (Let I 0 and 0 classically complete I and . If D is empty, there are no x variants of 0 , so 9x(x = a) is false under I 0 and 0 . If D is non-empty and 0 is any x-variant of 0 , then x = a is false under I 0 and 0 by vb 2; so 9x(x = a) is false under I 0 and 0 :) Moreover, P t ! 9xP x is not logically supertrue. (If t does not refer under I = hD; di and , but d(P ) is empty, then P t ! 9xP x is supervalueless under I and . For 9xP x is false under each classical completion I 0 = hD0 ; d0 i of I , whereas d0 (P ) may be de ned so as to include or to exclude the referent of t under I 0 and 0 :) Supervaluations do turn out desired results, subject to the limitations revealed by Woodru [1984]. Though certain sentences (such as Pa and t = s) may be supervalueless, the classical laws that free logicians like (such as t = t, P t _ :P t, and (Pt & 9x(x = t)) ! 9xP x) are logically supertrue, while those they dislike (such as P t ! 9xP x) are not. However, anyone who regards logical properties and relations as fundamentally semantic will regard such a justi cation of laws as circular. Bencivenga's appeal to classical completions with non-standard valuation rules is designed to provide a semantic rationale for supervaluations, which otherwise appear to be merely a \technical instrument".28 Bencivenga's case is as follows: (1) Where terms refer (as in `Caesar wore a white tunic when he crossed the Rubicon'), truth or falsity may be identi ed with the outcome of an ideal practical experiment that compares what the sentence says with the way the world is. (2) In most cases where terms do not refer (as in `Pegasus has a white hind leg'), such practical experiments are out of the question; but we may nonetheless identify truth or falsity with the outcome of mental experiments (represented by classical completions of partial interpretations) that assign such terms non-existent referents. (3) However, no mental experiment can override the facts, in the sense of altering the outcome of an ideal practical experiment (e:g:, that `Pegasus exists' is false); hence, the non-standard valuation rules vb 1 and vb 2. (4) Where all mental experiments (so constrained by the facts) agree on a truth-value for a sentence (as with `Pegasus is Pegasus'), it is reasonable to assign it that value; where they disagree (as with `Pegasus has a white hind leg'), it is reasonable to regard it as truth-valueless.29 Together, these conditions give us what Bencivenga [1986, p. 406], and [this volume, p. 179], calls the counterfactual theory of truth: \a sentence containing nondenoting singular terms is true (false) if and only if it would be true (false) in case these terms were denoting, no matter what their denotations were but provided that they were non-existent objects". 28 The phrase 29 Bencivenga

is Bencivenga's [1986, p. 405], and [this volume, p. 178]. [1980a, p. 225].

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Unfortunately, the counterfactual theory of truth seems merely to restate the diÆculty. Why should truth, which is ordinarily regarded as correspondence to fact, be reckoned in terms of what is contrary to fact? Why should we reckon that `Pegasus is Pegasus' is true because it would be true if, contrary to fact, `Pegasus' did refer? We do sometimes decide what is the case by considering what would be the case if things were dierent, as when we apply the semantic de nition of a valid argument. But usually this is not a good idea. The fact that Milosevic would agree to autonomy for Kosovo if he were reasonable does not, unfortunately, tell us that he will do so. Why is truth more like validity than Balkan politics? If partial interpretations merely re ected incomplete information about referents, lack of truth-value would represent ignorance of truth-value and supervaluations would make sense. I don't know whether `The rst person born in China in 1999 was a boy' is true, but clearly `The rst person born in China in 1999 is the rst person born in China in 1999' is true no matter who this person turns out to be. But lack of information about the referent of `Odysseus' is not what leads Frege to deny a truth-value to `Odysseus was set ashore at Ithaca while sound asleep'. I think we know everything there is to know about the referent of `Odysseus': there is no such thing. If supervaluations make sense in free logic, I believe we do not yet know why. Before leaving supervaluational semantics, let us note a connection with Kripke-style modal semantics established by Barba [1989]. The introduction to Barba's paper suggests that we will be shown how to translate sentences A of an ordinary rst-order language L with identity into sentences tr(A) of the corresponding modal language L and how to associate with a partial interpretation I of L a modal interpretation I 0 of L so that A is supertrue (superfalse) under I i tr(A) is true (false) under I 0 . But no such scheme is possible, since standard modal semantics is bivalent and supervaluational semantics is not. Instead, Barba shows how to associate with a partial interpretation I of L a class KI of modal interpretations so that A is supertrue under I i 3A is true under each interpretation in KI . Modal interpretations here are non-standard in some respects. For example, they are partial: a need not refer at world w | but if it refers at w to ,  exists in w and in every world w0 accessible from w, and a refers at w0 to . However, Barba's [1989, p. 134] valuation rules VL are bivalent: if a does not refer at w, Pa is true (!) at w:

3.6 Non-bivalent Semantics: Neutral Free Semantics Supervaluations are the last stop before neutral free semantics, where even t = t will lack truth-value if t does not refer, and lack of truth-value at the atomic level is inherited by at least some compounds, among them such classical logical truths as P t _ :P t and s = t ! (P s ! P t), when neither s nor t refers. This may not appear to be a very promising destination for the

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logician: if logical truths are conceived as those sentences which are always true, there are not going to be many | or perhaps any | logical truths. However, the situation may not be as bad as it appears to be, for two reasons. First, quanti cation may restore truth-values: if 9x(x = y) is to express `y exists', then 9x(x = a) should be false when a does not refer under I and , although x = a will lack truth-value under I and any x variant of . This can be achieved by understanding 9xA to be true under I and  if A is true under I and some x variant of  and to be false otherwise. Second, a weaker notion of logical truth | A is logically true i A is never false | may serve as well in many applications. t = t, P t _ :P t, and s = t ! (P s ! P t) are logically true in this weaker sense, as are such laws of free logic as (A(t) & 9x(x = t)) ! 9xA(x), where x does not occur in t: The underlying semantic rationale for neutral free semantics is Frege's functional view of reference: predicates and `=' name functions from individuals to truth-values. If functions are operations, as Frege seems to have thought, then the semantic rules governing subject-predicate and identity constructions are vf 1 and vf 2; for where there is no input to an operation, there is no output either. The truth-functional connectives name truthfunctions, so the same line of thought dictates the weak tables for them.30 v3 and v4 become: vf 3: :A is true if A is false; :A is false if A is true; :A lacks truth-value if A lacks truth-value. vf 4: A ! B is false if A is true and B is false; A ! B is true if A is true and B is true, or A is false and B is true, or A is false and B is false; A ! B lacks truth-value if either A or B lacks truth-value.

In classical semantics, 9 and 8 may be regarded as naming functions from `propositional functions' to truth-values. Under I and , A(x) names the 1-ary propositional function A: D ! fT; F g whose value at 0 (x), where 0 is an x-variant of , is the truth-value of A(x) under I and 0 . Then 9(A) = T if A() = T for some  2 D and 9(A) = F otherwise, while 8(A) = T if A() = T for each  2 D and 8(A) = F otherwise. If these clauses are carried over to the present case, where A may be a partial function D ! fT; F g, we have: vf 5: 9xA is true if A is true for some x-variant of ; otherwise, 9xA is false; 8xA is true if A is true for each x-variant of ; otherwise, 8xA is false. Note that 9xA and :8x:A are no longer equivalent in the sense of being true, false, or truth-valueless together. If a does not refer, 9xP xa is false 30 Woodru [1970, p. 128] argues that the strong tables | which dictate replacing the second clause of vf 4 by `A ! B is true if A is false or B is true' and the third by `A ! B lacks truth-value otherwise' | are required by Frege's view that reference is a function of sense. For skepticism, see [Lehmann, 1994, p. 326].

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(because P xa lacks truth-value for any assignment to x) but :8x:P xa is true (8x:P xa is false, since :P xa is always truth-valueless). The valuation rules vf 1{vf 5 are those of Lehmann [1994], except that he de nes 8xA as :9x:A, so that the universal clause of vf 5 becomes `8xA is false if A is false for some x-variant of ; otherwise 8xA is true' and hence 8xP xa is true (!) when a does not refer. Smiley's [1960, p. 126] rules dier only in taking 9xA(8xA) to lack truth-value if A lacks truth-value for some assignments to x but is otherwise false (true). Thus, 9x(a = fx) will lack truth-value under hD; di if a refers, d(f ) is partial, but d(a) 6= d(f )() for any  2 D at which d(f ) is de ned. I have noted that logical truth may be understood in a strong or a weak sense. Similarly, logical consequence may be de ned in a number of ways, depending upon whether we want valid inference (1) to lack counterexamples, (2a) to preserve truth, or (2b) to preserve non-falsehood: X j=1 A i there are no I and  such that: each X formula is true while A is false. X j=2a A i there are no I and  such that: each X formula is true while A is not true. X j=2b A i there are no I and  such that: A is false while no X formula is false.

j=1 supports contraposition (:B j= :A provided A j= B ) but not transitivity (X j= A provided X 0 j= A and X j= B for each B 2 X 0) : :9x(x = a) j=1 a = a and a = a j=1 9x(x = a), but :9x(x = a) 6j=1 9x(x = a). Both j=2a and j=2b support transitivity, but neither supports contraposition: P a j=2a 9x(x = a) but :9x(x = a) 6j=2a :P a, while :9x(x = a) j=2b :P a but P a 6j=2b 9x(x = a). Both transitivity and contraposition can be had by combining (2a) and (2b), as in Blamey [1986, pp. 5 and 58], to require that valid inference preserve (3) both truth and non-falsity. That is, X j=3 A i X j=2a A and X j=2b A. j=3 is obviously stronger than j=2a or j=2b , each of which is stronger than j=1 . Note that A is strongly logically true i j=2a A and weakly logically true i j=1 A (or j=2b A): Each of these consequence relations can be expressed in terms of a notion of satis ability, which in turn can be represented syntactically by a variant of Jerey's [1991] tree method. Add a marker  to L, and call A a -formula. Let Y range over sets of formulae and -formulae. Y is -satis able i there is some I and  for which each -formula of Y is true and no formula of Y is false. The basic free consequence relations de ned above may be expressed in terms of -satis ability as: X 1 A i X  [ f:A g is not -satis able X 2a A i X  [ f:Ag is not -satis able X 2b A i X [ f:A g is not -satis able

As in the classical case, a tree for nite Y is obtained by rst listing the members of Y vertically and then extending this list downward in a branching array by application of reductive rules.31 Here, each classical 31 The tree method may be modi ed to accommodate in nite sets of formulae. For a sketch of the argument applied to the free case, see [Lehmann, 1994, Section 4].

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rule for a connective is replaced by two rules: one governing formulae, the other governing -formulae. ::A ::A

A

A

(A ! B )

(A ! B )

:A

B

A B

A :A

B

B :B

A :B 

A(s) s = t

A(s) s = t

A(t)

A(t)

if A is elementary Quanti er rules apply only to -formulae:

8xA(x)

:A :B 

:(A ! B )

:(A ! B ) A :B

:A

:9xA(x)

:A(t) A(t) if t occurs in an elementary -formula above A(t) or :A(t) :8xA(x) y = y :A(y)

9xA(x) y = y A(y) if y does not occur free above y = y

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In addition, we need a rule for converting formulae into -formulae:

A A if A is (i) a quanti ed formula or the negation thereof; or (ii) an elementary formula, each term of which occurs in an elementary -formula32 above A*. As the rule for :(A ! B ) will suggest, branches are not closed when they contain A and :A, as they are in the classical case, because both may lack truth-value. Instead, at least one of A and :A must be a -formula. Similarly, t 6= t , not the classical t 6= t, closes a branch. A tree is closed if each of its branches is closed. Let Y ` i some tree for some nite subset of Y is closed. It can be proved that Y ` i Y is not -satis able; for details, albeit for a slightly dierent system of rules, see [Lehmann, 1994]. A Hilbert-style axiomatization of j=2a or j=2b would probably require introducing a non-Fregean connective t, as in [Smiley, 1960], [Woodru, 1970], or [Robinson, 1974]: tA is true if A is true and is false otherwise.33 Many classical tautologies are only weakly logically true, rules like ADD do not preserve truth, while rules like MP do not preserve non-falsehood. A Hilbert-style axiomatization of j=1 seems beyond reach, since j=1 is not transitive. 4 FREE DESCRIPTION THEORIES The requirement that singular terms denote something in the range of the variables constrains classical description theory, just as it constrains classical logic. In natural languages there are many singular terms that may be regarded as descriptions having the form `the (one and only) x such that : : : x : : :', where `: : : x : : :' is some condition on x : if `: : : x : : :' is true only of , then `the (one and only) x such that : : : x : : :' refers to . For example, `the least prime' refers to 2 because `x is prime and no prime is less than x' is true only of 2. Indeed, one might want to maintain, as does Quine [1997, p. 103], that the \universal form of singular terms" is `the (one and only) x such that : : : x : : :' in the sense that any constant or variable singular term may be regarded as having this form. Names, such as `Socrates', can be handled by introducing singular predicates, such as `x Socratizes'. Variable terms, 32 Assuming v 1. If subject-predicate formulae with referring terms can lack truthf value, replace `elementary -formula' here with `identity -formula or negated identity -formula'. 33 If A is identi ed with tA, then X is -satis able i some I and  fails to falsify any member of X . Smiley [1960] does not give any proof method, and Woodru's [1970] natural deduction system is unsound.

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such as `the least prime greater than y', may be construed as descriptions whose referents will generally depend upon the values of their variables. We may represent descriptions by adding to a rst-order language L the description operator  with formation rule: if A is a formula, xA is a term. Let L be the resulting language. The corresponding semantic rule will specify that under I and ; r4. xA refers to whatever uniquely satis es A where  uniquely satis es A i (i) A is true under I and 0 , where 0 is the x-variant of  for which 0 (x) = , and (ii) A is false under I and any other x-variant of . This understanding we might hope to capture prooftheoretically by adding to CL a schema sometimes termed `Lambert's Law': LL 8y(y = xA $ 8x(A $ y = x))) where y is not free in A (and a free occurrence of y in A is now any occurrence not in a part of A of the form 8yB or yB ):34 If A has no free variables other than x, then whatever xA designates under I and  will be an individual of D that is independent of , so we may treat xA as a constant. If in addition y is free, then whatever xA designates will be an individual of D that depends upon the value (y) of y, and we may regard xA as giving the value of some function at y. Thus, having descriptions xA available would permit de ning constants c by c = xA and function-names f by 8y(fy = xA): However, if xA is to be a singular term, classical semantics demands that it refer to something in the range of the variables. There is no problem if 9!xA is true. For then some individual  of D will uniquely satisfy A, and xA will refer to it by r4; such descriptions are said to be proper. But r4 tells us nothing about the referent of an improper description xA. If 9!xA is false, so that nothing uniquely satis es A, we must nonetheless specify a referent in D for xA. Moreover, LL is false if 9!xA is false, since 8y(y = xA $ 8x(A $ y = x))) j= 9!xA : both 8x(A ! x = xA) and A(xA) follow classically from LL, so 9!xA follows as well. Indeed, some instances of LL, as when A is x 6= x or P x&:P x, are logically false. The corresponding instances of A(xA) | x(x 6= x) 6= x(x 6= x) or P (x(P x&:P x))&:P (x(P x&:P x)) | are sometimes called Meinong's paradox, after Russell's derivation of them from Meinong's principle that `: : : x : : :' is true of the x such that : : : x : : :, a principle expressed by A(xA(x)): Lambert [1991b; 1995] shows that Russell's paradox may also be derived from LL and that ways of evading it in set theory parallel ways of evading Meinong's paradox in description theory. 34 Lambert [1991b; 1995] labels this system `NTDD' (for `naive theory of de nite descriptions').

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4.1 Classical Fixes Accordingly, if we wish to take descriptive terms seriously within the classical framework, we must either modify the formation rule to exclude improper descriptions, or we must specify a referent in D for improper descriptions and modify LL. The rst approach is associated with Hilbert and Bernays. Recall that we may extend a rst-order theory T to T 0 by adding a new constant c with the de ning axiom 8x(x = c $ Ac (x)), provided x is the only free variable of Ac (x) and 9!xAc (x) is a theorem of T . Similarly, we may introduce a new function name f with the de ning axiom 8x8y(y = fx $ Af (x; y)), provided x and y are the only free variables of Af and 8x9!yAf (x; y) is a theorem of T . Each addition is really only a notational change. We may eliminate c from a formula B 0 by replacing atomic parts Pc by 9x(Ac (x)&P x) and the resulting formula B will be a theorem of T i B 0 is a theorem of T 0. Similarly, we may eliminate f from a formula B 0 by replacing atomic parts Pft by 9y(Af (t; y)& Py), where y does not occur in t | and the resulting formula B will be a theorem of T i B 0 is a theorem of T 0: Since we may think of c as xAc (x) and fx as yAf (x; y), the conditions on de ning c and f give the Hilbert-Bernays conditions for considering descriptive terms well-formed: 9!xAc (x) and 8x9!yAf (x; y). Essentially, this amounts to saying that a descriptive term xA is well-formed only under (consistent) assumptions X | the axioms of T | that entail 9!xA, assumptions which accordingly function as additional premises in any argument involving xA: The Hilbert-Bernays approach has the awkward consequence of making the question of whether xA is a term undecidable, since logical consequence is not decidable. Normally, of course, the syntactical categories of term, formula, and sentence are decidable, provided the basic categories of constant, variable, (k-place) function-name, and (k-place) predicate are decidable. But that is not the case here. Instead of limiting attention to proper descriptions, we can instead follow Frege and stipulate a referent in D for the improper descriptions, modifying LL accordingly. To L we add a constant e for the designated (`empty') element. An interpretation of L is just an interpretation hD; di of L, which accordingly will assign e a referent d(e) in D. The reference rule for xA will now read: rf 4: If some individual  of D uniquely satis es A, then xA refers to ; otherwise, xA refers to d(e): Thus, all instances of the schema F B (xA) $ (9y(8x(A $ y = x) & B (y)) _ (:9y8x(A $ y = x) & B (e)))

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are logically true. To axiomatize logical truth, we may add LLf

8y(y = xA $ ((9!xA & A(y)) _ (:9!xA & y = e)))

in place of LL to CL. Assigning improper descriptions an arbitrary referent is like arbitrarily completing a partial function, and here as there we must be prepared for some weird results: if some other constant c refers to d(e), then c = x(x 6= x), will be true, though it will represent sentences like `Zero is the non-selfidentical number'. Moreover, 9y(y = x(x 6= x)) and x(x = x) = x(x 6= x) are logically true. An alternative to treating descriptions xA as genuine terms is to follow Russell and to regard formulae B (xA) in which they appear as giving only the surface form of corresponding sentences of natural language, their logical form being obtained by a transformation of B (xA) that eliminates descriptions. A sentence like `The present King of France is bald' looks like a subject-predicate sentence with form B (xKx), where Kx represents `x is King of France at present', but xKx is not a genuine singular term, according to Russell. Why? Because (1) genuine singular terms are meaningful, (2) the meaning of a meaningful singular term is its denotation, and (3) `the present King of France' has no denotation. In Russell's view, the logical form of `The present King of France is bald' is not subject-predicate, but existential: 9y(8x(Kx $ y = x) & Bx), which says that one and only one thing is King of France at present, and that thing is bald. For Russell, we have R

P (xA) $ 9y(8x(A $ y = x) & Py)

for predicates P , but not in general. `The present King of France is not bald' looks like a negated subjectpredicate sentence with form :B (xKx); what is its logical form? If we think of xKx as occurring in B (xKx), the form of :B (xKx) will be :9y(8x(Kx $ y = x) & Bx); if we think of xKx as occurring in :B (xKx), the form of :B (xKx) will be 9y(8x(Kx $ y = x) & :Bx). In the former narrow-scope reading, the sentence is true: it is being read as `it's not the case that: the present King of France is bald'. In the latter wide-scope reading, the sentence is false: it is being read as `the present King of France is non-bald'. If A is description-free, LL is logically true if xA is taken to have narrowest scope:

8y(9z (8x(A $ z = x) & y = z ) $ 8x(A $ y = x)) But LL is false if xA is improper and is taken to have wider scope:

8y9z (8x(A $ z = x) & (y = z $ 8x(A $ y = x))) 9z (8x(A $ z = x) & 8y(y = z $ 8x(A $ y = x)))

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Accordingly, the transformation of a formula B containing descriptions into a  free formula tr(B ) requires an indication of the scope of the descriptions in B . This may be done by using the notation [xA]B to indicate that the scope of xA in B is B . The wide scope reading of xKx in :B (xKx) would then be indicated by [xKx]:B (xKx) and the narrow scope reading by :[xKx]B (xKx). More formally, we may add to the formation rules of L two clauses: 1. If B is a formula in which xA appears, [xA]B is a formula 2. If B is a formula in which (a) each description xA that occurs in B occurs in a subformula [xA]C of B and (b) any subformula [xA]C of B is such that some occurrence of xA in C is not in a subformula [xA]C 0 of C , then B is a [] formula. For the purposes of (2), subformulae of B include formulae that appear in descriptions in B . Scope indicators [xA] are like quanti ers, and (2) is analogous to a clause de ning sentences as formulae lacking free occurrences of variables. (a) rules out unscoped occurrences of descriptions, and (b) rules out vacuous scope indicators. Let this language be L[]. tr then maps []-formulae of L[] into  free formulae of L : t1. tr(A) = A if A is not a [] formula. t2. tr commutes with connectives and quanti ers. t3. tr([xA]B (xA)) = 9y(8x(tr(A) $ y = x) & tr(B (y))): Despite its enormous in uence, Russell's treatment of descriptive terms is ill-motivated and cumbersome. The denotative theory of meaning which led Russell to banish improper descriptions from the realm of terms has little plausibility, especially in view of the many failed attempts to capture intension in extension (e:g:, the meaning of a sentence is the proposition it expresses, and that is a set of possible worlds). Although scope indicators may be useful to disambiguate constructions like `The present King of France isn't bald', they introduce a complication that in many cases is of no use. For example, `Winston Churchill or the present King of France is bald' appears to be unambiguous in English, though Russell's formalism provides two readings: Bc _ [xKx]B (xKx), which is true, and [xKx](Bc _ B (xKx)), which is false. There are no scope indicators in English, and assigning Russellian forms is an ad hoc business. The validity of `The janitor is guilty, so the janitor or the accused is guilty' requires a narrow-scope construal of `the accused'; the validity of `The accused is not guilty, those who are not guilty are innocent, so the accused is innocent' requires a wide-scope construal of `the accused' in the premise.

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A treatment of descriptions that enables us to handle them as we would other terms, while avoiding the problems of the other two approaches is obviously worth pursuing. Insofar as these problems arise from the straightjacket of classical semantics, we may hope to avoid them either by permitting improper descriptions to refer to individuals of an outer domain or not to refer at all.

4.2 Outer Domain Free Description Theory An outer domain interpretation for descriptions may be obtained from an outer domain interpretation hDi + Do ; di by adding a denotation function d0 for descriptions: io 5: d0 (xA) 2 Do :

d0 (xA) will be the referent of xA, if xA turns out to be improper. Thus, r4 is rewritten: ro4: If some individual  of Di uniquely satis es A, then xA refers to ; otherwise, xA refers to d0 (xA): Relative to this semantics, LL is logically true. If LL is added to PFL, we obtain a complete axiomatization of this semantics. This free description theory is rather weak; let us call it mFD (`m' for `minimal').35 mFD permits many individuals in Do and places no restrictions on their assignment as referents to improper descriptions. The limiting case appears to be the one in which Do consists of a single individual, which accordingly must be the referent of any improper description. If interpretations of L require as much, then FD2

(:9x(x = s)&:9x(x = t)) ! s = t

is logically true. If s and t do not refer to existents, i.e., to individuals in the inner domain, then they must refer to individuals of the outer domain, but there is just one of these. This semantics may be axiomatized by adding FD2 to mFD. Alternatively, we could add :9y(y = xA) ! xA = x(x 6= x), where y is not free in xA, as in [Scott, 1967, p. 35]. Let us term this theory MFD (`M' for `maximal'). MFD has obvious aÆnites with the Fregean treatment of improper descriptions in classical semantics. The classical principle F holds with x(x = 6 x) in place of e, and we must identify the present King of France with the unicorn in the closet. However, 9y(y = x(x 6= x)) and x(x = x) = x(x 6= x)) are no longer logically true. 35 This theory is called FD by Lambert and van Fraassen [1972, p. 160]] and MFD by Lambert [1997, p. 118].

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Lambert [1997, p. 118] notes that between mFD and MFD lie various free description theories, partially ordered by inclusion. One linear progression is: T1

mFD + x(x = t) = t

T2

mFD + (A(t) & 8x(A ! x = t)) ! A(xA)

T3

mFD + (:9x(x = t) & A(t)) ! A(xA)

These theories tell us when A(xA) is true. LL j= 9y(y = xA) ! A(xA), if y is not free in xA, so 9y(y = xA) ! A(xA) is a theorem of mFD and thus of T1 , T2 , T3 , and MFD: A(xA) holds when xA refers to an existent. If A is false of every individual, as when A is x 6= x or P x&:P x, then of course A is not true of xA. Otherwise, mFD is non-commital about whether A(xA) is true, and theories T1 , T2 , and T3 give us more information. T1 : If A(x) is x = t and xA refers to a non-existent, so does t. x(x = t) = t identi es these nonexistents and thus makes A(xA) true when A is t = x. Hence, the one and only thing that is Vulcan is Vulcan. T2 : If t refers to a non-existent and A is true of it, then 8x(A ! x = t) will hold only if A is not true of any existent. Thus (A(t) & 8x(A ! x = t)) ! A(xA) tells us that A(xA) provided A is true of some non-existent but not true of any existent. This will be the case if A is t = x and t does not refer, so T2 contains T1 . But it also makes the unicorn in the closet a unicorn in the closet, since Ux will not be true of any existent. T3 : (:9x(x = t) & A(t)) ! A(xA) says that A is true of xA provided (i) A is true of some non-existent. LL assures that A is true of xA provided A is uniquely true of some existent, so the extra content of T3 over mFD is that A is true of xA provided (i) and either (ii) A is not true of any existent or (iii) A is true of more than one existent. So the extra content over T2 is that A is true of xA provided (i) and (iii). Hence, the lost treasure is a lost treasure, since there are mythical lost treasures and more than one real one. Models of MFD have just one non-existent, so if xA refers to a non-existent, A(xA) will be true provided A is true of some non-existent. However, MFD is stronger than T3 , since the latter permits more than one non-existent.

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Another linear progression is: T01

mFD + 8cx(A(x) $ B (x)) ! xA = xB

T02

mFD + 8x(A(x) $ B (x)) ! xA = xB

8c is the comprehensive universal quanti er with the classical valuation rule v5, where D is Di + Do and x-variants may assign values in D0 . 8cx(A(x) $ B (x)) ! xA = xB says that xA and xB are co-referential provided A

and B are co-extensive in the strong sense of being true of the same existents and non-existents. Thus, the present King of France needn't be the unicorn in the closet. 8x(A(x) $ B (x)) ! xA = xB says that xA and xB are co-referential provided A and B are co-extensive in the weaker sense of being true of the same existents. Thus, the present King of France is the unicorn in the closet, though the lost treasure needn't be identi ed with either: T20 , like T2 , is weaker than MFD. Note that 8x(A(x) $ B (x)) ! 8y(y = xA $ y = xB ), and hence 8cx(A(x) $ B (x)) ! 8y(y = xA $ y = xB ), is a theorem of mFD. Outer domain semantics for descriptions permits a formal representation of Anselm's ontological argument. Here individuals that exist in re belong to the inner domain, while individuals that exist in intellectu populate the outer domain. If Gx represents `nothing greater than x can be conceived', then a simple version of the argument is:

:9y(y = xGx) ! :G(xGx) G(xGx) 9y(y = xGx)

This argument is valid by modus tollens. In mFD, the premises are falsi able, and both are required for validity. By contrast, as Mann [1967] has observed, a Russellian treatment of the descriptions collapses the argument into something trivial and question-begging. The antecedent of the rst premise is equivalent to 9x(x 6= x) if its form is taken to be [xGx]:9y(y = xGx) and to :9!xGx if its form is taken to be :[xGx]9y(y = xGx) or :9y[xGx](y = xGx); the consequent is equivalent to 9x(Gx&:Gx) if its form is taken to be [xGx]:G(xGx) and to :9!xGx if its form is taken to be :[xGx]G(xGx). Hence the rst premise is either logically true or logically false. The second premise is equivalent to 9!xGx, as is the conclusion, read either as [xGx]9y(y = xGx) or as 9y[xGx](y = xGx). So the argument becomes: A, 9!xGx =9!xGx, where A is logically true or logically false.

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4.3 Russellian Free Description Theory To obtain a `Russellian' free description theory for L, we need only supplement negative free semantics with rp4. If some individual  of D uniquely satis es A, then xA refers to ; otherwise, xA does not refer. While this treatment of descriptions diers from Russell's in taking xA to be a genuine term, it does make P (xA) false if xA is improper, just as Russell insisted: the Russellian biconditionals R are logically true. Every instance of LL is also logically true, and indeed logical truth relative to this semantics may be axiomatized by adding LL to NFL, as in [Burge, 1974]. Let us call this theory rFD. Principles de ning extensions of mFD generally do not carry over to rFD. Both (A(t) & 8x(A ! x = t)) ! A(xA) and (:9x(x = t) & A(t)) ! A(xA) are OK: if xA refers, then A(xA); if xA does not refer, then A(t) and A(xA) will have the same truth-value if t does not refer, and t cannot refer if :9x(x = t) is true or (A(t) & 8x(A ! x = t)) is true while xA does not refer. The other principles are not OK. The \cancellation" principle x(x = t) = t is false when t does not refer. FD2 is not always true; indeed, some instances, such as :9y(y = x(x 6= x)) ! x(x 6= x) = x(x 6= x), are logically false, since its antecedent is logically true and its consequent is logically false. 8x(A(x) $ B (x)) ! 8y(y = xA $ y = xB ), which says that if A and B are co-extensive, xA and xB do not dier in denotation, is always true. But 8x(A(x) $ B (x)) ! xA = xB is not always true; indeed, 8x(x 6= x $ x 6= x) ! x(x 6= x) = x(x 6= x) is logically false. rFD does not capture Russell's scope distinctions. We have schema R [xA]P (xA) $ 9y(8x(A $ y = x) & P y) but not, for example, [xA]:P (xA) $ 9y(8x(A $ y = x) & :P y) In a sense, only descriptions with narrowest scope are treated as genuine singular terms. By introducing machinery for forming complex predicates xB from formulae B , Scales [1969] is able to represent scoped descriptions as genuine singular terms satisfying the more general schema S

yB (y)(xA) $ 9y(8x(A $ y = x)&B (y))

Obtain L from L by adding the predicate-forming operator  with formation rule: x1 : : : xk A is a k-place (complex) predicate if the free variables of A are x1 ; : : : ; xk . \Russellian" interpretations of L are obtained from those of L by stipulating that the extension d(x1 : : : xk A) of the complex

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predicate x1 : : : xk A is the set of k-tuples h1 ; : : : ; k i of elements of D of which A is true, where A is true of h1 ; : : : ; k i i A is true when xi is assigned i . Note that if t does not refer, x:P x(t) is false while :P t is true. L thus embodies Aristotle's view that truly attributing a property requires an existing subject, though truly denying an attribution does not; `Pegasus is wingless' is false, while `Pegasus does not have wings' is true. Schema S holds because (1) if xA refers to , A is true only of , and  is in the extension of yB (y) i B (y) is true of , and (2) if xA does not refer, yB (y)(xA) is false and, because 9!xA is false, so is the right side of S. Logical truth in L may be axiomatized by adding LL and

x1 : : : xk A(t1 ; : : : ; tk ) $ (9x1 (x1 = t1 )& . . . & 9xk (xk = tk )&A(t1 ; : : : ; tk ))

to NFL; see Scales [1969, p. 11] and Lambert [1997, p. 112]. Recall that for Russell, [] formulae A of L[] are abbreviations for -free formulae tr(A) of L: tr(A) gives the Russellian meaning of A. We may also translate [] formulae A into formulae of L by tr0 , where tr0 is de ned by t1, t2 and t30 . tr0 ([xA]B (xA)) = y tr0 (B (y))(x tr0 (A)):

Under any interpretation I of L and assignment , tr0 (A) gets the same value as tr(A): in the basis case where both A and B are  free, tr(A) is the left side of schema S , while tr0 (A) is the right side. Accordingly, scoped descriptions can be regarded as genuine singular terms without altering Russellian truth-values, provided their contexts are treated as complex predicates. Kroon [1991, p. 24] has observed that \the Russellianizing of de nite descriptions is a clumsy and unnatural business | far more clumsy and unnatural than its defenders seem to realize". The problematic constructions Kroon has in mind are those in which we refer to something by describing a description, as in `The man denoted by the description John just used is bald.' To represent constructions of the general form `what's denoted by the description that s has P ' a la Russell, we would need a description predicate DES true of descriptions `x(x)', a correspondingopen-sentence predicate COS true of pairs h`x(x)'; `(x)'i, and a satisfaction predicate SAT true of pairs h; `(x)'i i  satis es `(x)'. Then the ugly Russellian analysis would be: 9x(9y(8z ((DES (z ) & (x)) $ z = y) & COS (y; x)) & 9w(8z (SAT (z; x) $ z = w) & P (w))). How much simpler it would be if we could write P (den(`x(DES (x)&(x))'), where den represents the denotation function. Kroon develops a modi ed free Russellian semantics for such constructions. Imagine that L has been supplemented with vocabulary that permits naming its terms and formulae (so that DES may be de ned) and let L0

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result from L by adding semantic predicates TRUE and DEN (so that den may be de ned by den(x) = y $ DEN (x; y) ). Interpretations I of L are as in negative free semantics, except that Kroon assumes that every individual of D is named by some constant of L so that reference and valuation rules need be given only for constant terms and sentences. d(TRUE) and d(DEN ) are de ned by a xed point construction over I . The extensions of I used in this construction assign semantic predicates P both a set dt (P ) of which they are true and a set df (P ) of which they are false; the initial extension I0 of I makes TRUE false of every non-sentence and true of nothing and DEN false of every pair h; i where  is not a constant term and true of nothing. If t refers to  and  is neither in dt (TRUE) nor in df (TRUE), then TRUE(t) lacks truth-value; if s refers to  and t refers to and h; i is neither in dt (DEN ) nor in df (DEN ), then DEN (s; t) lacks truth-value. rp4 is modi ed so that xA is unde ned if 9!xA lacks truth-value. As usual in negative free semantics, subject-predicate and identity sentences containing non-referring terms are false; however, if no constituent term fails to refer and at least one is unde ned, they lack truth-value. Strong tables are used for the connectives; 8xA lacks truth-value if A(c) is not false for any c but lacks truth-value for some c:

4.4 Non-bivalent Free Description Theories Finally, it is possible to give L a supervaluational or a neutral free semantics, so that non-referring descriptions xA generate truth-value gaps.

