it is very educational and very informative in relation to philosophy.It is a guide to learners of philosophy.it is very very educational.Guaranteed.
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Handbook of Philosophical Logic 2nd Edition Volume 4
edited by Dov M. Gabbay and F. Guenthner
CONTENTS Editorial Preface Dov M. Gabbay
Conditional Logic
D. Nute and C. B. Cross
Dynamic Logic
D. Harel, D. Kozen, and J. Tiuryn
Logics for Defeasible Argumentation H. Prakken and G. Vreeswijk
Preference Logic S. O. Hansson
Diagrammatic Logic E. Hammer
Index
vii 1 99 219 319 395 423
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 Expressive power for recurrent events. Speci cation of temporal control. Decision problems. Model checking.
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-
Negation by failure and modality
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
DONALD NUTE AND CHARLES B. CROSS
CONDITIONAL LOGIC
Prior to 1968 several writers had explored the conditions for the truth or assertability of conditionals, but this work did not result in an attempt to provide formal models for the semantical structure of conditionals. It had also been suggested that a proper logic for conditionals might be provided by combining modal operators with material conditionals in some way, but this suggestion never led to any widely accepted formal logic for conditionals.1 Then Stalnaker [1968] provided both a formal semantics for conditionals and an axiomatic system of conditional logic. This important paper eectively inaugurated that branch of philosophical logic which we today call conditional logic. Nearly all the work on the logic of conditionals for the next ten years, and a great deal of work since then, has either followed Stalnaker's lead in investigating possible worlds semantics for conditionals or posed problems for such an approach. But in 1978, Peter Gardenfors [1978] initiated a new line of inquiry focused on the use of conditionals to represent policies for belief revision. Thus, two main lines of development appeared, one an ontological approach concerned with truth or assertability conditions for conditionals and the other an epistemological approach focused on conditionals and change of belief. With these two major lines of development, the material which has appeared on conditionals is prodigious. Consequently, we have had to focus upon certain aspects of conditional logic and to give other aspects less attention. We have followed the trend set in the literature and given the most attention to the analysis of so-called subjunctive conditionals as they are used in ordinary discourse and to triviality results for the Ramsey test. Accordingly, our discussion of conditionals and belief revision will be more heavily technical than our discussion of subjunctive conditionals. Other topics are discussed in less detail. Some of the important papers which it has not been possible to review are included in the accompanying bibliography, but the bibliography itself is far from complete. 1 ONTOLOGICAL CONDITIONALS
1.1 Introduction Conditional logic is, in the rst place, concerned with the investigation of the logical and semantical properties of a certain class of sentences occurring 1 Another suggestion which has never been fully developed (but see Hunter [1980; 1982] is that an adequate theory of ordinary conditionals may be derived from relevance logic. We will say no more about this suggestion than it seems to us that conditional logic and relevance logic are concerned with very dierent problems, and it would be a tremendous coincidence if the correct logic for the conditionals of ordinary usage should turn out to resemble some version of relevance logic at all closely.
2
DONALD NUTE AND CHARLES B. CROSS
in a natural language. We will draw our examples from English, but much of what we have to say can be applied, with due caution, to other natural languages. Paradigmatically, a conditional declarative sentence in English is one which contains the words `if' and `then'. Examples include 1. If it is raining, then we are taking a taxi. and 2. If I were warm, then I would remove my jacket. We could delete the occurrences of `then' in (1) and (2) and we would still have perfectly acceptable sentences of English. In the case of (2), we can omit both `if' and `then' if we change the word order. Example (2) surely says the same thing as 3. Were I warm, I would remove my jacket. Other conditionals in which neither `if' nor `then' occur include 4. When I nd a good man, I will praise him. and 5. You will need my number should you ever wish to call me. Notice that all of these examples involve two component sentences or clauses, one expressing some sort of condition and another expressing some sort of claim which in some way depends upon the condition. The conditional or `if' part of a conditional sentence is called the antecedent, and the main or `then' part its consequent even when `if' and `then' do not actually occur. Notice that the antecedent precedes the consequent in (1){(4), but the consequent comes rst in (5). These examples should give the reader a fair idea of the types of sentences with which conditional logic is concerned. While the verbs in (1) are in the indicative mood, those in (2) are in the subjunctive mood. Researchers often rephrase (2), forming a new conditional in which the verbs contained in antecedent and consequent are in the indicative mood. This practice implicitly assumes that (2) has the same content as 6. If it were the case that I am warm, then it would be the case that I remove my jacket. Even without the rephrasing, it is sometimes said that `I am warm' is the antecedent of both (2) and (6). Thus the mood of the verbs in the grammatical antecedent and consequent of (2) are taken logically to be a component of the conditional construction, while the logical antecedent and consequent
CONDITIONAL LOGIC
3
are viewed as containing verbs in the indicative mood. Seen in this way, the conditional constructions in (1) and (2) look quite dierent and investigators have as a consequence made a distinction between indicative conditionals like (1) and subjunctive conditional like (2). This distinction is important because it appears that these two kinds of conditionals have dierent logical and semantical properties. Much of the work done in conditional logic has focused on conditionals having antecedents and consequents which are false. Such conditionals are called counterfactuals. In actual practice, little distinction is made between counterfactuals and subjunctive conditionals which have true antecedents or consequents. Authors frequently refer to conditionals in the subjunctive mood as counterfactuals regardless of whether their antecedents or consequents are true or false. Another special kind of conditional is the so-called counterlegal conditional whose antecedent is incompatible with physical law. An example is 7. If the gravitational constant were to take on a slightly higher value in the immediate vicinity of the earth, then people would suer bone fractures more frequently. Also recognized are counteridenticals like 8. If I were the pope, I would support the use of the pill in India. and countertemporals like 9. If it were 3.00 a.m., it would be dark outside. Analysis of these special conditionals may involve special diÆculties, but we can say very little about these special problems in a paper of this length. Two other interesting conditional constructions are the even-if construction used in 10. It would rain even if the shaman did not do his dance. and the might construction used in 11. If you don't take the umbrella, you might get wet. We might paraphrase (10) using the word `still' to get 12. It would still rain if the shaman did not do his dance. even-if and might conditionals have somewhat dierent properties from those of other conditionals. It is believed by many, though, that these two kinds of conditionals can be analyzed in terms of subjunctive conditionals once we have an acceptable analysis of these. The strategy in this
4
DONALD NUTE AND CHARLES B. CROSS
paper will be to concentrate on the many proposals for subjunctive conditionals, returning later (brie y) to the topics of indicative, even-if and might conditionals. We will use two dierent symbols to represent indicative and subjunctive conditionals. For indicative conditionals we will use the double arrow ), and for the subjunctive conditional we will use the corner >. (Where context makes our intention clear, we will sometimes use symbols and formulas autonomously to refer to themselves.) With these devices we may represent (1) as 13. It is raining ) I am taking a taxi. and represent (2) as 14. I am warm > I remove my jacket. Frequently we will have no particular antecedent or consequent in mind as we discuss one or the other of these two kinds of conditionals and as we examine forms which arguments involving these conditionals may take. In these cases we will use standard notation for classical rst-order logic augmented by our symbols for indicative and subjunctive conditionals to represent the forms of sentences and arguments under discussion. We assume, as have nearly all investigators, that conditional have truth values and may therefore appear as arguments for truth-functional operators. Students in introductory symbolic logic courses are normally taught to treat English conditionals as material conditionals. By material conditionals we mean certain truth-functional compounds of simpler sentences. A material condition ! is true just in case is false or is true. There can be little doubt that neither material implication nor any other truth function can be used by itself to provide an adequate representation of the logical and semantical properties of English conditionals or, presumably, the conditionals of any other language. Consider the following two examples. 15. If I were seven feet tall, then I would be over two meters tall. 16. If I were seven feet tall, then I would be less than two yards tall. In fact one of the authors is more than two yards tall but less than two meters tall, so for him the common antecedent and the two consequents of (15) and (16) are all false. Yet surely (15) is true while (16) is false. When both the antecedent and the consequent of an English subjunctive conditional are false, the conditional may be either true or false. Now consider two more examples. 17. If I were eight feet tall, I would be less than seven feet tall.
CONDITIONAL LOGIC
5
18. If I were seven feet tall, I would be over six feet tall. Here we have two conditionals each of which has a false antecedent and a true consequent. but the rst of these conditionals is false and the second is true. The moral of these examples is that when the antecedent of an English subjunctive conditional is false, the truth value of the conditional is not determined by the truth values of the antecedent and the consequent of the conditional alone. Some other factors must be involved in determining the truth values of such conditionals. But what about English conditionals with true antecedents? It is generally accepted that any conditional with a true antecedent and a false consequent is false, but the situation is more controversial where the conditionals with true antecedents and true consequents are concerned. Some researchers have maintained that all such conditionals are true while others have claimed that such conditionals are sometimes false. Later we will consider some of the issues involved in this controversy. For now we simply recognize that there are some very good reasons for rejecting the view that all English conditionals can be represented adequately by material implication or by any other truth function.
1.2 Cotenability theories of conditionals Chisholm [1946], Goodman [1955], Sellars [1958], Rescher [1964] and others have proposed accounts of conditionals which share some important features. Borrowing a term from Goodman, we can call these proposals cotenability theories of conditionals. The basic idea which these proposals share is that the conditional > is true in case , together with some set of laws and true statements, entails . A crucial problem for such an analysis is that of determining the appropriate set of true statements to involve in the truth condition for a particular conditional. If the antecedent of the conditional is false, then of course its negation is true. But any proposition together with its negation will entail anything. The set of true statements upon which the truth of the conditional is to depend must at least be logically compatible with the antecedent of the conditional or the conditional will turn out to be trivially true on such an account. But logical compatibility is not enough either. We can have a true proposition such that and are logically compatible but such that > : is also true. Then we should not wish to include in the set of propositions upon which the evaluation of > depends. Goodman said of such a that it is not cotenable with . So Goodman's ultimate position is that > is true just in case is entailed by together with the set of all physical laws and the set of all true propositions cotenable with , i.e. with the set of all true propositions such that no member of that set counterfactually implies the negation of and the negation of no member
6
DONALD NUTE AND CHARLES B. CROSS
of that set is counterfactually implied by . Such an account is obviously circular since the truth conditions for counterfactuals are given in terms of cotenability, while cotenability is de ned in terms of the truth values of various counterfactual conditionals. Although this is certainly a serious problem, it is not the only problem which theories of this type encounter. As a result of the role which law plays in such a theory, all counterlegal conditionals are counted as trivially true, and this is counterintuitive. Furthermore, even if we could provide a noncircular account of cotenability, another problem arises for conditionals which are not counterlegal. Suppose two true propositions and are each cotenable with , but that ^ is not. In selecting the set of propositions upon which the evaluation of > shall rest we must omit either or since otherwise our conditional will be trivially true once again. But which of these two propositions shall we omit? Most recent work in conditional logic is compatible with cotenability theory even though no attempt is made to de ne and use the notion of cotenability. We might view the resultant theories at least in part as attempts to determine, without ever specifying exactly what cotenability is, the logical and semantical properties which conditionals must have if the cotenability approach is essentially correct for conditionals without counterlegal antecedents. Indeed, the vagueness deliberately built into many of these recent theories suggests that our notion of cotenability, if we have one, varies according to our purposes and the context in which we use a conditional.2
1.3 Strict Conditionals We have seen that the truth value of a conditional is not always determined by the actual truth values of its antecedent and consequent, but perhaps it is determined by the truth values which its antecedent and consequent take in some other possible worlds. One way such an analysis might be developed is suggested by the role laws play in the cotenability theories. Perhaps we should look not only at the truth values of the antecedent and the consequent in the actual world, but also at their truth values in all possible worlds which have the same laws as does our own. When two worlds obey the same physical laws, we can say that each is a physical alternative of the other. The proposal, then, is that > is true if is true at every physical alternative to the actual world at which is true. Suppose we say a proposition is physically necessary if and only if it is true at every physical alternative to the actual worlds, and suppose we express 2 Bennett [1974] and Loewer [1978] arrive at opposite conclusions concerning the question whether Lewis's semantics is compatible with cotenability theory. Their discussions are instructive for other semantics as well.
CONDITIONAL LOGIC
7
the claim that a proposition is physically necessary by . Then the proposal we are considering is that the following equivalence always holds: 19. ( > ) $ ( ! ). Another way of arriving at (19) is the following. English subjunctive conditionals are not truth-functional because they say more than that the antecedent is false or the consequent is true. The additional content is a claim that there is some sort of connection between the antecedent and the consequent. The kind of connection which seems to occur to people most readily in this context is a physical or causal connection. How can we represent this additional content in our formalization of English subjunctive conditionals? One way is to interpret > as involving the claim that it is physically impossible that be true and false. Once again we come up with (19). A proposal resembling the one we have outlined can be found in [Burks, 1951], although we do not wish to suggest that Burks arrived at his account by exactly the same line of reasoning as we have suggested. We can generalize the proposal represented by (19). We might suppose that the basic form of (19) is correct but that the short of necessity involved in English subjunctive conditionals is not pure physical necessity. One reason for suspecting this is that the notion of cotenability has been ignored. It is not simply a consequence of physical law that Jane would develop hives if she were to eat strawberries; it is also in part a consequence of her having a particular physical make-up. In evaluating the claim that Jane would become ill if she were to eat strawberries, we do not count the fact that in some worlds which share the same physical laws as our own but in which Jane has a radically dierent physical make-up, she is able to eat strawberries with impunity, as a legitimate reason for rejecting this claim. Another reason for seeking a dierent kind of necessity for the analysis of conditionals is that some conditionals may be true because of connections between their antecedents and consequents which are not physical connections at all. Consider, for example, conditionals such as `If you deserted your family you would be a cad', which seems to be founded on normative rather than physical connections. The general theory we are considering, then, is that English subjunctive conditionals are strict conditionals of some sort, i.e. that their logical form is given by the equivalence (19). There remains the problem of determining which kind of necessity is involved in these conditionals. Regardless of the kind of necessity we choose in such an analysis of conditionals, we should expect our modal logic to have certain minimal properties. By a modal logic we mean any set L of sentences formed from the symbols of classical sentential logic together with the symbol in the usual ways, provided that L contains all tautologies and is closed under the rule modus ponens. We should expect that for any tautology our modal logic will contain . We should also expect our modal logic to contain all substitution
8
DONALD NUTE AND CHARLES B. CROSS
instances of the following thesis: 20.
( !
) ! ( ! ).
But when we de ne our conditionals according to (19), our logic will then also contain all substitution instances of the following theses: Transitivity: [( > ) ^ ( > )] ! ( > ) Contraposition: ( > : ) ! ( > :) Strengthening antecedents: ( > ) ! [( ^ ) > ]. But none of these theses seem to be reliable for English subjunctive conditionals. As a counterexample to Transitivity, consider the following conditionals: 21. If Carter had not lost the election in 1980, Reagan would not have been President in 1981. 22. If Carter had died in 1979, he would not have lost the election in 1980. 23. If Carter had died in 1979, Reagan would not have been President in 1981. (21) and (22) are true, but is far from clear that (23) is true. As a counterexample to Contraposition, consider: 24. If it were to rain heavily at noon, the farmer would not irrigate his eld at noon. 25. If the farmer were to irrigate his eld at noon, it would not rain heavily at noon. And nally, for Strengthening Antecedents, consider: 26. If the left engine were to fail, the pilot would make an emergency landing. 27. If the left engine were to fail and the right wing were to shear o, the pilot would make an emergency landing. Since even very weak modal logics will contain all substitution instances of these three theses, and since most speakers of English nd counterexamples of the sort we have considered convincing, most investigators are convinced that English conditionals are not a variety of strict conditional.
CONDITIONAL LOGIC
9
1.4 Minimal Change Theories While treating ordinary conditionals as strict conditionals does not seem too promising, investigators have still found the possible worlds semantics often associated with modal logic very attractive. The basic intuition, that a conditional is true just in case its consequent is true at every member of some set of worlds at which its antecedent is true, may yet be salvageable. We can avoid Transitivity, etc. if we allow that the set of worlds involved in the truth conditions for dierent conditionals may be dierent. But we do not wish to allow that this set of worlds be chosen arbitrarily for a given conditional. Stalnaker [1968] proposes that the conditional > is true just in case is true at the world most like the actual world at which is true. According to Stalnaker, in evaluating a conditional we add the antecedent of the conditional to our set of beliefs and modify our set of beliefs as little as possible in order to accommodate the new belief tentatively adopted. Then we consider whether the consequent of the conditional would be true if this revised set of beliefs were all true. In the ideal case, we would have a belief about every single matter of fact before and after this operation of adding the antecedent of the conditional to our stock of beliefs. Possible worlds correspond to these epistemically ideal situations. Stalnaker's assumption, then, is that at least when the antecedent of a conditional is logically possible, there is always a unique possible world at which the antecedent is true and which is more like the actual world than is any other world at which the antecedent is true. We will call this Stalnaker's Uniqueness Assumption. On some fairly reasonable assumptions about the notion of similarity of worlds, Stalnaker's truth conditions generate a very interesting logic for conditionals. Essentially these assumptions are that any world is more similar to itself than is any other world, that the -world closest to world i (that is, the world at which is true which is more similar to i than is any other world at which is true) is always at least as close as the ^ -world closest to i, and that if the - world closest to i is a -world and the -world closest to i is a -world, then the -world closest to i and the -world closest to i are the same world. The model theory Stalnaker develops is complicated by his use of the notion of an absurd world, a world at which every sentence is true. This invention is motivated by the need to provide truth conditions for conditionals with impossible antecedents. Stalnaker's semantics can be simpli ed by omitting this device and adjusting the rest of the model theory accordingly. When we do this, we produce what could be called simpli ed Stalnaker models. Such a model is an ordered quadruple hI; R; s; [ ]i where I is a set of possible worlds, R is a binary re exive (accessibility) relation on I , s is a partial world selection function which, when de ned, assigns to sentence and a world i in I a world s(; i) (the -world closest to i), and [ ] is a
10
DONALD NUTE AND CHARLES B. CROSS
function which assigns to each sentence a subset [] of I (all those worlds in I at which is true). Stalnaker's assumptions about the similarity of worlds become a set of restrictions on the items of these models: (S1)
s(; i) 2 [];
(S2)
hi; s(; i)i 2 R;
(S3)
if s(; i) is not de ned then for all j j 62 [];
(S4) (S5) (S6)
2I
such that hi; j i
2 R,
if i 2 [] then s(; i) = i;
if s(; i) 2 [ ] and s( ; i) 2 [], then s(; i) = s( ; i);
i 2 [ > ] if and only if s(; i) 2 [ ] or s(; i) is unde ned.
Until otherwise indicated, we will understand by a conditional logic any set L of sentences which can be constructed from the symbols of classical sentential logic together with the symbol >, provided that L contains all tautologies and is closed under the inference rule modus ponens. The conditional logic determined by Stalnaker's model theory is the smallest conditional logic which is closed under the two inference rules RCEC: from $ , to infer ( > ) $ ( > ) RCK:
from (1 ^ : : : ^ n ) ! , to infer [( > 1 ) ^ : : : ^ ( > n )] ! ( > ), n 0
and which contains all substitution instances of the theses ID:
Together with modus ponens and the set of tautologies, these rules and theses can be viewed as an axiomatization of Stalnaker's logic, which he calls C2. While Stalnaker supplies a rather dierent axiomatization for C2, these rules and theses enjoy the advantage that they allow easy comparison of C2 with other conditional logics. Several of these rules and theses are due to Chellas [1975]. It can be shown that a sentence is a member of C2 if and only if that sentence is true at every world in every simpli ed
CONDITIONAL LOGIC
11
Stalnaker model. Thus we say that the class of simpli ed Stalnaker models determines or characterizes the conditional logic C2. None of Transitivity, Contraposition, and Strengthening Antecedents is contained in C2. A variation of the semantics developed by Stalnaker treats the function s as taking sets of worlds rather than sentences as arguments and values. In this variation, s is a function which assigns to each subset A of I and each member i of I a subset s(A; i) of I . Then > will be true at i just in case s([]; i) [ ]. By setting our semantics up in this way, we ensure that we can substitute one antecedent for another in a conditional provided that the two antecedents are true at exactly the same worlds, and we can do this without any additional restrictions on the function s. Since many authors have called sets of worlds propositions, we could call Stalnaker's original semantics a sentential semantics and the present variation on Stalnaker's semantics a propositional semantics to represent this dierence in the kind of argument the function s takes. As we look at alternatives to Stalnaker's semantics we will always consider the sentential forms of these semantics although equivalent propositional forms will often be available. Equivalence of the two versions of a particular semantics is guaranteed so long as the conditional logic characterized by the sentential version is closed under substitution of provable equivalents, i.e. so long as it is closed under both RCEC and RCEA: from $ to infer ( > ) $ ( > ). C2 is closed under RCEA as is any conditional logic closed under RCK and containing all substitution instances of CSO. The dierence between sentential and propositional formulations of a particular kind of model theory becomes important if we wish to consider conditional logics which are not closed under RCEA. Reasons for considering such `non-classical' logics are discussed in Section 1.7 below. For parallel development of sentential and propositional versions of certain kinds of model theories for conditional logics, see [Nute, 1980b]. Lewis [1973b; 1973c] questions Stalnaker's assumptions about the similarity of worlds and thus his semantics for conditionals. It is Stalnaker's Uniqueness Assumption which Lewis rejects. Lewis argues that there may be no unique - world which is closer to i than is any other -world. As an example, Lewis asks us to consider a straight line printed in a book and to suppose that this line were longer than it is. No matter what greater length we choose for the line, there is a shorter length which is still greater than the actual length of the line. The conclusion is that worlds which dier from the actual world only in the length of the sample line may be more and more like the actual world as the length of the line in those worlds comes closer to the line's actual length. But none of these worlds is the closest world at which the line is longer. In fact, examples of this sort can also be oered against an assumption about similarity of worlds which is
12
DONALD NUTE AND CHARLES B. CROSS
weaker than Stalnaker's Uniqueness Assumption. This assumption, which Lewis calls the Limit Assumption, is that, at least for a sentence which is logically possible, there is always at least one -world which is as much like i as is any other -world. Both the Uniqueness Assumption and the weaker Limit Assumption are highly suspect. If we follow Lewis's advice and drop the Uniqueness Assumption, we must give up Conditional Excluded Middle (CEM). But this is exactly the feature of Stalnaker's logic which is most often cited as objectionable. Both disjuncts in CEM will be true if is impossible and hence s is not de ned for and the actual world. On the other hand, if is possible, then must be either true or false at the nearest -world. Lewis ([1973b], p. 80) oers the following as a counterexample to CEM: 28a It is not the case that if Bizet and Verdi were compatriots, Bizet would be Italian; and it is not the case that if Bizet and Verdi were compatriots, Bizet would not be Italian; nevertheless, if Bizet and Verdi were compatriots, Bizet either would or would not be Italian. Lewis [1973b] admits that (28a) sounds, ohand, like a contradiction, but he insists that the cost of respecting this ohand opinion is too high: However little there is to choose for closeness between worlds where Bizet and Verdi are compatriots by both being Italian and worlds where they are compatriots by both being French, the selection function still must choose. I do not think it can choose|not if it is based entirely on comparative similarity, anyhow. Comparative similarity permits ties, and Stalnaker's selection function does not.3 Van Fraassen [1974] has employed the notion of supervaluation in defense of CEM. The suggestion is that in actual practice we do not depend upon a single world selection function s in evaluating conditionals. Instead we consider a number of dierent ways in which we might measure the similarity of worlds, each with its appropriate world selection function. Each world selection function provides a way of evaluating conditionals. A sentence can also have the property that it is true regardless of which world selection function we use. We can call such a sentence supertrue. If we accept Stalnaker's semantics together with a multiplicity of world selection functions, it turns out that every instance of CEM is supertrue even though it may be the case that neither disjunct of some instance of CEM is supertrue. In fact, all the members of C2 are supertrue when we apply Van Fraassen's method of supervaluation, and the method mandates the following reinterpretation of the Bizet-Verdi example: 3 [Lewis,
1973b], p. 80.