A slight obstacle to extending Bencivenga's [1980] supervaluational semantics to L is that some terms, such as x(x 6= x), are now not going to refer under any classical extension I 0 and 0 of a partial interpretation I and assignment . Thus, valuation rules v3-v5 need to be rewritten to allow for formulae that are neither true nor false under I 0 and 0 . Bencivenga [1980b, p. 396] speci es Kleene's strong tables for the connectives and stipulates that 8xA is truth-valueless if A is never false but lacks truth-value for some assignment to x : 8xA is true (false) under I 0 and 0 if A is true (false) under I 0 and 0x for each (some) x-variant 0x of ; otherwise, 8xA lacks truth-value under I 0 and 0 : The reference rule for descriptions will be If some individual uniquely satis es A under I and , then xA refers to it under I 0 and 0 ; if no individual uniquely satis es A under I and  but some individual uniquely satis es A under I 0 and 0 , then xA refers to it under I 0 and 0 ; otherwise xA does not refer under I 0 and 0 :

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Supervaluational semantics for L renders t = t and P t _ :P t logically true, but that is not the case for L . Since x(x 6= x) does not refer under any classical extension I 0 of I , neither x(x 6= x) = x(x 6= x) nor P x(x 6= x) _ :P x(x 6= x) is true or false under I 0 , so both are supervalueless under (any) I . xP x = xP x will also be supervalueless under I = hD; di if d(P ) is empty (as when Px represents `x is a unicorn in the closet'), since xP x will fail to refer under some extension I 0 of I . If you want `the unicorn in the closet is the unicorn in the closet' to be true, then you can follow Bencivenga [1980b, p. 398] and modify the notions of supertruth and superfalsity so that A is supertrue (superfalse) under I and  i A has a truth-value under some some classical extension I 0 and 0 of I and , and A is true (false) under each such extension. The semantic rationale for this manoeuvre, however, is unclear. All instances of LL are logically supertrue for this semantics. Under I 0 and 0 , xA refers, if at all, to something in D that uniquely satis es A, whereas if xA does not refer, both y = xA and 8x(A $ y = x) are false of each individual of D in virtue of vb 2. The principles generating positive free description theories stronger than mFD are, in general, not logically supertrue. Counterexamples to x(x = t) = t, (A(t) & 8x(A ! x = t)) ! A(xA), (:9x(x = t) & A(t)) ! A(xA), (:9x(x = t) & :9x(x = s)) ! s = t, and 8x(A $ B ) ! (xA = xB ) are provided by t = s = x(x 6= x) and A = B = x 6= x. No axiomatization of logical supertruth is given, since Bencivenga establishes that no axiomatization is possible. Description theories that incorporate neutral free semantics have been developed by Stenlund [1973] and Robinson [1974]. Indeed, their systems probably deserve to be regarded as the rst complete neutral free logics. Both theories essentially identify referenceless terms with improper descriptions: interpretations of L are classical for Robinson and Stenlund.36 Thus, constants | and variables under assignment37 | refer via i1{i2 and r1{r2 to individuals of D, and function-names designate total functions on D by i3. Robinson's treatment of free semantics leaves a good deal to the imagination | it must be inferred from his proof system.38 Except as noted below, however, he appears to be committed to the same rules of reference 36 Stenlund [1973, p. 63] permits D to be empty. However, this does not appear to be consistent with his Theorem 6.2.1 [p.66], which states that t # is provable i t refers under each interpretation, and the fact that c # is an axiom [p.17]. 37 Both Stenlund and Robinson treat quanti cation substitutionally. 38 For example, Robinson's [1974, p. 498] rule viii allows us to derive 8xA ! A(t) provided we have derived formulae, written below as 8xA # and t #, to the eect that 8xA has a truth-value and t refers. This seems to require the understanding of the quanti ers given below, since if 8xA is false and t refers, we cannot be sure that A(t) | and therefore 8xA ! A(t) | is not truth-valueless unless A(x) has a truth-value for any assignment to x:

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and valuation that Stenlund [1973, p. 64] gives explicitly. The classical r3 is modi ed to: If ti refers to i , then ft1 : : : tk refers to d(f )(1 ; : : : k ); otherwise, ft1 : : : tk does not refer. Descriptions are governed by rp4, and subject-predicate formulae, identities, and negations by the Fregean valuation rules vf 1{vf 3. Robinson appears to endorse the weak reading of ! embodied in vf 4. Stenlund, however, amends it so that A ! B is true if A is false and B is truth-valueless; the weak table would render his system unsound, since 9!x(x 6= x) ! (x(x 6= x) = x(x 6= x)) is provable in it. Stenlund regards 8xA as truth-valueless if A is truth-valueless for some assignment to x, as when A is P y(f (y) = x) and d(f ) is not 1-1. Thus, the Fregean valuation rule vf 5 needs to be modi ed to:

8xA is true if A is true for each x

variant of , and 8xA is false if A is false for some x-variant of  and not truth-valueless for any x variant of ; otherwise, 8xA is truth-valueless.

A peculiar consequence of this understanding of the quanti ers is that sentences like :9x(x = y(y 6= y)) are not true, but truth-valueless. As one might expect of a neutral semantics, not all instances of LL are logically true in the sense of being true under every interpretation | 8y(y = x(x 6= x) $ 8x(A $ y = x)) is truth-valueless under any interpretation | though none are false under any interpretation. The same substitutions that generate supervalueless instances of the principles that extend mFD will generate truth-valueless instances of them here. Stenlund supplies a natural deduction system of rules for this semantics, Robinson a Hilbert-style system. Both employ notation for indicating that terms refer and formulae have truth-value.39 Let us use Beeson's [1985, p. 98] operator # for this purpose. De ne a #{formula as e #, where e is a term or formula, and extend the valuation rules to #{formulae by:

t # is true if t refers; otherwise, t # is false. A # is true if A is true or false; otherwise, A # is false. Both systems have axioms and rules of three kinds. Those of the rst kind, such as `c# t # ` ft # 9!xA ` xA # t # ` Pt # s # t #` s = t # A # ` :A # 39 Stenlund uses t 2 I for t # and A 2 F for A #; Robinson uses
e for e # :

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permit us to prove #{formulae. Rules of the second kind license moving from #{formulae to formulae, as by

t#

`t=t

More interesting examples are Robinson's

A1 #; : : : ; Ak #

` B (A1 ; : : : ; Ak );

provided B (A1 ; : : : ; Ak ) is a classical tautology,40 and Stenlund's rule of conditional proof:

X ` A ! B provided X ` A # and X; A ` B: More familiar rules of the third kind permit deriving formulae from formulae; they include MP and

9!xA ` A(xA): Both Stenlund and Robinson provide completeness proofs. Robinson sketches a proof that K ` A i K j=2a A, where A is a formula or #-formula, K is a set of  free sentences, and each non-logical symbol of A occurs somewhere in K . Stenlund claims only weak completeness (` A i j=2a A), though his proof may be generalizable to a result like Robinson's. 5 OTHER APPLICATIONS

5.1 Predication Again Recall the Russell-Meinong view that predication presupposes a subject in the strong sense that a subject-predicate form cannot be ascribed to a sentence unless the subject exists. Quine [1960, p. 96] is more liberal: \Predication joins a general term and a singular term to form a sentence that is true or false according as the general term is true or false of the object, if any, to which the singular term refers."41 This is really just a special case, since Quine counts any open sentence with purely referential occurrences of variables as a predicate: such open sentences are true or false of (tuples of) objects. The generalization to open sentences obliterates Aristotle's distinction between `Socrates is ill' and `Socrates is not well', but Quine's account of predication is like Aristotle's in allowing for irreferential terms. Strangely enough, as Lambert [1986, p. 277] observes, Quine's preferred logical idiom has no singular terms at all, except for variables, which do not challenge 40 This is presumably what Robinson [1974, p. 498] intends by rule vii; the paper is marred by an unusually large number of printing errors and omissions. 41 See also [Quine, 1953, p. 163], where Quine argues that \the notion that `F a' and ` F a' implies `a exists' " is rooted in the \familiar confusion" of meaning with denotation.

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the Russell-Meinong view if their range is non-empty. The others, \a major source of theoretical confusion" in Quine's view [1953, p. 167], are to be eliminated in favor of predicate constructions a la Russell. Lambert argues that Quine ought instead to have embraced a free semantics, and to a limited extent Quine [1997] has recently done so. In any case, his expressed view of predication needs a free setting, where its development is not entirely obvious. In classical semantics, Quinean predications are extensional in two senses: (1) s = t j= A(s) $ A(t) and (2) 8x(A(x) $ B (x)) j= A(t) $ B (t). (1) says that if s and t are co-referring, then A(s) is true i A(t) is true; it realizes Quine's condition that the terms in predications occur purely referentially. (2) says that if A(x) and B (x) are co-extensive predicates, then A(t) is true i B (t) is true. Free semantics will generally support (1), but not necessarily (2). `x rotates', `x exists and x rotates', and `if x exists, then x rotates' are co-extensive predicates, but `Vulcan rotates', `Vulcan exists and rotates', and `if Vulcan exists, it rotates' will not end up with the same truth-values in bivalent free semantics. If v does not refer in outer domain semantics, we can make Rv true or false, but 9y(y = v)&Rv is false and 9y(y = v) ! Rv is true regardless. If v does not refer in negative free semantics, Rv is false, but 9y(y = v)&Rv is false and 9y(y = v) ! Rv is true. Extensionality of type (2) may be restored by following Scales [1969] and regarding predications A(t) as the result of applying a complex predicate xA to t. Recall that the extension d(xA) of xA in L consists of those individuals of D (or of Di , if we employ outer domain semantics) of which A(x) is true. Thus, we have 8x(xA(x) $ xB (x)) j= xA(t) $ xB (t). For discussion, see Lambert [1986; 1997a; 1998]. For a supervaluational treatment of L that supports \general-term extensionality" of this kind, see Lambert and Bencivenga [1986]. On any of these free semantic treatments, L embodies two kinds of predication: ordinary predication, which does not have existential import (P t 6j= 9x(x = t)), and complex predication, which does (xP x(t) j= 9x(x = t)). Of course, we needn't introduce complex predicates to achieve this; we could simply de ne two types of subject-predicate constructions in L, say, P (t) for ordinary predication and P [t] for predication with existential import. Lambert and Simons [1994] suggest that ordinary predication P (t) corresponds to characterization, while P [t] corresponds to classi cation. The latter (`Catso is a tuxedo cat') presupposes an individual to classify, the former (`Catso is hungry') traditionally does not.

5.2 De nitions The fact that neither outer domain nor Russellian free semantics supports 8x(A(x) $ B (x)) j= A(t) $ B (t) creates problems for introducing de nitions in theories based on them. To take the simplest case, we may wish to

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add a new predicate P to the language of T with a de ning axiom Df

8x(P x $ A(x))

and then regard P t as an abbreviation of A(t). But when t does not refer to an existent, this may not be possible. If t refers to something in the outer domain, Df does not tell us whether P t $ A(t), because the bound variables range only over the inner domain. If t does not refer in Russellian free semantics, P t will be false but A(t) may be true, as when A is x 6= x: If outer domains consist of some xed nite number of individuals 1 ; : : : ; k , we can name them e1 ; : : : ; ek and use Lambert and Scharle's [1967] trick, noted above in Section 2.2a, to extend quanti cation to the outer domain: let 8cxA abbreviate 8xA&A(e1 )& : : : &A(ek ): If we then give what Gumb and Lambert [1997] call a \full explicit de nition" of P by Df c

8cx(P x $ A(x));

we may regard P t as an abbreviation of A(t), since 8cx(P x $ A(x)) j= P t $ A(t). Gumb and Lambert develop this approach to de nitions for outer domains with just one individual err, which could represent the `error object' of certain programming languages. A proof of Beth's de nability theorem is sketched. Dwyer's [1988] approach is somewhat more general. In outer domain semantics, partial functions Di ! Di are represented by total functions Di ! Di + Do : if  2 Di but d(f )() 2 Do, then f represents a function that is unde ned at . More precisely, from an outer domain interpretation hDi + Do ; di we may extract a partial interpretation hDi ; dp i: Let dp (c) = d(c) if d(c) 2 Di ; otherwise, dp is not de ned at c: If i 2 Di , then let dp (f )(1 ; : : : ; k ) = d(f )(1 ; : : : ; k ) if d(f )(1 ; : : : ; k ) 2 Di ; otherwise, dp (f ) is not de ned at h1 ; : : : ; k i. If i 2 Di , let h1 ; : : : ; k i 2 dp (P ) i h1 ; : : : ; k i 2 d(P ): Outer domain interpretations that coincide when restricted to the inner domain generate the same partial interpretation I . The class C (I ) of such \internally invariant" outer domain interpretations can be regarded as representing I . Dwyer exploits this connection to characterize de nability for partial functions from an outer domain perspective. The problem here again is that the classical conditions on de nitions do not carry over. We cannot allow just any formula A(x) in de nition Df of P , because not every open sentence is stable in the sense of being true of the same individuals of Di as we move from one interpretation of C (I ) to another. For example, let I be a partial interpretation in which D is the set of real numbers, so that under assignment neither x=0 nor x + 1=0 refers.

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Under any assignment relative to I 0 2 C (I ), however, x=0 and x + 1=0 will designate \unreal" numbers  and in Do, where  and may or may not coincide. Accordingly, as we move from one C (I ) interpretation to another, x=0 = x + 1=0 will be true of dierent sets of reals; it is not stable. If A(x) is x=0 = x + 1=0, then Df does not characterize any property of reals. Dwyer [1988, p. 31] develops a suÆcient syntactic condition on stability (viz., atomic constituents of A contain no more than one non-logical symbol), which is used in reformulating the classical conditions on admissible de nitions. Free versions of Robinson's joint consistency theorem, Craig's interpolation lemma, and Beth's de nability theorem are proved to establish the adequacy of this account.

5.3 Modality As noted at the end of Section 3.5, Barba [1989] has shown how to understand the supertruth of A in terms of something like the logical truth of 3A. Two additional connections between free and modal logic are described in this section. a. Garson [1991] develops a general system of quanti ed intensional logic based on free logic. By a general system, he means one (1) from which particular systems can be obtained by specifying (a) constraints on interpretations and (b) additional axioms or rules and (2) for which completeness can be established by a general proof | one easily modi ed to establish the completeness of these particular systems. Let us assume a rst order language L0 without function-names, but with operators  (if A is a formula, so is A) and E ! (if t is a term, E !t is a formula). QS-interpretations I = hW; w0 ; R; D; E; di of L0 are generalizations of Kripke interpretations. As usual, W is a set of possible worlds, w0 2 W represents the actual world, R is a binary accessibility relation on W , D is a (non-empty) set of possible individuals, and d assigns intensions d(P ) : W ! P (Dk ) to k place predicates P , d(P )(w) being the extension of P at w. In Kripke semantics, E (w)  D is the set of individuals that exist in w; in Garson's generalization, E (w) is a set of individual intensions W ! D, and  exists in w if  = f (w) for some f 2 E (w). In Kripke semantics, d also assigns possible individuals d(a) to constants a; in Garson's version, d assigns individual intensions d(a) to constants a. Like constants, variables x are assigned individual intensions (x); an x-variant of  at w is an assignment that diers from  at most at x, where (x) 2 E (w). Kripkean interpretations and assignments, in which designation is rigid, are the special case where individual intensions are constant functions.

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Under interpretation I and assignment : ri 1. x refers at w to (x)(w): ri 2. a refers at w to d(a)(w): vi 1. If ti refers at w to i , P t1 : : : tk is true at w i h1 ; : : : ; k i 2 d(P )(w): vi 2. If ti refers at w to i , then t1 = t2 is true at w i 1 = 2 : vi 3. :A is true at w i A is false at w: vi 4. A ! B is false at w i A is true at w and B is false at w: vi 5. 8xA is true at w i A is true at w for each x-variant of  at w: vi 6. A is true at w i A is true at each w0 such that wRw0 : vi 7. E !t is true at w i the intension of t 2 E (w): vi 8. A is true i A is true at w0 : As noted in Section 2.2b, Kripke-semantics is basically outer domain free semantics, with the individuals that exist in w constituting w's inner domain, while the rest of D functions as w's outer domain. In Kripke semantics, E !t can be de ned by 9x(x = t), but not here: vi 7 treats E ! as a predicate of intensions. If R is universal, E !t is equivalent to 9x(x = t); but in general existence is not de nable from identity. For this semantics, Garson sketches a complete Hilbert-style system GS, consisting of (a) propositional modal axioms and rules appropriate to the accessibility relation R, (b) identity axioms and rules that we may (with the stipulation that E !t is not atomic) identify with the identity axioms of PFL, and (c) quanti er rules which are generalizations of those of free logic:42 GUI

G[8xA] ` G[E !t ! A(t)]

GUG If ` G[E !t ! A(t)] and t does not occur in G[8xA], then

` G[8xA]

Here G[B ] is any formula of one of the following forms

B A!B (A1 ! ::. (Ak ! B ) : : :) A ! (A1 ! ::. (Ak ! B ) : : :)

42 This is apparently what Garson [1991, p. 134] intends by rules GUI and GUG, which are not clear as stated.

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After sketching a Henkin-style completeness argument for GS, Garson illustrates the generality of QS+GS by showing how to obtain several familiar systems (including Kripke's) as special cases. b. Schweizer [1990] develops Skryms' [1978] defense of the metalinguistic reading of necessity claims against Montague's [1963] argument that such readings are incoherent. The metalinguistic reading treats necessity not as an operator  (for `necessarily') applying to sentences, but as a predicate N (for `is necessary') applying to names of sentences. M interpretations I of such languages are pairs < I0 ; C >; where C is a set of interpretations and I0 2 C . A is true under I i A is true under I0 , and d(N ) is such that if pAq names A, N pAq is true under I 0 2 C i A is true under each C interpretation. Godel-numbering allows us to develop the metalinguistic interpretation of necessity in an extension T of formal arithmetic, whose language includes the predicate N . If pAq is the numeral for g(A), we want to read N pAq as `pAq is necessary'. To support this reading, Montague argues, T should be such that for any sentence A, (1) If `T A; then `T N pAq (2) `T N pAq ! A

Assuming T 's proof method is complete, (1) says that if pAq is necessary in the sense of being true in every model of T , then it is provable that pAq is necessary. (2) says that the standard modal principle `if pAq is necessary, then A is true' is provable. Now diagonalization gives us a sentence B such that (3) `T :N pB q $ B (2) and (3) imply `T :N pB q. But `T :N pB q and (3) imply `T B , which with (1) implies `T N pB q. So T is inconsistent. Let L be a rst-order language that includes the language of formal arithmetic, let L be the standard modal extension of L, and let LN result from L by adding the necessity predicate N . Assuming a Godel-numbering of LN , we may translate L formulae A into LN formulae tr(A) by: tr(A) = N pAq; tr commutes with :, !; and 8x; and tr(A) = A for L formulae A. If I = hW; w0 ; R; D; E; di is a Kripke interpretation of L with universal accessibility relation R;43 Schweizer shows how to obtain an M interpretation tr(I) of LN so that for sentences A, A is true under I i tr(A) is true under tr(I). The 43 Schweizer [1990, p, 165] stipulates only that I is an S5 interpretation (R is an equivalence relation), but his construction assumes that R is universal. This is a stronger assumption: (Pa & :8xP x) can be true if R is an equivalence relation but not if R is universal, assuming as usual that D = [w2W E (w):

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construction evades Montague's problem because the Godel-sentence B that issues from diagonalization is not tr(A) for any sentence A, whereas the modal axiom holds only in the form N ptr(A)q ! tr(A): Since tr(I) is an M interpretation hI0 ; C i, C is a set of interpretations such that N pAq is true under tr(I) i A is true under each interpretation in C . The connection with free logic is that these C interpretations are outer domain interpretations. At each w 2 W , I induces an outer domain interpretation I (w) = hDi + Do; dw i of L, where Di = E (w), Do = D E (w), and dw is the restriction of d to w: dw (a) = d(a) and dw (P ) = d(P )(w). When dw is appropriately extended to N , we can identify I0 with I (w0 ) and C with fI (w)jw 2 W g:44 The extension of dw is in stages corresponding to the number of nested occurrences of  in A: we put g(tr(A)) in dw (N ) at stage k + 1 provided tr(A) is true under I (w) at stage k. At stage 0, A is  free, so tr(A) = A, which gets a truth-value under each outer domain interpretation I (w): Schweizer [1990, p. 170] states that \analogous equivalence results can be obtained for the other normal systems of quanti ed modal logic, by simply utilizing the relevant accessibility relation R . . . within the eligible set of models." Presumably, his suggestion is that M interpretations I now be conceived as triples hI0 ; C; Ri, where C is a set of interpretations, R is a binary relation on C , and I0 2 C . Then A is true under I i A is true under I0 and N pAq is true under I 0 2 C i A is true under each I 00 such that I 0 RI 00 . The ordinary notion of a metalinguistic interpretation is then the special case where R is universal. ACKNOWLEDGEMENTS I thank Karel Lambert for directing my attention to many of the works cited in this survey. University of Connecticut, USA.

BIBLIOGRAPHY In cases where an essay has been reprinted, sometimes with cuts or other changes, pagereference citations are to the reprinted version. [Barba, 1989] J. Barba. A modal version of free logic, Topoi, 8, 131{5, 1989. [Bencivenga, 1980] E. Bencivenga. Free semantics. In Italian Studies in the Philosophy of Science, M. Dalla Chiara, ed. pp. 31{48. Reidel, Dordrecht, 1980. Reprinted in [Lambert, 1991, pp. 98{110].

44 All of the a-variants of I (w ), which dier from I (w ) at most at d (a) 2 E (w ), must w also be included in C:

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[Bencivenga, 1980a] E. Bencivenga. Truth, correspondence, and non-denoting singular terms. Philosophia, 9, 219{229, 1980. [Bencivenga, 1980b] E. Bencivenga. Free semantics for de nite descriptions, Logique et analyse, 23, 393{405, 1980. [Bencivenga, 1986] E. Bencivenga. Free logics. In Handbook of Philosophical Logic, Vol.III, D. Gabbay and F. Guenthner, eds. pp. 373{426. Reidel, Dordrecht, 1986. Reproduced in this volume. [Beeson, 1985] M. Beeson. Foundations of Constructive Mathematics. Springer-Verlag, Berlin, 1985. [Blamey, 1986] S. Blamey. Partial logic. In Handbook of Philosophical Logic, Vol.III, D. Gabbay and F. Guenthner, eds. pp. 1{70. Reidel, Dordrecht, 1986. [Burge, 1974] T. Burge. Truth and singular terms. Nous, 8, 309{325, 1974. Reprinted in [Lambert, 1991, pp. 189{204]. [Church, 1956] A. Church. Introduction to Mathematical Logic, Vol.I. Princeton University Press, Princeton, 1956. [Cocchiarella, 1991] N. Cocchiarella. Quanti cation, time, and necessity. In [Lambert, 1991, pp. 242{256]. [Dwyer, 1988] R. C. Dwyer. Denoting and De ning: A Study in Free Logic. University Micro lms International, Ann Arbor, 1988.  [Ebbinghaus, 1969] H.-D. Ebbinghaus. Uber eine Pradikatenlogik mit partiell de nierten Pradikaten und Funktionen. Archiv fur mathematische Logik und Grundlangenforschung, 12, 39{53, 1969. [Evans, 1979] G. Evans. Reference and contingency. The Monist, 62, 161{189, 1979. [Farmer, 1995] W. M. Farmer. Reasoning about partial functions with the aid of a computer. Erkenntnis, 43, 279{294, 1995. [Feferman, 1995] S. Feferman. De nedness. Erkenntnis, 43, 295{320, 1995. [Frege, 1892] G. Frege. On sense and reference. In Translations from the Philosophical Writings of Gottlob Frege, P. Geach and M. Black, eds. pp. 56{78. Basil Blackwell, Oxford, 1966. [Garson, 1991] J. Garson. Applications of free logic to quanti ed intensional logic. In [Lambert, 1991, pp. 111{142]. [Gumb, 1998] R. D. Gumb. Does identity precede existence? Read at World Congress of Philosophy, Boston, 1998. [Gumb and Lambert, 1997] R. D. Gumb and K. Lambert. De nitions in nonstrict positive free logic. Modern Logic, 7, 25{55, 1997. [Jerey, 1991] R. Jerey. Formal Logic: Its Scope and Limits. McGraw Hill, New York, 1991. [Kleene, 1950] S. C. Kleene. Introduction to Metamathematics. D. van Nostrand, Princeton, 1950. [Kleene, 1967] S. C. Kleene. Mathematical Logic. John Wiley & Sons, New York, 1967. [Kroon, 1991] F. W. Kroon. Denotation and description in free logic. Theoria, 57, 17{41, 1991. [Lambert, 1986] K. Lambert. Predication and ontological commitment. In Die Aufgaben

der Philosophie in der Gegenwart: Aktien des 10. Internationalen Wittgenstein Symposiums, W. Leinfellner and F. N. Wuketits, eds. pp. 281{287. Holder-Pichler=Temsky,

Wien, 1986. Reprinted in [Lambert, 1991, pp. 273{284]. [Lambert, 1991] K. Lambert, ed. Philosophical Applications of Free Logic. Oxford University Press, New York, 1991. [Lambert, 1991a] K. Lambert. The nature of free logic. In [Lambert, 1991, pp. 3{14]. [Lambert, 1991b] K. Lambert. A theory about logical theories of \expressions of the form `the so and so', where `the' is in the singular". Erkenntnis, 35, 337{346, 1991. [Lambert, 1995] K. Lambert. On the reduction of two paradoxes. In Physik, Philosophie und die Einheit der Wissenschaften, L. Kruger and B. Falkenburg, eds. pp. 21{32. Spektrum Academischer Verlag, Heidelberg, 1995. [Lambert, 1997] K. Lambert. Free Logics: Their Foundations, Character, and Some Applications Thereof. Academia Verlag, Sankt Augustin, 1997. [Lambert, 1997a] K. Lambert. Nonextensionality. In Das weite Spektrum der analytischen Philosophie, W. Lenzen, ed. pp. 135{148. de Gruyter, Berlin, 1997.

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[Lambert, 1998] K. Lambert. Fixing Quine's theory of predication. Dialectica, 52, 153{ 160, 1998. [Lambert, 2001] K. Lambert. From predication to programming. Minds and Machines, 11, 2001. [Lambert and Bencivenga, 1986] K. Lambert and E. Bencivenga. A free logic with simple and complex predicates. Notre Dame Journal of Formal Logic, 27, 247{256, 1986. [Lambert and Scharle, 1967] K. Lambert and T. Scharle. A translation theorem for two systems of free logic. Logique et Analyse, 39{40, 328{341, 1967. [Lambert and Simons, 1994] K. Lambert and P. Simons. Characterizing and classifying: explicating a biological distinction. The Monist, 77, 315{328, 1994. [Lambert and van Fraassen, 1972] K. Lambert and B. C. van Fraassen. Derivation and Counterexample: An Introduction to Philosophical Logic. Dickenson, Encino, 1972. [Leblanc, 1968] H. Leblanc. On Meyer and Lambert's quanti cational calculus FQ. Journal of Symbolic Logic, 33, 275{280, 1968. [Lehmann, 1994] S. Lehmann. Strict Fregean free logic. Journal of Philosophical Logic, 23, 307{336, 1994. [Lin, 1983] Y. Lin. Replacement-Closed Rules for Free and for Classical Logic. University Micro lms International, Ann Arbor, 1983. [Mann, 1967] W. E. Mann. De nite descriptions and the ontological argument. Theoria, 30, 211{229, 1967. Excerpted in [Lambert, 1991, pp. 257{272]. [Mendelsohn, 1989] R. L. Mendelsohn. Objects and existence: re ections on free logic. Notre Dame Journal of Formal Logic, 30, 604{623, 1989. [Meyer and Lambert, 1968] R. Meyer and K. Lambert. Universally free logic and standard quanti cation theory. Journal of Symbolic Logic, 33, 8{26, 1968. [Montague, 1963] R. Montague. Syntactical treatments of modality, with corollaries on re exion principles and nite axiomatizability. Acta Philosophica Fennica, 16, 153{ 167, 1963. Reprinted in Formal Philosophy: Selected Papers of Richard Montague, R. H. Thomason, ed. pp. 286{302. Yale University Press, New Haven, 1978. [Posy, 1982] C. J. Posy. A free IPC is a natural logic: strong completeness for some intuitionistic free logics. Topoi, 1, 30{43, 1982. Reprinted in [Lambert, 1991, pp. 49{ 81]. [Quine, 1948] W. V. O. Quine. On what there is. Review of Metaphysics, 2, 21{38, 1948. Reprinted in [Quine, 1963, pp. 1{19]. [Quine, 1953] W. V. O. Quine. Meaning and existential inference. In [Quine, 1963, pp. 160{167]. [Quine, 1960] W. V. O. Quine. Word and Object. MIT Press, Cambridge, 1960. [Quine, 1963] W. V. O. Quine. From a Logical Point of View. Harper Torchbooks, New York, 1963. [Quine, 1969] W. V. O. Quine. Set Theory and its Logic. Harvard University Press, Cambridge, 1969. [Quine, 1997] W. V. O. Quine. Free logic, description, and virtual classes. Dialogue, 36, 101{108, 1997. [Robinson, 1974] A. Robinson. On constrained denotation. In Nonstandard Analysis and Philosophy (vol. 2 of Selected papers of Abraham Robinson, H. Keisler, et al., eds.), W. Luxemburg and S. Korner, eds. pp. 493{504. Yale University Press, New Haven, 1979. [Scales, 1969] R. D. Scales. Attribution and Existence. University Micro lms International, Ann Arbor, 1969. [Schutte, 1960] K. Schutte. Beweistheorie, Springer-Verlag, Berlin, 1960. [Schweizer, 1990] P. Schweizer. A Metalinguistic Interpretation of Modality, University Micro lms International, Ann Arbor, 1990. [Scott, 1967] D. Scott. Existence and description in formal logic. In Bertrand Russell: Philosopher of the Century, R. Schoenman, ed. pp. 181{200. Little, Brown and Company, Boston, 1967. Reprinted in [Lambert, 1991, pp. 28{48]. [Shoen eld, 1967] J. R. Shoen eld. Mathematical Logic. Addison-Wesley, Reading, 1967. Re-issued in paperback by A. K. Peters, 2001. [Simons, 1991] P. M. Simons. Free part-whole theory. In [Lambert, 1991, pp. 285{305].

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[Skyrms, 1968] B. Skyrms. Supervaluations: identity, existence, and individual concepts. Journal of Philosophy, 65, 477{482, 1968. [Skyrms, 1978] B. Skyrms. An immaculate conception of modality. Journal. of Philosophy, 75, 77{96, 1978. [Smiley, 1960] T. Smiley. Sense without denotation. Analysis, 20, 125{135, 1960. [Stenlund, 1973] S. Stenlund. The Logic of Description and Existence. Filoso ska Studier Nr. 18, Uppsala Universitet, 1973. [Strawson, 1952] P. F. Strawson. Introduction to Logical Theory. Methuen, London, 1952. [Tidman and Kahane, 1999] P. Tidman and H. Kahane. Logic and Philosophy: A Modern Introduction. Wadsworth, Belmont, 1999. [Trew, 1970] A. Trew. Nonstandard theories of quanti cation and identity. Journal of Symbolic Logic, 35, 267{294, 1970. [van Fraassen, 1966] B. C. van Fraassen. Singular terms, truth-value gaps, and free logic. Journal of Philosophy, 63, 481{495, 1966. Reprinted in [Lambert, 1991, pp. 82{97]. [van Fraassen, 1968] B. C. van Fraassen. Presupposition, implication, and self-reference. Journal of Philosophy, 65, 136{152, 1968. Reprinted in [Lambert, 1991, pp. 205{221]. [Walton, 1990] K. L. Walton. Mimesis as Make-Believe: On the Foundations of the Representational Arts. Harvard University Press, Cambridge, 1990. [Woodru, 1970] P. W. Woodru. Logic and truth-value gaps. In Philosophical Problems in Logic, K. Lambert, ed. pp. 121{142. Reidel, Dordrecht, 1970. [Woodru, 1984] P. W. Woodru. On supervaluations in free logic, Journal of Symbolic Logic, 49, 943{950, 1984. [Woodru, 1991] P. W. Woodru. Actualism, free logic, and rst-order supervaluations. In Existence and Explanation, W. Spohn, et al. eds. pp. 219{231. Kluwer Academic Publishers, Boston, 1991.

STEPHEN BLAMEY

PARTIAL LOGIC

INTRODUCTION When I was originally asked to write about `partial logic' for the rst edition of the Handbook , I was a little puzzled: I was taken to be an expert in an apparently well de ned subject area that I didn't know existed. But it turned out to be the sort of thing I had written about in my D.Phil. thesis, so I had somewhere to start. Nowadays the label `partial logic' is much more familiar, and a lot of work is being done in the area it covers. The bulk of my own work, though|most of it dating right back to thesis days|has not yet been published: I have been bewilderingly bad about this. In particular, the various promises made in the rst edition about forthcoming work have still not been ful lled. In spite of this, I have resisted the temptation just to shove in more material of my own for the second edition|except in small ways here and there. Additions are largely in response to what has newly appeared in print. A wide range of work will be surveyed (much more now than in the rst edition), but the backbone of this chapter is the development of what I call `simple partial logic'. It is against this backbone that other more sophisticated projects are discussed. Simple partial logic results from the simple-minded following through of the idea that classical logic may be loosened up to cater for non-denoting singular terms and neither-true-nor-false sentences|to cater for them in a uniform way as semantically `unde ned' items|and at the same time to cater for `partially de ned' functors: termforming functors, predicates, and sentence connectives. These functors have to accommodate unde ned arguments, but they may also produce unde ned compounds even when all their arguments are fully de ned. In particular, we shouldn't ignore sentence connectives of this kind: once loosened up, classical propositional logic needs to be lled out with connectives such as interjunction and transplication . The uniformity behind all this comes from the idea of representing partial functions by monotonic functions| as explained in Section 1|and using monotonically representable partial functions to interpret functors of whatever logical category. All sections have undergone some stylistic revision for the second edition, and most of them have been expanded. Note that Section 2 now has more subsections: there is a new introductory subsection, which means that subsections 2.1 to 2.5 have become subsections 2.2 to 2.6; and the old subsection 2.6 has split into three|2.7 to 2.9|so that the old 2.7 is now 2.10. Section 4 has been disrupted in a similar way: the old subsection 4.1 has split into 4.1 and 4.2; subsection 4.3 is new; and the old subsection 4.2 has

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split into 4.4 and 4.5. There has been a more straightforward reorganization to Sections 6 and 7: a new subsection has been introduced as 6.3, which means that the old subsections 6.3 and 6.4 become 6.4 and 6.5; and the old subsection 7.2 has split into two: 7.2 and 7.3. The other Sections retain their original structure.

_^ _^ _^ interjunction :{ In the rst edition an interjunction sign was formed by juxtaposing two `'s: . This was a pity, because it made the symbol a bit too at. Interjunction is a squadging of conjunction and disjunction, and so the symbol for it should be a simultaneous occurrence of `^' and `_': ^ _. Sadly, the notation `' has found its way into the literature, and|much worse|this has sometimes become just two `x's: xx. I urge anyone who wants to write an interjunction sign in the future to avoid `xx' at all costs: `' is tolerable, but I recommend `^ _'. Notation for

1 A SKETCH OF SIMPLE PARTIAL LOGIC

1.1 Classical Semantics as Partial Semantics

In classical logic sentences are either true (>) or false (?) and the interpretation of the standard sentence connectives can be given in the following way:

:

is

^

is

_

is

!

is

$

is



> ?  > ?  > ?  > ?  > ?

i  is ? i  is >; i  is > and i  is ? or

is > is ?;

i  is > or i  is ? and

is > is ?;

i  is ? or i  is > and

is > is ?;

i ( is > and i ( is > and

is >) or ( is ? and is ?) or ( is ? and

is ?) is >):

For simple partial logic we shall adopt precisely these classical >/? conditions; only we give up the assumption that all sentences have to be classi ed either as > or as ?. This leaves room for the classi cation neither->-nor-?. At present we are concerned merely to highlight a parallel with classical semantics, and under the parallel we can think of the third classi cation as a `truth-value gap'. This thought is taken a little further in Sections 1.2

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263

and 3. But the point, if any, of seeing the third classi cation as dierent in philosophical kind from > and ? will of course depend on what particular motivation we consider for adopting the forms of partial logic. (See, especially, Sections 2 and 5.) To interpret universal and existential quanti ers over a given domain D, we shall again exploit the fact that the classical interpretation leaves room for a gap between > and ? when we write out >-conditions and ?-conditions separately. Assuming that a language has|or can be extended so as to have|a name a for each object a in D,



> ?  9x(x) is >?