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28b `If Bizet and Verdi were compatriots, Bizet would be Italian' is not supertrue; and `If Bizet and Verdi were compatriots, Bizet would not be Italian' is not supertrue; nevertheless, `If Bizet and Verdi were compatriots, Bizet either would or would not be Italian' is supertrue. (The relevant instance of CEM is also supertrue: `Either Bizet would be Italian if Bizet and Verdi were compatriots, or Bizet would not be Italian if Bizet and Verdi were compatriots.') In the Bizet-Verdi example, what Lewis accounts for as a tie in comparative world similarity, the method of supervaluation accounts for as a case of indeterminacy in the choice of a closest compatriot-world. Lewis [1973b] admits that ohand opinion seems to favor CEM, but, Stalnaker [1981a] shows that there is systematic intuitive evidence for CEM: the apparent absence of scope ambiguities in conditionals where Lewis' theory predicts we should nd them. Consider the following dialogue (see [Stalnaker, 1981a], pp. 93{95): X:
President Carter has to appoint a woman to the Supreme Court.
Y:
Who do you think he has to appoint?
X:
He doesn't have to appoint any particular woman; he just has to appoint some woman or other.
There is a clear scope ambiguity in X's statement, and this scope ambiguity explains why X's response to Y makes sense: Y reads X as having intended `a woman' to have wide scope, and X's response corrects Y by making it clear that X intended `a woman' to have narrow scope. Now compare this dialogoue to another, in which necessity is replaced by the past-tense operator: X:
President Carter appointed a woman to the Supreme Court.
Y:
Who do you think he appointed?
X:
He didn't appoint any particular woman; he just appointed some woman or other.
In this case X's response does not make sense. There is no semantically distinct narrow scope reading that X could have had in mind, so there is no room for Y to have misunderstand X's statement. Finally, consider a dialogue involving a conditional instead of a necessity or past tense statement: X:
President Carter would have appointed a woman to the Supreme Court last year if there had been a vacancy.
Y:
Who do you think he would have appointed?
14
X:
DONALD NUTE AND CHARLES B. CROSS
He wouldn't have appointed any particular woman; he just would have appointed some woman or other.
If Lewis' analysis of counterfactuals is correct, then in this dialogue, as in the rst dialogue, one should perceive an ambiguity in the scope of `a woman' in X's statement, and X's response should make sense as a correction of Y's misinterpretation. In fact there is no room for Y to have misunderstood X's statement, and X's response simply doesn't make sense. In this respect, the third dialogue parallels the second dialogue, not the rst, and the apparent lack of a scope ambiguity in X's statement in the third dialogue is evidence for CEM.4 If Stalnaker's example does not convince one to accept CEM, it is quite possible to formulate a logic and a semantics for conditionals which resembles Stalnaker's but which does not include CEM. Lewis [1971; 1973b; 1973c] suggests more than one way of doing this. The rst way is to replace the Uniqueness Assumption with the weaker Limit Assumption. Instead of looking at the closest antecedent-world, we look at all closest antecedent-worlds. These functions might better be called class selection functions rather than world selection functions. It is also not necessary to incorporate the accessibility relation into our models for conditionals if we use class selection functions since, if we make a certain reasonable assumption, we can de ne such a relation in terms of our class selection function. The assumption is that if is possible at all at i, then there is at least one closest -world for our selection function to pick out. Our models are then ordered triples hI; f; [ ]i such that I and [ ] are as before and f is a function which assigns to each sentence and each world i in I a subset of I (all the - worlds closest to i). By restricting these models appropriately, we can characterize a logic very similar to Stalnaker's C2. This logic, which Lewis calls VC, is the smallest conditional logic which is closed under the same rules as those listed for C2 and which contains all those theses used in de ning C2 except that we replace CEM with CS:
( ^ ) ! ( > ).
CS is contained by C2 although CEM is not contained by VC.5 A sentence 4 A dierent sort of argument for CEM can be found in [Cross, 1985], which adopts Bennett's [1982] analysis of `even if' conditionals and argues for the validity of CEM based on the intuitive validity of the following formulas: (e > ) ! ( > ) ( ^ :( > : )) ! (e > ); where (e > ) means `Even if , '. The argument turns on the fact that in any system of conditional logic that includes classical propositional logic and RCEC, CEM is a theorem i ( ^ :( > : )) ! ( > ) is a theorem. 5 This and other independence results cited in this paper are provided in Nute [1979;
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15
is a member of VC if and only if it is true at every world in every class selection function model which satis es the following restrictions: (CS1): (CS2): (CS3): (CS4): (CS5): (CS6):
if j 2 f (; i) then j 2 [];
if i 2 [] then f (; i) = fig;
if f (; i) is empty then f ( ; i) \ [] is also empty;
if f (; i) [ ] and f ( ; i) [], then f (; i) = f ( ; i); if f (; i) \ [ ] 6= ;, then f ( ^ ; i) f (; i);
i 2 [ > ] i f (; i) [ ].
Although Lewis endorses VC as the proper logic for subjunctive conditionals, he nds the Limit Assumption and, hence, the version of class selection function semantics we have developed, to be no more satisfactory than the Uniqueness Assumption. Consequently, Lewis proposes an alternative semantics for subjunctive conditionals. This alternative is also based on the similarity of worlds. The dierence is in the way Lewis uses similarity in giving the truth conditions for conditionals. A conditional > with a logically possible antecedent is true at a world i, according to Lewis, if there is a ^ -world which is closer to i than is any ^ : -world. Lewis uses nested systems of spheres in his models to indicate the relative similarity of worlds. A system-of-spheres model is an ordered triple hI; $; [ ]i such that I and [ ] are as before and $ is a function which assigns to each i in I a nested set $i of subsets of I (the spheres about i). If there is some sphere S about i such that j is in S but k isn't in S , then j is closer to or more similar to i than is k. To characterize the logic VC, we must adopt the following restrictions of system-of-spheres models: (SOS1): fig 2 $i ;
(SOS2): i 2 [ > ] if and only if $i \ [] is empty or there is an S such that S \ [] is not empty and S \ [] [ ].
2 $i
While Lewis rejects the Limit Assumption, it should be noted that in those cases in which there is a closest -world to i the conditions for a conditional with antecedent being true at i are exactly the same for system-of-spheres models as for the type of class selection function model we examined earlier. For this reason we classify Lewis's semantics as a minimal change semantics to contrast it with other accounts which lack this feature. Pollock [1976] also develops a minimal change semantics for conditionals. In fact, Pollock's semantics is a type of class selection function semantics. There are two primary reasons why Pollock rejects Lewis's semantics and the 1980b].
16
DONALD NUTE AND CHARLES B. CROSS
conditional logic VC. First Pollock rejects the thesis CV, a thesis which is unavoidable in Lewis's semantics. Second Pollock embraces the Generalized Consequence Principle: GCP:
If is a set of sentences such that > and if entails , then > is true.
is true for each
2
,
GCP does not hold in all system-of-spheres models, but it does hold in all class selection function models.6 The conditional logic SS which Pollock favors is the smallest conditional logic closed under the rules listed for VC and containing all those theses used in de ning VC except that we replace CV with CA:
[( > ) ^ ( > )] ! [( _ ) > ].
This again gives us a weaker system since CA is contained by VC while CV is not contained by SS. Obviously, SS is not determined by the class of class selection function models which satisfy conditions (CS1){(CS6) since this class of models characterizes the logic VC. Let's replace the condition (CS5) with (CS50 ) f ( _ ; i) f (; i) [ f ( ; i). Then SS is determined by the class of all class selection function models which satisfy this new set of conditions. One reason for Pollock's lack of concern for Lewis's counterexamples to the Limit Assumption may be that Pollock conceives of what would count as a minimal change quite dierently from the way Lewis does. Pollock [1976] oers a detailed account of the notion of a minimal change, an account 6 To see how GCP might fail in Lewis's semantics, consider the example Lewis uses to show that for a particular there may be no -world closest to i. The example, which we considered earlier, involves a line printed on a page of [Lewis, 1973b]. Lewis invites us to consider worlds in which this line is longer than its actual length, which we will suppose to be exactly one inch. If the only way in which these worlds dier from the actual world is in the length of Lewis's line, then it is plausible that we rank these worlds in their similarity to the actual world according to how close to one inch Lewis's line is in each of these worlds. But no matter how close to one inch the line is, so long as it is longer than one inch there will be another such world in which it is closer to an inch in length. This means that for any length m greater than one inch, there is a world in which the line is longer than one inch and in which the line does not have length m which is nearer the actual world than is any world in which the line has length m. but then Lewis's truth conditions for conditionals dictate that if the line were longer than one inch, its length would not be m, and this is true for any length m greater than the actual length of the line. then consider the set of sentences of the form `Lewis's line is not length m' where m ranges over every length greater than the actual length of Lewis's line. But entails the sentence `Lewis's line is not greater than one inch in length'. Applying GCP, we conclude that if Lewis's line were greater than one inch in length, then it would not be greater than one inch in length. This conclusion is not intuitively reasonable nor is it true at any world in the system-of- spheres model which Lewis describes in his discussion.
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17
which is later modi ed in [Pollock, 1981]. The later view, which avoids many problems of the earlier view, will be discussed here. While Stalnaker, Lewis and others maintain that the notions of similarity of worlds and of minimal change are vague notions which may change given dierent purposes and contexts, thus accounting for the vagueness we often nd in the use of conditionals, Pollock claims that the similarity relation is not vague but quite de nite. Pollock's account rests upon his use of two epistemological notions, that of a subjunctive generalisation and that of a simple state of aairs. Subjunctive generalisations are statements of the form `Any F would be a G;. The truth of some subjunctive generalisations like `Anyone who drank from the Chisholm's bottle would die' depends upon contingent matters of fact, in this case the fact that Chisholm's bottle contains strychnine and the fact that people have a certain physical makeup. Other subjunctive generalisations like `Any creature with a physical make up like ours who drank strychnine would die' do not depend for their truth on contingent matters of fact in the same way. Pollock calls the former `weak' subjunctive generalisations and the latter `strong' subjunctive generalisations. Some subjunctive generalisations are supposed by Pollock to be directly con rmable by their instances, and these he calls basic. The problem of con rmation is discussed in [Pollock, 1984]. The second crucial ingredient in Pollock's analysis is the notion of a simple state of aairs. A state of aairs is simple if it can be known non-inductively to be the case without rst coming to know some other state(s) of aairs which entail(s) it. The actual world is supposed by Pollock to be determined by the set of true basic strong subjunctive generalisations together with the set of true simple states of aairs. The justi cation conditions for a subjunctive conditional > are stated in terms of making minimal changes in these two sets in order to accommodate . The rst step is to generate all maximal subsets of the set of true basic strong subjunctive generalisations which are consistent with . For each such maximally -consistent set N of true basic strong subjunctive generalisations, we then generate all sets of true simple states of aairs which are maximally consistent with N [ fg. Finally, we consider every possible world at which , every member of some maximally -consistent set N of true basic strong subjunctive generalisations, and every member of some set S of true basic strong subjunctive generalisations, and every member of some set S of true simple states of aairs maximally consistent with N [ fg are all true. If is true at all such worlds, then > is true at the actual world. The set of worlds determined by this procedure serves as the value of a class selection function. If we try to de ne a relative similarity relation for worlds based upon Pollock's analysis of minimal change, we come up with a partial order rather than the `complete' order assumed by Lewis and, apparently, by Stalnaker. Because we can have two worlds j and k such that their similarity to a
18
DONALD NUTE AND CHARLES B. CROSS
third world i is incomparable, the thesis CV does not hold for Pollock's semantics.7 A simple model of Pollock's sort which rejects CV as well as another thesis which has been attributed to Pollock's conditional logic SS is developed in [Mayer, 1981]. Several authors have proposed theories which resemble Pollock's in important respects. One of these is Blue [1981] who suggests that we think of subjunctive conditionals as metalinguistic statements about a certain semantic relation between an antecedent set of sentences in an object language and another sentence of the object language viewed as a consequent. A theory (set of sentences of the object language) and the set of true basic (atomic and negations of atomic) sentences of the language play roles similar to those played by laws (true basic strong subjunctive generalisations) and simple states of aairs in Pollock's account. One problem with Blue's proposal is that treating conditional metalinguistically as he does prevents iteration of conditionals without climbing a hierarchy of metalanguages. Another problem concerns the role which temporal relations between the basic sentences plays in his theory, a problem for other theories as well. (This problem is discussed in Section 1.8 below.) For a more detailed discussion of Blue's view, see [Nute, 1981c]. The similarity of an account like Pollock's or Blue's to the cotenability theories of conditionals should be obvious. A conditional is true just in case its consequent is entailed (Blue uses a somewhat dierent relation) by its 7 Pollock has oered various counterexamples to CV, the most recent of which involves a circuit having among its components two light bulbs L1 and L2 , three simple switches A; B , and C , and a power source. These components are supposed to be wired together in such a way that bulb L1 is lit exactly when switch A is closed or both switches B and C are closed, while bulb L2 is lit exactly when switch A is closed or switch B is closed. At the moment, both bulbs are unlit and all three switches are open. Then the following conditionals are true: (5a) :(L2 > :L1 ) (5b) :[(L2 ^ L1 ) > :(B ^ C )] The justi cation for (5a) is that one way to bring it about that L2 (i.e. that bulb L2 is lit) is to bring it about that A (i.e. that switch A is closed), but A > L1 is true. The justi cation for (5b) is that one way to make L1 and L2 both true is to close both B and C . Pollock claims that the following counterfactual is also true: (5c) L2 > :(B ^ C ) If Pollock is correct, then these three counterfactuals comprise a counterexample to CV. Pollock's argument for (5c) is that L2 requires only A or B , and to also make C the case is a gratuitous change and should therefore not be allowed. But this is an oversimpli cation. It is not true that only A; B , and C are involved. Other changes which must be made if L2 is to be lit include the passage of current through certain lengths of wire where no current is now passing, etc. Which path would the current take if L2 were lit? We will probably be forced to choose between current passing through a certain piece of wire or switch C being closed. It is diÆcult to say exactly what the choices may be without a diagram of the kind of circuit Pollock envisions, but without such a diagram it is also diÆcult to judge whether closing switch C is gratuitous in the case of (5c) as Pollock claims.
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19
antecedent together with some subset of the set of laws or theoretical truths and some (cotenable) set of simple states of aairs or basic sentences. Veltman [1976] and Kratzer [1979; 1981] also propose theories of conditionals which resemble Pollock's in important respects. We will discuss Kratzer's view, although the two are similar. Kratzer suggests what can be called a premise or a partition semantics for subjunctive conditionals. Like Pollock, she associates with each world i a set Hi of propositions or states of aairs which uniquely determines that world. The set Hi is called a partition for i or a precise set for i. Kratzer proposes that we evaluate a subjunctive conditional > by considering - consistent subsets of Hi . > is true at a world i if and only if each -consistent subset X of Hi is contained in some -consistent subset Y of H such that Y [ fg entails . Kratzer points out that if every - consistent subset of Hi is contained in some maximally - consistent subset of Hi , then this truth condition is equivalent to the condition that > is true at i just in case is entailed by X [ fg for every maximally -consistent subset X of Hi . If we assume that every -consistent subset of Hi is contained in some maximally -consistent subset of Hi , the specialized version of Kratzer's semantics we obtain looks very much like Pollock's. Lewis [1981a] notes that this assumption plays the same role in premise semantics that the Limit Assumption plays in class selection function semantics. In fact, Lewis shows that on this assumption Kratzer's premise semantics is formally equivalent to Pollock's semantics. Given this equivalence, these two semantics will determine exactly the same conditional logic SS. Even if we assume that the required maximal sets always exist and adopt a version of premise semantics which is formally equivalent to Pollock's semantics, Kratzer's position would still dier radically from Pollock's since she does not assign to laws and simple states of aairs a privileged role in her analysis. Nor does she prefer Blue's object language theory and basic sentences for such a role. The set of premises which we associate with a world and use in the evaluation of conditionals varies, according to Kratzer, as the purposes and circumstances of the language users vary. Thus Kratzer reintroduces the vagueness which so many investigators have observed in ordinary usage and which Pollock and Blue would deny or at least eliminate. Apparently Kratzer does not accept the Limit Assumption, in her case the assumption that the required maximal sets always exist. Yet in [Kratzer, 1981] she describes what she calls the most intuitive analysis of counterfactuals, saying that The truth of counterfactuals depends on everything which is the case in the world under consideration: in assessing them, we have to consider all the possibilities of adding as many facts to the antecedent as consistency permits.
20
DONALD NUTE AND CHARLES B. CROSS
This certainly suggests maximal antecedent-consistent subsets of a premise set (the Limit Assumption) and a minimal change semantics. But if the Limit Assumption is unacceptable, this initial intuition must be modi ed. Kratzer's modi cation takes the form of the truth condition reported earlier. Besides the Limit Assumption, Kratzer's semantics also fails to support the GCP. One principle which does remain, a principle common to all the semantics discussed in this section, is the thesis CS. Beginning with some sort of minimalist intuition, all of these authors claim subjunctive conditionals with true antecedents have the same truth values as their consequents. When the antecedent of the conditional is true, the actual world is the unique closest antecedent world and hence the only world to be considered in evaluating the conditional. If Lewis's counterexamples to the Limit Assumption are conclusive, we must conclude that all the semantics for subjunctive conditionals which we have discussed in this section must be inadequate except for Lewis' system-of-spheres semantics and the general version of Kratzer's premise semantics. And if the GCP is a principle which we wish to preserve, then Lewis's semantics and Kratzer's semantics are also inadequate. Besides these diÆculties, minimal change theories have been criticized because they endorse the thesis CS. As was mentioned in Section 1.1, many researchers claim that some conditionals with true antecedent and consequent are false. For an excellent polemic against the minimal change theorists on this issue, see [Bennett, 1974].
1.5 Small Change Theories Aqvist [1973] presents a very interesting analysis of conditionals in which the conditionals in which the conditional operator is de ned in terms of material implication and some unusual monadic operators. Simplifying a bit, Aqvist's semantics involves ordered quintuplets hI; i; R; f; [ ]i such that I and [ ] are as in other models we have discussed, i is a member of I; R is an accessibility relation on I , and f is a function which assigns to each sentence a subset f () of [] such that for every member j of f (); hi; j i 2 R. A sentence whose primary connective is the monadic star operator is true at a world j in I just in case j 2 f (). The usual truth conditions are provided for a necessity operator, so that is true at j in I just in case for every world k such that hj; ki 2 R, is true at k, i.e. just in case k 2 []. Finally, a conditional > is true at a world j just in case ( ! ) is true at i. Aqvist modi es this semantics in an appendix. The modi cation involves a set of models of the sort described, each with the same set of possible worlds, the same accessibility relation, and the same valuation function, but each with its own designated world and selection function. The resulting semantics turns out once again to be equivalent to a version of class selection semantics.