8x(x)

is

i (a) is > for every a in D i (a) is ? for some a in D; i (a) is > for some a in D i (a) is ? for every a in D:

Most treatments of classical logic stipulate that the domain be non-empty. We shall not be so restrictive: D may be empty. These >/?-conditions for 8x and 9x of course presuppose a semantic account of predicate/singular-term composition. And this mode of composition deserves some attention, since it is the most familiar place to locate the cause of a sentence's being neither `true' nor `false'. It has been considered to give rise to a truth-value gap in two dierent ways: either (i) because a term t may lack a denotation and may, for this reason, make a sentence (t) neither true nor false; or (ii) because a predicate (x) may be only `partially de ned'|not either true or false of some object or objects|so that, if t denoted such an object, (t) would be neither true nor false. We shall want to accommodate both these ideas in one uniform account of predicate/singular-term composition. Our approach will be sketched in Section 1.2, along with an approach to functors which form singular terms from singular terms. But there is one particular atomic predicate to consider immediately: the identity predicate. Once again we can adopt classical >-conditions and ?-conditions verbatim for a sentence t1 = t2 :  i t1 and t2 denote the same thing t1 = t2 is > ? i t1 and t2 denote dierent things: This means that if either t1 or t2 is non-denoting, then t1 = t2 is neither > nor ?. Identity is an untypically straightforward case. At least, so it is if we restrict attention to a determinate relation over a discrete domain of objects|as we shall.

_^ _^ _^ Whatever general framework we set up for predicate/singular-term composition, our logic has so far been revealed as `partial' only in the weak sense

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that it accommodates value-gaps that might arise from the interpretation of non-logical terms or predicates. This is because the interpretation of classical logical vocabulary is classical. But there is a stronger sense of `partial logic': a logic will be partial in the stronger sense if it provides the resources for explaining why a sentence may be neither > nor ? in terms of logical vocabulary|vocabulary, that is, with a xed meaning in the logic. We should look for modes of logical composition whose interpretation can give rise to truth-value gaps, even when any classical sentence constructed out of the same non-logical vocabulary (with the same interpretation) would have to be either > or ?. Assuming that we have worked out the general account of how nondenoting terms can give rise to truth-value gaps, a term-forming descriptions operator would be an example of gap-introducing logical vocabulary. This is because a term x(x) may turn out not to denote, even when (x) is totally de ned. Assuming that (x) is in fact totally de ned, then the denotation conditions for x(x) must be that if a is an object in the domain, then:





x(x) denotes a i 8x[x = a $ (x)] is >; where, as before, a is a name|pre-existing or specially introduced|for a. In other words, x(x) denotes an object if and only if that object uniquely satis es (x) and is non-denoting if there is no such object. Of course, we also have to consider the case where (x) is not totally de ned, but the denotation conditions stated will continue to make sense. Furthermore, given the general constraint to emerge in Section 1.2, they will turn out to be the only possible ones for a determinate relation of identity over a discrete domain of objects (see Section 6.4). These -terms involve a rather complicated route to neither->-nor-? sentences. There is a much more straightforward, and no less interesting, kind of gap-introducing vocabulary: sentence connectives. Consider the following >/?-conditions for the connectives ^ _ and =, the rst of which we shall call interjunction and the second transplication :   is > and is > _ ^ is >? i i  is ? and is ?; 







is > is ?: Notice that ^ _ has the >-conditions of ^ and the ?-conditions of _, while = has the >-conditions of ^ but the ?-conditions of !. And so these connectives clearly meet our desideratum of introducing value gaps: we do not necessarily have to look to predicate/singular-term composition to nd a logical explanation why a sentence may be neither > nor ?. The particular usefulness of ^ _ and = will be touched upon in Section 2.2 and several later sections.

=

is

> i  is > and ? i  is > and

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265

Among our logical vocabulary we shall also include a constantly true sentence >, and a constantly false one ?. Thus we are using `>' and `?' both as truth-value labels and to stand for logical constants; and, in a similar way, we shall use `' both to label the classi cation `neither->-nor-?' and to stand for a sentence which is logically neither > nor ?. There will also be a logically non-denoting singular-term, denoted by `~'|which will be used also to denote the classi cation `non-denoting'. In the presence of the term ~, we shall then be able to abandon -terms without any loss in expressive power: this is explained in Section 6.4.



_^ _^ _^

Finally, we must consider the relation of (logical) consequence. Our semantical de nition of ` is a consequence of ' is, loosely stated, that (i) whenever  is >, is >, and (ii) whenever is ?,  is ?. And so, yet again, we are using a de nition which conjoins two formulations of the classical de nition, one involving > and the other ?|formulations which are equivalent in total logic, but not in partial logic. To illustrate the idea, consider for the moment just a propositional calculus with formulae built up from atomic sentences using the connectives we have introduced. Then `interpretations' will simply be partial assignments of > and ? to atomic sentences, and formulae may be evaluated according to our >/?clauses for the connectives. We shall use `' for the relation of logical consequence, and so   if only if (i) and (ii) above both hold when `whenever' is understood to mean `under any partial assignment under which'. (By `partial assignment' I do not mean to exclude total assignments: here, as elsewhere, `partial' means `not necessarily total'.) The tendency among authors on partial logics of one sort or another is to take condition (i) on its own to de ne logical consequence; and sometimes (i) and (ii) are used to frame two separate notions|for example, in [Dunn 1975], [Hayes 1975] and, in disguised form, in [Woodru 1970]. In [Cleave 1974], on the other hand, there is a (rather algebraic) version of our double-barrelled de nition. And across the literature of the last twenty years the picture has not greatly changed. But perhaps making a choice between these alternatives is not such a fundamental matter. After all, we can de ne the two halves of our single notion:  > i    _ ;  ? i  ^   : And, putting them back together again,  i  > and  ? : Or, if we invoke negation, either one of the halves on its own would do:  i  > and : > : i  ? and : ? ::

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The issue might be set in a more interesting context if thought were given to the connection between these de nitions and inferential practice; but this question goes far beyond our semantics-orientated essay. To motivate working with the double-barrelled de ntion we can adduce some arguments from theoretical neatness. First, the law of contraposition holds:  i :  :: Secondly, logical equivalence|a relation which must be taken to obtain between two formulae if and only if they take the same resultant classi cation under any interpretation|turns out as mutual consequence. Using `'' for equivalence, ' i   and  : Thirdly, equivalence and consequence t together with conjunction and disjunction in the natural (at least the classical) way:

'

^

'  _ i   : These properties of  break down for > and for ? . i

Neatness aside, some interesting dierences between working with  and working just with > (equally just with ?) can be extracted from [Langholm 1988]. In particular, it emerges that in a rst-order logic without non-denoting terms some interpolation results for > are much cheaper than corresponding results for . (On interpolation for  in a full rst-order language, see Sections 6.5, 7.2, and 7.3.) In Section 6.5 we shall present a rigorous de nition of (double-barrelled) consequence for rst order languages, and there will be two generalisations. First, we shall be interested not merely in logical consequence, but in relations of consequence determined by a given range of interpretations|to match a proof theoretical notion of consequence in a given theory (presented in Section 7.1). Secondly, consequence will be de ned between sets of formulae, rather than individual formulae: not only will several premises be allowed, but also several `conclusions'|to be understood disjunctively. This will match our sequent-style proof theory; and another advantage of the double-barrelled de nition will then emerge: we shall be able to frame fewer and simpler rules, since sequent principles will be able to constrain the >-conditions and ?-conditions of logical vocabulary at one go. There is, nally, a dierent kind of generalization to consider: more-thantwo-place `consequence' relations. For example, > and ? are combined into a four-place relation in [Langholm 1989, Fenstad 1997, Bochman 1998]. If, for simplicity's sake, we restrict attention to single formulae rather than sets of formulae, then the relation|call it C |can be de ned as follows: C (1 ; 1 ; 2 ; 2 ) if and only if whenever 1 is > and 2 is ?, then either

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1 is > or 2 is ?. Notice that we could de ne C , using negation, in terms of either > or ? : C (1 ; 1 ; 2 ; 2 ) i 1 ^ : 2 > 1 _ :2 i : 1 ^ 2 ? :1 _ 2 :

Alternatively|and I have myself found this more useful to work with| we could adopt a four-place relation C 0 that just conditionalizes the two place : C 0 (1 ; 1 ; 2 ; 2 ) if and only if whenever 1 is > and 1 is ?, then 2  2 . In terms of  this relation could be de ned as follows: C 0 (1 ; 1 ; 2 ; 2 ) i 1 ^ : 1 ^ 2  2 _ :1 _ 1 : In Section 7.1 we shall use the the proof-theoretical correlate of  to de ne a three-place consequence relation along these lines|one that ignores the 1 argument place. Some of the quanti er and identity rules are most perspicuously presented in terms of this relation. (Compare the three- and fourplace relations used for systems of modal logic in [Blamey and Humberstone 1991].)

1.2 Partial Semantics as Monotonic Semantics

To interpret sentence connectives we have speci ed >-conditions and ?conditions for formulae constructed by means of them: -conditions then take care of themselves. Even so,  is a semantic classi cation, and the apparatus of 3-valued logic is at our disposal: our >/?-conditions are summed up in the following matrices. (The constant sentences >,  and ? can be thought of as 0-place connectives, but their matrices are trivial).

: > ?   ? >





> > >    ? ? ?

^

_

_ ^

$

!

=

> > > > > > >   >     ? ? >  ? ? ? >  >   >         ? ?      > ? >  ? >   ?    >  ? ? ? ? > >  Partial assignments of > or ? to atomic constituents can now be replaced by total assignments of >,  or ?. And, if we take it that each assignment

assigns a classi cation to all of a denumerable stock of atomic formulae, then everything will t neatly into place when we just assign  to any vocabulary we are not interested in.

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Let us now impose a simple ordering v on f>; ; ?g: >

?

x v y i either x =  or x = y. 

Equivalently: x v y i both, if x = >, then y = >, and, if x = ?, then y = ?. Then we can extend the use of `v' to de ne a `degree-of-de nedness' relation between assignments v and w:

v v w i v(p) v w(p) for every atomic formula p: In other words, v v w if and only if wherever v assigns the value > or ?, w assigns that value also. If v() is the result of evaluating a formula  under v, it is then easy to deduce the following monotonicity of evaluation: if v v w; then v() v w(); for every formula . An intuitive way to think about this is that if a formula has taken on a value (> or ?), then this value persists when any atomic gaps () are lled in by a value (> or ?) (cf. Lemma 3 in section 6.2). Here we have a global monotonicity condition, but we might direct attention to individual formulae. If all atomic formulae occurring in  are among p1 ; : : : ; pn, then we can specify a 3n-row matrix for , which describes a function f from f>; ; ?gn into f>; ; ?g, where f (x1 ; : : : ; xn ) is the classi cation of  under the assignment of xi to pi , 1  i  n. And f will then be a monotonic function. That is to say if xi v yi for all i, then f (x1 ; : : : ; xn ) v f (y1 ; : : : ; yn ). Observe that this is equivalent to monotonicity in each coordinate separately. What lies behind both forms of monotonicity is that the matrix for each sentence connective describes a monotonic function and that the class of monotonic functions is closed under composition. The question then arises: Is our logic expressively adequate for all monotonic functions? It is. In Section 4.1 we shall show that :, ^, _, ^ _, >, and ? form a neatly complete bunch of connectives. Our `partial' propositional logic could, then, simply be seen as the total logic of 3-valued monotonic modes of sentence composition|modes (p1 ; : : : ; pn ) that are interpreted by monotonic functions. The connection between the two ways of looking at it is made by the idea that monotonic functions from f>; ; ?gn into f>; ; ?g can be taken to represent partial functions from f>; ?gn into f>; ?g. Modes of composition in the logic

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can then be taken to be interpreted by partial functions. On this understanding of the mathematical semantics, > and ? are obviously the only `truth values' that there are:  plays a role merely in the representation of partial functions by monotonic total ones. Thus the idea that a sentence classi ed  suers from a `truth-value gap' is given immediate, but in itself uninteresting, sense.

_^ _^ _^ The use of monotonic functions to represent partial ones has nothing specifically to do with truth functions. Given any domain D, we can pick on an extraneous object ~ and consider functions from (D [ f~g)n into D [ f~g which are monotonic|in exactly the same sense as before|with respect to an order relation v given by: D:






~ x v y i either x = ~ or x = y: Equivalently: x v y if and only if, for any a 2 D, if x = a then y = a. These functions can be taken to represent partial functions from Dn into D. And we can just as easily consider a range of dierent domains D1 ; : : : ; Dn+1 , each xed up with their own extraneous objects ~1 ; : : : ; ~n+1 , and represent a system of partial functions from D1  : : :  Dn into Dn+1 by functions from (D1 [ f~1 g)  : : :  (Dn [ f~n g) into Dn+1 [ f~n+1 g which are monotonic with respect to the respective orderings. A simple example would be the system of partial n-place relations on a domain D, represented by monotonic functions from (D [ f~g)n into f>; ; ?g. If n = 1, these would be `partial subsets' of D. The functions represented are partial not only in that they may be unde ned for some n-tuple of arguments, but also in that they allow for `empty argument places':  and ~ stand equally for the gap of an empty argument place and for the gap of no output value. This suggests that these partial functions might aptly be deployed to provide the uniform account of linguistic composition that we demanded in Section 1.1|to handle partially de ned functors that may embrace non-denoting terms. But what kind of sense does it makes to say that monotonic functions represent partial ones? The notion of representation is itself unproblematic: it is just the same as when we say that ordinary total functions can be represented in set theory by sets of a certain kind. Still, when it is observed that an `empty argument place' does not necessarily mean no output value

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(consider for example the matrices for ^ and _), it may be objected that it is nonsense to talk of a function which can yield an output value form an incomplete, possibly total vacuous, array of input values. This thought, only thinly veiled in talk about functors, seems to have gured in some discussions of Frege, and we shall tackle it in this context in Section 3.2. A dierent|and opposite|reaction would be to question all the fuss about monotonicity: granted the idea of  and ~ representing gaps in both input and output, why restrict the range of representing functions at all? In Section 2 we shall see how some speci c applications for partial functions in semantics call for the monotonicity constraint, and a more general view will emerge when we discuss the rst reaction. For the moment we can put the point intuitively: the output value, if any, of a monotonically representable partial function can be seen to depend, and depend only, on the input values in occupied argument places (and not on the gaps of empty ones), precisely because of the monotonicity condition that if a gap is ` lled in', then the output value remains xed. The degree-of-de nedness ordering v becomes more interesting than merely a gap versus an object when we push the idea of representing partial functions up to higher-level categories|to functions with systems of partial functions as their domain (and possibly also as their range). Consider, the simple example of the system of partial subsets of a domain D, represented by monotonic functions form D [ f~g into f>; ; ?g. Between two such functions f and g we can de ne f v g to mean that f (x) v g(x) for any x in D [ f~g. Then, to represent partial subsets of the system of partial subsets of D, we can use functions on the monotonic functions|functions F into f>; ; ?g which are themselves monotonic: if f v g; then F (f ) v F (g): Intuitively, the point of this higher-level monotonicity is that if F yields a value when applied to f , then this depends, and depends only, on the range of output values of f , not on its gaps. This means that if g behaves like f except possibly that it is more de ned, then F must send g to the same value it sends f to. A full hierarchy will emerge for higher-level categories of monotonicallyrepresentable partial functions, and a non-trivial study of its characteristics can be found in [Lepage 1992]. In [Muskens 1989] and in [Lapierre 1992], on the other hand, there are special hierarchies designed to interpret intensional partial logic. Muskens has a cunning reduction of functional application and abstraction to operations on partial relations, which are what his hierarchy is actually a hierarchy of. But Lepage and Lapierre adopt a more familiar style of reduction: they take hierarchies of just one-place functions as primitive. Nothing is lost, because a domain of partial functions from D1  : : :  Dn into Dn+1 is isomorphic to, and can be modelled by, the domain of partial functions from D1 into the domain of partial functions from

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D2 into . . . into the domain of partial functions from Dn into Dn+1. And so, in particular, if Dn+1 = f>; ?g, then we have a modelling of partial n-place relations. In [Tichy 1982] it had been argued that such a reduction to one-place functions was possible only with domains of total functions, but Lepage exposes the error in Tichy's argument. _^ _^ _^ To provide a semantics for rst-order languages we need neither go very far up the hierarchy nor reduce all functions to one-place ones. Predicates will be interpreted by monotonically-representable partial sets and relations over a domain D. Similarly, n-place functors which form singular terms out of singular terms will be interpreted by monotonically representable partial functions from Dn into D. And in a model theory, conceived of as a theory developed in some standard set theory, we can expect to work with the representing monotonic functions. A model will directly assign such a function to unstructured predicate symbols and term-functor symbols, but we are no less interested in the complex predicates that arise as formulae (x1 ; : : : ; xn ), with free variables x1 ; : : : ; xn signaling the argument places, and in the complex term-functors that arise as compound terms t(x1 ; : : : ; xn ). If we take free variables to range over D [ f~g, are we guaranteed that these complex modes will be monotonic? We are, given that every unstructured functor|logical and non-logical alike|is interpreted via a monotonic function of the appropriate category, since combining monotonic functions invariably leads to a monotonic function. Straightforward functional composition lies behind all linguistic combinations except for the variable-binding quanti ers 8 and 9 (and also the variable-binding operator , if we include it: see Section 6.4). In the simplest case quanti ers are just second-level predicates, taking a one-place predicate (x) to a sentence 8x(x) or 9x(x). Disentangling them from the apparatus of variable-binding, it is easy to see that the >=?-conditions we gave for 8 and 9 match an interpretation via monotonic second-level functions F8 and F9 on the domain of partial subsets of D:  i f (a) = > for every a in D F8 (f ) = > ? i f (a) = ? for some a in D; 

F9 (f ) =



> i f (a) = > for some a in D ? i f (a) = ? for every a in D:

But quanti ers play a general role in converting any (n + 1)-place predicate (x1 ; : : : ; xi ; : : : ; xn+1 ), into an n-place predicate 8xi (x1 ; : : : ; xi ; : : : ; xn+1 ) or 9xi (x1 ; : : : ; xi ; : : : ; xn+1 ), and we have to check that monotonicity will always be preserved in this move. This is easy enough. Notice that variables bound by a quanti er will `range over' just the domain of objects D|not, as free variables do, over the whole of D [ f~g.

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Finally, what about the interpretation of singular terms?|`closed' terms, which contain no free variables? To t in with the model-theoretic apparatus for functors, we should expect to be able to assign an object in the domain of quanti cation to a term to mean that the term denotes that object, and to assign ~ to a non-denoting term. If we stipulate the classi cation of all unstructured singular terms in this way, the apparatus of monotonic functions will then yield an appropriate classi cation for compound closed terms. The reader who is eager for formal details could now skip on to Section 6. But a few further remarks are prompted, if we want seriously to understand a term's denoting an element of D in a way that matches the informal idea of a term's standing for an object. A sharp contrast must be drawn with the assignment of ~ to a term. For ~ is not the nonsense of an object which doesn't exist; nor is it a special object picked on (Frege-style) to be the actual denotation for terms that should really be non-denoting: ~ has been introduced simply as part of the apparatus for representing partial functions. It does then make sense to see ~ playing a derived model theoretic role as the semantic classi cation `non-denoting', but it would be courting confusion if we then went on to think of the monotonic functions of the model theory just as functions on semantic classi cations. The classi cation of a denoting term would then turn out to be the very object denoted, but to keep semantic levels straight, we should distinguish the object a that a term denotes from the classi cation `denoting-a': such a classi cation is not an object in the domain and can be aligned with ~. Of course, objects and the corresponding classi cations do correspond one-to-one, and so it is in fact open to us to adopt an alternative understanding of the semantics right from the start|as a semantics that operates throughout on classi cations. And this could either be thought of as a total monotonic semantics on all classi cations or as a partial semantics on the range of classi cations `denoting-so-and-so' (see Section 3). Observe that a parallel nickiness over sentences and  would be called for only if the assignment of > or ? to a sentence were intended to be more than a model-theoretic device for classifying sentences|as it would, for example, according to Frege's uni ed theory of reference, where the truth-values > and ? are seriously thought of as objects denoted by sentences. Otherwise, it is harmless to take the monotonic functions that represent partial ones simply as (total) functions on semantic classi cations.

1.3 Comparisons with Supervaluations The preceding remarks bring our partial logic very much in line with traditional truth-table approaches. The most notable dierence is simply in the choice of connectives. We have the novelty of gap-introducing modes, such as interjunction, but we have not introduced any of the familiar gap-closing

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vocabulary, which tends to have a metalinguistic avour. There is no `it is true that: : :' connective, for example, which is often introduced to turn gappy sentences into false ones. Nor can we de ne such a mode: it would not be monotonic. In Section 4 we take up the theme of non-classical vocabulary, but now we contrast simple partial logic with an altogether more sophisticated approach, viz. supervaluations. See [Van Fraassen 1966]. To illustrate the basic|but by no means the theoretically most general| idea, consider the question of evaluating a classical propositional formula under a given partial assignment of the truth values > and ? to atomic constituents. First we are to evaluate the formula in the ordinary classical way, under all total assignments which extend the partial assignment. Then the formula is taken to be > if all these total assignments make it >; ?, if they all make it ?; and  otherwise. In other words, using the de nitions we have already introduced, the supervaluational evaluation vs () of a formula  can be given by:

vs ()

=



> ?

i w() = > for all total w such that v v w i w() = ? for all total w such that v v w:

It is easy to see that this scheme of evaluation yields global monotonicity of evaluation, just as well as simple partial logic (see Section 2.5): if v v w; then vs () v ws (); for every (classical) formula . However, since the basic evaluation of formulae is just classical, the idea of using monotonic functions to give the interpretation of sentence modes has no role to play. In simple partial logic the monotonicity of a mode (p1 ; : : : ; pn ) can be stated in terms of a substitutivity condition: given any particular assignment v, and any formulae 1 ; : : : ; n ; 1 ; : : : ; n , if v( i ) v v(i ) for all i, then v(( 1 ; : : : ; n )) v v((1 ; : : : ; n )). But clearly there is nothing parallel for the supervaluational scheme. Say, for example, that vs (p) = v(p) =  and vs (q) = v(q) = >, then vs (p _ :p) = > but vs (p _ :q) = . This example points up in a particularly startling way the `intensional' character of supervaluational semantics, which is a departure from the spirit of classical logic. It is, however, a price that supervaluation theorists are willing to pay in order to preserve what is considered to be a more important feature of classical logic, viz. the stock of classical tautologies. More exactly, it is considered important to be able to capture the `logical truths' of classical logic|formulae true under any total assignment|as `logical truths' of partial logic|formulae true under any partial assignment. The supervaluational scheme makes this work, because, if  is a classical formula, then  is a classical tautology if and only if vs () = > for any partial assignment v.

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This contrasts markedly with our naive scheme of evaluation: logical truths of any kind are very thin on the ground. Indeed, only formulae containing some occurrence of one of the constant sentences > or ? can ever be true under all partial assignments. But why should we be interested in logical truth? In [Thomason 1972, p. 231], where the author is arguing in favour of supervaluational techniques in spite of their intensionality, the suggestion seems to be that the truths of a logic are supposed to capture distinctions between good and bad reasoning. But why so? Can we not leave it to the laws of logical consequence|or perhaps to a more encompassing theory of logical relationships between formulae|to capture cannons of correct reasoning? Then we might still be in a good position to show that classical tautologies are indeed `preserved' in partial logic. Consider, for example, the relation  which we de ned in Section 1.1 (or > would serve equally well). It is easy to check that, assuming  is a classical formula,  is a classical tautology if and only if [p1 _ :p1 ] ^ : : : ^ [pn _ :pn ]  ; where p1 ; : : : ; pn are the atomic constituents of . Does this not set classical tautologies in exactly their rightful place? The formula to the left of `' could never be ?, but it is not trivially >, as it would be under the supervaluational scheme: it is > precisely when all the pi are either > or ?. Observe that it would be vain to expect the logic of monotonic matrices to capture even its own relation of logical consequence in terms of truth: there can be no mode of composition (p; q) such that   if and only if ( ; ) is logically true. For if there were, then ( ; ) would be >, but (>; ?) would not be, which violates monotonicity. And this has nothing speci cally to do with our double-barrelled de nition of : it is exactly the same with either > or ? . If we wanted to introduce some special conditional connective to play the role of (
;
), then either it would have to have a non-monotonic matrix (see Section 4.4), or else it would lead to an intensional semantics of the kind we discuss in Section 2.7. However, the exercise we have set ourselves is to use the framework of consequence to set up logic without any such connective. It would be a mistake to suppose that the theory of supervaluations is not actually concerned with logical relations. On the contrary, there is much sophisticated work involved with comparing and contrasting relations of `implication', `necessitation', `presupposition', etc., etc.|for example in [Van Fraassen 1967, Van Fraassen 1971]. But here the theory quickly becomes rather abstract and we lose sight of any particular formal language. In contrast, simple partial logic puts emphasis on a particular logical vocabulary, and this includes gap-introducing connectives such as interjunction and transplication. These connectives actually prove something of a nuisance to the supervaluational idea: the de nition we gave

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for vs ( ) continues to make sense when ^ _ and = are allowed to occur in , but the point of the exercise is rather spoilt, since there will be formulae of the overall form of classical tautologies which do not come out true. For example, if  is p _ ^ :p, then there can be no v|not even a v which is already total|such that vs ( _ :) = >. In the face of this problem various supervaluational manoeuvres might be prompted: consider, for example, [Van Fraassen 1975], where Belnap's connective of `conditional assertion' (see Sections 2.3 and 4.5) is supervaluationalized. The supervaluational evaluation of a formula  under an assignment v is a boosting-up of its simple evaluation, in that v() v vs (). The question then arises what other kinds of boost-up evaluation may be de ned|in particular, what kinds k such that v() v vk () v vs ()|and [Langholm 1988] experiments with various de nitions. So long as we remain with propositional logic, these in fact turn out to yield the same result as supervaluational semantics, but corresponding de nitions of the evaluation of rstorder formulae in partial relational structures give rise to non-trivial dierences. Aside from any intrinsic interest in varying the de nition of evaluation, this proves to be a useful model-theoretic technique for investigating extensions of a classical language. However, Langholm's partial relational structures do not capture the full semantics of monotonically-representable partial functions. And, as far as I know, it remains uninvestigated how his work ts in with the model theory we introduce in Section 6 and use in Section 7. 2 SOME MOTIVATIONS AND APPLICATIONS

2.1 Varieties of Partiality In classical logic a sentence, or the assertion of a sentence in a particular context, is classi ed as either true or false: the classi cation is an assessment of propositional content against how things are|or maybe against a possible way for things to be. And the propositional content is xed as what it is precisely by conditions for its assessment. Specifying such conditions is then a way of specifying meaning for a sentence, due account being taken, in one way or another, of contextual parameters. This, roughly, is the picture that standardly goes along with classical logic. What about partial logic? Dierent concerns prompt dierent partial-logic pictures: these are not necessarily intended to surplant the classical picture, but may oer a modi cation of a part of it, or may simply oer something to complement it or to esh it out in some way. Among the variety of motivations for adopting partial logic, some will wear on their sleeves a picture they t, but others leave it a contentious matter what picture to t them into. As an introduction to this variety, I want to draw two rough and ready distinctions to

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be discerned between dierent accounts of the point of classifying sentences as > (`true'), or ? (`false'), or neither->-nor-?, rather than just true or false. First, let us distinguish between a one-tier and a two-tier framework for assessment. The one-tier framework is something like this: (1) The classi cation `neither->-nor-?' is, like > and ?, a way of assessing content expressed in (the assertion of) a sentence (in a context)|a way of assessing it against how things actually are, or against a possible way for things to be. This framework lends itself to a straightforward scheme of meaningspeci cation: a speci cation of content- xing conditions for assessment as either >, or ?, or neither->-nor-?, will be a speci cation of meaning. But it leaves open how, as an assessment of content, to understand what `neither>-nor-?' means. In what sense, if any, is this a `gap' rather than just a third truth value? How do the three classi cations >, ?, and neither->nor-? mesh with the two classical truth values, if they mesh at all?|in other words, how, if at all, does content xed by classi cation in partial logic mesh with classical propositional content? The two-tier framework, on the other hand, does not leave these questions open: (2) The classi cation `neither->-nor-?' is a way of assessing (the assertion of) a sentence (in a context) to signify that no content is expressed | nothing to be either > or ?. Then > and ? may themselves just be taken to be the classical truth values true and false. But in this framework for assessment the account of meaning-speci cation will be complicated. We seem to need both a speci cation of conditions for assessing when there is content, and a speci cation of content- xing conditions (which will be classical truth/falsity conditions). But how exactly these two tiers t together, or whether they can somehow be wrapped up into one, is left open. The two-tier framework will suggest itself most obviously|though not exclusively|when things have to do with the contribution of a context in determining propositional content. For example, it might be said of an assertion of the sentence `This is blue' that it is a precondition for there being any content to be either > or ? that there is something which, in the context of the assertion, can be understood to be what `this' stands for. The second distinction is between two dierent choices for what a sentence is to be assessed against. The contrast between a one-tier and a twotier framework was formulated with the following `global' kind of set-up in mind:

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(A) The assessment of (the assertion of) a sentence (in a context) as either >, or ?, or neither->-nor-?, is against (a formal representation of) the whole way things are, or a possible whole way for things to be. But there may be reasons to invoke a `local' kind of set-up: (B) The assessment of (the assertion of) a sentence (in a context) as either >, or ?, or neither->-nor-?, is against (a formal representation of) some part of the way things are, or some possible part of a way for things to be. The wholeness of a global set-up is not meant to rule out relativity to a particular domain of discourse, or to the vocabulary of a particular language. For example, there would be nothing non-whole about the standard model for a rst-order language of arithmetic. But in a local set-up we might be working with a mere `part' of this model which, say, consisted just of the information that 10 to 31 are natural numbers and that 10 < 30 and 11 < 29, but nothing more. In a global set-up the classi cation neither->-nor-? will arise|whether in the one-tier or the two-tier framework|in virtue of some speci c feature of a sentence, perhaps in conjunction with a feature of a particular context of assertion. But in a local set-up a dierent sort of explanation arises for the classi cation neither->-nor-?. The classi cations > and ? may be thought of as `positive' truth values that an assessment can determine, leaving `neither->-nor-?' to mean that no positive truth value is determined: a sentence may be neither > nor ? because the mere part against which it is assessed does not have enough in it to determine anything positive. Local set-ups, will not appear standing on their own: they will be constitutive of some wider semantic system which invokes assessment against partial states or stages of information in one way or another. And it will only be within the wider system that questions about propositional content and sentence meaning can be raised and answered. Three dierent ways have emerged to understand `neither > nor ?', and there would be nothing but confusion if we tried to assimilate them. But in an overall semantic enterprise more than one of these ways may be in play at the same time|perhaps independently of one another, or perhaps interdependently: there will then be issues about criss-crossing or meshing. (And to complicate things further, our characterization of a one-tier framework describes a general kind of understanding of `neither > nor ?' of which there may be various instances.) Criss-crossing would arise, for example, if we were working with a notion of content determined by conditions for (global) assessment as either > or ? or neither > nor ?, but if we also wanted a classi cation for there being no content: then, presumably, sentences would have to be classi ed as either > or ? or neither > nor ? or neither > nor ? nor neither->-nor-?. An example of meshing, on the

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other hand, will arise if the global assessment of sentences as either > or ? or neither > nor ? is to be explained as the outcome of a succession|or some more complicated structure|of set-ups for local assessment. We shall come across meshing of this sort in several places, and the question will arise whether the resulting global assessment is to be taken in a one-tier or a two-tier framework. Maybe, though, this distinction is not as cut and dried as my over-neat schematizing would suggest. We shall be scratching only the surface of the possible complexity of things. The rst few applications we consider are ones that assume global assessment, but a role for local set-ups will become increasingly more prominent as we move through the list. Some of the issues raised by the examples in this section will be discussed in subsequent sections; though the discussions still leave a lot of loose ends.

2.2 Presupposition In the context of a logic which admits of sentences which are neither `true' (>) nor `false' (?), the `presupposition' of a sentence can simply be thought of as its `either->-or-?' conditions. Then, whether we are working with a one-tier or a two-tier framework in which to specify the overall >/?conditions of a sentence, its presupposition will be constitutive of these >=?conditions. Such a notion makes quite general sense, but the terminology is usually associated with a particular application: when triclassi catory logic is deployed in an account of a particular linguistic phenomenon called `presupposition'. A paradigm example sentence would be one containing a de nite description, such as (1) The present King of France is sane. It might be said that if this sentence were used to make an assertion, then the existence of a (unique) present King of France is not thereby asserted as a straightforward `conjunctive constituent'|as it would be in an assertion of `There's someone who (alone) is presently King of France and who is sane'|but gures in some other, subtler, way: it is presupposed. Theoretical approaches to the linguistic phenomenon vary widely: see Scott Soames's chapter of the Handbook. But the kind of approach that partial logic has relevance to is that according to which the presupposition associated with (the assertion of) a sentence is to be captured semantically as a presupposition in the sense we began with. Of course, to explain what it is that is being captured in this way, we would still have to look to a wider theory of meaning|an issue we shall touch upon in some later sections. Anyhow, if we wished to construe the description `The present King of France' as a singular term, then we might be prompted to treat (1) along the lines introduced in Section 1.1. Such a treatment would make it a

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case of a `truth-value gap' caused by a denotationless term|an idea which authors on presupposition like to trace back to [Frege 1892] but associate more strongly with Strawson in his attack on Russell's theory of descriptions: [Russell 1905, Russell 1959], [Strawson 1950, Strawson 1964]. This is an encounter we ought to consider. At a super cial level it may simply be seen as a debate between someone who is sensitive to presupposition, and therefore wants to say that a sentence such as (1) is neither true nor false (Strawson) and someone who takes a conservative line that classical logic is to apply and that the sentence is just plain false (Russell). However, there are deeper stands which confuse this simple contrast. According to Russell, de nite descriptions are not properly construed as singular terms at all, but are to be de ned away in terms of identity and the quanti ers 8 and 9. Strawson, on the other hand, not only construes descriptions as singular terms but suggests a particular theory of reference for them according to which they function much like demonstratives: conditions to determine whether or not they have a denotation and, if so, what it is, cannot be schematized outside a theory about how they are used in particular contexts to refer to particular things. But then, with partial logic at hand, we might actually be prompted to side very much with Russell and against Strawson. Let us consider three progressive stages of becoming more Russellian and less Strawsonian. First, we might agree to consider descriptions as singular terms, but abandon the Strawsonian account of reference. Partial logic provides a semantics for `logically pure' terms x(x) whose denoting-conditions depend solely on the way (x) determines its extension over a given domain of objects. Perhaps we could work with such a semantics? As a residue from the Strawsonian account, we should recognize that description terms call for a contextually determined restriction on the range of the bound variable; but contextual dependence of this sort is a quite general phenomenon, in no way speci c to de nite descriptions, and it might best be treated separately|in some suitably general account of such dependence. The second stage away from Strawson towards Russell is the thought that perhaps we might not always want to construe de nite descriptions as singular terms. They share many features with quanti er phrases of the form `every F ', `most F ', and so on. And it is perhaps a virtue of Russell's analysis that it casts `the F ' as a quanti er phrase along with these other forms: the Russellian formula 9x[8y[x = y $ F y] ^ Gx]|or anything equivalent will do equally well|can be seen as an analysis of a scheme of complex quanti cation Ix[F x; Gx] for `the F is G', just as 8x[F x ! Gx] is the familiar analysis of a scheme 8x[F x; Gx] for `every F is G'. This analysis imposes classical total >=?-conditions on Ix[F x; Gx], but why not impose presuppositional >=?-conditions instead? Universal quanti cation has now come into the picture, and so it is pertinent to observe that a sentence such as



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(2) All Jack's children are bald. provides another standard example of presupposition: (2) presupposes that Jack is not childless. Hence we should think of imposing presuppositional >=?-conditions on 8x[F x; Gx] as well as on Ix[F x; Gx]. What we need for these schemes is something along the following lines:

Ix[F x; Gx] is > i there is just one F , which is G; Ix[F x; Gx] is ? i there is just one F , which is not G.

8x[F x; Gx] is > 8x[F x; Gx] is ?

i something is an F and any F is G; i something is an F and some F is not G.