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21
While the interesting formal details of Aqvist's theory are quite dierent from those of other investigators, the most signi cant feature of his account may be his suggestion that a class selection function might properly pick out for a sentence and a world i all those -worlds which are `suÆciently' similar to i rather than only those -worlds which are `most' similar to i. By changing the intended interpretation for the class selection function, we avoid the trivialisation of the truth conditions for conditionals in all those cases where the Limit Assumption in either of its forms fails. At the same time, class selection function semantics supports the GCP. Aqvist's suggestion looks very promising. A similar approach is taken by Nute, [1975a; 1975b; 1980b], but the semantics Nute proposes is explicitly a version of class selection function semantics. This model theory diers from versions of class selection function semantics we examined earlier in two important ways. First the intended interpretation is dierent, i.e. there is a dierent informal explanation to be given for the role which the class selection functions play in the models. Second the restriction (CS2) is replaced by the weaker restriction (CS20 ) if i 2 [] then i 2 f (; i). The second change is related to the rst. Surely any world is more similar to itself than is any other. Thus, if f picks out for and i the -worlds closest to i, and if i is itself a -world, then f will pick out i and nothing else for and i. The objection to the thesis CS, though can be thought of as a claim that there may be other worlds suÆciently similar to the actual world so that in some cases we should consider these worlds in evaluating conditionals with true antecedents. When we modify our earlier semantics for Lewis's system VC by replacing (CS2) with (CS20 ), the resulting class of models characterizes the logic which Lewis [1973b] calls VW. VW is the smallest conditional logic which is closed under all the rules and contains all the theses listed for VC except for the thesis CS. By weakening our semantics further we can characterize a logic which is closed under all the rules and contains all the theses of VW except CV. This, of course, would give us a logic for which Pollock's SS would be a proper extension. Although many count it as an advantage of small change class selection function semantics that such theories allow us to avoid CS, it should be noted that such semantics do not commit us to a rejection of CS. As we have seen, both Lewis's VC and Pollock's SS are characterized by classes of class selection function models. For those who favor CS, it is still possible to avoid the diÆculties of the Limit Assumption and embrace the GCP by adopting one of these versions of the class selection function semantics but giving a small change interpretation of the selection functions upon which such a semantics depends. It is possible to avoid CS within the restrictions of a minimal change semantics. We can do this by `coarsening' our similarity relation, to use
22
DONALD NUTE AND CHARLES B. CROSS
Lewis's phrase, counting worlds as equally similar to some third world despite fairly large dierences in these worlds. For example, we might count some worlds other than i as being just as similar to i as is i itself. When we do this for a minimal change version of class selection function semantics, the formal results are exactly the same as those proposed earlier in this section and the resulting logic is VW. Of course, we must still cope with Lewis's objections to the Limit Assumption. But it is even possible to avoid CS within Lewis's system-of-spheres semantics. All we need to do is replace the restriction (SOS1) with the following: (SOS10 ) i 2 \si . The class of all those system-of-spheres models which satisfy (SOS10 ) and (SOS2) determines the conditional logic VW. While such a concession to the critics of CS is possible within the con nes of Lewis's semantics, Lewis does not favor such a move. We should also remember that the resulting semantics still does not support the GCP. Since we can formulate a kind of minimal change semantics which avoids the controversial thesis CS, the only advantage we have shown for small change theories is that they avoid the problems of the Limit Assumption while giving support for the GCP. But this advantage may be illusory. Loewer [1978] shows that for many versions of class selection function semantics we can always view the selection function as picking out closest worlds. For a model of such a semantics, we can de ne a relative similarity relation R between the worlds of the model in terms of the model's selection function f . It can then be shown that for a sentence and a world i; j 2 f (; i) if and only if j is a -world which is at least as close to i with respect to R as is any other -world. Consider such a model and consider a proposition which is true at world j just in case there are in nitely many worlds closer to i with respect to R than is j . There will be no -world closest to i. Consequently f (; i) will be empty and any conditional with as antecedent will be trivially true. How seriously we view this example depends upon our attitude toward the assumption that there exists a proposition which has the properties attributed to . If we take propositions to be sets of worlds, then the existence of such a proposition is very plausible. We should also note that this argument involves a not so subtle change in our semantics. Until now we have been thinking of our selection functions as taking sentences as arguments rather than propositions. If we restrict ourselves to sentences it is very unlikely that our language for conditional logic will contain a sentence which expresses the troublesome proposition. Nevertheless, it is not entirely clear that every small change version of class selection function semantics will automatically avoid the problems associated with the Limit Assumption. There is another advantage which can be claimed for small change theories which doesn't involve the logic of conditionals. If, for example, VW
CONDITIONAL LOGIC
23
is the correct logic for conditionals, we have seen that it is possible to take either the minimal change or the small change approach to semantics for conditionals and still provide a semantics which determines VW. But even if agreement is reached about which sentences are valid, these two approaches are still likely to result in dierent assignments of truth values to contingent conditional sentences. Suppose for example that Fred's lawn is just slightly too short to come into contact with the blades of his lawnmower. Thus his lawnmower will not cut the grass at present. Suppose further that the engine on Fred's lawnmower is so weak that it will only cut about a quarter of an inch of grass. If the height of the grass is more than a quarter of an inch greater than the blade height, the mower will stall. Then is the following sentence true or false? 29. If the grass were higher, Fred's mower would cut it. On the minimal change approach, whether we use class selection function semantics or system-of-spheres semantics, the answers to this question must be `yes' for there will be worlds at which the lawn is higher than the blade height but no more than a quarter inch higher than the blade height, which are closer to the actual world than is any world at which the grass is more than a quarter inch higher than the blade height. But the correct answer to the question would seem to be `no'. If someone were to assert (29) we would likely object, `Not if the grass were much higher'. This shows that we are inclined to consider changes which are more than minimally small in our evaluations of conditionals. We might avoid particular examples of this sort by `coarsening' the similarity relation, but it may be possible to generate such examples for any similarity relation no matter how coarse. All of the small change theories we have considered propose semantics which are at least equivalent to some version of class selection function semantics. There is, however, at least one small change theory which does not share this feature. Warmbrod [1981] presents what he calls a pragmatic theory of conditionals. This theory is based on similarity of worlds but in a radically dierent way than are any of the theories we have yet examined. According to Warmbrod, the set of worlds we use in evaluating a conditional is determined not by the antecedent of that particular conditional but rather by all the antecedents of conditionals occurring in the piece of discourse containing that particular conditional. Thus a conditional is always evaluated relative to a piece of discourse rather than in isolation. For any piece of discourse D and world i we select a set of worlds S which satis es the following conditions: (W1) if > occurs in D and is logically possible, then some world j in S is a -world; (W2) for some > occurring in D; j 2 s if and only if j is at least as close to i as are the closest - worlds to i.
24
DONALD NUTE AND CHARLES B. CROSS
Condition (W1) ensures that S is what Warmbrod calls normal for D and (W2) ensures that S is what Warmbrod calls standard for some antecedent occurring in D. (Warmbrod formulates his theory in terms of an accessibility relation, but the semantics provided here is formally equivalent.) Then a conditional > is true at i with respect to D if and only if ! is true at every world in S . The resulting semantics resembles both class selection function semantics and an analysis of conditionals as strict conditionals, but it diers from each of these approaches in important respects. Like other proposals which treat subjunctive conditionals as being strict conditionals, Warmbrod's theory validates Transitivity, Contraposition, and Strengthening Antecedents. Warmbrod argues that the evidence against these theses can be explained away. Apparent counterexamples to transitivity, for example, depend according to Warmbrod on the use of dierent sets S in the evaluation of the sentences involved in the putative counterexamples. Consider the example (21){(23) in Section 1.3 above. According to Warmbrod, this example can be a counterexample to Transitivity only if there is some set of worlds S which contains worlds at which Carter did not lose in 1980, contains some worlds at which Carter died in 1979, which is normal for these two antecedents, and for which the material conditional corresponding to (21) and (22) are true at all members of S while the material conditional corresponding to (23) is false at some world in S . But this, Warmbrod claims, is exactly what does not happen. The apparent counterexample depends upon an equivocation, a shift of the set S during the course of the argument. Warmbrod's theory has a certain attraction. It is certainly true that Transitivity and other controversial theses are harmless in many contexts, and it is certainly true that these theses are frequently used in ordinary discourse. The problem is to provide an account of the dierence between those situations in which the thesis is reliable and those in which it is not. Warmbrod's strategy is to consider the thesis to be always reliable and then to provide a way of falsifying the premises in unhappy cases. An alternative approach is to count these theses as being invalid and then to look for those features of context which sometimes allow us to use them with impunity. We think the second strategy is safer. It is probably better to occasionally overlook a good argument than it is to embrace a bad one. Or to put a bit dierently, it is better to force the argument to bear the burden of proof rather than to consider it sound until proven unsound. Another problem with Warmbrod's theory is that it suggests that we should nd apparent counterexamples to certain theses which have until now been considered uncontroversial. For example, we should nd apparent counterexamples for CA. (See [Nute, 1981b] for details.) Warmbrod's semantics also runs into diÆculty with the Limit Assumption. The requirement (W2) that S be standard for some antecedent in D involves the Limit Assumption explicitly. although Warmbrod's semantics
CONDITIONAL LOGIC
25
may tolerate small, non-minimal changes for some of the antecedents in a piece of discourse, it demands that only minimal changes be considered for at least one such antecedent. Of course, we might be able to modify (W2) in such a way as to avoid this problem. There remains, though, the nagging suspicion that none of the small change theories we have considered will in the end be able to escape the Limit Assumption, with all its diÆculties, in some form or other.
1.6 Maximal Change Theories Both minimal change theories and small change theories of conditionals are based on the premise that a conditional > is true at i just in case is true at some - world(s) satisfying certain conditions. The dierence, of course, is that for the one approach it is suÆcient that be true at all closest -worlds while the other requires that be true at all -worlds which are reasonably or suÆciently close to i. There is a third type of theory which shares the same basic premise as these two but which does not require that the worlds upon which the evaluation of > at i depends be very close or similar to i at all. According to this way of looking at conditionals, all that is required is that the relevant worlds resemble i in certain very minimal respects. Otherwise the relevant worlds may dier from i to any degree whatever. We might even think of this approach as requiring us to consider worlds which dier from i maximally except for the narrowly de ned features which must be shared with i. One theory of this sort is developed by Gabbay [1972]. To facilitate comparison, we will simplify Gabbay's account of conditionals rather drastically. When we do this, Gabbay's semantics for conditionals resembles the class selection function semantics we have discussed, but there are some very important dierences. A simpli ed Gabbay model is an ordered triple hI; g; [ ]i such that I and [ ] are as in earlier models, and g is a function which assigns to sentences and and world i in I a subset g(; ; i) of I . A conditional > is true at i in such a model just in case g(; ; i) [ ! ]. The dierence between this and class selection function semantics of the sort we have seen previously is obvious: the selection function g takes both antecedent and consequent as argument. This means that quire dierent sets of worlds might be involved in the truth conditions for two conditionals having exactly the same antecedent. This change in the formal semantics re ects a dierence in Gabbay's attitude toward conditionals and toward the way in which we evaluate conditionals. When we evaluate > , we are not concerned to preserve as much as we can of the actual world in entertaining ; instead we are concerned to preserve only those features of the actual world which are relevant to the truth of , or perhaps to the eect would have on the truth of . In actual practice the kind of similarity which is required is supposed by Gabbay to be determined by , by , and
26
DONALD NUTE AND CHARLES B. CROSS
also by general knowledge and particular circumstances which hold in i at the time when the conditional is uttered. What this involves is left vague, but it is not more vague than the notions of similarity assumed in earlier theories. When we modify Gabbay's semantics in this way, we must impose three restrictions on the resulting models: (G1)
i 2 g(; ; i);
(G2)
if [] = [ ] and [] = [] then g(; ; i) = g( ; ; i);
(G3)
g(; ; i) = g(; : ; i) = g(:; ; i).
With these restrictions Gabbay's semantics determines the smallest conditional logic which is closed under RCEC and the following two rules: RCEA: from $ , to infer ( > ) $ ( > ). RCE: from ! , to infer > .
We will call this logic G. At the end of [Gabbay, 1972], a conjectured axiomatisation of G is presented, but it was later shown to be unsound and incomplete in [Nute, 1977], where the axiomatisation of G presented here was conjectured to be sound and complete (see [Nute, 1980b]). Working independently, David Butcher [1978] also disproved Gabbay's conjecture, and proved the soundness and completeness of G for the Gabbay semantics (see [Butcher, 1983a]). It is obvious that G is the weakest conditional logic we have yet considered. We can characterize a stronger logic if we place additional restrictions on our Gabbay models, but we may not be able to guarantee a suÆciently strong logic without restricting our models to the point where they become formally equivalent to models we examined earlier. Consider, for example the theses CC: [( > ) ^ ( > )] ! [ > ( CM: [ > (
^ )] ! [( >
^ )]
) ^ ( > )].
To ensure that our conditional logic contains CC and CM, we could impose the following restriction on Gabbay's semantics: (G4) g(; ; i) = g(; ; i). Once we do this we have eliminated the most distinctive feature of Gabbay's semantics. According to David Butcher [1983a], it is possible to ensure CC and CM by adopting conditions weaker than (G4). However, Butcher has indicated that these conditions are problematic for other reasons.8 8 Many
of these isues are also discussed in Butcher [1978; 1983a].
CONDITIONAL LOGIC
27
A rather dierent and very specialized maximal change theory has been developed in two dierent forms by Fetzer and Nute [1979; 1980] and by Nute [1981a]. Both forms of this theory are intended not as analyses of ordinary subjunctive conditionals as they are used in ordinary discourse, but rather as analyses of scienti c, nomological, or causal conditionals, i.e. of subjunctive conditionals as they are used in the very special circumstances of scienti c investigation. Formally the two theories propose class selection function semantics for scienti c conditionals, but the intended interpretation is quite dierent from that of any theory we have yet considered. In the version of the theory developed by Fetzer and Nute the selection function f is intended to pick out for a sentence and a world i the set of all those -worlds at which all the individuals mentioned in possess, in so far as the truth of will allow, all those dispositional properties which they permanently possess in i. This forces us to ignore all features of worlds except those assumed by the underlying theory of causality to aect the causal eÆcacy of the situation, events, etc., described in . We are forced, in other words, to consider worlds which preserve only these features and otherwise dier maximally from the world at which the scienti c conditional is being evaluated. In this way we can ensure that the conditional in question is true if but only if the antecedent and the consequent are related causally or nomologically in an appropriate manner. Physical law statements are then analysed as universal generalisations or sets of universal generalisations of such scienti c conditionals. The view subsequently developed in [Nute, 1981a] departs a bit from the requirement of maximal speci city which we seek in our scienti c pronouncements and in doing so comes closer to representing a kind of conditional used in ordinary discourse. Nute suggests that the selection function f selects for a sentence and a world i all those -worlds at which all those individuals mentioned in possess, so far as the truth of allows, not only all those dispositional properties which they permanently possess in i but also all those dispositional properties which they accidentally or as a matter of particular fact possess in i. For example, a particular piece of litmus paper permanently possesses the tendency to turn red when dipped in an acidic solution, since it could not lose this tendency and still be litmus paper, but it only accidentally possesses the tendency to re ect blue light, since it could certainly lose this disposition through being dipped in acid and yet still be litmus paper. Where it is impossible to accommodate without giving up some of the dispositional properties possessed by individuals mentioned in , preference is given to dispositions which are possessed permanently. On this account, but not on the account developed by Fetzer and Nute conjointly, the following conditional is true where x is a piece of litmus paper which is in fact blue: 30. If x were cut in half, it would be blue.
28
DONALD NUTE AND CHARLES B. CROSS
Nute [1981a] suggests that many ordinary conditionals may have such truth conditions, or may be abbreviations of other more explicit conditionals which have such truth conditions. Each of the theories presented in this section is in fact only a fragment of a more complex theory. It is impossible to discuss the larger theories in any greater detail and the reader is encouraged to consult the original publications. What allows us to consider them under a single category is their departure from the premise that the truth of a conditional depends upon what happens in antecedent- worlds which are very much like the actual world. Each of these theories assumes and even requires that the divergence from the actual world be rather larger than minimal or small change theories would indicate.
1.7 Disjunctive Antecedents One thesis in particular has caused considerable controversy among the investigators of conditional logic. This thesis is Simpli cation of Disjunctive Antecedents: SDA: [( _ ) > ] ! [( > ) ^ ( > )]. The intuitive plausibility of SDA has been suggested in [Fine, 1975], in [Nute, 1975b] and in [Ellis et al., 1977]. Unfortunately, any conditional logic which contains SDA and which is also closed under substitution of provable equivalents will also contain the objectionable thesis Strengthening Antecedents. If we add SDA to any of the logics we have discussed, then Transitivity and Contraposition will be contained in the extended logic as well.9 Ellis et al. suggest that the evidence for SDA is so strong and the problems involved in trying to incorporate SDA into any account of conditionals based upon possible worlds semantics is so great that the possibility of an adequate possible worlds semantics for ordinary subjunctive conditionals is quite eliminated. With all the problems which the various theories encounter, the possible worlds approach has still proven to be a powerful tool for the investigation of the logical and semantical properties of conditionals and we should be unwilling to abandon it without rst trying to defend it against such a charge. The rst line of defence has been a `translation lore' approach to the problem of disjunctive antecedents. It is rst noted that, despite the intuitive appeal of SDA, there are examples from ordinary discourse which show that SDA is not entirely reliable. The following sentences comprise one such example: 9 As further evidence of the problematic character of SDA, David Butcher [1983b] has shown that any logic containing SDA and CS will contain ! , where is de ned as : > .
CONDITIONAL LOGIC
29
31a. If the United States devoted more than half of its national budget to defence or to education, it would devote more than half of its national budget to defence. 31b. If the United States devoted more than half of its national budget to education, it would devote more than half of its national budget to defence. Contrary to what we should expect if SDA were completely reliable, it looks very much as if (31a) is true even though (31b) cannot be true. Fine [1975], Loewer [1976], McKay and Van Inwagen [1977] and others have suggested that those examples which we take to be evidence for SDA actually have a quite dierent logical form from that which supporters of SDA suppose them to have. While a sentence like (31a) really does have the form ( _ ) > , a sentence like 32. If the world's population were smaller or agricultural productivity were greater, fewer people would starve. has the quite dierent logical form ( > ) ^ ( > ). According to this suggestion, the word `or' represents wide scope conjunction rather than narrow scope disjunction in (32). Since we can obviously simplify a conjunction, this confusion about the logical form of sentences like (32) results in the mistaken commitment to a thesis like SDA. This would be a neat solution to the problem if it would work, but the translation lore approach has a serious aw. According to the translation lorist, the two sentences (31a) and (32) have dierent logical forms even though they share the same surface or grammatical structure. We can point out an obvious dierence in surface structure since one of the apparent disjuncts in the antecedent of (31a) is also the consequent of (31a), a feature which (32) lacks. But we can easily produce examples where this is not the case. Suppose after asserting (31a) a speaker went on to assert 33. So if the United States devoted over half its national budget to defence or education, my Lockheed stock would be worth much more than it is. It would be very reasonable to accept this conditional but at the same time to reject the following conditional: 34. So if the United States devoted over half of its national budget to education, my Lockheed stock would be worth much more than it is. The occurrence of the same component sentence in antecedent and consequent is not a necessary condition for the failure of SDA and cannot be used as a criterion for distinguishing those cases in which English conditional with `or' in their antecedents are of the logical form ( _ ) > from
30
DONALD NUTE AND CHARLES B. CROSS
those in which they are of the logical form ( > ) ^ ( > ). We cannot decide on purely syntactical grounds which of the two possible symbolisations is proper for an English conditional with `or' in its antecedent. Loewer [1976] suggests that this decision may be made on pragmatic grounds, but it is diÆcult to see what the distinguishing criterion is to be except that English conditionals with disjunctive antecedents are to be symbolized as ( > ) ^ ( > ) when simpli cation of their disjunctive antecedents is legitimate and to be symbolized as ( _ ) > when such simpli cation is not legitimate. Until Loewer's suggestion concerning the pragmatic pressures which prompt one symbolisation rather than another can be provided with suÆcient detail, the translation lore account of disjunctive conditional does not provide us with an adequate solution to our problem. We nd an interesting variation on the translation lore solution in [Humberstone, 1978] and in [Hilpinen, 1981]. Both suggest the use of an antecedent forming operator like Aqvist's . We will discuss Hilpinen's theory here since it diers the most from Aqvist's view. Hilpinen's analysis utilizes two separate operators which we can represent as If and Then. The If operator attaches to a sentence to produce an antecedent If . the Then operator connects an antecedent and a sentence to form a conditional Then . The role of the dyadic truth functional connectives is expanded so that _, for example, can connect two antecedents and to form a new antecedent _ . An important dierence between Hilpinen's If operator and Aqvist's is that for Aqvist is a sentence or proposition bearing a truth value while for Hilpinen If is not. Finally Hilpinen proposes that sentences like (31a) be symbolized as If ( _ ) Then while sentences like (32) be symbolized as (If _ If ) Then . Hilpinen then accepts a rule similar to SDA for sentences having the latter form but not for sentences having the former. This proposal allows us to incorporate a rule like SDA into our conditional logic while avoiding Strengthening Antecedents, etc., and, unlike other versions of the translation lore approach, Hilpinen's proposal seems to suggest how it might be possible for sentences like (31a) and (32) to have a legitimate scope ambiguity in their syntactical structure, like the scope ambiguity in `President Carter has to appoint a woman'. In fact, however, the ambiguity postulated by Hilpinen's proposal does not seem simply to be a scope ambiguity. The sentence `President Carter has to appoint a woman' is ambiguous with respect to the scope of the phrase `a woman', but the phrase `a woman' has the same syntactical function and the same semantics on both readings of the sentence. The same cannot be said of the word `or' in Hilpinen's account of disjunctive antecedents: on one resolution of the ambiguity, what `or' connects in examples like (31a) and (32) are sentences; on the other resolution of the ambiguity, `or' connects phrases that are not sentences. It is diÆcult to see how the ambiguity in (31a) and (32) can be simply a scope ambiguity if `or' does not have the same syntactical role in both readings of a given sentence.