These interpretation clauses remain rather informal, but it easy enough to see that Ix[F x; Gx] will be neither > nor ? unless there is exactly one F , and 8x[F x; Gx] will be neither > nor ? unless there is at least one F . In [Thomason 1979] the presupposition of universal sentences is handled in this way, though de nite descriptions remain singular terms; in [Keenan 1973], on the other hand, descriptions are handled with a scheme of quanti cation. Note that if G is a straightforward unstructured predicate, then the >=?-conditions of Ix[F x; Gx] should turn out to match those of G xF x, but Ix[F x; : : : x : : :] promises greater scope for scope distinctions than the singular term xF x (see Section 6.4). The third stage of Russellianization should now be obvious: why not provide an analysis for the scheme Ix[F x; Gx] in terms of identity and the quanti ers 8 and 9? This, of course, should be an analysis in partial logic, which captures the presuppositional >=?-conditions. And, while we are about it, why not give an analysis of 8x[F x; Gx] as well? In Section 4.2 we shall show how interjunction and transplication may be used to do this. If we work with connectives of this sort, perhaps we shall then have progressed some way towards the ideal expressed in [Thomason 1979] of a formal language `rich enough that every genuine instance of presupposition is formalizable'? Various kinds of presuppositional idiom might be tackled, since with a simple semantics for languages enriched with ^ _ or = we can produce formulae which actually exhibit non-trivial presuppositions in virtue of `logical structure' of a very basic kind. This provides something to complement abstract theorising about relations of presupposition, such as what occurs in some of the literature on supervaluations, where there is a baroque formal semantics for no particular language at all. For we should, I think, object to the contrast made in [Van Fraassen 1971, p. 138]. According to van Fraassen some non-classical logics, such as modal logic, contain `non-classical connectors', while others, such as the `logic of presuppositions', are where `one studies non-classical relations among (sets of) sentences'. No: the logic of presuppositions should be non-classical in the rst sense.





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Of course, it is easy enough in simple partial logic to de ne a formal relation of presupposing|if we want to. We can say that  (logically) presupposes if and only if  is > whenever is either > or ?. And once we have interjunction and transplication in our language, then even this simpleminded de nition becomes interesting|and even when we restrict attention to propositional logic: for example, = presupposes , and  _ ^ presupposes  $ . On the other hand, observe that we could use transplication to de ne presupposing in terms of equivalence, in a way that matches the use of conjunction in a de nition of entailment:  logically presupposes if and only if  ' =. But all this is of parenthetical interest only, since a formal relation of presupposing will have no essential role to play when a semantic theory is set up in our logic.

2.3 Conditional Assertion Related to the idea of a truth-value gap for sentences whose presupposition fails to obtain is the thought that naturally occurring conditional sentences of the form `if ; ' are neither true nor false when  is false. And in [Belnap 1970] a possible world semantics is developed for a connective `= ' of `conditional assertion' according to which, if  is false, then = is neither true nor false because it makes no assertion, in a depragmatized (sic ) sense of assertion. Otherwise = `asserts' what `asserts' (unless itself makes no assertion). In Section 4.5 we shall consider this semantics and contrast Belnap's `= ' with transplication in simple partial logic. But observe straightaway that Belnap's project is manifestly to provide a partial logic for what we called the two-tier framework for the assessment of sentences: `no assertion' means no propositional content to be either true or false. This prompts us to ask whether partial logic for presupposition should be understood in the same way. Well, any formal treatment of a Strawsonian context-involving account of presupposition would slip naturally enough into a two-tier framework (though Strawson himself might eschew a formal enterprise). But I want to suggest that such a framework would be less happy for the `logically pure' treatment we outlined for the presuppositional schemes of quanti cation Ix[F x; Gx] and 8x[F x; Gx]|or indeed for description terms xF x. For example, the presupposition of the sentence `All Jack's children are bald' is taken simply to be the condition that Jack has children: whether or not this presupposition obtains is an objective fact of the matter, and in an assertion of the sentence it may be contextually quite remote, so that it would be something of a mystery how it might be supposed to eect the question whether or not there is assessible content in the assertion. To elaborate the point, say I know Jack, and say it is taken to be `mutual knowledge' between us that Jack is a father; and say you announce `All Jack's children are bald'. Let us assume, furthermore, that only



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yesterday you had seen all Jack's children, and they were as bald as coots. Even so, if they had subsequently taken a wonder drug and had in the meantime sprouted hair, then we would say that you had made a false assertion, viz. an assertion with false content. If, on the other hand, they had all been run over by a bus, would this mean that your assertion was stripped of any content? What would the dierence be between the two cases to the success of your linguistic performance as an expression of content? In particular, what dierence to my understanding of your performance? Any attempt to explain a two-tier framework for presuppositional semantics would need to counter these re ections. At least so far as sentences like our example sentence are concerned, it would seem to make more sense to espouse a one-tier framework and to seek an account of `true', `false' and `neither-true-nor-false' simply as three dierent ways of assessing the content of assertions that sentences can be used to make, whatever status the classi cation `neither-true-nor-false' might then turn out to have (see Sections 2.4 and 5.2).

2.4 Sortal Incorrectness Some basic examples of `category mismatch', or `sortal incorrectness' motivate allowing predicate/singular-term composition to give rise to a truthvalue gap in the second of the two ways mentioned in Section 1.1, viz. because the predicate is not considered to be either true or false of a given object. For example, we might want to say that (1) The moon is sane. is neither true nor false, on the grounds that the moon is just not the kind of thing to be either sane or insane. A logically conservative response would be that this simply means the sentence is false|very obviously so. But there is a counter-response that appeals to the behaviour of negation. In the sentence (2) The moon is not sane. the negation seems naturally to `go with the predicate', just as much as it would have if we had had `insane' in place of `not sane'. If (1) is false, so should (2) be, and a certain tension then arises, since (1) seems to be the straightforward negation of (2). Precisely this tension is familiar, of course, from logically conservative treatments of paradigm presuppositional sentences, according to which presupposition failure is a straightforward case of falsity. For example, both of the following sentences would be said to be false, yet one is the natural negation of the other: (3) The present King of France is sane.

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(4) The present King of France is not sane. There is room here for considerable discussion concerning negation and ambiguity, but the fact remains that on its most natural reading (4) both appears to play a role as the direct negation of (3) and yet fails to be true for precisely the same reasons as (3). In partial logic there is no tension with negation, since failure-to-be-true is subdivided between the classi cations ? and , and we have a mode of negation which switches ? with truth (>) but leaves  xed. And so, if (1) and (3) are cast as , (2) and (4) fall into place. Indeed, a desire to do justice to the naturalness of natural negation might alone be suÆcient to motivate the apparatus of `partial' semantics. Then > and ? might be considered `proper truth values', as opposed to the `gap' , just because they are the classi cations that negation switches about. Saying this does not in itself preclude regarding  as a case of falsity (see Section 5.2). In other words, we may have an application for partial logic in a one-tier framework, along with a clear answer to the question how the three sentence classi cations mesh with the classical truth values truth and falsity: > coincides with truth, while falsity spans both ? and . However this may be, the idea of sortal incorrectness presents its own special issues, and in [Thomason 1972] the behaviour of negation is just one strand in a highly developed semantic theory. Thomason rejects three-entry matrices for giving the meaning of standard connectives and adopts a logical framework of a supervaluational kind. One reason for his doing is this is the thought that sentences of the form of classical tautologies ought to be true. In Section 1.3 we discussed|and found fault with|the general argument behind this thought; now we should consider the particular example sentence that is chosen to back up the argument. This is `What I am thinking of is shiny or not shiny'. Thomason points out that if we were using three-entry matrices, it would be necessary to nd out what is being thought of before we can say whether or not the sentence is true. It would be true if I were thinking of an apple, say, but sortally incorrect, and hence neither true nor false, if I were thinking of the number 2: this is because on any matrix approach|at least, on any non-eccentric one| _ : would be  if  were . However, it is not clear why this fact should constitute a special problem for matrices or provide any extra ammunition for the general argument, though it is presented as if it did. This is especially puzzling, given the way Thomason deploys the related sentence `What I am thinking of is shiny' against a `syntactic' account of sortal incorrectness, according to which sortally incorrect sentences are intrinsically ungrammatical. For he points out precisely that we cannot know just by looking at the sentence whether or not it is sortally incorrect: the answer depends on discovering what is being thought about. This is a neat argument, but it will be an

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uncomfortable one if it is considered to be a problem when we cannot tell a priori the sortal correctness or incorrectness of a sentence.

2.5 Semantic Paradox A partial-valued approach to the semantic paradoxes rivals the `orthodox' Tarskian account of a hierarchy of languages, in which the semantical predicates of a given language can apply only to the language immediately preceding it in the hierarchy. On this account, a simple paradoxical sentence such as `This sentence is false' would be ruled out as anomalous on the grounds that there can be no place for it in a hierarchy. But in [Kripke 1975] an argument is deployed against the Tarskian theory very similar to the one Thomason deploys against a syntactical account of sortal incorrectness. The point is that paradoxicality cannot be seen as an intrinsic anomaly of given sentences|or for that matter of given con gurations of sentences|since even the most innocent of truth-assertions and falsity-assertions can, in unfavourable circumstances, turn out to be paradoxical: examples of this involve people talking about one another's assertions. A lot of work has recently been done on the paradoxes|and a lot of that involves partiality in one way or another: see Visser's chapter in the Handbook (and see Section 2.10). Here I shall focus on Kripke. To replace a syntactical hierarchy of truth predicates in dierent languages, he proposed a single language containing its own partially de ned truth predicate. This idea had previously occurred in various authors (see [Martin 1970]), but Kripke took up the formal challenge of addressing particular interpreted languages, such as arithmetic, which are suÆciently rich already to provide the kind of self-reference that leads to paradox. Brie y described, his procedure is to graft a predicate symbol T onto a language and then to expand its interpretation so as to make T a truth predicate. It is a truth predicate in the sense that for any sentence  (of the expanded language), if  is a name in the language for , then

T  is true (>) i  is true (>); T  is false (?) i  is false (?).

We shall be able to de ne a `Liar sentence' , such that  is true if and only if :T  is true, and such that  is false if and only if :T  is false, but there is no contradiction:  and :T  will both be neither true nor false. The construction of a model to interpret T depends on the monotonicity of evaluation that partial logic can provide (see Sections 1.2 and 6.2). Kripke considers a supervaluational scheme of evaluation, but seems to prefer simply partial logic (see Section 5.1). The actual method of model construction is a trans nite induction similar to ones used, for example, in [Gilmore 1974], [Feferman 1975] and, most cunningly, in [Scott 1975]. And compare Aczel's induction in the appendix

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to [Aczel and Feferman 1980]. These references all have to do with systems of type-free class abstraction, where paradoxes are diused by going unde ned: in particular, Scott de nes truth/falsity conditions appropriate to turn a model for the -calculus into a partial-valued language of classes. From a set -theoretical point of view all these systems pay a rather high price, viz. the loss of extensionality, but some work has also been done using partial logic to set up extensional set theories: see [Hinnion 1994]. Truth theories and set theories are the obvious lairs for paradox, but it lurks too in quotational logic|logic set up in a language with explicit devices for talking about itself. For example, a sentence such as

M = \ 9p [\p" = M

^ :p ]"

may be thrown up, where M is a sentence name, and p is a sentence variable. If p is taken to range over all sentences, and if our background logic is classical, then we have a version of the Liar. One strategy for avoiding trouble is to impose a ranking on sentence variables, and a quotational logic with such a ranking is investigated in [Wray 1987a]. But Wray ends with a proposal for adopting partial logic as the background logic, so that variable-ranking can safely be dropped. And this proposal is carried through in [Wray 1987b]. In his article Kripke criticised other authors who had wanted to defuse the paradoxes by going partial, on the grounds that they did not provide `genuine theories'|no `precise semantical formulation of a language at least rich enough to speak of its own elementary syntax', and no `mathematical de nition of truth'. However, there is a sense of `theory' in which Kripke himself did not provide a theory: that is to say a formal theory in the language for which we have a `precise semantical formulation' and a `mathematical de nition of truth'. Kripke's de nition of truth is a metalinguistic model-theoretic construction and he left it at that. He provided no system in which a truth-language can express its own semantical principles, let alone any stock of basic `axioms' to generate such principles. I want to suggest that the way to ll in this gap is to use the de nition we shall give in Section 7 of what a `theory' is in partial logic. It is not clear, though, what Kripke himself would make of the suggestion, since he claimed that his logic is utterly classical. We shall pursue this thought a little way in Section 5.1.

2.6 Stage-by-stage Evaluation The bare existence of models for a semantically closed language is only half of Kripke's story about truth: the construction he employs to demonstrate the existence of such models is associated with an intuitive picture of how sentences can be evaluated as true or as false. In terms of this picture an

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account is given|along lines originally explored in [Herzberger 1970]|of `paradoxicality' and related notions. The monotonicity of evaluation now comes to life as a persistence condition governing a procedure of evaluation which runs through stages of increasing information. At a given stage the truth predicate has been de ned to a given extent and sentences can be evaluated at that stage in the ordinary way|according to simple partial logic or a supervaluational scheme. But this evaluation then determines the truth predicate for the next stage of evaluation. The truth predicate becomes more de ned, and as it becomes more de ned so more sentences become true or false, and the truth predicate becomes still more de ned . . . and so on. Monotonicity ensures that once a sentence has taken on the value `true' or `false', and the interpretation of the truth predicate has been strengthened accordingly, then it can neither become unde ned nor switch truth value at any later stage of evaluation. Recall the distinction we drew in section 2.1 between a `local' and a `global' set-up for assessing sentences. It would not seem inappropriate to think of the evaluation of sentences at each particular stage of information as a local set-up. But the succession of stages leads up to a global set-up, viz. a stable model to interpret semantically closed partial languages: this model can be seen as the result of pursuing a stage-by-stage evaluation process until it settles down and no new true or false sentences are produced. By general principles governing the inductive de nition behind this process it must settle down sooner or later, though in the case of interesting languages this will not be without trans nite leaps to limit-ordinal stages, where all previous truths and falsehoods are gathered up to de ne the new interpretation of the truth predicate. Assuming, then, that the model we end up with constitutes a global set-up, the question arises whether it provides a one-tier or a two-tier framework of assessment. `One-tier' would seem to be the obvious answer, but this seems to con ict with some of Kripke's own remarks, and we shall return to the question in Section 5.1.

_^ _^ _^ However this may be, another, and in some ways rather simpler, illustration of monotonicity as a constraint in the context of a stage-by-stage process evaluation is provided by the discussion of partial recursive predicates in [Kleene 1952, Section 64]. `Kleene's strong matrices' are introduced here|the same matrices that we presented in Section 1.2. A partial recursive predicate P (~x) may be unde ned for some n-tuple ~a of numbers, and, accordingly, Kleene rst oers the simple gloss `true', `false' and `unde ned' for the matrix entries >, ? and  (for which he used `t', `f ' and `u'). These classi cations are intended to apply to sentences built up out of partial recursive predicates, and the point of monotonic (which Kleene calls `regular') matrices can be described in terms of the derived role sentence modes play as modes which compound predicates. For, if (p1 ; : : : ; pn ) is

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a monotonic mode of sentence composition and P1 (~x); : : : ; Pn (~x) are partial recursive predicates, then (P1 (~x); : : : ; Pn (~x)) is partial recursive also; while, conversely, if (p1 ; : : : ; pn ) is not monotonic, then we can nd predicates P1 (~x); : : : ; Pn (~x) which are themselves partial recursive, but which are such that (P1 (~x); : : : ; Pn (~x)) is not. (See Kleene's Theorems XX and XXI.) Kleene explains and illustrates monotonicity in terms of a particular kind of algorithm for the interpretation of partial recursive predicates. For a given input ~a, one of these algorithms will either yield the output `true', or yield the output `false', or else go on for ever. A second, `computational', construal then emerges for the matrix entries: `true', `false' and `unknown (or value immaterial)'. These are classi cations for a sentence P (~a) which can be applied at successive stages in pursuing the algorithm for P (~x) with input ~a. The matrix for a given connective, _ say, re ects the way algorithms for predicates Q(~x) and R(~x) are to be combined to yield an algorithm for Q(~x) _ R(~x). The classi cation  means `unknown' because if the value > or ? has not been decided at a given stage, then we do not know what might or might not happen at a further stage. On the other hand, it can also be glossed `value immaterial', since we may be able to determine the value > or ? for a compound sentence independently of some constituent sentence which remains . For example, Q(~a) _ R(~a) can be evaluated as > if R(~a) has been decided as >, even if Q(~a) remains . The original objective construal of the matrix-entries now falls into place in the following way: `true' applies to sentences which are decided as > at some stage, `false' to those which are decided as ? at some stage, and `undecided' to sentences which are never decided as either > or ? at any stage|in other words, which remain  for ever. Thus Kleene's algorithms can never actually tell us that a sentence P (~a) is unde ned. (And since, if P (~x) is partial recursive, it is, in general, undecidable whether or not P (~a) is de ned, it would, in general, be vain to demand a dierent kind of algorithm which did tell us.) This explains why none but monotonic connectives are admissible: a resultant value > or ?, decided by a compound algorithm, is allowed to depend only on out-put values > or ? from constituent algorithms|never on the classi cation . (See Sections 1.2 and 3.2). Here we appear to have a paradigm for the use of monotonically representable partial truth-functions. But in [Haack 1974, Haack 1978] it is claimed that Kleene ought rather to have used a supervaluational scheme of evaluation|indeed that his own arguments dictate this. There is no space to do full justice to Haack's remarkable claim, but it would appear to depend primarily on two things. The rst is that Kleene mentions a secondary application for his matrices|to sentences built up from total predicates of a kind which are decidable (by one of his algorithms) on part of their domain and have their extension over the rest of the domain given by a separate stipulation. It seems that this enables Haack to misunderstand Kleene's gloss for  as `lack of information that a sentence is > or is ?' to mean

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lack of information which of either > or ? it is. Kleene does not mean this, however:  (under its computational construal) signi es lack of information whether a sentence is > or ? or  for ever. It is diÆcult to see what sense Haack can have made of Kleene's discussion of the `law of the excluded fourth', which is required to advance from the computational to the objective construal. Secondly, and connected in some not altogether clear way with the mistaken idea that all sentences under consideration are really either > or ?, there seems to be a confusion between the constraint of monotonicity (regularity) and a totally dierent point about the particular matrices chosen for classical connectives: that they are, in Kleene's words, `uniquely determined as the strongest possible regular extensions of the classical 2-valued truth-tables'. For Haack never actually mentions the notion of regularity, but she interprets Kleene's explanatory discussion of the constraint as if it were some kind of direct argument for a desideratum that modes of composition be as strong as possible. In [Haack 1974] she reports on Kleene's illustrative discussion of _ (which I sketched above), but she seems to get the point back-to-front. And, in conclusion, she is prepared to announce the `underlying principle' to be that `if F (A; B; : : :) would be > (?) whether A; B; : : : were true or false, then it is to be > (?) if A; B; : : : are '. If Kleene's principle were something like this, then perhaps we should consider supervaluational semantics. But it isn't and we shouldn't.

2.7 Stages, States, and Exotic Connectives Partial logic extends in various directions to more elaborate kinds of semantics than we shall be pursuing. In one direction the computational idea of a process of evaluation can actually be built into the interpretation of some of the logical connectives: consider for example the semantics in [Thomason 1969] for the theory of constructible falsity. This theory is a kind of two-sided intuitionism whose proper constructivist interpretation| handled in [Nelson 1949] and [Lopez-Escobar 1972]|would appeal to twin notions of `provability' and `refutability' in the way that intuitionists appeal just to provability. But for a model theory we can consider a two-sided version of Kripke's semantics for intuitionistic logic. For simplicity of illustration let us consider just a propositional language. Models can then be taken to consist of a set V, whose elements , are each associated with a partial assignment v of > and ? to atomic sentences, and a re exive transitive relation  on V, which satis es the condition that if   then v v v . The elements of V are to be thought of as stages of information; and the condition on  is meant to embody the idea that when   then has all the information at  but possibly more besides. Formulae are then evaluated at stages in V. For atomic sentences the persistence of truth value (> or ?) through stages of increasing information

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is constitutive of the model, and the guiding constraint on evaluation rules is that this persistence be extended to all formulae. In other words, our de nition of v () must be such that, for any  if   then v () v v (). The evaluation of negations, conjunctions and disjunctions, at a given stage, involves only the classi cation at that stage of their immediate constituents|according to the >=?-conditions of simple partial logic. But the evaluation of conditionals involves constituent classi cations at stages of further information. Thus we have a system of local set-ups for assessment with a special kind of interdependence between the set-ups: it resides in the actual assessment conditions of a logical connective. Thomason and Lopez-Escobar give the following >=?-conditions:

v ( ! ) = > i for every  , if v () = > then v ( ) = >; v ( ! ) = ? i v ( ) = > and v ( ) = ?. Notice that in fact it is only the >-conditions that appeal to further stages. But, in virtue of them, ! matches a truth-preservation consequence relation:  ! is true at any  in any model if and only if, in any model, is true at any  at which is true. We can take this to mean that  ! is logically true if and only if is a (single-barrelled) logical consequence of . This is how the theory has grown up, but the >-conditions for ! could easily be modi ed to match a double-barrelled notion of consequence|one which also requires preservation of falsity from conclusion to premiss. And we might also adopt stronger ?-conditions which, like the >-conditions, appeal to further stages of information, and which match the failure of consequence:

8 < if v () = > then v ( ) = > v ( ! ) = > i for every   : if v ( ) = ? and then v () = ?; 8 6 > < v () = > and v ( ) = or v ( ! ) = ? i for every   : v ( ) = ? and v () 6= ?.

The full point of adopting this strong interpretation of ! only emerges if we consider setting up non-logical theories in this sort of language:  ! will be true in all models of a theory if and only if follows from  in the theory; and  ! will be false in all models of a theory if and only if 's following from  is inconsistent with the theory. The details of this would take us too far a eld, but see Sections 6.5 and 7.1 for non-logical theories in simple partial logic|and for an indication how to spell out theory-relative notions of `following-from' and `(in)consistency'. Anyhow, in the framework of this kind of model there are various ways of ringing the changes on the

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interpretation of particular connectives, and obviously a variety of dierent connectives could be introduced. A similar framework is provided by the `data semantics' of [Veltman 1981]: `data sets' play the role of stages of information, and an increase-of-information ordering is given simply by the relation  between these sets. In this framework Veltman interprets a pair of operators for `it may be that' and `it must be that'. But analogous operators can be introduced into the two-sided Kripke models we have set up: let us write `3' and `'. For 3 the >=?-conditions will be that

v (3) = > i for some  , v () = >; v (3) = ? i for every  , v () = ?; and  is dual to 3:  is equivalent to :3:. In [Turner 1984] and [Wansing 1995] the consistency operator M of [Gabbay 1982] is translated into a partial-logic setting by giving it precisely the interpretation we have given 3. But observe that we have now introduced a crucial departure from the original models: the general persistence condition|that if   then v () v v ()|has now broken down. It is scuppered by the >-conditions for 3 (and dually by the ?-conditions for ). The search for exciting new operators can be continued by observing that 3 and  are a special case of something more general: `dynamic' operators hi and [], formed from a formula . For hi the >=?-conditions will be that

v (hi ) = > i for some  , v () = > and v ( ) = >; v (hi ) = ? i for every  , if v () = > then v ( ) = ?: And, again, [] is dual to hi: [] is equivalent to :hi: . The formulae hi and [] could in fact be thought of as kinds of conditional|`if , then it may be that ' and `if , then it must be that '. (Notice that the >-conditions of [] , though not the ?-conditions, are exactly the same as those we originally gave for  ! .) Anyhow, 3 and  can now be captured as h>i and [>]. In [Jaspars 1995] a logic is presented which not only contains these `upwardlooking' operators, but also a (mutually dual) pair of `downward-looking' ones|let us write hi0 and []0 |whose >-conditions and ?-conditions at  involve quantifying over  . The >=?-conditions for hi0 are:

v (hi0 ) = > i for some  , v () 6= > and v ( ) = >; v (hi0 ) = ? i for every  , if v () 6= > then v ( ) = ?. Jaspars glosses hi0 as meaning `it is possible to retract  from the current state [of information] in such a way that holds afterwards'. Now,

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this takes us even further away from the original idea of a two-sided intuistionistic system than 3 or hi does. Originally we were to think of the elements of V as representing progressive stages in a process of discovery, for which the quasi-ordering  represented possible advances in information| indefeasible advances, which once achieved remained rm. The idea of `losing' information had no role to play in interpreting the language, and the possibility that we might not only lose information, but subsequently `advance' in a dierent and incompatible way, would have been in clear con ict with the intended interpretation of the model. But this possibility is now envisaged: we have variable states, not progressive stages, of information. [Wansing 1993] is a comprehensive essay investigating the ups and downs of all this; and [Wang and Mott 1998] provides a discussion of how quanti ers t in. Jaspars emphasizes the dynamic character of his semantics by de ning two relations over the elements of V which a formula determines as its `dynamic meaning':

[ ] > i   and v () = >; [ ] > i   and v () 6= >. Thus [ ] > ([ ] > ) means that is a possible way of extending (reducing)  to include (remove) the information that  is true. The notation used here is mine; in particular, I have put in the subscript `>' to point up the one-sidedness of these de nitions: there is a complementary pair of relations, de ned by replacing `>' with `?'. These relations between states of information have been de ned in terms of  and the evaluation of a formula at a state of information (which is itself de ned in terms of ). But an alternative strategy would be to take relations that determine dynamic meaning as semantically primitive|to de ne them directly, by recursion on the complexity of formulae. De nitions of this kind, giving an explicit `dynamic semantics', are very popular nowadays: further examples appear at the end of Section 2.10 and in Section 4.3. In Section 4.3 there are also some general remarks on the very idea of a dynamic semantics.

2.8 Under-de ned and Over-de ned Another way to extend simple partial logic is to consider more truth-value classi cations than just >,  and ?. In particular, if  means `neither > nor ?', what about a classi cation for `both > and ?'? This might even make some sense in an application where `neither > nor ?' signi es a kind of unde nedness that is under de nedness: there might then be a correlative notion of overde nedness.

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There is in any case an irresistible temptation to add a top element| never mind what it could mean|to the degree-of-de nedness ordering on truth-value classi cations. This yields a four-element lattice: (>)

(t)

>

?

(f )



(?)

The labels in brackets are the ones used in [Scott 1973a]. Let us call this lattice D0 : the beauty of Scott's idea is that D0 can be naturally embedded into the domain D1 of monotonic functions from D0 into D0 |and this in a way which provides the basis for embedding D1 into its monotonic function space D2 , and so on. There is a sequence of nested domains, and a limit domain can be de ned which constitutes a system of type-free functions closed under application and abstraction|a model for the -calculus. But in fact a similar construction can be carried out if we start with our more modest semi -lattice of >,  and ?|see [Barendregt 1984], for example|and so there is no special motivation here for adding `over-de ned' as a fourth truth-value classi cation. What a -calculus model of this sort provides is a kind of higher-order, but type-free, partial propositional logic: truth values and truth functions inhabit a single uni ed domain. Quanti ers, however, would seem to present something of a stumbling block in attempts to provide a full-blown type-free partial logic by means of this sort of construction. Application in a limit domain is, loosely speaking, de ned in terms of approximations from preceding domains, and, even if we iterate the construction through trans nite stages, it is not clear how successive approximations could ever build up to any decent de nition of quanti cation as a function both ranging over and contained in a limit domain. (The workable de nitions I have discovered so far perhaps just about count as non-trivial, but they specify too weak a notion of quanti cation to be useful.) Furthermore, it does not seem that starting with the four-element lattice of truth-value classi cations would oer any advantage. Intensional type-free logic, on the other hand, is much easier to obtain: consider [Scott 1975] and the other similar work mentioned in Section 2.5. Anyhow, the idea that a sentence may be over-de ned, in being both true and false, is one that paraconsistent logicians would like to make serious sense of: see Priest's chapter in the Handbook . But in this chapter we need only advert to places in work we have already mentioned where the

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four-element lattice plays a role. First, then, it turns up in the type hierarchy of [Muskens 1989] (see Section 1.2). Secondly, the general framework set up in [Langholm 1988] allows both-truth-and-falsity as well as neithertruth-nor-falsity, though a de nition of `coherence' is immediately given to delineate those logics which run on just the three truth-value classi cations >, ? and  (see Section 1.3). And again in [Bochman 1998] partial logic turns out to be a special case in a more general four-valued framework (see Section 1.1). Compare, too, the work on the paradoxes in [Visser 1984].

2.9 Non-deterministic Algorithms But four is still a small number: there are even more truth-value classi cations in the `non-deterministic partial logic' developed for the semantics of programming languages in [Pappinghaus and Wirsing 1981]. This logic is applicable to the evaluation of sentences under `non-deterministic algorithms'. The algorithms are `non-deterministic' because at given stages in pursuing them a choice may be left of ( nitely many) dierent ways to proceed. Assuming a particular choice is always made, then a sentence will either be evaluated as > or as ?, or else remain unde ned () (either because the procedure grinds to a conclusionless halt or because it goes on for ever). But dierent choices might result in dierent resultant classi cations. And so, for a given non-deterministic algorithm, there is a spread of alternative classi cations. The seven values of Pappinghaus and Wirsing's logic are the dierent possible spreads: the non-empty subsets of f>; ; ?g. The authors explain various constraints on the interpretation of modes of sentence composition and provide a stock of connectives which is expressively complete for modes meeting these constraints. I am too out of touch properly to survey the role partial logic and its relatives have played in computer science. But I do know that in an extended version of [Blamey 1991], a degree-of-de nedness ordering derived from partial logic is called in to handle divergence|along with non-determinism|in models for CSP processes.

2.10 Situation Semantics `Situation Semantics' was introduced in [Barwise and Perry 1981a, Barwise and Perry 1981b] as a rival to the Fregean tradition in semantics according to which truth and truth conditions are central notions. Rather, it was argued, situations and truth-in-a-situation conditions are central. Quite an industry has subsequently developed, and there is now a chapter in the Handbook which is dedicated to the theory of situations. Here I shall restrict attention largely to the early foundational papers. In this work objects and relations are taken as metaphysically basic, and|suppressing complications to do with time and place|situations are then con guarations of objects and re-

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lations. They could be modelled over a given domain D of objects as partial functions from the set of all (n + 1)-tuples consisting of an n-place relation on D and n elements of D into the truth-values > and ?. (Empty argument places of the kind considered in Section 1.2 do not enter the picture here: we can take these functions, modelled set-theoretically, just as subsets of total functions.) Thus situations turn out to be a kind of partial model and provide a paradigm for the idea of a local set-up for the assessment of sentences. A simple sentence such as `John hits Mary', for example, would be true (false) in a situation s if and only if s(hits; John; Mary) = > (?). We might, then, think of the meaning of a sentence  as a predicate of situations|one which determines, as its truth-sided interpretation, the set `[[] > ' of situations in which it is true. (Barwise and Perry use `[[] ' for this set.) However, we can only think in this way once a number of parameters have been lled in. For the linguistic meaning of a sentence, just like that of any subsentential item, is given as a many-place relation with an array of argument places designed to reveal its sensitivity to both linguistic and non-linguistic context: and a great many of these argument places are for situations. For example, a de nite description is evaluated for a denotation relative to a situation|a situation which can cross-refer in various ways with situation slots elsewhere in the architecture of a sentence, possibly, but not necessarily, to be ultimately determined by the context of utterance. Furthermore, situations are taken to be the very objects of perception in certain `naked in nitive' constructions such as `Hilary sees Mary hit John': roughly, this would be true in a situation in which Hilary sees a situation in which Mary hits John. Along these lines Barwise and Perry oer an account of the `logical transparency' of such `: : : sees ' contexts, which contrasts with the opacity arising in sentences of the form `: : : sees that '. Anyhow, if we ignore the internal structure of situations, then they can be thought of just as `partial possible worlds'|points with respect to which sentences are to be evaluated as >, ? or neither > nor ?. This prompts comparison with other work: for example, in [Humberstone 1981] partial possible worlds are called `possibilities' and are used to provide a semantics for traditional modal logic. (And see [Van Benthem and Van Eijck 1982, Fenstad et al. 1987, Van Benthem 1988] for more exploration of interconnections.)

_^ _^ _^ Classically propositions are often modelled as sets of possible worlds, but what happens if we are working with partial possible worlds or situations? In the early work we are considering Barwise and Perry suggest modelling propositions as sets of situations satisfying the coherence condition that if s 2 P and s  s0 , then s0 2 P . And the interpretation sets [ ] > then turn out as propositions. This approach to propositions is later abandoned (see below), but it is worth pursuing a little way, if only as partial-

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possible-world theory. In particular, the question arises how to de ne logical operations over propositions|operations to interpret modes of sentence composition. Conjunction and disjunction, obviously enough, turn out to be just intersection and union, so that [  ^ ] > = [ ] > \ [ ] > and [  _ ] > = [ ] > [ [ ] > . But what about negation? Barwise and Perry do not actually treat negation as a mode of sentence composition: it turns up in more complicated categories. Even so, given a proposition P , another proposition P  = fs j s 2 P g is determined, where s is the situation obtained from s by reversing the values > and ?. And for basic sentences , such as `John hits Mary', [ ] > turns out to be the set [ ] ? of situations in which  is false: this looks to be a likely candidate for [ :] > . But to cater for the negation of complex sentences, we had better modify our representation of propositions so that they have their negative side explicitly built in. If we take pairs hP; P  i of Barwise-and-Perry propositions to interpret sententially atomic items, then to interpret compound sentences we can use the following clauses:

h[ :] > ; [ :] ? i h[  ^ ] > ; [  ^ ] ? i h[  _ ] > ; [  _ ] ? i

= = =

h[ ] ? ; [ ] > i; h[ ] > \ [ ] > ; [ ] ? [ [ h[ ] > [ [ ] > ; [ ] ? \ [

] ? i; ] ? i:

And we can add clauses for interjunction and transplication too:

h[  _^ h[  =

] >; [  _ ^ ] ?i = ] >; [  = ] ?i =

h[ ] > \ [ ] > ; [ ] ? \ [ ] ? i; h[ ] > \ [ ] > ; [ ] > \ [ ] ? i: These equations of course just model the >=?-conditions proposed in Sec-

tion 1.1. We should (in parenthesis) observe that the same equations will serve if we are interested in capturing not the local assessment of a formula in a system of situations, partial possible worlds, or whatever, but rather the global assessment of a formula against complete possible worlds|against whole possible ways for things to be. If v (p) is the (partial) evaluation of an atomic sentence p at a possible world , the following pair gives the interpretation of p:

hf j v (p) = >g; f j v (p) = ?gi: Then, given an arbitrary formula , we may invoke the displayed equations to x the sets of possible worlds [ ] > , in which  is >, and [ ] ? , in which  is ?. Assuming that this provides a one-tier framework of assessment| and it requires some ingenuity to see it providing anything else|the pair h[ ] > ; [ ] ? i then models the `partial proposition' that  expresses. Anyhow, in the framework of de nitions of this kind, a natural version of our double-barrelled consequence relation would be that   if and only

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if both [ ] >  [ ] > and [ ] ?  [ ] ? . Barwise and Perry use just the rst conjunct of this to de ne a notion of consequence (matching >)|and hence to de ne equivalence as bi-consequence. Thus de ned, consequence and equivalence are stronger, and so more discriminating, than relations which the authors grudgingly label `logical' and de ne as follows: is a `logical consequence' of (is `logically equivalent' to)  if and only if, if s is any total situation, then s 2 [ ] > only if (if and only if) s 2 [ ] > . (Total situations are just situations that are total functions.) It is then one strand in their argument against the Fregean tradition that trouble results if `logical' equivalence is expected to play a role which should rather be played by the more discriminating relation. This involves ringing the changes on the problem, if substitutively of `logical' equivalents is allowed, of non-truthfunctional modes of composition which create extensional contexts. The general aim here is to point up oddities which result from thinking directly in terms of truth values (and truth conditions), rather than situations (and truth-in-a-situation conditions). But it's far from clear that this aim is met. The more discriminating relation of equivalence has nothing specially to do with the local set-ups of situation theory: it is available in any partial semantics. Oddities may equally well be avoided by going partial with a global set-up for assessment|and thinking directly in terms of the truth values > and ? (and >/?-conditions).