CONDITIONAL LOGIC
31
Another approach to disjunctive antecedents is developed by Nute [1975b; 1978b] and [1980b]. Formally the problem with SDA is that it together with substitution of provable equivalents results in Strengthening Antecedents and other unhappy results. The translation lorist's suggestion is that we abandon SDA. Nute's suggestion, on the other hand, is that we abandon substitution of provable equivalents, at least for antecedents of subjunctive conditionals. One fairly strong logic which does not allow substitution of provably equivalent antecedents is the smallest conditional logic which is closed under RCEC and RCK and contains ID, MP, MOD, CV, and SDA. Logics of this sort have been called `non- classical' or `hyperintensional' to contrast them with those intensional logics which are closed under substitution of provable equivalents. Classical logics (those closed under substitution of provable equivalents) are preferred by most investigators. Besides the fact that non-classical logics are much less elegant than classical logics, Nute's proposal has other very serious diÆculties. First, substitution of certain provable equivalents within antecedents appears to be perfectly harmless. For example, we can surely substitute _ for _ in ( _ ) > with impunity. How are we to decide which substitutions are to be allowed and which are not? Non-classical conditional logics which allow extensive substitutions are developed in Nute [1978b] and [1980b]. But these systems are extremely cumbersome and there still is the extra-formal problem of justifying the particular choice of substitutions which are to be allowed in the logic. Second, we are still left with the apparent counterexamples to SDA like (31a). Nute suggests a pluralist position, maintaining that there are actually several dierent conditionals in common use. For some of these conditionals SDA is reliable while for others it is not. The conditional involved in (31a), it is claimed, is unusual and should not be represented in the same way as other subjunctive conditionals. While there is good reason to admit a certain pluralism, to admit, for example, the distinction between subjunctive and indicative conditionals, Nute's proposal is little more than a new translation lore in disguise. The translation lore we discussed earlier at least has the virtue that it attempts to explain the perplexities surrounding disjunctive antecedents in terms of a widely accepted set of logical operators without requiring the recognition of any new conditional operators. Non-classical logic appears to be a dead end so far as the problem of disjunctive antecedents is concerned. A completely dierent solution is suggested in [Nute, 1980a], a solution based upon the account of conversational score keeping developed in [Lewis, 1979b]. Basically, the proposal concerns the way in which the class selection function (or the system-of-spheres if Lewis-style semantics is employed) becomes more and more de nite as a linguistic exchange proceeds. During a conversation, the participants tend to restrict the selection function which they use to interpret conditionals in such a way as to accommodate claims made by their fellow participants. This growing set of restrictions on the se-
32
DONALD NUTE AND CHARLES B. CROSS
lection function forms part of what Lewis calls the score of the conversation at any given stage. Some accommodations, of course, will not be forthcoming since some participant will be unwilling to evaluate conditionals in the way which these accommodations would require. Each restriction on the selection function which the participants implicitly accept will also rule out other restrictions which might otherwise have been allowed. Nute's suggestion is that our inclination is to restrict the selection function in such a way to make SDA reliable, but that this inclination can be overridden in certain circumstances by our desire to accommodate the utterance of another speaker. When we hear the utterance of a sentence like (31a), for example, we restrict our selection function so that SDA becomes unreliable for sentences which have `the United States devotes more than half its national budget to defence or education' as antecedent. Once (31a) is accommodated in this way, this restriction on the selection function remains in eect so long as the conversational context does not change. Nute completes his account by formulating some `accommodation' rules for class selection functions. By oering a pragmatic account of the way in which the selection function becomes restricted during the course of a conversation, and by paying attention to the inclination to restrict the selection function in such a way as to make SDA reliable whenever possible, it may be possible to explain the fact that SDA is usually reliable while at the same time avoiding the many diÆculties involved in accepting SDA as a thesis of our conditional logic. This proposal is similar in certain respects to Loewer's [1976]. Like Loewer, Nute is recognising the important role which pragmatic features play in our use of conditionals with disjunctive antecedents. However, Nute's use of Lewis's notion of conversational score keeping results in an account which provides more details about what these pragmatic features might be than does Loewer's account. We also notice that Nute's suggestions might provide the criterion which Loewer needs to distinguish those conditionals which should be symbolized s ( _ ) > from those which should be symbolized as ( > ) ^ ( > ). But once the distinction is explained in terms of the evolving restrictions on class selection functions, there is no need to require that these conditionals be symbolized dierently. The point of Nute's theory is that all such conditionals have the same logical form, but the reliability of SDA will depend on contextual features. There is also considerable similarity between Nute's second proposal and Warmbrod's semantics for conditionals which was discussed in Section 1.5. In fact, Warmbrod's semantics is oered at least in part as an alternative to Nute's proposed solution to the problem of disjunctive antecedents. The important similarity between the two approaches is that both recognize that the interpretation of a conditional is a function not of the conditional alone but also of the situation within which the conditional is used. The important dierence is that Warmbrod's semantics makes SDA, Transitivity, Contraposition, Strengthening Antecedents, etc. valid and uses pragmatic
CONDITIONAL LOGIC
33
considerations to explain and guard us from those cases where it seems to be a mistake to rely upon these principles, while Nute ultimately rejects all of these principles, but uses pragmatic considerations to explain why it is perfectly reasonable to use at least one of these theses, SDA, in many situations. Warmbrod also oers a translation lore as part of his account. His suggestion about the way in which we should symbolize English conditionals with disjunctive antecedents is essentially that of Fine, Lewis, Loewer, and others, but he oers purely syntactic criteria for determining which symbolisation is appropriate in a particular case. His semantics is oered as a justi cation for his translation lore in an attempt to make his rules for symbolisation appear less ad hoc. Warmbrod points out some diÆculties with Nute's rules of accommodation for class selection functions, and his translation rules might be used as a model for improving the formulation of Nute's rules. Nute's theory of disjunctive antecedents in terms of conversational score might also be proposed as an alternative justi cation for Warmbrod's translation rules.
1.8 The Direction of Time We turn now to a problem alluded to in Section 1.4, a problem which concerns the role temporal relations play in the truth conditions for subjunctive conditionals. Actually, there are two dierent sets of problems to be considered. One of these involves the use of tensed language in conditionals and the other does not depend essentially on the use of tense and conditionals together. We will consider the latter set of problems in this section and save problems concerning tense for the next section. A particularly thorny problem for logicians working on conditionals has to do with so-called backtracking conditionals, i.e. conditionals having antecedents concerned with events or states of aairs occurring or obtaining at times later than those involved in the consequents of the conditional. It is widely held that such conditionals are rarely true, and that when they are true they usually involve much more complicated antecedents and consequents than do the more usual true non-backtracking conditionals. Consider, for example, the two conditionals: 35. If Hinckley had been a better shot, Reagan would be dead. 36. If Reagan were dead, Hinckley would have been a better shot. The rst of these two conditionals is an ordinary non-backtracking conditional, while the second is a backtracking conditional. the rst is very plausible and perhaps true, while the second is surely false. The problem with (36) which makes it so much less plausible than (35) is that Reagan might have died subsequent to the assassination attempt from any number
34
DONALD NUTE AND CHARLES B. CROSS
of causes which would not involve an improvement of Hinckley's aim. The problem for the logician or semanticist is to explain why non-backtracking conditionals are more often true than are backtracking conditionals. The primary goal of Lewis [1979a] is to explain this phenomenon. Lewis's proposal makes explicit, extensive use of the technical notion of a miracle. In a certain sense miracles do not occur at all in Lewis's analysis: rather a miracle occurs in one world relative to another world. No event ever occurs in any world which violates the physical laws of that world, but events can certainly occur in one world which violate the physical laws of some other world. These are the kinds of miracles Lewis relies upon. Assuming complete determinism, which Lewis does at least for the sake of argument, any world which shares a common history with the actual world up to a certain point in time but which diverges from the actual world after that time cannot obey the same physical laws as does the actual worlds. Basically Lewis proposes that the worlds most similar to the actual world in which some counterfactual sentence is true are those worlds which share their history with the actual world up until a brief transitional period beginning just prior to the times involved in the truth conditions for . In the case of (35) this might mean that everything happens exactly as it did except that Hinckley miraculously aimed better than he actually did. this might only require something as small as a neuron ring at a slightly dierent time than it actually did. This is about as small a miracle as we could hope for. Once this miracle occurs, events are assumed by Lewis to once again follow their lawful course with the result, perhaps, that Reagan is mortally wounded. In the case of (36), on the other hand, Reagan might be dead if the FBI agent miraculously failed to jump in front of Reagan, if Reagan miraculously moved in such a way that the bullet struck him dierently, or even if Reagan miraculously had a massive stroke at any time after the assassination attempt. Even if events followed their lawful course after any of these miracles, Hinckley's aim would not be improved. Lewis notes that the vagueness of conditionals requires that there may be various ways of determining the relative similarity of worlds, dierent ways being employed on dierent occasions. There is one way of resolving vagueness which Lewis considers to be standard, and it is this way which provides us with the explanation of (35) and (36) given above. This standard resolution of vagueness is expressed in the following guidelines for determining the relative similarity of worlds: (L1) It is of the rst importance to avoid big, complicated, varied, widespread violations of law. (L2) It is of the second importance to maximize the spatio- temporal region throughout which perfect match of particular fact prevails.