_^ _^ _^

In [Barwise and Etchemendy 1987] the apparatus of situations is invoked to address semantic paradox. This involves subjecting the notion of a proposition to some scrutiny, and we are oered two conceptions|`Russellian' and `Austinian'. Under either conception, the formal modelling of propositions is very dierent from the one presented above. First we have to have `states of aairs': these are the basic constituents of situations, which, working with the de nition we set out at the start, can be taken just to be the members of the sets representing the partial functions that model situations, viz. (n + 2)-tuples consisting of an n-place relation, n objects, and a truth value (> or ?). Russellian propositions are then de ned as constructs built up from states of aairs, in much the way formulae of a formal language are built up from atomic sentences. At bottom we have basic propositions which just are|or directly correspond to|individual states of aairs, and these will be true in a situation if and only if they are contained in it: truth conditions for arbitrary propositions can then be given by recursive clauses that follow their construction, in just the way that clauses are given for evaluating formulae. Austinian propositions, on the other hand, have a particular situation built in as a kind of contextual parameter|what the proposition is `about'. A construction from states of aairs gives a `proposition type ', which needs to be paired with a situation to model an actual

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proposition. So an Austinian proposition contains within itself a situation with respect to which it is true or not. To give an adequate perspective on the Liar sentence, Barwise and Etchemendy espouse Austinian propositions. The Liar sentence is to be taken in a situation-supplying context and will express a proposition about that situation. If s is the situation supplied and ps is the proposition expressed, then ps will be a constituent of itself: its proposition type will consist just of the state of aairs hT; ps ; ?i, where T is (an item to represent) the property of being true. (Aczel's theory of non-well-founded sets is invoked as the framework in which to de ne such self-re exive propositions.) Thus ps will be true if and only if hT; ps ; ?i 2 s. But assuming that no situation can be unfaithful to semantic facts, so that hT; ps ; ?i 2 s only if ps is not true, it follows that ps will not be true|in other words, hT; ps; ?i 62 s. And, since s cannot be unfaithful to this fact, hT; ps ; >i 62 s. But the modelling of propositions leaves no room for the conclusion that ps is therefore neither true nor false: separate falsity conditions are not de ned, and so, because ps is not true, it's counted simply false. Rather than admitting a neither-true-nor-false proposition, we are invited to appreciate the inevitable partiality of the situation . This means we could always extend s to a situation s0 = s [ fhT; ps ; ?ig, which includes information about the proposition the Liar sentence expressed|though of course in a context supplying this situation the Liar sentence will express a dierent proposition ps0 , and hT; ps0 ; ?i will not be contained in s0 . In [Groeneveld 1994] the idea that the Liar sentence actually drives us on from situation s to situation s0 is taken up and built into a semantics for languages in which the Liar sentence can be formulated. Partial logic now comes back into the picture|a dynamic partial logic, for which a pair of relations [ ] + and [ ] are de ned between situations (`+' for >, and ` ' for ?). These may be glossed as follows: s[ ] + s0 if and only if `s0 is the weakest extension of s that covers the information of '; s[ ] s0 if and only if `s0 is the weakest extension of s that rejects the information of '. 3 FREGEAN THEMES

3.1 Reference Failure In Section 1.1 we announced that we should, in partial logic, be able to do justice to the idea that a sentence (t) can be neither > nor ? because some constituent term t is non-denoting. This calls to mind Frege's theory of reference (Bedeutung ), according to which the truth value `true' or `false' is the reference of a true or false sentence, just as the object denoted by a singular term is its reference, and according to which there is a general principle of reference failure that any compound expression lacks a reference

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whenever any constituent expression lacks a reference. This principle would then explain particular claims that (t) is neither > or ? `because' t is nondenoting. Of course, our partial logic does not obey this strict principle: if the range of interpretation for predicates (x) is the system of monotonically representable partial subsets of a domain (see Section 1.2), then, since an empty argument place does not necessarily mean no output value, (t) could be > or ? even if t is non-denoting. But can we argue that our semantics provides some other, subtler, general principle to give more than ad hoc content to particular claims that (t) is neither > nor ? because t is nondenoting? This question leads to thoughts that are in any case prompted if we pursue a Fregean parallel and think of f>; ?g as the range of reference for sentences and of a domain of objects, or indeed the corresponding classi cations `denoting so-and-so', as the range of reference for singular terms. And it is diÆcult to avoid the parallel. This is not because of any conception external to systematic semantics of what the `reference' of a sentence or singular term is to consist in, but simply because it is a central strand in Frege's theorising that compound reference be (functionally) dependent on constituent reference: the parallel points up precisely the dependence that must obtain according to the idea that modes of composition are interpreted by partial functions. But then there might seem to be a problem, since the dependence of reference on reference is supposed to be intimately connected with the strict Fregean principle of reference failure, which our logic does not obey. The connection is made (in rather dierent styles) in [Woodru 1970, Dummett 1973, Haack 1974, Haack 1978], for example| and a host of more recent references could equally well be given. Haack even presents a deductive-looking argument to the eect that the principle actually follows from the idea of dependence. To defend our framework from the charge that its range of modes is too liberal for it to be understood as a semantics of partial functions, we have to argue that, on the contrary, the dependence of reference on reference does not in itself dictate the crude principle that a compound () lacks a reference whenever any constituent  lacks a reference. Such an argument will be attempted in Section 3.2. It is not, of course, just a matter of predicate/singular-term composition: either () or  could be either a singular term or a sentence. And at the end of Section 3.2 we shall generalize the question even further. Frege himself regarded reference failure as a defect of ordinary language, and in his systematic logical language he went to great, and often arti cial, lengths to avoid any kind of unde nedness arising. In [Frege 1891] the suggestion seems to be that logical laws could not be given otherwise. Perhaps this was because he tended to assimilate any kind of unde nedness into an intractable kind of `vagueness', but it might anyway have seemed rather impractical to do with logic with so many gaps. In our semantics, however, there are not so many gaps. Moreover, what gaps there are will not ham-

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per our formal development as they would have hampered Frege's, because we shall be presenting logic in terms of consequence rather than truth (see Sections 6.5 and 7.1.)

_^ _^ _^ However this might be, let us brie y consider some meta-semantical discussion of the Fregean idea that (t) seriously `lacks' a reference (truth value) when t `lacks' a reference (objected denoted). [Dummett 1973, Chapters 10 and 12] approaches the matter by discerning dierent strands in Frege's notion of reference, and the possession of a `semantic role' is taken to be the only strand in common between sentences and singular terms: the semantic role of an item turns out to be what we have been calling its `semantic classi cation', though the notion of semantic role is anchored to more fundamental ideas (see Section 5.2). First, then, we should distinguish the realm of objects that can be denoted by terms from the realm of semantic roles, which includes the classi cation `non-denoting'. Secondly, Dummett also insists on a distinction between the notion of `truth-value' in the sense of semantic role, viz. classi cation or matrix entry in whatever semantics there is reason to adopt, and notions of truth and falsify applicable in the evaluation of what someone asserts using a sentence. Hence, no purchase is to be gained on the idea of sentences actually lacking a truth value by drawing a parallel with names lacking a bearer. Moreover, according to Dummett, whenever anyone ever asserts anything, one or other of the truth values in the second sense must apply (see Section 5.2). There is, though, room for the idea that a sentence may be neither `true' nor `false' if these labels apply to two, among more than two, semantic classi cations. Dummett takes bearer-less names to be a paradigm source for the problems with negation that we discussed in Section 2.4, and, as we saw, these problems motivate a triclassi catory semantics. According to Dummett we are concerned throughout with singular terms possessing a Fregean sense (Sinn ), understood as a cognitive content which determines, but is independent of, the object, if any, denoted. In that case, there is no question of denotation failure in any way infecting what a sentence can express, and the right foundations for the use of partial logic to handle possibly-non-denoting singular terms will then be what in Section 2.1 we called a `one-tier' framework for assessment. However, it would be contentious to assume that all singular terms can properly be treated in such a framework. It has been argued that the function of at least some singular terms is to introduce denoted objects so intimately into what their containing sentence is used to express that, should such a term in fact not denote, then nothing could have been expressed at all: there would be nothing to be either true or false. This is how we glossed `neither > nor ?' in the `two-tier' framework, and it echoes the Strawsonian approach to the presupposition of de nite descriptions, which we put on one side in Sec-

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tions 2.2 and 2.3. But recent theorizing along these lines has become more concerned with demonstratives and proper names: some classic references are [Wiggins 1976], [McDowell 1977], and [Evans 1982], where it is argued that it is, in fact, an important strand in some of Frege's own thinking that bearerless proper names cannot be used to express anything|any Fregean thought. A rich debate has subsequently developed, encompassing both exegetical questions and the question what it is right to say: for example, [McDowell 1984, McDowell 1986, Bell 1990, Wiggins 1995, Sainsbury 1999, Wiggins 1999]. This is not the place to disentangle the debate, but we have to consider how it impinges on our theme about reference. First, it would seem that at least pure description terms must fall within the scope of Dummett's account. (I eschew|though I cannot here provide a proper rebuttal of|the Russellian view that there should in principle be no such singular terms in a properly constituted logical language.) In that case, if we take descriptions as a paradigm for the singular terms that our logic is to accommodate, then it might be supposed that any problem about the dependence of reference on reference will have evaporated: surely we can simply extract from Dummett's account the picture of a total-valued semantics operating throughout on semantic classi cations? But, even if we do this, the problem will reappear. Given our particular semantics, with monotonic functions interpreting modes of composition, we can ask what sense, if any, it makes to say of that semantics that it exhibits functional dependence just among the classi cations > and ? and the classi cations `denoting-so-and-so'. This is precisely the question what sense it makes to say that monotonic functions represent partial ones. Itself the question remains internal to the mathematical semantics, but it becomes interesting in connection with at least some applications, if we want a general explanation behind the speci c need for, or usefulness of, monotonic forms. But what if we hanker after taking proper names as the paradigm for singular terms in partial logic? And what if we espouse the two-tier position that when a barerless name makes a sentence neither true nor false, this is because there can be no Fregean thought expressed by the sentence? It might then be supposed that the kind of infection a barerless name causes will be so radical that, however it occurs in a sentence, it must block the expression of a thought|so that the crudely Fregean principle will be inevitable. But I want tentatively to suggest that it is perhaps not so obviously inevitable. Central to arguments for the two-tier position is Frege's characterization of the sense of an expression as the `mode of presentation' (Art des Gegebenseins ) of a reference. It seems to follow from this that if there is no reference, then there can be no sense: there will be nothing for there to be any mode of presentation of. In particular, if a name has no bearer to be its reference, then it will have no mode of presentation of a bearer to be its sense. Now, suppose we espouse this characterization of sense, and accept the inference from it. Still, does it follow that a sentence containing

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a bearerless name can express no thought? It might be supposed to follow, because a thought is the sense of a sentence and, as such, will be dependent on the sense of constituent expressions|in a way that somehow or other matches the dependence of the reference of the sentence (its truth value) on the reference of the constituents. But our thesis concerning reference is that such dependence does not entail the principle that a compound expression must lack a reference whenever any constituent does. If this is right, then a matching thesis concerning sense cannot be dismissed out of hand. There may be room for a sentence that can be used to express a thought even when a constituent name lacks a bearer. For there may be sense for the sentence even when there is no sense to the name. This remains the mere mooting of a possibility: a thorough investigation is called for into the compositionality of sense, and this is no place for my inchoate thoughts on the matter.

3.2 Functional Dependence The problem, recall, is to provide an account of functional dependence which makes sense of saying that the `reference' if any, of a compound () depends on the `reference', if any, of a constituent |an account which can explain why () may sometimes lack a reference because  lacks a reference, but one which is not subject to the crude Fregean principle that it is always the case that (1) if  lacks a reference, then () lacks a reference. Here ( ) is a functor, and for the moment we shall assume that both  and () are either sentences or singular terms, though our remarks will be suÆciently general for it not to matter which. Frege himself wished actually to con ate these categories, but we will not be committed to that: indeed, we could envisage a many-sorted semantics with more than just two distinct domains of reference for basic, non-functor, categories. ([Wiggins 1984], for example, needs this.) Now, when reference failure is not the issue, the principle that, with respect to given ranges of reference, `compound reference depends on constituent reference' is familiar as an `extensionality' condition|to pick out modes of composition as extensional predicates or truth-functional sentence functors, for example. Here the idea of dependence is actually being put to work, and what is important is not that each constituent reference has to pull its weight as something on which compound reference depends| a thought that would indeed suggest the crudely Fregean principle|but rather that compound reference depends only on constituent reference, not on anything else. This is often spelt out with the following substitutivity condition:

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(2) If  and have the same reference, then () and ( ) have the same reference. As formulated, (2) presupposes that ; ; () and ( ) each have a reference; but we are considering the possibility that expressions lack a reference, and the question naturally arises how (2) might be modi ed so as to allow for this. In fact is an appropriate answer to this question not precisely what we are looking for? Presumably, then, we must adopt at least the following constraint on modes of composition: (3) If  has a reference and () has a reference, then, if has the reference of , ( ) has the reference of (). But what if () has a reference, though  lacks one? We do not want to rule out this possibility, but, to preserve the idea of dependence, it must be constrained. An obvious thought is that if () has a reference even when  lacks one, then  must occur in () in a slot that happens to be irrelevant to determining the reference of ()|given the reference of all other constituents. But in that case, whatever we care to substitute for ; ( ) must have the reference of (). Hence for any (given any ): (4) If  lacks a reference but () has a reference, then ( ) has the reference of (). And now, to replace (2), the conjunction of (3) and (4) can be logically manipulated into the following substitutivity condition: (5) If has the reference, if any, of , then ( ) has the reference, if any, of (). Here, of course, we have to understand the antecedent in a way that makes it trivially true for any  that lacks a reference. We are now in a position to explain why it is sometimes apt to say that () lacks a reference `because'  lacks one. For (4) yields a conditional form of Frege's principle (1): (1) obtains when 's slot in () is relevant to determining compound reference. It is a mark of relevance that there exist expressions and such that ( ) and ( ) take on a dierent reference, or such that one of them has a reference but not the other. And it follows from (4) that if such and do exist, then condition (1) does obtain. This discussion was originally prompted simply as a defence of our partial semantics against the strict Fregean principle (1). But the criterion of functional dependence embodied in condition (5) in fact does more: it dictates precisely a semantics of monotonically representable partial functions. Our semantics is not just not too liberal, but it is as liberal as it can be|given the criterion of dependence. To see this, consider a domain of reference D1 for constituent expressions  and , and a domain of reference D2 for compounds () and ( ). Then (5) means precisely that

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, for any a1 in D1 , if  refers to a1 , refers to a1 , , for any a2 in D2 , if () refers to a2 , ( ) refers to a2 . And so, if we assume that for any item in D1 there is|or can be introduced| an expression whose reference it is, then we may deduce from (50 ) that the interpretation of ( ) can be given as a partial function from D1 into D2 of the kind that is representable by a monotonic function from the xed-up domain D1 [ f~1 g into the xed-up domain D2 [ f~2 g: recall Section 1.2. This deals with one-place modes of composition, but the idea generalizes easily enough to arbitrary n-place ones, since monotonicity coordinate by coordinate is equivalent to monotonicity across all coordinates. It is interesting to contrast the discussion here with that in [Woodru 1970, pp. 128-9], where the speci c question is raised how to reconcile the use of Kleene's `strong' matrices for ^ and _ (in other words the matrices we have adopted) with a generally Fregean way of thinking. Woodru does not argue, as we have, that there is no trouble over the dependence of compound reference on constituent reference; rather, he argues that dependence may break down|for example when  _ is > because  is >, though is | but that this does not matter. The idea seems to be that, provided the constituent items of a sentence all have a sense, including ones without a reference, then we at least have a compound sense for the whole sentence, and this sense can be considered as determining a reference. However, according to our criterion of dependence, this detour through sense is unnecessary. And, to avoid entanglement with the debate that gured at the end of Section 3.1, the detour is in any case best not taken. if

then

_^ _^ _^

So far we have been thinking of the function which interprets a functor simply as what exhibits dependence of compound reference on constituent reference, but, in Fregean theory, the interpreting functions are themselves the reference of functors, and compound reference `depends' no less on this kind of reference than on the reference of a constituent singular term or sentence. What then of our monotonically representable partial functions? Can we see them as constituting a range of reference|or a range of `partial reference'|which is subject to some suitable principle of dependence? It seems we can set them in this Fregean light by considering appropriate generalisations of principle (5) for higher-level functors ( ) which take functors  for arguments. If ( ) is a simple second-level predicate, for example, (such as a rst-order quanti er) the principle would be one which linguistically embodied the intuitive idea of dependence that we sketched in Section 1.2 in connection with partial subsets of the system of partial subsets of a given domain. But in fact we can cater for a complete hierarchy of functor categories|one which includes not only functors which take functors as arguments, but also (though this unFregean) functors which make functors.

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There is no space to pursue these thoughts, but we should point out that it would be inadequate to think of the `partial reference' of partial functors as a `partially speci ed' (total) reference. This is the idea that [Dummett 1973, p. 170] would like to oer Frege, but it could not explain the subtlety of monotonically representable partial functions. The reason is that rst-level functors accommodate empty argument places for reference-less terms in a way which is subject only to the constraint of principle (5). Full account has to be taken of this in our generalization of (5) to higher-level functors. 4 NON-CLASSICAL CONNECTIVES

4.1 Interjunction and Transplication: Expressive Adequacy Let us begin with the proof of expressive adequacy. We argued in Section 1.2 that, since the matrices for the connectives of simple partial logic all describe monotonic functions, any propositional formula, however complex, must also have a matrix which describes a monotonic function. We now show that, conversely, given any monotonic function f from f>; ; ?gn into f>; ; ?g, we can nd a formula f (p1 ; : : : ; pn )|f for short|whose matrix describes f : in other words, f will take the classi cation f (x1 ; : : : ; xn ) under the assignment of xi to pi . We shall use just :, ^, _, ^ _, > and ? to de ne f . The case when n = 0 is easy: there are three 0-place functions, which are described by the trivial matrices for the logically constant sentences (or 0-place connectives) >,  and ?. And  can be de ned away as > _ ^ ?. Otherwise, when n > 0, we can proceed as follows. First, for any n-tuple ~x 2 f>; ; ?gn and any number i from 1 to n. Let the formulae >(~x; i) and ?(~x; i) be de ned by cases|by cases within cases|as follows:

>(~x; i) ?(~x; i)

= = = = = = = =

9

pi if xi = > = :pi if xi = ? ; > otherwise

?

















otherwise; 9 pi if xi = ? = :pi if xi = > ;


if f (~x) = ?, ? otherwise >

















otherwise.

Then we can de ne f to be

"

WW

VV

~x2f>;;?gn 1in




if f (~x) = >,

#

>(~x; i) _^

It is now not diÆcult to check that:

"

VV

WW

~x2f>;;?gn 1in

#

?(~x; i) :

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(i) if the left-hand right-hand interjunct of >fis of xi to pi , then f (xn ; : : : ; xn ) = ? ;

 

305

  > under the assignment ?

(ii) if f (xn ; : : : ; xn ) = > ? , then both interjuncts are assignment of xi to pi .

  > under the ?

Given the >=?-conditions of ^ _, it follows from (i) and (ii) that the matrix of f does indeed describe the function f . It also follows that that the left-hand interjunct gives the >-conditions of f , while the right-hand interjunct gives the ?-conditions. And so these formulae provide interesting `normal forms' for monotonic modes of sentence composition. In Section 6.3 we shall show that interjunctive normal forms of this kind exist in quanti er logic too. As speci ed f is likely to contain many otiose occurrences of > and ?, but there are obvious ways of obtaining a more economical formula. We have shown that f:; ^; _; ^ _ ; >; ?g is a set of connectives adequate to express any monotonic function from f>; ; ?gn into f?; ; >g. The question now arises what other sets of connectives are expressively adequate. In particular, given the classical connectives (including ! and $, which can be de ned in terms of :; ^ and _ in the usual way), what are the variations on ^ _ ? First, then, observe that transplication has equal expressive power. Not only is = de nable in terms of ^ _, but also conversely:

' [ ^ ] ^_ [ ! ]; ' [ $ ] =  ' [ $ ] = : Or we could take the logically unde ned sentence . We observed above that  can be de ned as > ^ _ ?; now observe that ^_ can be de ned in terms of : _ ^ ' [ ^  ] _ [ ^ ] _ [  ^ ] ' [ _  ] ^ [ _ ] ^ [  _ ]: Hence each of ^ _, = and  has the same expressive power as either of the = _ ^

others. But to give a more complete answer to our question, rst consider the subclass of monotonic functions satisfying the following condition (a converse to the crude Fregean principle that we eschewed in Section 3): if xi 6=  for all i, then f (x1 ; : : : ; xn ) 6= . In [Van Benthem 1988] such functions are called `closed'. Thus the matrix of a formula  will describe a closed function if and only if, for all total assignments v, either v() = > or v() = ?; and in [Langholm 1988] such formulae are called `determinable'. Clearly the matrix for any formula which

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contains no connectives beyond :; ^; _; > and ? will describe a closed function, since closed functions are closed under composition; furthermore|and less trivially|any such function is described by the matrix of some such formula: in other words, a formula is determinable if and only if it equivalent to a classical formula. There are proofs of this|all dierent|in [Blamey 1980] and in the two works referred to above. We are now in a position to provide a general answer to the adequacy question for monotonic modes of composition: the set f:; ^; _; >; ?; 1g is expressively adequate if and only if 1 is a connective (of any arity) whose matrix describes a non-closed monotonic function. `Only if' is immediate: compounding closed functions will never reach ^ _, for example. On the other hand, we can deduce `if'|the claim that anything monotonic and non-closed will do|from the fact that the constant sentence  will do. First,  itself is the one and only 0-place non-closed monotonic connective. Secondly, if n > 0 and 1 is an n-place connective whose matrix describes a non-closed monotonic function f1 , then f1 (x1 ; : : : ; xn ) = , for some x1 ; : : : ; xn such that either xi = > or xi = ? for each i. And so, together with the constant sentences > and ?, 1 will be suÆcient to de ne |and hence any monotonic mode. For some particular applications of partial logic, the determinability of all formulae in the language may be a desideratum , so that non-closed connectives would be out of place. But in [Jaspars 1995] there is a more general claim, which, in the light of the discussion in Sections 1.2 and 3, would seem to be incorrect. He claims that it follows from the idea that being neither > nor ? means being `genuinely unde ned', rather than having a third truth value, that `whenever all the parts of some proposition have obtained a truth value, then the proposition ought to get a truth value as well'. However, without some question-begging assumption about the possible structure of propositions|or the sentences that express them|I cannot see why it follows. You might just as well say that it follows from the idea of a singular term's being genuinely unde ned, rather than denoting some specially introduced object, that whenever the constituent terms of a compound term are all de ned, then the compound term must be too. But in that case `0 1 ', for example, wouldn't be unde ned. No doubt Jaspars has particularly in mind the kind of unde nedness that arises from what in Section 2.1 I called a local set-up for assessment, so that being neither > nor ? means that so-and-so information is not suÆcient to determine the value > or ?. But, even if so, this does not dictate any principle that information which is suÆcient to determine a value for all constituents must also be suÆcient to determine a value for the compound.

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4.2 Interjunction and Transplication: Logical Analysis

The two formulae given to de ne ^ _ in terms of  are each other's dual: and ^ _ is self-dual. This means that negation, when applied to an interjection, can be driven through to rest equally on both interjuncts. Applied to transplication, on the other hand, negation can be driven past the left-hand constituent|which we may call the transplicator |to rest on the right-hand constituent|which we may call the transplicand :

:[ _^ ] ' : _^ : ; :[ = ] '  = : : If > and ? are thought of as the classi cations which negation switches,

then these equivalences reveal how it is that interjunction and transplication give rise to non-trivial either->-or-? conditions. Notice, then, that a transplicator can be taken to introduce a presupposition, in the sense that 's being > is a necessary condition for = 's being either > or ?. But interjunctions are more interesting:  _ ^ can be thought of as expressing  and `as standing or falling together', or|as the de nition of ^ _ in terms of = makes explicit|under the presupposition that they are equivalent. Recall that in Section 2.2 we gave informal >=?-conditions for the schemes of presuppositional quanti cation Ix[F x; Gx] and 8x[F x; Gx]. We can now show how to capture these >=?-conditions by analysis under interjunction and transplication. This is a project that could be generalized|see [Van Eijck 1995] and [Sandu 1998] for general frameworks in which to handle modes of quanti cation in partial logic|but Ix[F x; Gx] and 8x[F x; Gx] will do to illustrate the use of interjunction and transplication. For the moment we shall adopt the simplifying assumption that F and G are unstructured predicates, totally de ned over a given domain: we can then assume that classical principles govern all classical-looking formulae. First, then, the scheme Ix[F x; Gx], for `the F is G', admits the following interjunctive analysis (where F !x abbreviates 8y[x = y $ F y]):

9x[F !x ^ Gx] _^ 8x[F !x ! Gx]: Clearly the left-hand interjunct had the desired >-conditions, and whenever it is in fact >, the right-hand interjunct must also be >; similarly, the righthand interjunct has the desired ?-conditions and, whenever it is in fact ?, the left-hand interjunct must also be ?; while the conditions under

which the two interjuncts take on opposing truth-values are precisely the required -conditions. Hence the interpretation of ^ _ guarantees that we have the right >=?-conditions for Ix[F x; Gx]. Under presuppositional >=?conditions :Ix[F x; Gx] must be equivalent to Ix[F x; :Gx]: the scheme is self-dual. This is revealed by our analysis, since the negation of the formula above is equivalent to

9x[F !x ^ :Gx] _^ 8x[F !x ! :Gx]:

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To see this, rst drive negation through onto the interjuncts, and thence onto Gx, and nally switch the interjuncts around. This analysis of Ix[F x; Gx] is just the interjunction of formulae giving a classical Russellian analysis of Ix[F x; Gx] and of :Ix[F x; :Gx]. But there are other versions of classical analysis which contain 9xF !x as a distinct conjunctive component. On a presuppositional interpretation this component is a presupposition, and the simple strategy of replacing conjunction by transplication yields the following formulae, either of which may serve to analyse Ix[F x; Gx]:

9xF !x = 8x[F x ! Gx]; 9xF !x = 9x[F x ^ Gx]:

Notice that these formulae are equivalent because given that 9xF !x is > the >=?-conditions of the two transplicands must coincide. Notice too that when we apply negation it slips past the tranplicator onto the transplicand, and thence through onto Gx, to give

9xF !x = 9x[F x ^ :Gx]; 9xF !x = 8x[F x !:Gx]: So again our analysis reveals that :Ix[F x; Gx] is equivalent to Ix[F x; :Gx]. To provide a transplicative analysis for the scheme 8x[F x; Gx] of universal quanti cation, we can follow a similar pattern:

9xF x = 8x[F x ! Gx]: It is easy to check, given our simplifying assumption concerning F and G, that this formula captures the right presuppositional >=?-conditions. And we should also consider a scheme 9x[F x; Gx]|to be dual to 8x[F x; Gx], in having the same ?=>-conditions as :8x[F x; :Gx]. The obvious analysis is:

9xF x = 9x[F x ^ Gx]: We could use this to symbolize a sentence such as `Some of Jack's children are bald', which, no less than `All Jack's children are bald', carries the presupposition that Jack is not childless. I shall leave it as an exercise to provide an interjunctive analysis for 8x[F x; Gx] and for 9x[F x; Gx]. We cannot, of course, rest with the assumption that F and G are unstructured and totally de ned: if our schemes of analysis are any good, then they should continue to make appropriate sense when applied to arbitrary formulae (x) and (x) in place of F x and Gx. And so we should consider what happens when one scheme of presuppositional quanti cation occurs embedded in another. Horrendously complicated formulae can arise if a number of quanti ers are analysed out together: in particular, occurrences of = or ^ _ will be obscurely embedded not only within the scope of

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sentence connectives (including other occurrences of themselves) but also within the scope of the quanti ers 8 and 9. Yet it turns out that any formula, however complex, is in fact equivalent to one of the form = where  and themselves contain no occurrence of either = or ^ _. Furthermore, we can specify rules systematically to transform an arbitrary formula into an equivalent formula of this form; and these rules can be framed so that the transplicator  will capture the `overall presupposition' of the formula: 's >-conditions will be precisely the either >-or-? conditions of = |and hence too of the original formula. These transformation rules, which we shall present in Section 6.3, can be seen as a logician's version of `projection rules' for presupposition. The examples presented here reveal only a small fraction of what interjunction and transplication have to oer in the analysis of presupposition: I hope there will very soon be a publication telling more of the story.

4.3 Static versus Dynamic Semantics The idea of a `dynamic' semantics that emerged rather abstractly at the end of Section 2.7, and turned up again in Section 2.10, has gured prominently in the linguistics literature: in particular, presupposition has been given a dynamic treatment. The questions therefore arise whether our use of transplication and interjunction in the analysis of presupposition can be captured in a dynamic semantics, and whether it has to be to provide an adequate foundation. The answers, I want to argue, are respectively `yes' and `no'. Approached dynamically, the meaning of a sentence is seen as captured by its potential to change contextual information states. These states might be taken to be cognitive states of an individual participant in linguistic exchange, or perhaps to be something more communal and complicated; and they might be represented in the form of a partial model of some kind, or as a set of total models or of possible worlds, or as structures that are formulae of some elaborate formal language, or whatever. The general idea can be traced back to work such as [Stalnaker 1972] and [Seuren 1976], and has been developed in [Kamp 1981, Kamp and Reyle 1993, Heim 1982, Seuren 1985, Veltman 1996], and so on. (See [Van Benthem 1991, Muskens et al. 1997], and so on, for illuminating surveys.) In such work the old-fashioned idea of giving meaning in terms of truth/falsity conditions is pushed aside|just as it is in situation semantics. Or, at least, it is pushed back, for we must come down to earth at some stage and actually give the meaning of the expressions of any particular language: this is the fundamental message of [Lewis 1972]. And presumably the way to come down to earth, via the dynamic apparatus, is to give conditions for the correctness of information states.

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Anyhow, the presuppositional characteristics of a sentence seem always to be considered context-involving in some special way. But in [Blamey 1980] it was argued against [Karttunen 1973, Karttunen 1974] that a contextinvolving account of the meaning of presuppositional idioms was unnecessary and something of a distortion: contextual phenomena could best be accounted for on the basis of a semantical account|using the forms of partial logic|which was itself independent of a theory of context. A dynamic approach will not be set up in quite the same way as Karttunen's, but can we make an analogous point? In [Beaver 1997] dynamic clauses are given to interpret a language with :, ^, and = (though Beaver writes `  ' for `= '|notation which he adopts from the work in [Krahmer 1995]); and so let us consider his propositional semantics. We may describe the underlying models as consisting of a set V of possible worlds , each  determining a classical total assignment v of > or ? to atomic sentences. States of information are then represented by sets of possible worlds (all those possible worlds compatible with the state of information represented), and to interpret a formula  there is a relation [ ]  between states  and  |glossed as meaning `it is possible to update  with  to produce  '. The de nition of [ ]  has the following dynamic clauses: [ p]  i  =  \ f j v (p) = >g, [ :]  i for some , [ ]  and  =  r , [  ^ ]  i for some , [ ]  and [ ]  , [  = ]  i [ ]  and [ ]  . The question we should now ask is whether this de nition for [ ]  has to be taken as primitive, or whether the relation can be de ned in terms of something which is static and arguably more basic. Well, any formula  can obviously be evaluated in simple partial logic under a (total) assignment v to yield a value v (). And so if we de ne [ ] > = f j v () = >g; [ ] ? = f j v () = ?g; then 's content (in V) under partial semantics may be represented by the pair h[ ] > ; [ ] ? i. Alternatively, and equivalently, we could use the equations displayed in Section 2.10 directly to de ne content-evaluation for . It then turns out that this content is suÆcient to de ne the relation [ ]  : a straightforward inductive argument shows that [ ]  i  =  \ [ ] > =  r [ ] ? : (Hence, observe, the relation is actually a function, though not a total one.) The right-hand side is equivalent to saying that (i) for all  2 , either v () = > or v () = ?, and (ii)  is got from  by taking away all those  such that v () 6= >|equivalently, given (i), such that v () = ?. This argument is essentially the same as the one presented in [Muskens et al. 1997] concerning dynamic clauses formulated in a slightly dierent way.

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This brief commentary on Beaver's apparatus falls short of a full justi cation for my answers to the opening questions, but it does show that a natural possible world semantics for presuppositional analysis in partial logic is suÆcient to determine natural dynamic clauses. These clauses do not have to be taken as the foundation. It would, though, be more natural still if the world-relative assignments v were not restricted to total ones: to function smoothly the atomic formulae of a logical syntax ought to be schematic for arbitrary sentences, and so not subject to any special semantic restriction.

4.4 Non-Monotonic Matrices Non-monotonic matrices provide the most obvious examples of what our languages cannot express. In [Woodru 1970], for instance, there are several of the `metalinguistic' sort of connective that we mentioned in Section 1.3. These are obtained by semantic descent from metalinguistic predicates or relations:





T F

> >  ? ? ?

? ? >

 + > > > ? ? >

> > >    ? ? ?

>  ? >  ? >  ?

 =

> ? ? ? > ? ? ? >

)

> ? ? > > > > ? ?

 7!

> ? ? > > > > > >

Thus  =; ), and 7! (for which Woodru uses `!') are obtained from relations of equivalence, presupposition and single-barrelled consequence (the relation > of Section 1.1) respectively. Woodru comments that the `distinctive feature' of these connectives is that they yield compounds which are de ned even when every constituent is unde ned. However, a mode t(p) which is just constantly >, whatever the classi cation of p, would have this feature, and yet it is monotonic. From our point of view, `not monotonic' is a more fundamental feature. But is there any natural way of classifying more nely among additional connectives? It is a well-known result that the T connective, together with our :; ^; _ and , is expressively adequate for arbitrary matrices. And, given :; ^, and _, any of the other connectives listed above can de ne T . Hence together with monotonic modes they would each yield a full-blown 3-valued logic. This fact about Woodru's connectives is rather more interesting than

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the simple fact that they are not monotonic, since it raises the question: are there non-monotonic connectives which would not provide a full-blown 3valued logic if they were included with the monotonic modes? In other words: are there any logics whose expressive range is intermediate between the logic of monotonic matrices and the logic of arbitrary matrices? It turns out that there is precisely one. To complement the relation v on f>; ; ?g we can de ne a relation , which might be thought of as a relation of `compatibility', in the following way:

x  y i neither (x = > and y = ?) nor (x = ? and y = >): This relation will be of interest in Sections 6 and 7, but in the present context it provides a characterization of the intermediate logic: it is the logic of those matrices which describe functions f that are `-preserving' in the following sense: if xi  yi for all i, then f (x1 ; : : : ; xn )  f (y1 ; : : : ; yn). To see that -preserving logic ts in as we claim, notice rst that monotonic functions are -preserving, though there are -preserving functions which are not monotonic: for example, f such that f (>) = >, f () = > and f (?) = . And there are also functions which are not -preserving| including all the functions described by the matrices listed above. We now need two facts whose proofs are omitted, because they are tedious (though not diÆcult): (i) if we add to the monotonic sentence modes any non-monotonic preserving mode, then we can express all -preserving functions. (ii) if we add to the monotonic sentence modes any non--preserving mode, then we can express all three-valued functions. It is easy to check that the class of -preserving functions is closed under composition, and so it follows from (i) that the -preserving modes do indeed provide an intermediate logic. And then it follows from (ii) that this is the only one. As a corollary of this argument we also have a general answer to the adequacy question for -preserving modes of composition: f:; ^; _; ^ _ ; >; ?; 1g is expressively adequate if and only if 1 is a connective (of any arity) whose matrix describes a non-monotonic -preserving function.

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4.5 Two-Tier Semantics We now turn to something more exotic, viz. the semantics of [Belnap 1970], which is intended to model a two-tier framework for assessment in which the classi cation  means `no assertion': see Section 2.3 above. This is not to say it is intended to be a general modelling of any two-tier framework; nor is it plausibly taken as such: for example, it would not seem to be appropriate for developing any account of Fregean thoughts of the kind mooted at the end of Section 3.1. Anyhow, in Belnap's semantics propositions are rst modelled as sets of possible worlds in the usual classical way, so that a proposition is true at at world if and only if it contains that world, and then interpretation clauses are given which either assign a proposition to a formula at a world| for it to `assert' at that world|or else leave a formula `unassertive' at a world, with no proposition assigned to it. With this apparatus Belnap's connective `=' is interpreted by stipulating that at a world in which  asserts a false proposition = is unassertive, and at any other world = asserts what asserts, unless itself is unassertive, in which case = is again unassertive. Thus `=' turns out very like transplication; though to match it up properly we should have to modify its interpretation so that = is unassertive not only when  is false, but also when  is unassertive. This is a minor modi cation and would not disrupt Belnap's idea. But we should stress that our (monotonic) interpretation of transplication in simple partial logic is in no way committed to further explication with Belnap's apparatus. If we want to consider a possible-world semantics, then we have the alternative, and simpler, one-tier option of modelling propositions as `partial propositions' of the kind rst introduced in Section 2.10 and later invoked in Section 4.3|that is, as pairs of sets of possible worlds which just model our talk of >=?-conditions. Any formula would then express a proposition at any world: either->-or-?-conditions would be constitutive of this proposition rather than being conditions for the existence of a proposition expressed. The simpler one-tier option would certainly be more appropriate for a logic of presuppositional analysis: recall Section 2.3. But there is a further special point about the use of transplication in analysis which shows that Belnap's interpretation for `/' makes it crucially dierent. It would not just be a mistake to think that the role of a transplicator  in = is to determine whether or not anything is `asserted', but it would be an even worse mistake to take on its own to represent what is asserted, if anything is. We can think of the transplicand in an assertion-specifying role only if we take it ltered through the transplicator, so to speak. Recall that we used 8x[F x ! Gx] as a transplicand to analyse both Ix[F x; Gx] and 8x[F x; Gx]: thus we may lter the same transplicand through dierent transplicators to get something entirely dierent. Furthermore, dierent transplicands may be ltered through the same transplicator to yield the same thing|the same

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>=?-conditions. For if 9xF !x is taken as the tranplicator, then we saw that either 8x[F x ! Gx] or 9x[F x ^ Gx] does equally well as a transplicand in an

analysis of Ix[F x; Gx]|and there are plenty of other inequivalent formulae we could just as well have chosen: 8x[F !x ! Gx] or 9x[F !x ^ Gx], for example. In [Beaver 1997] there is some ambivalence over a formula `  ', which in Section 4.3 we assimilated to a transplication = . He glosses  as `the assertion of carrying the presupposition that ', but this is ambiguous. Does it mean (i) the assertion of , carrying the presupposition that ; or (ii) the assertion of -carrying-the-presupposition-that-? The wording is more likely to convey reading (i), though apparently Beaver actually wants to leave both readings open. But as a gloss on our use of transplication only reading (ii) is admissible, where -carrying-the-presupposition-that- is understood to mean - ltered-through-, in the way that our examples of analysis illustrate. This is the content of any assertion that = represents: the presupposition that  is constitutive of this content, not a separate item just stuck on alongside. This point about the undetachability of a transplicator could in fact be made independent of our espousal of a one-tier rather than a two-tier framework for presuppositional semantics. For even if we wanted to gloss the `neither->-nor-?' of presupposition failure to mean no assertion, what is asserted when  is true and = represents an assertion could not be speci ed by on its own. If, as in Belnap's semantics, classical propositions are the only candidates for the content of assertions, then, to put it in Belnap's language, what = asserts when it asserts anything|that is, when  is true and asserts something|cannot be what asserts, but can only be the conjunction (intersection) of what  asserts and what asserts. Indeed, it would be easy enough to revise Belnap's clauses for `/' along these lines. This is not a point against Belnap, of course; for recall that his `/' is not intended for presuppositional analysis at all, but rather to construe conditionals. Anyhow, as in an alternative to going to meet Belnap among the possible worlds, we could in fact unravel his semantics into simple >=?-conditions. Clauses for evaluating a formula at a given world|clauses which make no appeal to any other world|are given in [Dunn 1975]. The following matrices for ^; _, and / then emerge:

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> > >    ? ? ?

>  ? >  ? >  ?

^

> > ? >  ? ? ? ?

_

> > > >  ? > ? ?

315

=

>  ? >  ?    Thus, quite apart from `/', the matrices for ^ and _ show a dierence form

simple partial logic: conjunction and disjunction are not monotonic (nor even -preserving). This prompts a question: If we started out with our monotonic matrices for ^ and _, then could we sensibly convert them into a Belnap-style two-tier semantics? This becomes a pertinent question in Section 5.1, where we shall address it.

_^ _^ _^ But rst we should observe that the above non-monotonic, and prima facie rather odd, matrix for _ also arises in [Ebbinghaus 1969], where a rstorder semantics is oered to handle the kind of unde nedness that arises from natural modes of mathematical expression. Ebbinghaus presents his semantics by rst giving clauses for when a formula is de ned|in a given model|and then building truth conditions on top of this. The rules for disjunction are:

 _ is de ned  _ is true

i  is de ned or i  is true or

is de ned, is true.

Hence, if  means unde ned, > means true, and ? means de ned but not true, then Belnap's matrix for _ results. Negation is taken to work in the same way that it does in simple partial logic, and Ebbinghaus de nes
() as  _ :, to yield a sentence-mode expressing ` is de ned'. Hence
() yields , if  is  (just as it would if we had de ned it in simple partial logic). Contrast Woodru's +. The interpretation of the existential quanti er is analogous to disjunction: 9x(x) is taken to be de ned just in case (x) is de ned for at least one element in the domain of quanti cation, and to be true just in case (x) is true of at least one element. This interpretation is motivated by the desire to allow existential statements to come out false, even when the quanti ed predicate is unde ned for some elements|and so not false of everything: for example, in the domain of rationals or reals, 9x[x 1 = 0] is to be false, though 0 1 = 0 is unde ned. Clearly this would not be possible

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in monotonic logic. However, since (unlike Ebbinghaus) we envisage setting up all nonlogical theories directly in terms of consequence, we are not under the same pressure to assign such existential statements a truth value. Disjunction and existential quanti cation thus turn out to be much `stronger' than in simple partial logic. But conjunction and universal quanti cation are much `weaker'. For conjunction we have:  ^ is de ned i  is de ned and is de ned,  ^ is true i  is true and is true. and so  ^ is unde ned whenever either  or is. Then 8 matches ^ just as 9 matched _. These interpretations do not, therefore, yield the classical duality between ^ and _ and between 8 and 9; but they allow Ebbinghaus to frame neat rules for
( ) in a natural deduction system which is designed to axiomatize a truth-preservation notion of consequence. This system falls squarely under the heading `partial logic', but in much recent work there seems to be something of a division of interest. On the one hand, partial logicians tend to ignore unde ned singular terms|perhaps because they are primarily concerned with partial states of information, or situations, or the like (see Sections 2.7 and 2.10); though this is certainly not a de nitive reason for ignoring unde ned terms. On the other hand, those setting up systems to accommodate unde ned singular terms tend to prefer a logic which at the level of sentences is totally de ned and two valued. See [Feferman 1995] for a magisterial exposition of doctrine|and for a survey of work; and for work speci cally in the `free logic' tradition, see Bencivenga's chapter of the Handbook . But the system in [Lehmann 1994], for example, is an exception to the trend: it is a partial logic with unde ned terms. This is work in the Fregean tradition, and I would want to take issue with it because it espouses the principle of functional dependence that in Section 3 I argued was unnecessarily crude. 5 PARTIAL LOGIC AS CLASSICAL LOGIC

5.1 Partial Truth Languages A proper discussion of the idea of `alternative' logics is far beyond the scope of this essay. But, via some themes we have touched upon already, we shall brie y puzzle over two particular accounts of how the triclassi catory semantics of partial logic can play a role which does not, in any interesting sense, give rise to an alternative to classical logic. First consider [Kripke 1975] which we discussed in Sections 2.5 and 2.6. His remarks about logic are, in fact, rather sketchy and largely centred in footnotes, but nonetheless they are forcefully expressed. In footnote 18, for example, he claims that in adopting Kleene's monotonic matrices for evaluating sentences he is doing no more than adopting `conventions for handling

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sentences that do not express propositions' and that these conventions `are not in any philosophically signi cant sense changes in logic'. For logic is supposed to apply primarily to propositions which are all either true or false. Kripke draws a parallel between handling possibly non-denoting (numerical) terms and handling sentences which are unde ned (), and this parallel calls to mind our account (in Sections 1.2 and 3) of the partial-functional interpretation of functors. However, the parallel there was the Fregean one between objects denoted and the truth-values > and ?, whereas Kripke's parallel is between objects denoted by (numerical) terms and propositions expressed by sentences. And in the text he presents us with an explicitly two-tier picture of the meaning of a sentence: gapless truth conditions determine propositions, but sentences, which might turn out to be paradoxical and hence neither true nor false, are not directly interpreted by truth conditions, but by conditions for truth conditions. Clearly these conditions must not only determine when a sentence expresses a proposition|has gapless truth conditions|but also what proposition a sentence expresses when it does express one. Kripke is vague at this point, but his picture of the interpretation of sentences looks to be of the same general kind that Belnap's semantics is intended to model. And so we return to the question raised in Section 4.5: can Kleene's monotonic matrices be made to t with such a semantics? Kripke seems (in footnote 30) to suggest that they stand a better chance than a supervaluational scheme of evaluation. This is presumably because, according to this scheme, there would be the diÆculty of sentences none of whose constituents expressed a proposition, but which are true, just because they are of the form of a tautology. The problem would be to say what proposition such a sentence expresses, in a way which does justice to ideas of compositionality whereby a compound proposition is in some sense determined by constituent propositions. However, even on the Kleene scheme we may have a sentence which is true even though one of its constituent sentences is neither true nor false, and so, according to Kripke, expresses no proposition: for example, something of the form  _ , where  is a straightforward truth and is paradoxical. What proposition does  _ then express? And, in general, what are the rules which tell us what proposition a compound sentence expresses? Let us assume we can make suitable sense of saying that propositions are closed under boolean operations (perhaps, but not necessarily, because we have modelled them as sets of possible worlds). And let us, by way of example, compare Belnap's and Kleene's matrices for disjunction:

_ >  ?

> > > >

 >  ?

Belnap

? > ? ?

_ >  ?

> > > >

 >  

Kleene

? >  ?

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The four corners of Belnap's matrix are accounted for by saying that if both disjuncts of a disjunction express (or `assert') a proposition, then the disjunction expresses the corresponding disjunction of the propositions. If, on the other hand, neither disjunct expresses a proposition, then the disjunction expresses none: this explains the centre of the matrix. So far the two matrices coincide, but what happens when one disjunct expresses a proposition but not the other? The prima facie oddity of Belnap's matrix is explained by his stipulation that the disjunction expresses the same proposition as the proposition-expressing constituent. But what could Kripke say about Kleene's matrix? The only obvious course would be to make _ the same kind of connective as Belnap's `/' of conditional assertion and to say that the existence of a proposition expressed by the disjunction depends on the truth value of the disjunct which expresses a proposition (the truth value of that proposition): if it is true, then this is the true proposition expressed, and if it is false, then no proposition is expressed. It might, then, be possible to make sense of things along these lines, treating conjunction in a parallel way and, of course, extending it all to handle quanti ers. And some such elaboration of partial semantics would have to be given, if Kripke ever wants to set up logic for his truth languages so that it can be seen to apply to classical propositions that sentences might or might not express. But then we might ask what role these propositions would play in his account of truth and paradoxicality. We are invited to see the monotonicity-dependent construction of models in some way re ecting an intuitive evaluation process of sentences, in a progression of succesive stages: as the process is pursued more sentences receive truth values. But we can hardly think of this process as evaluating sentences for the propositions, if any, they express. For, though monotonicity guarantees persistence of truth value, there would not be persistence of propositions. If, for example,  were true and neither true nor false, but at some stage of evaluation took on a truth value, then the proposition originally expressed by  _ would disappear as a disjunctive constituent of the later proposition. Or are classical propositions meant to be there from the start, in some sense, so that they can determine the process of evaluation? This is a picture it seems diÆcult to make sense of. So what theoretical role would classical propositions play? The oddity is that they seem to have no role. But why should we envisage a two-tier semantics at all? The alternative is to give a direct account of meaning in terms of (partial) >=?-conditions, so that sentences have `partial propositions' as their meaning: see Section 2.10 above, and compare the remarks in Section 4.5. This would mesh naturally with Kripke's account of the stage-by-stage evaluation of sentences: as the evaluation progresses, so propositions become progressively `more de ned'. The idea of partial propositions is crying out for further elucidation, but if it can be provided, then we have the most straightforward way to gloss the formal construction of models for semantically closed languages. As

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we explained in Section 2.6, a succession of partial, but progressively less partial, models culminates in a model which is still partial but which is stable: it throws up no new true or false sentences in terms of which to de ne (the truth predicate of) any less partial model. There are >/?conditions for all sentences in each model in the succession, and in the nal stable model they give the nal stable meaning of sentences of the language. The natural logical apparatus to adopt would then be, or be something similar to, what we shall outline in Sections 6 and 7. And there is surely nothing to stop us interpreting this apparatus as delivering a logic that is esssentially classical|richer than usual simply because it embodies rules for handling varieties of unde nedness. The presentation of partial logic in Section 1 was meant to reveal this interpretation as a coherent option.

5.2 Natural Negation If we turn to Dummett's views on presupposition and the role a logic such as ours might play in providing a semantics, then the debate becomes a very dierent one. The idea that a sentence classi ed as  expresses no proposition, or that no assertion can be made using it, does not enter the picture at all. Thus Dummett's account is in what we have been calling a one-tier framework. But it does invoke two dierent aspects of meaning, and these give rise to two dierent levels of content. Sentences are semantically classi ed as > or  or ?, and there is a notion of the `semantic content' of a sentence as its >-versus--versus-? conditions; but assertions made using sentences are to be classi ed exhaustively into true ones and false ones, and the `assertoric content' of a sentence matches truth-versus-falsity conditions. Semantic classi cations then divide into the `designated', for sentences which can be used to make true sentences, and the `undesignated', for sentences which can be used to make false ones. Presuppositional  will side with ? as a case of falsity. With this framework at hand, Dummett is polemical|for example in the introduction to [Dummett 1978]|against theorists who would deploy notions of `truth' and `falsity' matching the semantic classi cations > and ? in a way which he reserves exclusively for truth and falsity. For according to Dummett, so long as we concern ourselves with the linguistic activity of making assertions and with the meaning a sentence manifests in this linguistic practice, then a basic notion of objective truth and falsity leaves no room for anything but an exhaustive dichotomy into the true and the false. There is an exclusion clause for `vagueness' and `ambiguity'|which Dummett thinks of as cases where an assertion would have no fully determinate content (and which he supposes have nothing to do with presupposition)| but, otherwise, the way things are is either incorrectly ruled out by an assertion, in which case it is false, or else it is not, in which case it is true.

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This thesis emerges in various places in [Dummett 1973], but is crispest in [Dummett 1959]. (Note that `anti-realist' worries are not at issue here.) Why then bother with a semantics that operates on the classi cations >; , and ?? The point, it is suggested, will simply be to obtain a smooth account of how sentences are composed from their constituents. To interpret modes of linguistic composition|not just sentence composition|a system of semantic classi cations reveals how the meaning (semantic content) of a complex expression is determined by the meaning (semantic content) of its constituents; but the point of a systematic semantics of this sort is just to lead up in an appropriate way to a correct speci cation of true-versusflase conditions|assertoric content. It is here that the notion of `semantic role', alluded to in Section 3.1, ts in: the classi cations of a semantics capture one strand in the Fregean notion of reference because they play a role|a semantic role|in determining the truth or falsity of (assertions made using) sentences. Thus the subtleties of a presuppositional semantics are taken to derive just from structural features we are prompted to discern in a language. The most salient feature would seem to be negation. We saw in Section 2.4 that, to account for natural modes of negation as straight-forward sentence functors, we need to split non-truth (falsity) into ?, which negation switches with > (truth), and , which it leaves xed. This is a standard example of Dummett's to illustrate the role of triclassi catory semantics, and he uses it also to explain our naive inclination to apply the labels `true', `false' and `neither-true-nor-false' directly to the evaluation of assertions themselves. For we are inclined, he suggests, to call the assertion of a sentence `false' only if the assertion of the (natural) negation of that sentence would have been true (true). This seems to provide an explanation of the three-fold scheme of semantic classi cation|and hence of the phenomenon of presupposition|in terms of the true/false dichotomy and natural negation. But, as Dummett himself points out, natural negation is not a purely syntactical notion. Just consider the complex variety of forms: for example, `Some of Jack's children are not bald' is just as much a natural negation of `All Jack's children are bald' as `Not all Jack's children are bald' is. Hence natural negation is not identi able as such in a meaning-independent way. Yet as natural speakers we do recognise it, and as theorists it is handy for us to do it justice. So, what is it? It is not unreasonable to call, in turn, for an explanation of this mode of sentence modi cation. Furthermore, why is natural negation negation at all? The classical truth values true and flase are taken to be fundamental, but natural negation takes some false sentences to ones that are again false (when there is presupposition failure). At this point Dummett's overall picture might leave us restless. For it does not seem to leave much room to answer these questions|or not without going round in a circle. For what can we say about natural negation other than that

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it is a mode of sentence modi cation which is to be called in to spell out the way we talk about presupposition and its treatment in triclassi catory semantics? To break out of the circle, we might be prompted to look to an account of presupposition in the theory of assertion|to mesh with the semantic notion cast in triclassi catory logic. And, whatever we think of the particular accounts on oer in the literature, there is surely something to be said along these lines. Dummett's response to this would probably be that we would just have decorated the circle with super cial aspects of meaning, unless it had been shown that presupposition can make a distinctive contribution to the cognitive adjustments that people undergo when they understand what is said to them; and that this could never be shown. Even so, in the work referred to at the end of Section 4.2 I'm foolhardy enough to attempt an account which is intended to provide more than super cial decoration. 6 FIRST-ORDER PARTIAL SEMANTICS

6.1 Languages and Models In this section we outline a model-theoretic semantics to match the sketch of rst-order partial logic given in Section 1. A few facts about the logic will emerge, and their proofs will be outlined in Section 7, after we have presented an axiomatization of logical laws. (I hope that a much fuller account of things will soon appear.) The languages we work with will contain no description terms, though Section 6.4 deals with how they would t in. Let us, then, take a language L to consist of the following. (a) Logical vocabulary: (1) (2) (3) (4) (5)

sentence connectives :, ^, _, ^ _, >, and ?, quanti er symbols 8 and 9, an identity predicate symbol =, a constant symbol ~, a set Var of denumerably many variables.

(b) Non-logical vocabulary: (1) a set Prd (L) of predicate symbols, (2) a set Fnc (L) of function symbols, (3) a set Cns (L) of constant symbols. The elements P of Prd (L) and f of Fnc (L) are taken to come along with xed numbers (P ) and (f ) to give their number of argument places. Accordingly, a model for L is to be a structure M consisting of

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(0) a set DM (which does not have to be non-empty), (1) for each P 2 Prd (L), a monotonic function PM : (DM [ f~g)(P ) ! f>; ; ?g, (2) for each f 2 Fnc (L), a monotonic function fM : (DM [ f~g)(f ) ! DM [ f~g, (3) for each c 2 Cns (L), an element cM 2 DM [ f~g. For assignments to variables we shall just use functions s : V ar ! DM [ Then, if we de ne the terms of a language L in the usual inductive way, the classi cation Ms (t) of a term t under an assignment s is given as follows:

f~g.

Ms (x) Ms (~) Ms (c) Ms (ft1


t(f ) )

= = = =

s(x); for all x 2 Var ; ~; cM ; fM (Ms (t1 );


; Ms (t(f ) )):

We can now build on this to de ne the formulae of L and their interpretation in a model. Formulae|like terms|are taken to be de ned by functor rst construction throughout. But we shall be writing ` ^ ', `c = d', etc., rather than `^ ', `= cd', etc., and so be helping ourselves to brackets when necessary. This is just so much notation. And we can regard the following `de nitions' in the same light:

! $ =



=df > _ ^ ?; =df : _ ; =df [ ! ] ^ [ ! ]; =df [ ! ] _ ^ [ ^ ]:

Given an assignment s, a variable x and an element a in the xed-up domain DM [ f~g of a model M , let s(xja) be the assignment such that s(xja)(x) = a and s(xja)(y) = s(y) if y is a variable distinct from x. Then the classi cation Ms () of a formula  under an assignment s can be speci ed as follows:

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Ms (>) = >; Ms (?) = ?;



i Ms (t1 ); Ms (t2 ) 2 DM , and Ms (t1 ) = Ms (t2 ) Ms (t1 = t2 ) = > ? i Ms (t1 ); Ms (t2 ) 2 DM , and Ms (t1 ) 6= Ms (t2 );



i PM (Ms (t1 ; : : : ; t(P ) ) = > Ms (P t1 : : : t(P ) ) = > ? i PM (Ms (t1 ; : : : ; t(P ) ) = ?;



i Ms () = ? Ms (:) = > ? i Ms () = >;



i Ms () = > and Ms ( ) = > Ms ( ^ ) = > ? i Ms () = ? or Ms ( ) = ?;



i Ms () = > or Ms ( ) = > Ms ( _ ) = > ? i Ms () = ? and Ms ( ) = ?;



Ms () = > and Ms ( ) = > Ms ( _ ^ ) = >? i i Ms () = ? and Ms ( ) = ?;



i Ms(xja)() = >; for every a 2 DM Ms (8x) = > ? i Ms(xja)() = ?; for some a 2 DM ;



i Ms(xja)() = >; for some a 2 DM Ms (9x) = > ? i Ms(xja)() = ?; for every a 2 DM : These are the conditions for > and ?: Ms () is  if it is neither > nor ?. Observe how it is that variables have nothing more to do with ~, once they are bound by a quanti er. The classi cation of a formula has been de ned relative to an assignment, but we can neatly advance to a non-relative de nition: let M () be Ms (), where s assigns ~ to all variables. It will then follow (from Lemma 3) that M () = >(?) if and only if Ms () = >(?) for every assignment s. A free occurrence of a variable in a formula can be de ned in the usual way, and sometimes we shall call free variables parameters. Sentences are parameter free formulae and, as we should expect, their classi cation is in any case quite independent of assignments. This is a corollary of the following standard semantical lemma: LEMMA 1 (Relevant Variables). (1) If s1 (x) = s2 (x) for every x in t, then Ms1 (t) = Ms2 (t). (2) If s1 (x) = s2 (x) for every x free in , then Ms1 () = Ms2 ().

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Let us use the notation `u(t=x)' for the term obtained from a term u by substituting t for x throughout. Similarly, let us use `(t=x)' for the formula obtained from  by substituting t for all free occurrences of x in . And we shall say that t is substitutable for x in  when no occurrence of a variable in t becomes a bound (i.e., not free) occurrence in (t=x). Then there is a second standard lemma: LEMMA 2 (Substitution for Variables). (1) Ms (u(t=x)) = Ms(xjMs(t)) (u). (2) Ms ((t=x)) = Ms(xjMs (t)) (), provided that t is substitutable for x in .

6.2 Monotonicity and Compatibility Now for something more interesting: the monotonicity of evaluation (cf. Section 1.2). First we need to de ne a `degree-of-de nedness' relation, v, between models for a given language L: this consists in the appropriate `v'relation holding between the respective interpretations of the vocabulary of L. Writing it all out explicitly, in terms only of the basic relations on f>; ; ?g and on a xed-up domain D [ f~g, we have: M v N if and only if M and N have a common domain D and, for all P 2 Prd (L), f 2 Fnc (L) and c 2 Cns (L), (1) PM (~a) v PN (~a), for all ~a 2 (D [ f~g)(P ) , (2) fM (~a) v fN (~a), for all ~a 2 (D [ f~g)(f ) , (3) cM v cN . We also need to extend v, in the natural way, to assignments: s1 v s2 i s1 (x) v s2 (x), for all x 2 Var . Then for terms as well as formulae: LEMMA 3 (Monotonicity of Evaluation). If M1 v M2 and s1 v s2 , then (1) M1 s1 (t) v M2 s2 (t), (2) M1s1 () v M2 s2 (). The proof of this lemma is just a matter of checking|by induction on the complexity of terms and formulae. To set alongside `degree-of-de nedness' there is also a `compatibility' relation between models. In Section 4.4 we de ned a relation  on f>; ; ?g: neither >  ? nor ?  >, but otherwise  holds. And, analogously, we can de ne  on a xed-up domain D [ f~g by:

a  b i a and b are not distinct elements of D.

Then to de ne compatibility between models: M  N if and only if M and N have a common domain D and for all P 2 Prd (L), f 2 Fnc (L) and c 2 Cns (L),

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(1) PM (~a)  PN (~a), for all ~a 2 (D [ f~g)(P ) , (2) fM (~a)  fN (~a), for all ~a 2 (D [ f~g)(f ) , (3) cM  cN . And, as in the case of v, a natural compatibility relation is induced between variable assignments: s1  s2 i s1 (x)  s2 (x), for all x 2 Var . We could now prove a lemma parallel to Lemma 3, got by replacing `v' by `'; but this result will shortly be generalized, at least so far as formulae are concerned (part (2)), to something usefully stronger: Lemma 6. Observe now that if M  N , then we can coherently stick M and N together to de ne a model M t N , which is the least upper bound of M and N with respect to the v ordering: if D is the common domain of M and N , then, the interpretation of P 2 Prd (L), f 2 Fnc (L) and c 2 Cns (L), is given by stipulating that, (1) for any ~a 2 (D [ f~g)(P ) :  i either PM (~a) = > or PN (~a) = > PM tN (~a) = > ? i either PM (~a) = ? or PN (~a) = ?; (2) for any ~a 2 (D [ f~g)(f ) , and any b 2 D: fM tN (~a) = b i either fM (~a) = b or fN (~a) = b, (3) for any b 2 D: cM tN = b i either cM = b or cN = b. Similarly, if s1 and s2 are assignments D [ f~g ! Var , and if s1  s2 , then an assignment s1 t s2 is coherently de ned by stipulating that for any x 2 Var , and any a 2 D, s1 t s2 (x) = a i either s1 (x) = a or s2 (x) = a. We shall also be interested in purely `elementary' relations ve and e between models|and also a relation of elementary equivalence e |which can indierently be characterised either in terms of the classi cation of arbitrary formulae , or sentences , as follows:

M ve N i M () v N (); for any ; M e N i M ()  N (); for any ; M e N i M () = N (); for any : Notice that M e N if and only if M ve N and N ve M , just as M = N if and only if M v N and N v M . Notice, too, that the relations v and |and indeed the identity relation| can be characterized in terms of the evaluation of formulae: LEMMA 4. M v N i DM = DN and Ms () v Ns (), for any  and any s; M  N i DM = DN and Ms ()  Ns (), for any  and any s; M = N i DM = DN and Ms () = Ns (), for any  and any s:

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`Only if' follows trivially from Lemma 3 and the parallel result for ; `if' can easily be checked by considering atomic formulae. Relations of `degree-of-de nedness' and `compatibility' also arise in a natural way between formulae . Let us restrict attention to `purely logical' relations, de ned by generalising over all the models for a given language; then, including also a relation ' of equivalence:

v  '

i Ms () v Ms ( ); for any M and any s; i Ms ()  Ms ( ); for any M and any s; i Ms () = Ms ( ); for any M and any s:

Notice that  ' if and only if  v and v . The relation of compatibility between formulae gives rise to an interesting question. If   , then  and never take on con icting truth values: can we then stick  and together to yield a more de ned formula  which takes the value > or ? whenever either one of  and does? In other words, is there for compatible formulae any thing analogous to M t N for compatible models M and N ? Let us call  a joint for  and if and only if, for any model M and assignment s,

Ms () =



> i either Ms () = > or Ms ( ) = > ? i either Ms () = ? or Ms ( ) = ?:

There is clearly no monotonic mode of sentence composition which we could use to compound  and and thereby produce such a , but in fact joints for compatible formulae always exist. In the restricted case of propositional logic this is an immediate corollary of `expressive adequacy' (see Section 4.1 above), but it holds in quanti er logic too: THEOREM 5 (Compatibility Theorem). Any two logical compatible formulae have a joint. To prepare for our proof of this result in Section 7.3, we need two lemmas. The rst is the promised generalization of the compatibility result parallel to Monotonicity of Evaluation (Lemma 3): LEMMA 6. If   , M1  M2, and s1  s2 , then M1 s1 ()  M2 s2 ( ). To see this, consider M1 t M2 and apply part (2) of Lemma 3. (Note that part (2) of Lemma 3 can itself be generalized along the lines of this lemma: replace `' by `v'.) The second lemma could be thought of as saying that  and have a `least upper bound', viz. a joint, when and only when they have an `upper bound'. (Indeed, this makes quite literal sense if we think of the relation induced by v on the Lindenbaum algebra of a language.) LEMMA 7.  and have a joint if and only if there is a formula  such that  v  and v .

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`Only if' is trivial. Conversely, given , the following formula is obviously a joint: [[ _ ] ^ ] _ ^ [ _ [ ^ ]].

6.3 Interjunctive and Transplicative Normal Forms In section 4.1 we promised normal forms in quanti er logic to match the in propositional normal forms that derive from our proof of expressive adequacy. Let us, then, say that a formula is in interjunctive normal form when it is an interjunction _ ^  such that neither nor  contains any occurrence of ^ _ and such that, for any model M and any assignment s, Ms ( _ ^ ) = > if and only if Ms ( ) = >, and Ms ( _^ ) = ? if and only if Ms () = ?. Logical consequence has not yet been oÆcially de ned for our rst-order languages, but from the outline in Section 1.1 it is easy to see that this condition will turn out equivalent to saying that  . (The precise de nition of  is in section 6.5.) We can now show that an arbitrary formula  is logically equivalent to a formula in interjunctive normal form: in fact we can describe a procedure to transform  into normal form. The procedure relies on the fact|easy to check|that our language admits `substitutivity of equivalents': when a subformula is replaced by something equivalent, then the resulting formula is equivalent to the original one. This means we can rst replace any atomic subformula 0 of a formula  by 0 _ ^ 0 |which itself is clearly in normal form|and, since 0 ' 0 _^ 0 , the resulting formula will be equivalent to . Then we can progressively pull ^ _ out of the scope of the logical operators in |both connectives and quanti ers|working up from those with narrowest scope to the one with widest scope. What makes this possible is that if _ ^  is in normal form, or if both 1 _ ^ 1 and 2 _^ 2 are in normal form, then the following equivalences hold, and the formula on the right of `'' will again be in normal form: : ( _^ ) ' : _^ : ( 1_ ^ 1 ) ^ ( 2 _^ 2 ) ' ( 1 ^ 2 ) _^ (1 ^ 2 ) ( 1_ ^ 1 ) _ ( 2 _^ 2 ) ' ( 1 _ 2 ) _^ (1 _ 2 ) ( 1_ ^ 1 ) _^ ( 2 _^ 2 ) ' ( 1 ^ 2 ) _^ (1 _ 2 ) 8x( _^ ) ' 8x _^ 8x 9x( _^ ) ' 9x _^ 8x: Thus we can pull ^ _ out of the scope of an operator by replacing a subformula of one of the forms displayed on the left by the equivalent formula on the right. At each stage equivalence to  is preserved; and at each stage the replacement subformula is in normal form: and so we end up with an equivalent formula in normal form. The displayed equivalences do not of course hold unconditionally, except for the rst. We could alternatively use ones that did, but the formulae

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on the right would then be double the length. For example, to specify how to pull _ ^ out of the scope of a quanti er, when it governs an arbitrary interjunction, we need the following:

8x( ^_ ) ' 8x( ^ ) _^ 8x( _ ) 9x( _^ ) ' 9x( ^ ) _^ 9x( _ ): Suitable equivalences for ^, _, and ^ _ I leave as an exercise.

Let us now pretend that = is a primitive connective|and ! and $ as well. And let us say that a formula is in transplicative normal form when it is a transplication = such that neither nor  contains any occurrence of either = or ^ _ (so that there are only classical logical operators in and ) and such that, for any M and any s, Ms ( ) = > if and only if either Ms ( =) = > or Ms ( =) = ?. Then if we have a procedure, along the lines of the one above, for transforming an arbitrary formula into an equivalent one in transplicative normal form, this will yield projection rules for presupposition of the kind we were interested in at the end of Section 4.2. Such a procedure can be based on the following equivalences (which hold whether or not the constituents on the left are already in normal form):

( ( ( ( ( (

 :( = ) 1 =1 ) ^ ( 2 =2 ) 1 =1 ) _ ( 2 =2 ) 1 =1 ) ! ( 2 =2 ) 1 =1 ) $ ( 2 =2 ) 1 =1 ) = ( 2 =2 ) ^ ( 2 =2 ) 1 =1 ) _ 8x( = ) 9x( = )

' ( _ :) =  ' = : ' (( 1 ^ 2 ) _ ( 1 ^:1 ) _ ( 2 ^:2 )) = (1 ^ 2 ) ' (( 1 ^ 2 ) _ ( 1 ^ 1 ) _ ( 2 ^ 2 )) = (1 _ 2 ) ' (( 1 ^ 2 ) _ ( 1 ^:1 ) _ ( 2 ^ 2 )) = (1 ! 2 ) ' ( 1 ^ 2 ) = (1 $ 2 ) ' ( 1 ^ 2 ^ 1 ) = 2 ' ( 1 ^ 2 ^ (1 $ 2 )) = 2 ' (8x( ^ ) _ 9x( ^ :)) = 8x ' (9x( ^ ) _ 8x( ^ :)) = 9x

The rst equivalence gives us a way to transform atomic subformulae, and the rest show to pull = out of the scope of any logical operator|including other occurrences of = itself. If we have transformed a formula into transplicative normal form, then the resulting transplicator will be a summing up, in a =-and-^ _-free formula, of any presupposition introduced into the original formula by = or by ^_. (Some horrendously complicated transplicators can arise, but obvious simpli cations will be possible particular cases.) Furthermore, since the transplicand is also =-and-_ ^-free, we can see that a single occurrence of = is suÆcient for representing the overall content|the >=?-conditions|of the original formula.

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But if projection rules are the only thing you want to get, then observe that the equivalences for = and ^ _ may be brought in line with the others: ( 1 =1 ) = ( 2 =2 ) ( 1 =1 ) _ ^ ( 2 =2 )

' '

( (

1 ^ 2 ^ 1 ) = (1 =2 )

^ 2 ): 1 ^ 2 ^ (1 $ 2 )) = (1 _

A procedure based on these equivalences will transform a formula  into =, where sums up the overall presupposition, as before, but  is left to stand. On the other hand, we may want to pin down a =-and- _ ^ -free transplicand more tightly. Observe that a formula _ ^  in interjunctive normal form will be equivalent to ( ! ) = and to ( ! ) = , which are in transplicative normal form. (We can make do with  ! , rather than  $ , because  .) The transplicand then xes >-conditions, while the transplicand  xes ?-conditions. I shall leave it as an exercise to formulate equivalences on which to base a procedure for transforming a formula directly into a transplicative normal form of each of these special kinds: the equivalences given for ^, _, 8, and 9 can be kept, but the others need to be revised.

6.4 A Parenthesis on Description Terms If we expand our languages to contain a term-forming descriptions operator , and if we consider its interpretation in the kind of model we are working with, then the denotation conditions sketched in Section 1.1 turn out in the following way: for any model M , and any assignment s, if a 2 DM , then



Ms ( x) = a i Ms(yja) (8x[x = y $ ]) = >: 

And Ms ( x) = ~ if there is no such a. (We are here assuming that y is a variable distinct from x and extraneous to .) These denotation conditions can be spelt out to mean that if a 2 DM , then



Ms ( x) = a i



Ms(xja) () = >; and Ms(xj b) () = ?; for every b 2 DM not identical to a:



Hence, to be the denotation of x, a has to be determinately `the unique x such that ':  must be false, not just not true, when any other object in DM is assigned to x. But do we have to work with such a stringent form of uniqueness? In the present context we do, on pain of violating monotonicity. Notice that, according to our de nition, Ms ( x) is an element of DM only if Ms(xja)() is either > or ? for any a in DM . This guarantees monotonicity for -terms. If, for Ms ( x) to be an element a of DM , we were to require only that Ms(xja)() = > and that Ms(xjb) () 6= > for any b in DM distinct from a, then there might be a model N such that M v N and Ns(xjb) () = > for









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some such b, in which case Ns ( x) could not be a and monotonicity would have been violated. (For example, take M and N to be models interpreting a predicate symbol P over the domain f0; 1g, where PM (0) = PN (0) = PN (1) = > and PM (1) = | ll in other details as you like|and consider xP x.) Notice, then, that according to our de nitions x may be non-denoting for two dierent kinds of reason: either (i) because  is not suÆciently de ned to determine a denotation, or (ii) because  is suÆciently highly de ned to rule out there being one. Case (i) arises when the formula 9y8x(x = y $ ) is . and case (ii) when it is ?. If we had a subtler theory of identity and of the interpretation of `singular terms', then subtler interpretations for x would be available. But this leads far beyond the simple kind of model we are working with. The literature on description terms is vast and varied, but two approaches which it is interesting to compare and contrast with the present one occur in [Smiley 1960] and [Scott 1967]. Smiley entertains `neither-true-nor-false' sentences, but he is unconstrained by monotonicity; while Scott treats nondenoting terms in a logic which, at sentence-level, is classical and total. In [Czermak 1974], on the other hand, there is a theory more like the one here. But it should be emphasized that our de nitions do not involve any special ideas concerning the interpretation of description terms: they merely follow a path which was pre-determined once we embarked on partial logic as the logic of monotonic modes of composition. The standard semantical de nitions and lemmas of Sections 6.1 and 6.2 all extend in the obvious way to languages which contain |due account being taken of the fact that terms, as well as formulae, may now contain `bound' variables. And so we have a framework in which to address the question whether, having introduced -terms, we can after all `eliminate' them without decreasing the expressive power they provide. But what does this mean? There are various degrees of eliminability that we should distinguish. In a weak sense, would be eliminable provided that any formula were equivalent to an -free one. In a stronger sense of eliminability there would be some procedure which we could apply to transform a formula into an equivalent -free one. But we should really hope for something stronger still: to be in possession of a general scheme of scope-free elimination. And this is something we can indeed obtain. To signal one or more occurrence in a formula of a term x (possibly ignoring other occurrences of x) we can always pick on some extraneous variable y and describe the formula as ( x=y). And so we can take our goal to be to de ne a scheme I (x; ; y; ) which does not involve and which, for any  and , will yield a formula equivalent to ( x=y), provided only that x is `substitutable for y in '|i.e., that no free occurrence of a variable in x becomes a bound occurrence in ( x=y). Then we may read the scheme I (x; ; y; ) as `the x such that  is a y such that



































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', and it will provide for the `scope-free' elimination of -terms simply because -languages admit `substitutivity of equivalents': when a subformula is replaced by an equivalent one an equivalent formula results. The point is that to eliminate a term x from a formula we can apply the scheme to any subformula ( x=y) which binds no variables occurring free in x. Moreover, to transform a formula into an entirely -free one, we can apply the scheme to -terms in any order we like, and (variable-binding permitting) dierent occurrences of the same term can be eliminated all at once, or one at a time, or in any combination we choose. Such a scheme will then exhibit a semantical scope-freedom which exactly matches the scope-freedom possessed by an -term in virtue of its syntactic category. In Section 4.2 we presented a `Russellian' analysis for a de nite-description quanti er Ix[
;
], but any thought that this could serve as the required elimination scheme is soon dispelled. The >=?-conditions for Ix[
;
] certainly give de nite descriptions a fair degree of semantical scope-freedom|in particular, freedom with respect to negation|but it is not thorough-going. For example, if  is >, then Ix[; ] _  has to be >, though Ix[; _ ] might be . This is not a defect of our analysis for Ix[
;
], since scope sensitivity can be important if we are considering natural language description idioms, but we have to look elsewhere for a scheme to go proxy for de nite descriptions that are construed as terms. In fact, Ix[
;
] would not even serve to eliminate -terms from atomic formulae. This is because our monotonicity constraint is suÆciently liberal to allow sentences P t1


x


tn which are > or ? even when x is ~, though x is ~ only if 9y8x[x = y $ ] is not >, in which case Ix[; P t1


x


tn ] must be . It is not surprising, given this last observation, that our scheme of elimination will involve the logically non-denoting term ~. Let us abbreviate the formula 8x[x = y $ ] as (x!y), then we could use either of the following as de nitions of I (x; ; y; ):

























9y[(x!y) ^ ] _ [ 8y[(x!y) ! ] ^ (~=y)]; 8y[(x!y) ! ] ^ [ 9y[(x!y) ^ ] _ (~=y)]: To see that these formulae work, it is just a matter of checking >=?conditions (with the aid of an extended version of Lemma 2) to show that they are equivalent to ( x=y)|assuming, that is, that x is substitutable for y in . We have emphasized that an elimination scheme of this kind allows us to dispense with the syntax of description terms as terms without disrupting any of the characteristics they manifest as such. But in fact this could be achieved much more cheaply: simply introduce a primitive mode of complex quanti cation Dx[
;
] interpreted so that - - - Dx[;


x


]- - - will always mimic - - -(


x


)- - -. Stating explicit >=?-conditions for Dx[
;
] is routine. What we should now emphasize is that our de nitions for a scheme







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of elimination go a stage further than this: they show how a quanti er Dx[
;
] may be analysed in terms of simple and basic logical vocabulary. In other words, we can do for Dx[
;
] what in Section 4.2 we did for Ix[
;
]. In the basic languages presented in Section 6.1, the displayed elimination schemes can of course be viewed as de nitions|explicit de nitions for a complex quanti er or, `contextual de nitions' for an -term. And so we have a sense in which is de nable in terms of ~. Conversely, if we have , then ~ can be de ned directly|for example, as x?. Hence the presence of either ~ or provides equivalent expressive resources in a rst-order language subject to the kind of interpretation we are considering. However, we cannot dispense with ~ in -free languages without a decrease in expressive power: the atomic sentence P ~, for example, is equivalent to no ~-free formula. (To see this consider models M and N with the singleton domain f0g such that PM (0) = PM (~) = PN (0) and PN (~) = : if s(x) = 0, for all x 2 Var , then for any ~-free formula , Ms () = Ns (), though Ms (P ~) 6= Ns (P ~).) In the presence of ~, on the other hand, other vocabulary distinctive to partial logic could be dispensed with: given our interpretation of =,  could be de ned as ~ = ~, and hence|as we showed in Section 4.1|^ _ (and = ) could also be de ned. Although ~ is not logically eliminable, it remains a possibility that it is in some sense eliminable in particular non-logical theories set up in partial logic: we shall mention a theorem about this in Section 7.3.













6.5 Semantic Consequence To provide for a suitably powerful notion of semantic consequence, conceived along the lines suggested in Section 1.1, our basic de nition is of what it is for a model M for a language L, together with an assignment s, to reject a pair h ;
i of sets of formulae of L. We shall say that (M; s) rejects h ;
i if and only if either: (i) Ms () = > for all  2 and Ms ( ) 6= > for all or: (ii) Ms () 6= ? for all  2 and Ms ( ) = ? for all

2
, 2
: And let us say that M (on its own) rejects , or is a counter model to , h ;
i when there is an s such that (M; s) rejects h ;
i. Then, if M is any class of models for L, M |consequence in M|is de ned by M
i no model in M rejects h ;
i. When M is the class of all models for a given language, we just write `':

this is logical consequence . Following the common notational practice with turnstyles, we shall ignore squiggly brackets and the empty set, and replace union signs by commas: for example, ` >; ;
' means that ;  f>; g[
.

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In Section 1.1 we remarked on single-barrelled relations of consequence. Note the way in which  may now be deployed to capture such relations:

M ;
i no model in M satis es condition (i) above, ;  M
i no model in M satis es condition (ii) above. And M
if and only if both M ;
and ;  M
. In fact this biconditional is just an instance of a quite general principle: for any formula , M
if and only if both M ;
and ;  M
. In Section 7.1 we shall present logical laws using sequents : these will be understood to be pairs of nite sets, for which we use the special notation ` >
' instead of `h ;
i'. And we shall mention sequents in the same style that we state facts about consequence, writing ` > ; >; ;
', for example, to stand for ; > f>; g[
. When M is not a counter model to >
we shall say that M is a model of >
, or that >
holds in M . More generally, if  is a set of sequents, M will be said to be a model of  if and only if M is a model of every sequent in ; and `K()' will be the notation for the class of all such models. (Note: `model for L', `model of ' ). A sequent >
embodies a principle of consequence|
's following from . It is a principle of logical consequence if 
, in which case it holds in all models, but there are sequents which hold in some models but not in others; and there are also sequents, such as ; > ;, which hold in none. A set  of sequents then embodies a collection of such principles, and K() is the relation of consequence semantically determined by them:

K()
i no model of  rejects h ;
i. Observe, then, that K(;) is logical consequence; and that K(f; >

;g) is the universal relation between sets of formulae. Clearly, if >
is contained in , then K()
; but the converse does not generally hold: ; is an obvious counterexample. When it does hold|when  = f >
j K()
g|of sequents which is closed under the sequent principles it determines, and our proof theoretical apparatus will be designed to pick out precisely such sets of sequents as what `theories' are in partial logic. Thus we shall be adopting an extensional notion of a theory, not involving any particular axiomatization. Pure logic, for a given language, will be one such theory, viz. f >
j 
g. But K() is a full-blown consequence relation between arbitrary (not necessarily nite) sets of formulae, and we should demand of our proof system that it yield consequence relations ` to match K(). We shall produce a suitable de nition which is `sound and complete' in that, for any and
,

K()


i

`
:

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But then we shall be able to show that the relation K() does not actually go beyond the sequent principles|the nite principles of consequence| determined by |in fact not beyond those determined by some nite subset of . For the de nition of ` will guarantee that `
if and only if 0 `0
0 for some nite subsets 0 of ,
0 of
and 0 of ; so that K() too turns out to be nitary in this way. Contraposing, we could state the fact as a two-pronged form of compactness: THEOREM 8 (Compactness). There is a model of  which rejects h ;
i i, for every nite subset 0 of ,
0 of
and 0 of , there is a model of 0 which rejects h 0 ;
0 i. Two complementary parallels with standard treatments of classical logic are now emerging, which pervade the development of partial logic. First, pairs of sets of formulae and their rejectability (by a model and an assignment) play a role which single sets of formulae and their satis ability (by a model and an assignment) usually play in classical logic. Secondly, sets of sequents and their models play the part which sets of sentences and their models play in classical logic. But why should things turn out like this? It has already been explained|in Section 1.3|that principles of logical consequence cannot be summed up in terms of the truth of sentences, but the irreducibility of consequence to truth extends further than this. For, given a sequent >
, it is not in general possible to nd a sentence  such that M is a model of >
if and only if M () = T |equivalently, if and only if M is a model of > . (Moreover, if there is no sentence, then there is no formula of any kind to play this role; since, if there were a formula  then a suitable sentence could be obtained by substituting ~ for all parameters in .) This contrasts with classical logic, in which a sequent >
can always be summed up in the sentence 8~x[^^ ! __
], where ^^ is the conjunction of elements of , __
is the disjunction of elements of
, and 8~x binds all free variables. We can extend and strengthen this point about partial logic: given a set  of sequents it is not in general possible to nd a corresponding set 0 of sequents of the truth-expressing form >  such that M is a model of  if and only if M is a model of 0 . To see this observe that if we can nd such a 0 , then K() satis es the following closure condition|because K(0 ) obviously does and K() = K(0 ). (y)

If M

2 K() and M ve N , then N 2 K().

In fact we could use the Compactness Theorem to show that (y) is a suÆcient, as well as a necessary, condition for nding such a 0 . But the present point depends on its being necessary: we just have to produce a  such that K() does not satisfy (y). A simple example would be fP ~ > Q~; g: checking this is essentially an exercise in propositional logic.

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Although the principles of consequence that arbitrary sequents express cannot be reduced to the truth of sentences, still, can we at least make do with parameter-free sequents, which contain only sentences, not arbitrary formulae? No, we cannot. Let us argue in the same pattern as before: the following is obviously a necessary condition (and in fact also a suÆcient condition) for there being a set 0 of parameter free sequents such that K() = K(0 ). (z)

If M

2 K() and M e N , then N 2 K().

However, fP x > Qx; g, for example, does not satisfy (z)|though it is more involved to check out this example than the previous one. This is perhaps a little surprising: it means that the relation e of `elementary equivalence' between models is a strictly weaker relation that the relation of being a model of the same sequents. Anyhow, let us return to the relation  of logical consequence. This has been de ned relative to a particular language L, but, as in classical logic, it is in fact an absolute notion, in the sense that 
in L1 if and only if 
in L2 , whenever the formulae in and
are formulae of both L1 and L2 . In particular, 
in any given language if and only if 
in the language containing no non-logical vocabulary other than that occurring in or
. Observe too that the relations of equivalence ('), degree-ofde nedness (v) and compatibility (), which we de ned in Section 6.2, are absolute in this sense. These facts are easy to check, using the notion of the reduct M  L0 of a model M for L to a smaller language L0 : M  L0 is the model for L0 which has the same domain as M and interprets the vocabulary of L0 in the same way as M , just ignoring any vocabulary in L but not in L0 . We shall use this de nition later on, and we shall also talk of expanding a model M for L to a model N for a bigger language L+ when M = N L. The absoluteness of  means that we can state the following theorem without reference to any particular language (though its proof|in Section 7.3|will depend on being very nicky about languages). THEOREM 9 (Craig Interpolation). If   , then    and   for some formula  which contains no non-logical vocabulary which does not occur both in  and in . It is noteworthy that there is an analogous result for degree-of-de nedness: if  v , then  v  and  v for an interpolant  subject to the same constraint.

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7 FIRST-ORDER PARTIAL THEORIES

7.1 Logical Laws It will be neatest to take our logical laws as directly de nitive of what a `theory' is. The laws will be in the form of sequent axioms and sequent rules, and a theory , in a given language L, is de ned to be a set of sequents of L which contains the sequent axioms and is closed under the sequent rules, in the sense that if the `premise(s)' of a rule is (are) in the set then so is its `conclusion'. `Proofs' then enter the picture in the following way. If, given a set  of sequents of L, we de ne  to be the intersection of all theories in L which contain , then  will be a theory|the `smallest' theory in L containing |and a sequent will be contained in  if and only if there is a sequent proof of it from a nite subset of . That things t together in this way is just part of the general theory of inductive de nitions (see for example [Aczel 1977]). We shall call  the theory axiomatised by ; and  will already be a theory if and only if  = . Pure logic, for a given language L, then slots into place as the smallest theory in L, viz. ;. The rst three laws are general principles of consequence, which we label after [Scott 1973b]: a basic axiom scheme (R), a (double) rule of thinning (M), and cut (T). (R) >  (M) (T)

>
> ;


>
; >


> ;


; >


>
Clearly any instance of (R) will hold in any model, and if the `premise(s)' of an instance of (M) or (T) hold in a model, then the `conclusion' holds in that model. Hence individually these laws are `sound'. It will be left unsaid that all the remaining axioms and rules are individually sound in the same way: this can be checked using the de nitions and lemmas of Section 6.1. The next rule is a general rule of (S) of substitution. When  is a set of formulae, we use `(t=x)' to stand for f(t=x) j  2 g. >
(S) (t=x) >
(t=x) This holds provided that the term t is substitutable for x in all the formulae in and
(see Section 6.1). In the presence of this rule we shall be able to specify the quanti er and identity laws with parameters, instead of using schematic letters for terms.

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For propositional laws we can use the following. Double lines means the rule applies upwards as well as downwards, and `:' stands for f: j  2 g.

>

>

?>

: > 

 > :

; : >



 > :; 

: > :
>




>
>

; ;

>


> ; ;


;^ >


> _ ;


;
;^ _ > ;
; ;

:
:

;  > ; ;


>

; > _ ^ ;
Observe how  may be deployed to cancel one or the other half of our double-barrelled notion of consequence. Thus, in particular, the rules for interjunction match _ ^ with ^ for >-conditions and with _ for >-conditions. From these laws we can immediately deduce some further fundamental principles (which could be swapped in various obvious ways to provide alternative sets of propositional laws):  > :: :: >  ; : >

: ;

>


:
> : ^ >  ^ > ; >  ^

 > _ > _  _ > ;

_ ^ > ;  _ ^ > ; ; >  _ ^

;  >  _^ ; >  _^ _ ^ > ;

Let us now adopt the abbreviation ` >




' for ` ;  >

:;
'.

The

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force of such sequents can be expressed informally as `when  is true, then
follows from ': recall the discussion at the end of Section 1.1. Then for quanti ers we can use the following up-and-down rules, subject to the proviso that x does not occur free in any formula in or in
:

>

x=x

;


;>

x=x




> 8x;
; 9x >
The proviso is only of importance for the downward rules, but given (S) its presence does not hamper the upward ones, which are equivalent to the following axioms:

8x >

x=x



>

x=x

9x:

Notice how x = x is here playing the role of an `existence predicate'. Of course, x = x can never actually be false, and so we include the following axiom:  > x = x: And to capture the determinateness of identity:

x = x; y = y > x = y; :x = y: For the substitutivity of identicals we adopt the following scheme, which means that whenever x = y is true, then occurrences of x and y can be shued around in a formula in any way you like:

(x=u; y=v) >

x=y

(y=u; x=v):

However a further substitutivity principle is required to govern non-denoting terms: (x=z ) > x = x; (y=z ): Since parameters are schematic for terms, the force of this is that a nondenoting term can be replaced by any term without aecting the truth value of a formula, if it already has one. If we were envisaging subtler theories of identity these laws would need to be modi ed, but in the present context they capture our semantics of monotonic composition, once we include an axiom for the logically nondenoting term: x=~ > : There is room for variation in the choice of primitive laws for identity; but let us adopt these. We can then go on to derive a characteristic principle for ~, whose eect is that if a formula is true (or false), then it remains so on making any substitution for an occurrence of ~:

(~=x) > (y=x); :

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And other basic laws are easily obtained; for example, the symmetry of identity and distinctness:

x = y > y = x; and the transitivity of identity:

x = y; y = z > x = z; : Observe that  cannot be taken away here: if y assigned no object, then neither of the left-hand formulae can be false, even if x = z is. However, we can easily derive a general principle to handle distinctness as well as identity: y=y x = y; y = z > x = z: The laws we have given provide the de nition of a theory (in L) and of the theory  (in la) axiomatised by , in the way explained at the outset. Furthermore between arbitrary sets and
of formulae of a language L we can de ne the consequence relation ` , demanded in Section 6.5, by stipulating that `
if and only if, for some nite subsets 0 of and
0 of
, 0 >
0 2 . This will be if and only if there is a proof of 0 >
0 from some nite subset 0 of |hence if and only if 0 >
0 2 0 . Thus ` turns out to be nitary in the way announced in Section 6.5. Note that, although the de nitions of  and ` are relative to a particular language L, a given set  of sequents will always be a set of sequents of (in nitely) many dierent languages. This means that, on its own, our notation is radically ambiguous, and we need to be careful when more than one language is in play. Since our laws are individually sound, it is easy to check that no model of  can be a counter model to any sequent in : in other words, not just is it the case that K()  K(), but K() = K(). And the following theorem, which makes reference to arbitrary sets and
, is a trivial extension of this fact: THEOREM 10 (Soundness). If `
then K()
. The converse, guaranteeing that ` coincides with the semantically de ned relation K() , is rather more diÆcult to establish: THEOREM 11 (Completeness). If K()
then `
. We shall turn our attention to the proof of completeness in Sections 7.2 and 7.3. It is easy to see that  = , and so ` is the same relation as ` . Also, given soundness, K() is the same relation as K() . Hence we would lose nothing by stating Theorem 11 with  restricted to theories. We would lose something if we restricted and
to nite sets, viz. being able to deduce as a corollary the full version of compactness stated in Theorem 8. But note

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that when we do consider just nite sets of formulae, then Theorems 10 and 11 may be wrapped up together into the following equation:  = f >
j K()
g. These remarks put us in a position to convert the discussion in Section 6.5 of the conditions labelled (y) and (z) into facts about theories. We may deduce from that discussion that a theory  is axiomatizable by sequents of the form >  if and only if K() is closed under the relation ve , in the sense of condition (y), and that a theory  is axiomatizable by parameterfree sequents if and only if K() is closed under the relation e , in the sense of condition (z). And there are various other results along these lines: necessary and suÆcient conditions for a theory's being axiomatizable by sequents of a given kind are provided by specifying closure conditions on the class of its models. In connection with soundness and completeness we should also think about `consistency'. We have no use for a notion of the consistency of a set of formulae, but it makes sense to ask about the consistency of a set of sequents. Let us say that  is consistent if and only if ; > ; 62 . And `in consistent' will just mean not consistent. Hence we may also de ne relational notions: 1 is (in)consistent with 2 if and only if 1 [ 2 is (in)consistent (which in turn makes sense of the words ` 's following from  is inconsistent with : : :', used in Section 2.7: this means that f > g is inconsistent with : : : ). By rule (M), it follows that  is consistent if and only if  does not contain all sequents (of the language in question). It also follows, by Theorems 10 and 11, that  is consistent if and only if  has a model, since the statement that  is in consistent if and only if  has no model is just the special case of soundness and completeness when and
are both empty. On the other hand, the special case of Theorems 10 and 11 when  is empty gives the soundness and completeness of an axiomatization of the relation `; of logical consequence (for which we shall just write ``'). Happily the theory thus axiomatised, viz. ;, turns out to be consistent, according to our de nition, since there will be models of ;|and hence too of ;|in great abundance. It is noteworthy that to axiomatize pure logic we could abandon the system presented here and instead use a cut-free sequent calculus that has `introduction rules' only. (See Sundholm's chapter in Volume 2 of the second edition of this Handbook.) One way to proceed would be to have `negative' rules as well as `positive' rules|the negative rules for vocabulary in the immediate scope of negation. In [Cleave 1974] there are rules of this kind that we could use for classical vocabulary, but to handle interjunction we need to include the following three-premise rules.

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; ;  >


; ; >
; ^ _ >


; ; >


> ; ;


> ; ;
> ^ _ ;


>

; :;  >
>

341

:; ;


; :; : >
; :[ ^ _ ]>
> >

:; : ;
:[ ^_ ];


; ;


; ; : >
>

; : ;


7.2 Model-Existence Theorems Wrapping Soundness and Completeness up together, contraposing, and spelling out ` 2K()
' we have that 0
i there is a model of  which rejects h ;
i:

(The line through the turnstyles signi es negation.) We could then establish completeness (`only if') by adopting a Henkin-style strategy to boost up any pair h ;
i such that 0
to an exhaustive pair h + ;
+ i of sets of sentences of an extended language, from which we could then read o a model rejecting h ;
i. But this strategy can be elaborated to yield much more powerful model-existence results: kinds of interpolation theorem. We can then go on to deduce the Completeness Theorem and a lot more besides|facts both about pure logic and about non-logical theories. To introduce the idea, consider the following set up:{ 1 is a set of sequents of a language L1 , and 1 and
1 are sets of formulae of L1 ; 2 is a set of sequents of a language L2 , and 2 and
2 are sets of formulae of L2 ; and  is a set of formulae common to both L1 and L2 . We can then ask: Is there a  2  such that

1 `1

;
1 and

2 ;  `2


2 ?

(We may suppose that `1 is de ned relative to L1 and `2 relative to L2 .) Notice that, provided  is non-empty, this is a generalization of the question `Is it the case that `
?'. For if  = 1 = 2 , = 1 = 2 , and
=
1 =
2 , then, by rules (M) and (T), the two questions must have the same answer. And our interpolation theorems may be seen as generalizations of the Completeness Theorem, because they state that the answer `no' to certain questions of the displayed form entails the existence of a pair of models M1 of 1 and M2 of 2 such that M1 rejects h 1 ;
1 i, M2 rejects h 2 ;
2 i and

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M1 and M2 are related in a particular speci ed way: dierent ways for M1 and M2 to be related correspond to dierent assumptions about . We also have corresponding generalizations of the Soundness Theorem, since the non-existence of an interpolant will be necessary as well as suÆcient for the existence of a suitably related pair of models. But necessity is not as interesting as suÆciency; it gives us nothing new: it will always be immediately deducible from soundness. To give a taste for all this, I shall develop a little way the case where, in the set up described,  is the set of all formulae of some sublanguage L of L1 and of L2 . This is a simple and straightforward case, but even so we shall be able to deduce quite a lot from it. First, to specify appropriate relationships between models, we need a generalization of the relations v and  de ned in Section 6.2: if M1 is a model for L1 and M2 is a model for L2 , then there are relations of degreeof-de nedness (vL ) and of compatibility (L ) relative to the vocabulary of a common sublanguage L. With the notion of a reduct at hand (see Section 6.5), we can de ne the relations like this: M1 vL M2 i M1 L M1 L M2 i M1 L

v M2 L;  M2 L:

Next observe that the claim that an interpolant exists can be analysed as the conjunction of three separate interpolant-existence claims: LEMMA 12 (Combination Lemma). There is a  2  such that ; 1 `1 ;
1 and 2 ;  `2
2 i the following all hold: (1) there is a 1 2  such that 1 `1 ; 1 ;
1 and 2 ; 1 `2 ;
2; (2) there is a 2 2  such that 1 ;  `1 2 ;
1 and 2 ; 2 ;  `2
2 ; (3) there is a 3 2  such that 1 `1 ; 3 ;
1 and 2 ; 3 ;  `2
2 :

`Only if' is trivial: put 1 = 2 = 3 = . For `if' it is straightforward to check that we may take  = [[1 ^ 3 ] _ 2 ] _ ^ [1 ^ [3 _ 2 ]]. We shall sketch a proof of a model-existence result that is in fact split up into three parallel theorems, corresponding to the three cases above: Theorem 13. But the Combination Lemma will show how they can be combined into one: Theorem 14. So there are two theorems to state. The assumptions common to both are that L is a sublanguage of L1 and of L2 , and  is the set of all formulae of L; that 1 and
1 are sets of formulae and 1 a set of sequents of a language L1 ; and that 2 and
2 are sets of formulae and 2 is a set of sequents of a language L2 .

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THEOREM 13 (Interpolant-Excluding Model Pairs: split-up version). In each of the three cases

 2
1 \
2 ; there is no  2  such that (1)

(2) 1

2

1 \ 2;

`1 ;
1

(3)

 2
1 \ 2 ;

and 2 ;  `2
2

i there are models M1 of 1 and M2 of 2 , with a common domain and assignments s1 and s2 such that (M1 ; s1 ) rejects h 1 ;
1 i, (M2 ; s2 ) rejects h 2 ;
2 i, and in case (1), M1 vL M2 and s1 v s2 ; in case (2), M2 vL M1 and s2 v s1 ; in case (3), M1 L M2 and s1  s2 :

THEOREM 14 (Interpolant-Excluding Model Pairs: combined version). There is no  2  such that 1 `1 ;
1 and 2 ;  `2
2 i there are models M1 of 1 and M2 of 2 , with a common domain and assignments s1 and s2 such that at least one of the following holds:



(M1 ; s1 ) rejects h 1 ; fg [
1 i;  (M2 ; s2 ) rejects h 2 ; fg [
2 i; 1 ; s1 ) rejects h 1 [ fg;
1 i; (2) M2 vL M1 ; s2 v s1 ; and ((M M2 ; s2 ) rejects h 2 [ fg;
2i;  1 ; s1 ) rejects h 1 ; fg [
1 i; (3) M1 L M2 ; s1  s2 ; and ((M M2 ; s2 ) rejects h 2 [ fg;
2i: (1) M1 vL M2 ; s1 v s2 ; and

It is now easy to see that the split-up version together with the Combination Lemma entails the combined version; and it is easy to check directly| from basic de nitions|that the combined version entails the split-up version. Some applications can appeal directly to just one of the three cases of the split-up version, but most will invoke the combined one. Now we sketch a proof|in its bearest outlines|of Theorem 13. `If' follows easily from soundness in each of the three cases. `Only if' is nontrivial, but the main construction is the same in each case: distinguishing between them comes only at the very end. First, then, take two disjoint sets C and D of new constants, where C is denumerable, and the cardinality of D is the maximum of the cardinalities

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of the the two languages L1 and L2 ; and take some one-one function  from + Var onto C . Now let L+ 1 and L2 be the extensions of L1 and L2 got by taking C [ D as additional constants; and let + be the set of all sentences obtained from a formula in  by making any substitution of constants from C [ D for the parameters (so the sentences in + will be common to both L+1 and L+2 ). And nally, some notation: if  is a formula,  is the formula obtained by substituting (x) for all free occurrences of x; and if  is a set of formulae,  = f j  2 g. Assuming that there is no  2  such that 1 `1 ;
1 and 2 ;  `2
2 , it is now fairly easy to deduce that there is no  2 + such that 

1

`1 ; 
1

and  2 ; 

`2 
2 ,

where `1 and `2 are now de ned relative to the extended languages L+1 and L+2 , rather than L1 and L2 . The hard work is then to provide a construction that achieves the following. First,  1 , 
1 ,  2 , and 
2 are extended to sets +1 ,
+1 , +2 , and
+2 of sentences such that +1 [
+1 exhausts all the sentences of L1 , +2 [
+2 exhausts all the sentences of L2 , and there is no  2 + such that

+ 1

`1 ;
+1

and

+;  2

`2


+2 .

(Notice that, since ? 2 +, +1 01
+1 ; and, since > 2 + , +2 ; 02
+2 : thus +1 \
+1 = +2 \
+2 = ;.) Secondly, the construction de nes a subset D0 of D such that for all d 2 D0 , d = d 2 +1 \ +2 and :d = d 2
+1 \
+2 ; if 9x 2 +1 , then (d=x) 2 +1 , for some d 2 D0 , if 9x 2 +2 , then (d=x) 2 +2 , for some d 2 D0 , if 8x 2
+1 , then (d=x) 2
+1 , for some d 2 D0 , if 8x 2
+2 , then (d=x) 2
+2 , for some d 2 D0 .

(Thus quanti ers will be `witnessed' by elements of D0 |which the rst condition will guarantee are `de ned'.) Now we de ne relations 1 and 2 over D0 as follows:

d 1 e i d = e 2 d 2 e i d = e 2

+ 1 + 2

and :d = e 2
+1 , and :d = e 2
+2 .

These turn out to be equivalence relations, and we can use them to factor out D0 to provide domains for models M1+ for L+1 and M2+ for L+2 , such that M1+ is a model of 1 that rejects h +1 ;
+1 i, and M2 is a model of 2 that rejects h +2 ;
+2 i: the models can be de ned in terms of h +1 ;
+1 i and h +2 ;
+2 i in much the same way that a classical model is de ned from a consistent and complete set of sentences in a standard Henkin-style completeness proof.

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But, by axiom (R), +1 \ + \
+2 = ;, from which we can deduce that 1 and 2 are in fact the same relation, so that M1+ and M2+ have a common domain. Their reducts M1 and M2 to the original languages L1 and L2 then turn out to be models of 1 and of 2 such that (M1 ; s1 ) rejects h 1 ;
1 i and (M2 ; s2 ) rejects h 2 ;
2 i, where s1 and s2 are de ned by putting s1 (x) = M1+ ((x)) and s2 (x) = M2+((x)). Finally, to deduce the relationship between M1 and M2, and between s1 and s2 |which is peculiar to each of the three cases|we again make use of the fact that +1 \ + \
+2 = ;. This guarantees the following facts: in case (1), M1+() v M2+ () for any  2 + ; in case (2), M2+() v M1+ () for any  2 + ; in case (3), M1+ ()  M2+ () for any  2 + .

Hence, rst, we can deduce that in case (1), M1 s () v M2 s () for any  2  and any s; in case (2), M2 s () v M1 s () for any  2  and any s; in case (3), M1s ()  M2 s () for any  2  and any s. But  contains all formulae of L. And, for any  2 , M1 s () = (M1 L)s () and M2s () = (M2 L)s (). It therefore follows from Lemma 4 that the displayed conditions are equivalent, respectively, to (1) M1 vL M2;

(2) M2 vL M1;

(3) M1 L M2 :

Secondly, since, for any variable x and any d 2 D0 , (x) = d and :(x) = d are both in + , we can also deduce|from the facts about M1+ and M2+ | that (1) s1 v s2 ; (2) s2 v s1 ; (3) s1  s2 :

7.3 Some Proofs The Completeness Theorem (Theorem 11) can now immediately be established: we shall argue by contraposition. Assume, then, that 0
. By rule (T), it follows that there can be no formula  such that ` ;
and ;  `
. And so to show that some model of  rejects h ;
i we may apply Theorem 14, taking each of L1 , L2 , and L to be whatever language we're working with, and taking 1 = 2 = , 1 = 2 = , and
1 =
2 =
. This guarantees models M1 and M2, with assignments s1 and s2 , which satisfy at least one of the three conditions speci ed. But each of these conditions obviously entails that both M1 and M2 reject h ;
i|which is over-kill: pick either one. To establish the Compatibility Theorem (Theorem 5), we can appeal directly to case (3) of Theorem 13. Assume that   , and|aiming for a

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contradiction|assume that there is no joint for formulae  and . Then, by Lemma 7, there is no lambda such that both  v  and  v . But this is equivalent to the absence of any  such that

_

 ;  and ;    ^ : By Soundness we can replace  by `, and then we have something in the

right form to apply Theorem 13, case (3). Since we are working with pure logic in a single language, we take each of L1 , L2 , and L to be this language| so that L will just be |and we take 1 = 2 = ;. Then we take 1 = f _ g, 2 = fg,
1 = fg,
2 = f ^ g. This guarantees models M1 and M2 , with assignments s1 and s2 , such that (M1 ; s1 ) rejects h 1 ;
1 i, (M2 ; s2 ) rejects h 2 ;
2 i, M1  M2 , and s1  s2 . But the rejections mean that (M1 s1 () = > or M1 s1 ( ) = >) and (M2 s2 () = ? or M2s2 ( ) = ?) Distributing `and' across `or' there are then four possibilities, each of which, by Lemma 6, contradicts the assumption that   . To establish Craig Interpolation (Theorem 9) we now make use of the fact that in Theorem 14 L1 and L2 might be dierent languages. Given formulae  and , let L1 be the language whose non-logical vocabulary is precisely that occurring in , let L2 be the language whose non-logical vocabulary is precisely that occurring in , and let L be the language whose non-logical vocabulary is precisely that common to both L1 and L2 . Assume now that there is no Craig interpolant for formulae  and : we have to show that  2 . But, by Soundness, the absence of a Craig interpolant means that there is no formula  of L such that  `  and  ` . And so we may apply Theorem 14 taking 1 = 2 = ;, 1 = fg, 2 = ;,
1 = ;,
2 = f g. This guarantees models M1 for L1 and M2 for L2 , along with assignments s1 and s2 , such that at least one of three possible conditions obtains. We shall consider each in turn. In case (1), M1 vL M2, s1 v s2 , (M1 ; s1 ) rejects hfg; fgi, and (M2 ; s2 ) rejects h;; f; gi. But now let M be an expansion of M2 which gives vocabulary in L1 but not in L2 the interpretation that M1 gives it. Then M1 v M L1 . Thus, by Monotonicity of Evaluation (Lemma 3), and since M L1 treats formulae of L1 in the same way as M , it follows that

v (M L1 )s1 () = Ms1 () v Ms2 (): But (M1 ; s1 )'s rejecting hfg; fgi means that M1 s1 () = >, from which it follows that Ms2 () = >. On the other hand, (M2 ; s2 )'s rejecting h;; f; gi means that M2 s2 ( ) = 6 >, from which it follows that Ms2 ( ) = 6 >. Hence (M; s2 ) rejects hfg; f gi|showing that  2 . M1 s1 ()

In case (2) we can argue in an exactly parallel way.

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In case (3), M1 L M2 , s1  s2 , (M1 ; s1 ) rejects hfg; fgi, and (M2 ; s2 ) rejects hfg; f gi. But now let M1+ be an expansion of M1 which gives vocabulary in L2 but not in L1 the interpretation that M2 gives it; and let M2+ be an expansion of M2 which gives vocabulary in L1 but not in L2 in the interpretation that M1 gives it. Clearly M2+  M2+ , and if M = M2+ t M2+ and s = s2 t s2 , then, by Monotonicity of Evaluation,

M1 s1 () = M1+ s1 () v Ms () and M2 s2 ( ) = M2+s2 ( ) v Ms ( ): But the rejections mean, respectively, that M1 s1 () = > and M2 s2 ( ) = ?. It follows that Ms () = > and Ms ( ) = ?. Hence (M; s) rejects hfg; f gi|again showing that  2 . Finally we shall use the Interpolant-Excluding Model Pairs Theorem to prove a result, which has not been mentioned before, about non-logical theories: a model-theoretic criterion for when a piece of non-logical vocabulary is de nable in a theory . First we need a relation ' of equivalence in | or -equivalence . Now that we have soundness and completeness in place, we can indierently de ne this relation either in terms of ` or in terms of the models of :  '

i  ` and ` , i Ms () = Ms ( ) , for any M

2 K() and any s.

Then let us say that (i) a predicate symbol P , (ii) a function symbol f , (iii) a constant symbol c, is (explicitly) de nable in  if and only if there is a formula  that does not contain (i) P , (ii) f , (iii) c, such that (i) P x1 :::x(P )

' ;

(ii) y = fx1 :::x(f )

' ;

(iii) y = c ' ;

(where the displayed variables are assumed to be distinct from one another). The de nability theorem takes exactly the same form for each of these three cases, and so we can state it schematically for an item  of non-logical vocabulary. Say that L is the language of the theory , and let L6 be the language got from L by dropping , then THEOREM 15.  is de nable in  i, for any models M and N of , (a) if M vL6 N , then M

v N,

and (b) if M L6 N , then M  N .

In other words, it is necessary and suÆcient for the de nability of  that given a pair of models of , if (a) the relation v, or (b) the relation , obtains between the interpretations of vocabulary other than , then it also obtains between the interpretations of . It is easy enough to check `only if' directly. To establish `if', we can argue by contraposition and invoke

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Theorem 14. I shall sketch the case where  is a predicate letter P : the other cases are not too dierent. Assume, then, that P is not de nable in . This means that there is no formula  of L6 such that P x1 : : : x(P ) `  and  ` P x1 : : : x(P ) . Hence we can apply Theorem 14, taking both the L1 and L2 of that theorem to be the language L of this one, and taking the L of that theorem to be L6 . And we then take 1 = 2 = , 1 = fP x1 : : : x(P ) g,
1 = ;, 2 = ;,
2 = fP x1 : : : x(P ) g. This guarantees models M1 and M2 of , along with assignments s1 and s2 , such that at least one of three possible conditions obtains. We shall consider each in turn. In case (1), M1 vL6 M2 , but the rejection conditions, together with the fact that s1 v s2 , entail that M1 6v M2 . For (M1 ; s1 ) rejects hfP x1 : : : x(P ) g; fgi, so that M1 s1 (P x1 : : : x(P ) ) = >, and therefore M1 s2 (P x1 : : : x(P ) ) = >; but (M2 ; s2 ) rejects h;; f; P x1 : : : x(P ) gi, so that M2 s2 (P x1 : : : x(P ) ) 6= >. In case (2) we can argue in an exactly parallel way. In case (3), M1 L6 M2 , but the rejection conditions, together with the fact that s1  s2 , entail that M1 6 M2 . For (M1 ; s1 ) rejects hfP x1 : : : x(P ) g; fgi, so that M1 s1 (P x1 : : : x(P ) ) = >; and (M2 ; s2 ) rejects hfg; fP x1 : : : x(P ) gi, so that M2 s2 (P x1 : : : x(P ) ) = ?: and therefore M1s (P x1 : : : x(P ) ) = > and M2s (P x1 : : : x(P ) ) = ?, where s = s1 t s2 . There are two noteworthy comments on this de nability result. First, the condition on models of  is strictly stronger than the condition that whenever models agree exactly on vocabulary other than , then they also agree on . Secondly, it follows from the de nability of  in  that there will be a uniform procedure for transforming any formula into an -free -equivalent one. In the case of a predicate symbol this is just a matter of making the obvious substitution. In the case of a de nable function symbol f , on the other hand, there will be a scheme of elimination for terms ft1 : : : t(f ) that is scope-free in the same way that the descriptionscheme we speci ed in Section 6.4 is scope free. Given terms t1 ; : : : ; t(f ) , we shall always be able to de ne f using a formula  that contains no variables occurring in t1 ; : : : ; t(f ) : y = fx1 : : : x(f ) ' . (We can always rewrite variables as required.) Then, by rule (S),

y = ft1 : : : t(f )

' (t1 =x1 ) : : : (t(f ) =x(f ) )

((ti =xi )

for short).

It follows that, provided ft1 : : : t(f ) is substitutable for y in , (ft1 : : : t(f ) =y) will be -equivalent to each of the following:

9y[(ti =xi ) ^ ] _ [ 8y[(ti =xi ) ! ] ^ (~=y)]; 8y[(ti =xi ) ! ] ^ [ 9y[(ti =xi ) ^ ] _ (~=y)]: And a de nable constant symbol can be handled in a parallel way|without any need to fuss about variables.

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_^ _^ _^ Further model-theoretic results about non-logical theories can be derived from subtler versions of the Interpolant-Excluding Model Pairs Theorem(s). An example of this is the theorem we mentioned in Section 6.4 concerning the eliminability of ~ in a theory . By `eliminability' let us agree to mean simply that any formula  is equivalent in  to some ~-free formula :  ' . And let us de ne a new degree-of-de nedness relation v6~ between models M and N by taking over the de nition of `M v N ' given in Section 6.2, but restricting ~a, in clauses (1) and (2), to D(P ) and to D(f ) . D is the common domain of M and N , and so M v6~ N if and only if N is more de ned than M over objects in the domain. In general v6~ is a strictly weaker relation than v, but THEOREM 16. ~ is eliminable in a theory  if and only if, whenever M and N are non-empty models of  and M v6~ N , then M v N . Another result about non-logical theories arises from further consideration of the Compatibility Theorem (Theorem 5). This theorem was a result about pure logic, but the question arises concerning an arbitrary theory  whether formulae that are compatible in |i.e. never take on con icting truth values in models of |have a joint in the theory |i.e. a formula with the >=?-conditions of a joint in all models of . The answer is `no', but we can derive a model-theoretic criterion for when a theory is guaranteed joints for all compatible formulae. This result, however, requires more apparatus than we have developed|even to state, let alone to prove. St Edmund Hall, Oxford.

BIBLIOGRAPHY [Aczel 1977] P. Aczel. An Introduction to inductive de nitions. In Handbook of Mathematical Logic, J. Barwise (ed.), pp. 739{782. North Holland, Amsterdam, 1977. [Aczel and Feferman 1980] P. Aczel and S. Feferman. Consistency of the Unrestricted Abstraction Principle using an Intensional Equivalence Operator. In To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, J. P. Seldin and J. R. Hindley, eds., Academic Press, London, 1980. [Barendregt 1984] H. P. Barendregt. The Lambda Calculus, North Holland, Amsterdam, 1984, 2nd reprint edition in paperback, 1997. [Barwise 1981] J. Barwise. Scenes and other situations. J. Philosophy, 78, 369{397, 1981. [Barwise and Etchemendy 1987] J. Barwise and J. Etchemendy. The Liar, Oxford University Press, Oxford, 1987. [Barwise and Perry 1981a] J. Barwise and J. Perry. Situations and attitudes. J. Philosophy, 78, 668{691, 1981. [Barwise and Perry 1981b] J. Barwise and J. Perry. Semantic innocence and uncompromising situations. In Midwest Studies in Philosophy VI, French et al. (eds). pp. 387{403. University of Minnesota Press, Minneapolis, 1981. [Barwise and Perry 1983] J. Barwise and J. Perry. Situations and Attitudes, MIT Press, Cambridge, MA, 1983.

350

STEPHEN BLAMEY

[Beaver 1997] D. I. Beaver. Presupposition. In Handbook of Logic and Language, J. van Benthem and A. ter Meulen, eds. pp. 939{1008. Elsevier, Amsterdam, 1997. [Bell 1990] D. Bell. How Russellian Was Frege? Mind, 99, 267{277, 1990. [Belnap 1970] N. D. Belnap. Conditional assertion and restricted quanti cation. Nous, 4, 1{13, 1970. [Blamey 1980] S. R. Blamey. Partial-Valued Logic, D.Phil. thesis, Universtity of Oxford, 1980. [Blamey 1991] S. R. Blamey. The Soundeness and Completeness of Axioms for CSP. In Topology and Category Theory in Computer Science, G. M. Reed, A. W. Roscoe and R. F. Wachter, eds., pp. 29{56, Oxford Universtity Press, Oxford, 1991. [Blamey and Humberstone 1991] S. R. Blamey and L. Humberstone. A Perspective on Modal Sequent Logic. Publications of the Research Institute for Mathematical Sciences, 27, Kyoto University, 763{782, 1991. [Bochman 1998] A. Bochman. Biconsequence Relations: A Four-Valued Formalism of Reasoning with Inconsistency and Incompleteness. Notre Dame Journal of Formal Logic, 39, 47{73, 1998. [Cleave 1974] J. P. Cleave. Logical consequence in the logic of inexact predicates. Z. Math. Logik Grundlagen Math, 20, 307{324, 1974. [Czermak 1974] J. Czermak. A logical calculus with descriptions. J Philosophical Logic, 3, 211{228, 1974. [Dummett 1959] M. A. E. Dummett. Truth. Proc. Aristotelian Soc., 59, 141{162, 1959. [Dummett 1973] M. A. E. Dummett. Frege, Duckworth, London, 1973. [Dummett 1978] M. A. E. Dummett. Truth and Other Enigmas. Duckworth, London, 1978. [Dunn 1975] J. M. Dunn. Axiomatizing Belnap's conditional assertion. J. Philosophical Logic, 4, 383{397, 1975.  [Ebbinghaus 1969] H.-D. Ebbinghaus. Uber eine Pradikatenlogik mit partiell de nierten Pradikaten und Funktionen. Arch. Math. Logik Grundlagenforschung, 12, 39{53, 1969. [Evans 1982] G. Evans. The Varieties of Reference. Oxford University Press, Oxford, 1982. [Feferman 1975] S. Feferman. : 1975, Non-extensional type-free theories of partial operations and classi cations. In Springer Lecture Notes in Mathematics No.500. J. Diller and J. H. Muller, eds. pp. 73{118. Springer Verlag, 1975. [Feferman 1995] S. Feferman. De nedness. Erkenntnis, 43, 295{320, 1995. [Fenstad 1997] J. E. Fenstad. Partiality. In Handbook of Logic and Language, J. van Benthem and A. ter Meulen, eds. pp. 649{682. Elsevier, Amsterdam, 1997. [Fenstad et al. 1987] J. E. Fenstad, P. K. Halvorsen, T. Langholm and J. van Benthem. Situations, Language and Logic, Reidel, Dordrecht, 1987. [Frege 1891] G. Frege. 1891, Funktion und Begri. (Vortrag, gehalten in der Sitzung vom 9. Januar 1891 der Jenaischen Gesellschaft fur Medizin und Naturwissenschaften.), Verlag H. Pohle, Jena, 1891. Tr. in Translations from The Philosophical Writings of Gottlob Frege. 2nd revised edn., P. Geach and M. Black, eds. Blackwell, Oxford, 1960.  [Frege 1892] G. Frege. Uber Sinn, und Bedeutung. Zeitschrift fur Philosophie und Philosophische Kritik, pp. 25{50, 1892. Tr. in Translations from The Philosophical Writings of Gottlob Frege. 2nd revised edn., P. Geach and M. Black, eds. Blackwell, Oxford, 1960. [Gabbay 1982] D. Gabbay. Intuitionistic Basis for Non-Monotonic Logic. In Proceedings of the 6th Conference on Automated Deduction, Lecture Notes in CS 138, pp. 260{273, Springer-Verlag, Berlin, 1982. [Gilmore 1974] P. C. Gilmore. The consistency of partial set theory without extensionality. In Axiomatic Set Theory: 1967 UCLA Symposium Proceedings of Symposium in Pure Mathematics, Vol. 13, Part 1, T. Jech, ed. pp. 147{153. American Mathematical Society, 1974. [Groeneveld 1994] Dynamic Semantics and Circular Propositions. Journal of Philosophical Logic, 23, 267{306, 1994. [Haack 1974] S. Haack. Deviant Logic, Cambridge University Press, Cambridge, 1974.

PARTIAL LOGIC

351

[Haack 1978] S. Haack. Philosophy of Logics, Cambridge University Press, Cambridge, 1978. [Hayes 1975] P. Hayes. Three-valued logic and Computer Science, CSM-6, University of Essex, 1975. [Heim 1982] I. Heim. The Semantics of De nite and Inde nite Noun Phrases, PhD dissertaion, University of Massachusetts, Amherst, 1982. [Herzberger 1970] H. G. Herzberger. Paradoxes of grounding in semantics. J. Philosophy, 67, 145{167, 1970. [Hinnion 1994] R. Hinnion. Naive Set Theory with Extensionality in Partial Logic and in Paradoxical Logic. Notre Dame Journal of Formal Logic, 35, 15{40, 1992. [Humberstone 1981] L. Humberstone. From worlds to possibilities. J. Philosophical Logic, 10, 313{339, 1981. [Jaspars 1995] J. O. M. Jaspars. Partial Up and Down Logic. Notre Dame Journal of Formal Logic, 36, 135{157, 1995. [Kamp 1981] H. Kamp. A Theory of Truth and Semantic Representation. In Truth, Interpretaion and Information, J. Groenendijk et al., eds., pp. 1{41. Foris, Dordrecht, 1981. [Kamp and Reyle 1993] H. Kamp and U. Reyle. From Discourse to Logic, Kluwer, 1993. [Karttunen 1973] L. Karttunen. Presuppositions of Compound Sentences. Linguistic Inquiry, 4, 167{193, 1973. [Karttunen 1974] L. Karttunen. Presuppositions and Linguistic Context. Theoretical Linguistics, 1, 181{194, 1974. [Keenan 1973] E. L. Keenan. Presupposition in natural logic. Monist, 57, 334{370, 1973. [Kleene 1952] S. C. Kleene Introduction to Metamathematics, North Holland, Amsterdam, 1952. [Krahmer 1995] E. Krahmer. Discourse and Presupposition, PhD dissertation, ITK/TILDIL Dissertation Series, Universtity of Tilburg, 1995. [Kripke 1975] S. Kripke. Outline of a theory of truth. J. Philosophy, 72, 690{716, 1975. [Langholm 1988] T. Langholm. Partiality, Truth and Persistence, CSLI Lecture Notes no. 15, CSLI, Stanford, 1988. [Langholm 1989] T. Langholm. Algorithms for Partial Logic, COSMOS Report no. 12, Department of Mathematics, University of Oslo, 1989. [Lapierre 1992] S. Lapierre. A Functional Partial Semantics for Intensional Logic. Notre Dame Journal of Formal Logic, 33, 517{541, 1992. [Lehmann 1994] S. Lehmann. Strict Fregean Free Logic. Journal of Philosophical Logic, 23, 307{336, 1994. [Lepage 1992] F. Lepage. Partial Functions in Type Theory. Notre Dame Journal of Formal Logic, 33, 493{516, 1992. [Lewis 1972] D. Lewis. General Semantics. In Semantics of Natural Language, D. Davidson and G. Harman, eds., pp. 169{218, Reidel, Dordrecht, 1972. [Lopez-Escobar 1972] E. Lopez-Escobar. Refutability and elementary number theory. Koninkl. Nederl. Akademie van Wefenschappen Proceedingss, Series A, 75, 362{374, 1972. Also in Indag. Math., 34, 362{374. [McDowell 1984] J. McDowell. De Re Senses. In Frege: Tradition and In uence, C. Wright, ed., pp. 98{109, Blackwell, Oxford, 1984. [McDowell 1986] J. McDowell. Singular Thought and the Extent of Inner Space. In Subject, Thought and COntext, J. McDowell and P. Pettit, eds., Oxford University Press, Oxford, 1986. [McDowell 1977] J. McDowell. On the sense and reference of a proper name. Mind, 86, 362{374, 1977. [Martin 1970] R. L. Martin, ed. The Paradox of the Liar, Yale University Press, New Haven, 1970. [Muskens 1989] R. A. Muskens. Meaning and Partiality, Ph.D. Dissertation, University of Amsterdam, 1989. [Muskens et al. 1997] R. A. Muskens, J. van Benthem and A. Visser. Dynamics. In Handbook of Logic and Language, J. van Benthem and A. ter Meulen, eds. pp. 587{648. Elsevier, Amsterdam, 1997. [Nelson 1949] D. N. Nelson. Constructible falsity. J. Symbolic Logic, 14, 16{26, 1949.

352

STEPHEN BLAMEY

[Pappinghaus and Wirsing 1981] P. Pappinghaus and N. Wirsing. Nondeterministic Partial Logic: Isotonic and Guarded Truth-Functions, Internal Report CSR-83-81, University of Edinburgh, 1981. [Russell 1905] B. Russell. On denoting. Mind, 14, 479{493, 1905. [Russell 1959] B. Russell. Mr Strawson on referring. In My Philosophical Development, pp. 238{245. Allen and Unwin, London, 1959. [Sainsbury 1999] R. M. Sainsbury. Names, Fictional Names and Reality. Proceedings of the Aristotlelian Society, Supp. Vol., 73, 243{269, 1999. [Sandu 1998] G. Sandu. Partially Interpreted Relations and Partially Interpreted Quanti ers. Journal of Philosophical Logic, 27, 587{601, 1998. [Scott 1967] D. S. Scott. Existence and description in formal logic. In Bertrand Russell, Philosopher of the Century, R. Schoenman, ed. pp. 181{200. Allen and Unwin, London, 1967. [Scott 1973a] D. S. Scott. Models of various type-free calculi. In Logic, Methodology and Philosophy of Science IV, P. Suppes et al., eds. pp. 157{187. North Holland, Amsterdam, 1973. [Scott 1973b] D. S. Scott. Background to formalization. In Truth, Modality and Syntax, H. Leblanc, ed. pp. 244-273. North-Holland, Amsterdam, 1973. [Scott 1975] D. S. Scott. Combinators and classes. In -Calculus and Computer Science, C. Bohm, ed. pp. 1-26. Springer Verlag, Heidelberg, 1975. [Seuren 1976] P. Seuren. Tussen Taal en Denken, Oosthoek, Scheltema en Holkema, Utrecht, 1976. [Seuren 1985] P. Seuren. Discourse Semantics, Blackwell, Oxford, 1985. [Smiley 1960] T. J. Smiley. Sense without denotation. Analysis, 20, 125{135, 1960. [Stalnaker 1972] R. Stalnaker. Pragmatics. In Semantics of Natural Language, D. Davidson and G. Harman, eds., pp. 380{397, Reidel, Dordrecht, 1972. [Strawson 1950] P. F. Strawson. On referring. Mind, 59, 320{344, 1950. [Strawson 1964] P. F. Strawson. Identifying reference and truth values. Theoria, 30, 96{118, 1964. [Thijsse 1992] E. G. C. Thijsse. Partial Logic and Knowledge Representation, PhD Thesis, University of Tilburg, 1992. [Thomason 1969] R. H. Thomason. R.H.: 1969, ' A semantical study of constructible falsity. Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik, 15, 247{257, 1969. [Thomason 1972] R. H. Thomason. A semantic theory of sortal incorrectness. J. Philosophical Logic, 1, 209{258, 1972. [Thomason 1979] S. K. Thomason. Truth-value gaps, many truth-values and possible worlds. In Syntax and Semantics, Vol.10. C. Oh and D. Dinneen, eds. pp. 357-369. Academic Press, New York, 1979. [Tichy 1982] P. Tichy. Foundations of Partial Type Theory. Reports on Mathematical Logic, 14, 59{72, 1982. [Turner 1984] R. Turner. Logics for Arti cial Intelligence, Ellis Horwood, Chichester, 1984. [Van Benthem 1988] J. F. A. K. van Benthem. A Manual of Intensional Logic, CSLI Lecture Notes No. 1, CLSI, Stanford, 1988. [Van Benthem 1991] J. F. A. K. van Benthem. Logic and the Flow of Information. In

Proceedings of the 9th International Congress of Logic, Methodology and Philosophy of Science (Uppsala, Sweden), D. Prawitz, B. Skirms and D. Westerstahl, eds., 1991.

[Van Benthem and Van Eijck 1982] J. F. A. K. van Benthem and J. van Eijck. The dynamics of interpretation. J. Semantics, I, 1{20, 1982. [Van Eijck 1995] J. van Eijck. Quanti ers and Partiality. In Quanti ers, Logic and Language, Jaap van der Does and J. van Eijck, eds., Stanford University, 1995. [Van Fraassen 1966] B. C. van Fraassen. Singular terms, truth-value gaps and free logic. J. Philosophy, 63, 481{495, 1966. [Van Fraassen 1967] B. C. van Fraassen. Presupposition, implication and self-reference. J. Philosophy, 65, 136{152, 1967. [Van Fraassen 1971] B. C. van Fraassen. Formal Semantics and Logic, Macmillan, New York, 1971.

PARTIAL LOGIC

353

[Van Fraassen 1975] B. C. van Fraassen. , B.C.: 1975, ' Incomplete assertion and Belnap connectives. In Contemporary Research in Philosophical Logic and Linguistic Semantics, D. Hockney et al., eds. pp. 43{70. D. Reidel, Dordrecht, 1975. [Veltman 1981] F. Veltman. Data semantics. In Formal Methods in the Study of Language, J. Groenendijk and M. Stokhof, eds. pp. 541{565. Math. Centre Tract 136, Amsterdam, 1981. [Veltman 1996] F. Veltman. Defaults in Update Semantics. Journal of Philosophical Logic, 25, 221{261, 1996. [Visser 1984] A. Visser. Four Valued Semantics and the Liar. Journal of Philosophical Logic, 13, 181{212, 1984. [Wang and Mott 1998] X. Wang and P. Mott. A Variant of Thomason's First-Order Logic CF Based on Situations. Notre Dame Journal of Formal Logic, 39, 74{93, 1998. [Wansing 1993] H. Wansing. The Logic of Information Structures, Springer, Heidelberg, 1993. [Wansing 1995] H. Wansing. Semantics-based Nonmonotonic Inference. Notre Dame Journal of Formal Logic, 36, 44{54, 1995. [Wiggins 1976] David Wiggins. Frege's Problem of the Morning Star and the Evening Star. In Studies in Frege, M. Schirn, ed., Vol. II, pp. 221{255, Gunthner Holzboog, Stuttgart-Bad Cannstatt, 1976. [Wiggins 1984] David Wiggins. The Sense and Reference of Predicates: A Running Repair to Frege's Doctrine and a Plea for the Copula. In Frege: Tradition and In uence, C. Wright, ed., pp. 126{143, Blackwell, Oxford, 1984. [Wiggins 1995] David Wiggins. The Kant{Frege{Russell View of Existence: Toward the Rehabilitation of the Seccond-Level View. In Modality, Morality and Belief, W. Sinnott-Armstrong, D. Raman and N.Asher, eds., Cambridge University Press, Cambridge, 1995. [Wiggins 1999] David Wiggins. Names, Fictional Names and Reality. Proceedings of the Aristotlelian Society, Supp. Vol., 73, 271{286, 1999. [Woodru 1970] P. Woodru. Logic and truth value gaps. In Philosophical Problems in Logic, K. Lambert, ed. pp. 121-142. D. Reidel, Dordrecht, 1970. [Wray 1987a] D. O. Wray. Logic in Quotes. Journal of Philosophical Logic, 16, 77{110, 1987. [Wray 1987b] D. O. Wray. Algebraic Quotational Logics. Communication and Cognition, 20, 403{422, 1987.

INDEX

-satis able, 235 Anselm, ontological argument, 244 answer, 116 apartness relation, 63 argumentation form, 115 Aristotle, on predication, 209, 226, 250 attack, 116 automated theorem proving, 225 bar, 25 Barba, on supervaluation and modality, 233 Beeson, M., 7 Behmann, 48 being while lacking existence, 207 Bencivenga, on story semantics, 224 Bencivenga, on super valuation, 230 Beth model, 29 Beth, E. W., 22 bivalence, 220 Boolean algebra, 38 Brouwer, L. E. J., 2, 95 Brouwer, on partial functions, 208 Burge, on descriptions, 217 Burge, on ctional entitites, 215 Burge, on predication, 226

C -dialogues, 139 cancellation, 245 characterization, 251 choice sequences, 87 Church's Thesis, 51, 81 CL, 202 class abstracts, 217

classical description theory, 239 classical existence assumptions, 202 classical logic (CL), 202 classi cation, 251 closure under rules, 70 combinatory logic, 18 communication, 2 compactness, 229 complex predicate, 209, 245 complex predicate, and extensionality, 251 complex quanti ers, 279, 307, 313 comprehension axiom, 217 comprehensive quanti er, 207, 244, 252 conditional reading of free variables, 200 constant domain axiom, 35 constructible falsity, 288 contingent a priori truths, 208 continuum, 100 convention, 222, 230 creating subject, 97 Curry, H., 21 Curry{Howard isomorphism, 19, 22

D-dialogue, 118 Dalen, D. van, 25, 61, 63, 100 De Morgan's Law, 35 de nability paradoxes, 87 de ning axiom, 252 de nite description(s), 278, 279, 294 de nite descriptions(s), 264 de nitions, 238, 251 degree-of-de nedness, 268, 270 dense linear ordering, 65

356

descriptions, 237 inproper, 238 proper, 238 Diaconescu, 97 Dialectica Interpretation, 5 dialogue, 115 disjunction property, 44, 68 disjunction property for analysis, 94 double negation principle, 35 double negation shift, 35, 47 Dummett's axiom, 35 Dummett, M., 25, 51, 52, 102 Dwyer, on de nition, 252 dynamic operators, 290 dynamic semantics, 291, 297, 309

E -dialogue, 118 Ebbinghaus, on attribution, 220 Ehrenfeucht{Frasse games, 44 elementary formula, 198 elimination rules, 11 elimination theorems, 92 equivalence, axiomatisation of, 226 error object, 252 Evans, on contingent a priori truths, 208 existence predicate, 198, 207, 209, 216, 254 existence property, 45, 68 expressive adequacy for monotonic truth functions, 268, 304 extension principle, 92 extensionality, 214, 251 extensionality axiom, 217 Farmer, on partial functions, 220, 225, 226 FD, see mFD, 242 FD2, see MFD, 242 Feferman, on partial functions, 225 Fine, K., 59 nite Kripke models, 46 nite model property, 55

INDEX

Firedman translation, 72 Firedman, H., 73 xed point construction, 247 formal argumentation forms, 142 formal dialogue, 142 formal strategy, 142 formulas as types, 22 Fourman, M., 106 free description theory, neutral, 248 free description theory, outer-domain, 242 free description theory, Russellian, 245 free description theory, supervaluational, 247 free logic, 197 Frege, on descriptions, 239 Frege, on functions, 234 Frege, on non-referring terms and bivalence, 227 Friedman, H., 52, 69, 72 functional dependence, 301 Godel sentence, 256 Godel, K., 74 Gornemann, S., 60 Gabbay, D. M., 47, 49, 53, 55, 96 Gallier, J., 22 Garson, on intensional logic, 253 generality reading of free variables, 200 Gentzen, G., 4, 74 Girard, J. Y., 60 Glivenko's theorem, 53 Glivenko, V., 3, 75 gluing, 94 Goldblach's Conjecture, 3 Goldblatt, R., 66 Goodman, 97 Grayson, R., 99 Gumb, on de nition, 252 Harrop, R., 55 Herbrand Theorem, 48

INDEX

Heyting algebras, 37 Heyting's arithmetic, 67 Heyting's second-order arithmetic, 85 Heyting, A., 4, 7, 22 Hilbert and Bernays, on descriptions, 239 Howard, W., 21 identity, axiomatisation of, 217 implication, axiomatisation of, 226 inclusive, 197, 212 independence of premiss principle, 35 intension, 253 intensional logic, 253 interjunction, 262, 264, 274, 280, 281, 295, 304, 307 interjunctive normal forms, 305 intermediate logic, 53 internal validity, 51 interpolation theorem, 59 introduction rules, 11 intuitionism, 208 intuitionistic logic, 4, 208 IPC, 15 IQC, 15 Jankov, V. A., 59 Jaskowski sequence, 47 Jerey trees, for neutral semantics, 235 Johansson, I., 74 Johnstone, P., 38 Jongh, D. de, 49 Kleene slash, 70 Kleene, S., 5 Kolmogorov, A. N., 3 Komori, Y., 60 Kreisel, G., 7, 22, 69, 73, 76, 97 Kreisel, H., 49 Kripke frame, 42 Kripke model, 29, 67

357

Kripke's schema, 93, 99, 100 Kripke, S., 22, 97 Kroll, 99 Kroon, on descriptions, 246 Kroon, on ctional entitites, 215 Kroon, on logical form, 213

-calculus, 18 Lowenheim{Skolem theorem, 229 -calculus, 285, 292 Lambert's law, 238 Lambert, on de nition, 252 Lambert, on logical form, 213 Lambert, on negative semantics, 226 Lambert, on outer domains and Meinong, 222 Lambert, on predication, 209 Lambert, on story semantics, 224 Lambert, on theories between mFD and MFD, 243 lattice of intermediate logics, 59 law of the excluded fourth, 288 lawles sequence, 89 Leblanc, on PFL, 222 Lehmann, on neutral semantics, 235 Leivant, D., 70 Lejeweski, on identity, 207 Lemmon, E., 54 Lin, on equivalence and implication, 226 Lindenbaum algebra, 39 locally true, 105 logic of constant domains, 60 logical analysis in partial logic, 280, 307, 309, 311, 313 logical consequence, 200, 265, 266, 274, 296 logical consequence, in neutral semantics, 235 logical form, 211 logically neither true nor false sentence, 265

358

INDEX

logically neither true-nor-false sentence, 305 logically non-denoting singular-term, 265 loigc of existence, 102 Lorenzen, P., 123 Malmnas, P., 74 Mann, on ontological argument, 244 Markov's principle, 51, 70 Markov, A. A., 83 Martin-Lof type theories, 22 Martin-Lof's type theory, 7 Martin-Lof, P., 7, 22 mathematical language, 3 maximal free description theory (MFD), 242 Maximova, L., 59 McCarty, D., 83 McCarty, D. C., 52 McKinsey, J. C. C., 49 meaning, as denotation, 240, 241 Meinong's paradox, 238 Meinong, on being vs. existence, 207 Meinong, on bivalence, 222 Meinong, on predication, 209 Mendelson, on non-referring terms, 205 mereology, 210 Minc, G., 70 minimal free description theory (mFD), 242 minimal logic, 73 modal semantics, 207, 208, 233, 253 model existence lemma, 31 monadic fragment, 48 monotonically representable partial functions, 269{271, 287, 302 monotonicity of evaluation, 268, 273, 284, 286

Montague, on necessity, 255 more-tahn-two-place `consequence' relations, 266 Moschovakis, J., 95 Myhill, 97, 99 naive theory of de nite descriptions (NTDD), 238 Natural Deduction, 10, 11 natural deduction, 4 natural negation, 282, 283 necessity operator, 255 necessity predicate, 255 necessity, metalinguistic interpretation of, 255 negative free logic (NFL), 225 negative part of formula, 227 negative semantics, 221, 225 Negri, S., 18 neighbourhood, 23 neutral semantics, 221, 233 new foundations (NF), 217 Nishimura, T., 40 non-deterministic algorithms, 293 non-monotonic matrices, 311 non-strict function, 219 normal form theorem, 18 normalisation theorem, 18 objectual quanti cation, 200 objectual quanti cation, in story semantics, 223 Ono, H., 60 ontological argument and Russellian descriptions, 244 outer domain semantics, 218, 221 partial element, 102 partial interpretation, 218, 222 partial interpretation, completion of, 228 partial recursive predicates, 286 path, 25 PEM, 3

INDEX

Plato, J. von, 18 polar replacement, 227 positive free logic (PFL), 222 positive part of formula, 227 positive semantics, 221 possible worlds, 212 possible worlds and propositions, 241 Posy, C., 102 Prawitz, D., 17, 18, 49, 74, 85 predicate/singular-term composition, 263, 298 prenex fragment, 48 presupposition, 210, 220, 278, 281, 282, 307, 309, 311, 314 presuppositional analysis, 313 pretend objects, 202, 218, 224 principle of open data, 89 principle of the excluded third, 3 projection rules for presupposition, 309 proof interpretation, 6 proof-interpretation, 4 proof-terms, 19 propositional content, 214 provably recursive functions, 73 quanti cation, vacuous, 212 Quine's dictum, 197, 207 Quine, on classes, 217 Quine, on descriptions, 237 Quine, on eliminating singular terms, 251 Quine, on inclusive logic, 212 Quine, on predication, 250 Quine, on set theory, 217 quotational logic, 285 Rasiowa, H., 25, 38 Rautenberg, H., 53 Rautenberg, W., 59 realizability, 5 recursively axiomatisable, 229 reference failure, 297

359

referential opacity, 215 replacement, 226 Rieger{Nishimura lattice, 40 Robinson, on descriptions, 248 Russell's paradox, 218, 238 Russell, on descriptions, 204, 216, 240 Russell, on predication, 209 S5 semantics, 255 satis able, 200 Scales, on complex predicates, 209 Scales, on descriptions, 217, 245 Schutte, K., 25, 49 Schroeder-Heister, P., 18 Schweizer, on necessity, 255 Schwichtenberg, H., 49 scope in natural language, 241 indicators of, 241 narrow, 240 of Russellian descriptions, 240 wide, 240 Scott, D., 54, 105, 106 Scowcroft, P., 99 second-order logic, 84 second-order quanti cation, and supervaluation, 229 selective ltration, 55 semantic paradox, 284 sense, 299{301, 303 Sequent Calculus, 17 sheaf interpretation, 106 Sikorski, R., 25, 38 singular predicate, 204, 210, 237 situation semantics, 293 skeleton, 141 Skolem functions, 50 Skyrms, on necessity, 255 Skyrms, on supervaluation, 231 Smiley, on attribution, 220 Smiley, on quanti cation in netural semantics, 235

360

INDEX

Smorynski, C., 46, 50, 54, 55, 65, 71, 76 sortal incorrectness, 282 stable formula, 229 stable open sentence, 252 Statman, R., 63 Stenlund, on descriptions, 248 Stenlund, on ctional entitites, 215 story, 220, 222 actualist constraints on, 230 story interpretation, 222 story semantics, 220 story semantics and bivalence, 220 story semantics, equivalence to outerdomain semantics, 224 strategy, 115, 117 strati ed formula, 217 Strawson, on presupposition, 210 strong completeness, 200 strong continuity principle, 92, 93 strong negation, 77 strong normalisation theorem, 18 strong tables, 228 subformula property, 18 substitutional quanti cation, 200 Sundholm, G., 7 superfalsity, 228 supertruth, 228 supervaluation, 221, 228 supervaluations, 272, 283 Swart, H. C. M. de, 52 Tarski, A., 22, 61 term-forming descriptions operator, 264 theories in partial logic, 266 theory of apartness, 62 theory of equality, 61 theory of order, 65 topological interpretation, 4, 22 topological space, 22 topos theory, 83 transplicand, 307, 313

transplication, 264, 274, 280, 281, 295, 304, 307, 310, 313 transplicator, 307, 313 Trew, on free logics as rst-order theories, 202 Troelstra, A., 49, 82, 86 truth connective, 237, 249 truth, counterfactual theory of, 232 uniformity principle, 86 universal Beth model, 52 universally free, 197, 218 unsolved problem, 5 Veldman, W., 44, 52 virtual classes, 217 Visser, A., 72 Walton, on pretense, 224 weak completeness, 201 weak tables, 228 Woodru, on Frege, 234 Woodru, on supervaluation, 229