CONDITIONAL LOGIC
35
(L3) It is of the third importance to avoid even small, simple, localized violations of law. (L4) It is of little or no importance to secure approximate similarity of particular fact, even in matters that concern us greatly. Lewis would maintain that application of these guidelines together with his system-of-spheres semantics for subjunctive conditionals will have the desired result of making (35) at least plausible while making (36) clearly false. One major objection to Lewis's account is that once we allow miracles in order to produce a world which diverges from the actual world, there is nothing in Lewis's guidelines to prevent us from allowing another small miracle in order to get the worlds to converge once again. Since Lewis's guidelines place a higher priority on maximising the area of perfect match of particular facts over the avoidance of small, localized violations of law, we should prefer a small convergence miracle to a future which is radically dierent. Lewis's response to such a suggestion is that divergence miracles tend to be much smaller than convergence miracles or, what amounts to the same thing, that past events are overdetermined to a greater extent than are future events. If correct, then Lewis's guidelines would place greater importance on avoidance of a large convergence miracle than on maximising the area of perfect match of a particular fact. and careful consideration of examples indicates that Lewis's suggestion is at least plausible, although no conclusive argument has been provided. In [Nute, 1980b] examples of very simple worlds are given in which convergence miracles could be quite small and in which Lewis's guidelines would thus dictate that for some counterfactual antecedents the nearest antecedent worlds are those in which such small convergence miracles occur. In these examples, we get the (intuitively) wrong result when we apply Lewis's standard method for resolving the vagueness of conditionals. Lewis [1979a] warns that his guidelines might not work for very simple worlds, though, so the force of Nute's examples is uncertain. Lewis's guidelines may give an adequate explanation for our use of conditionals in the context of a complex world like the actual world, and since our intuitions are developed for such a world they may be unreliable when applied to very simple worlds. If we consider Lewis's proposal in the context of a probabilistic world, we discover that we no longer need employ the troublesome notion of a miracle. Instead of a miracle, we can accommodate a counterfactual antecedent in a probabilistic world by going back to some underdetermined state of aairs among the causal antecedents of the events or states of aairs which must be eliminated if the antecedent is to be true and change them accordingly. Since these states of aairs were underdetermined to begin with, they could have been otherwise without any violation of the probabilistic laws governing the
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DONALD NUTE AND CHARLES B. CROSS
universe. But if we do this, Lewis's emphasis on maximising the spatiotemporal area of perfect match of particular fact would require that we always change a more recent rather than an earlier causal antecedent when we have a choice. This consequence is very much like the Requirement of Temporal Priority in [Pollock, 1976], a principle which is superseded by the more complex account to be discussed below. Such a principle is unacceptable. Suppose, for example, that Fred left his coat unattended in a certain room yesterday. Today he returned to the room and found the coat had not been disturbed. Suppose that both yesterday and earlier today a number of people have been in the room who had an opportunity to take the coat. Then a principle like Lewis's L2 or Pollock's RTP will dictate that if the coat had been taken, it would have been taken today rather than yesterday. Other things being equal, the later the coat is taken the greater the area of perfect match of particular fact. But this is counterintuitive. (In fact, experience teaches that unguarded objects tend to disappear earlier rather than later.) While Lewis's theory is intended to explain why many backtracking conditionals are false, a consequence of the theory is that some very unattractive backtracking conditionals turn out to be true. In fact, this particular problem plagues Lewis's analysis whether the world is determined or probabilistic. As it is presented, Lewis's account does rely upon miracles. As a result, Lewis in eect treats all counterfactual conditionals as also being counterlegals. This is the feature of his account which most writers have found objectionable. Pollock, Blue, and others place a much higher priority on preservation of all law than on preservation of particular fact no matter how large the divergence of particular fact might be. Given such priorities, and given a deterministic world of the sort Lewis supposes, any change in what happens will result in a world which is dierent at every moment in the past and every moment in the future. If we adopt such a position, how can we hope to explain the asymmetry between normal and backtracking counterfactual conditionals? Probably the most sophisticated attempt to deal with these problems within the framework of a non-miraculous analysis of counterfactuals is that developed by John Pollock [1976; 1981]. Pollock has re ned his account between 1976 and 1981, but we will try to explain what we take to be his latest position on conditionals and temporal relations. Pollock says that a state of aairs P has historical antecedents if there is a set of true simple states of aairs such that all times of members of are earlier than the time of P and nominally implies P . nominally implies P just in case together with the set of universal generalisations of material implications corresponding to Pollock's true strong subjunctive generalisations entail P (or entail a sentence which is true just in case P obtains). Pollock next de nes a nomic pyramid which is supposed to be a set of states of aairs which contains every historical antecedent of each of its members. Then P
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undercuts another state of aairs Q if and only if for every set of true states of aairs such that is a nomic pyramid and Q 2 , nominally implies that P does not obtain. In revising his set S of true simple states of aairs to accommodate a particular counterfactual antecedent P , Pollock tells us that we are to minimize the deletion of members of S which are not undercut by P . (We hope the reader will forgive the vacillation here since Pollock talks about entailment and other logical relations holding between states of aairs where most authors prefer to speak of sentences or propositions.) Perhaps this procedure will give us the correct results for backtracking and non-backtracking conditionals as Pollock suggests it will if the world is deterministic, but problems arise if we allow the possibility that there may be indeterministic states of aairs which lack historical antecedents. Consider a modi ed version of an example taken from [Pollock, 1981]. Suppose that protons sometimes emit photons when subjected to a strong magnetic eld under a set of circumstances C , but suppose also that protons never emit photons under circumstances C if they are not also subjected to a strong magnetic eld. As a background condition, let us assume that circumstances C obtain. Now let be true just in case a certain proton is subjected to a strong magnetic eld at time t and let be true just in case the same proton emits a photon shortly after t. Suppose that both and are true. Assuming that no other states of aairs nomologically relevant to obtain, we would intuitively say that : > : is true, i.e. if the proton hadn't been subjected to the magnetic eld at t, then it would not have emitted a proton shortly after t. But Pollock cannot say this. Since has no historical antecedents in Pollock's sense, it cannot be undercut by :. Because Pollock does not recognize historical antecedents of states of aairs when the nomological connection involved is merely probable, he must say that : > is true. Pollock's earlier account, which included the Requirement of Temporal Priority, and Lewis's account with its principle L2, in either its original miraculous formulation or the probabilistic, non-miraculous version, both tend to make objectionable backtracking conditionals true when they are intended to explain why they should be false. Blue [1981] includes a feature in his analysis which produces the same result in much the same way. While Pollock's latest theory of counterfactuals avoids examples like that of the unattended coat, it nevertheless encounters new problems with backtracking conditionals in the context of a probabilistic universe. It makes certain backtracking counterfactuals false which our intuitions say are true while making others true which appear to be false. Yet these are the only positive proposals known to the authors at the time of this writing. Other work in the area such as [Nute, 1980b] and [Post, 1981] is essentially critical. An adequate explanation of the role the temporal order plays in the truth conditions for conditionals is still a very live issue.
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1.9 Tense There are relatively few papers among the large literature on conditionals which attempt an account of English sentences which involve both tense and conditional constructions. Two of the earliest are [Thomason and Gupta, 1981] and [Van Fraassen, 1981]. Both of these papers attempt the obvious, a fairly straightforward conjunction of tense and conditional operators within a single formal language. Basic items in the semantics for this language are a set of moments, an earlier-than relation on the set of moments which orders moments into tree-like structures, and an equivalence relation which holds between two moments when they are `co-present'. A branch on one of these trees plays the role of a possible world in the semantics. Such a branch is called a history, and sentences of the language are interpreted as having truth values at a moment-history pair, i.e. at a moment in a history. Note that a moment is not a clock time but rather a time-slice belonging to each history that passes through it. The tense operators in the language include two past-tense operators P and H , two future-tense operators F and G, and a `settledness' or historical necessity operator S . P is true at moment i in history h just in case is true at some moment j in h where j is earlier than i. H is true at some moment i in h if and only if is true at j in h for every moment j in h which is earlier than i. F is true at i in h if is true at a moment later than i in h, and G is true at i in h if is true at every moment later than i in h. S is true at i in h if and only if is true at i in every history h0 which contains i. For a further discussion of semantics for such tense operators, see Burgess [1984] (Chapter 2.2 of this Handbook). In both of these papers, that part of the semantics which is used to interpret conditionals is patterned after the semantics of Stalnaker. A conditional > is true at a moment i in a history h just in case is true at the pair hi0 ; h0 i at which is true which is closest or most similar to the pair hi; hi. Much of the discussion in the two papers is devoted to the eort to assure that certain theses which the authors favor are valid in their model theories. The measures needed to ensure some of the desired theses within the context of a Stalnakerian semantics are quite complicated, but the set of theses that represents the most important contribution of the account of [Thomason and Gupta, 1981], namely the doctrine of Past Predominance, turns out to be quite tractable model theoretically. According to Past Predominance, similarities and dierences with respect to the present and past have lexical priority over similarities and dierences with respect to the future in any evaluation of how close hi; hi is to hi0 ; h0 i, where i and i0 are co-present moments. This doctrine aects the interaction between the settledness operator S and the conditional. For example, Past Predominance implies the validity of the following thesis: (:S : ^ S ) ! ( > ):
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This thesis is clearly operative in the reasoning that leads to the two-box solution to Newcomb's Problem: `If it's not settled that I won't take both boxes but it is settled that there is a million dollars in the opaque box, then if I take both boxes there will (still) be a million dollars in the opaque box.'10 Cross [1990b] shows that since, concerning the selection of a closest momenthistory pair, Past Predominance places no constraints on what is true at past or future moments, Past Predominance can be formalized and axiomatized in terms of settledness and the conditional using ordinary possible worlds models in which relations of temporal priority between moments are not represented. The issue of how the conditional interacts with tense operators, such as P , H , F and G, is more problematic. The accounts presented by Thomason and Gupta and by Van Fraassen adopt the hypothesis that English sentences involving both tense and conditional constructions can be adequately represented in a formal language containing a conditional operator and the tense operators mentioned above. Nute [1983] argues that this is a mistake. Consider an example discussed in [Thomason and Gupta, 1981]: 37. If Max missed the train he would have taken the bus. According to Thomason and Gupta, this and other English sentences of similar grammatical form are of the logical form P ( > F ). Nute argues that this is not true. To see why, consider a second example. Suppose we have a computer that upon request will give us a `random' integer between 1 and 12. Suppose further that what the computer actually does is increment a certain location in memory by a certain amount every time it performs other operations of certain sorts. When asked to return a random number, it consults this memory location and uses the value stored there in its computation. Thus the `random' number one gets depends upon when one requests it. We just now left the keyboard to roll a pair of dice. If anyone cares, we rolled a 9. Consider the following conditional: 38. If we had used the computer instead of dice, we would have got a 5 instead of a 9. It is certainly true that there is a time in the past such that if we had used the computer at that time we would have got a 5, so a sentence corresponding to (38) of the form P ( > F ) is certainly true. Yet (38) itself is not true. Depending upon when we used the computer and what operations the computer had performed before we used it, we could have obtained any integer from 1 to 12. Perhaps we are simply using the wrong combination of operators. Instead of P ( > F ), perhaps sentences like (37) and (38) are of the form H ( > F ). A problem with this suggestion is that such conditionals do 10 See
[Gibbard and Harper, 1981].
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DONALD NUTE AND CHARLES B. CROSS
not normally concern every time prior to the time at which they are uttered but only certain times or periods of time which are determined by context. Suppose in a football game Walker carries the ball into the end zone for a touchdown. During the course of his run, he came very close to the sideline. Consider the conditional 39. If Walker had stepped on the sideline, he would not have scored. Can this sentence be of the form H ( > F )? Surely not, for Walker could have stepped on the sideline many times in the past, and probably did, yet he did score on this particular play. Perhaps we can patch things up further by introducing a new tense operator H which has truth conditions similar to H except that it only concerns times going a certain distance into the past, the distance to be determined by context. Once again, Nute argues, this will not work. Consider the conditional 40. If Fred had received an invitation, he would have gone to the party. This sentence might very well be accepted even though Fred would not have gone to the party had he received an invitation ve minutes before the party began. The period of time involved does not begin with the present moment and extend back to some past moment determined by context. Indeed if this were the case, for (40) to be true it would even have to be true that Fred would have gone to the party if he had received an invitation after the party ended. It would seem, then, that if a context-dependent operator is to be the solution to the problem Nute describes, then the contextually determined period of time involved in the truth conditions for English sentences of the sort we have been investigating must be some subset of past times, but one that need not be a continuous interval extending back from the present moment. This is the solution suggested by Thomason [1985].11 Nute [1991] argues for a dierent approach: the introduction of a new tensed conditional operator, i.e. an operator which involves in its truth conditions both dierences in time and dierences in world. Using a class selection function semantics for this task, we could let our selection function f pick out for a sentence , a moment or time i, and a history or world h a set f (; i; h) of pairs hi0 ; h0 i of times and histories at which is true and which are otherwise similar enough to hi; hi for our consideration. We would introduce into our formal language a new conditional operator, say iP F i, and sentences of the form iP F i would be true in an appropriate model at hi; hi if and only if for every pair hi0 ; h0 i 2 11 The following example may be linguistic evidence for this sort of context-dependence in tensed constructions not involving conditionals: a dean, worried about faculty absenteeism, asks a department chair, `Was Professor X always in his classroom last term?' the correct answer may be `Yes' even though Professor X was not in his classroom at times last term when his classes were not scheduled to meet.
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f (; i; h) such that there is a time j in h0 which is copresent with i and later than i0 ; is true at hj; h0 i. It appears that three more operators of this sort will be needed, together with appropriate truth conditions. These operators may be represented as iP P i, iF F i, and iF P i. These operators would be used to represent sentences like 41. If Fred had gone to the party, he would have had to have received an invitation. 42. If Fred were to receive an invitation, he would go to the party. 43. If Fred were to go to the party, he would have to have received an invitation. Notice that (41) and (43) are types of backtracking conditionals. Since such conditionals are rarely true, we may use the operators iP P i and iF P i infrequently. This may also account for the cumbersomeness of the English locution which we must use to clearly express what is intended by (41) and (43). A number of other interesting problems concerning tense and conditionals occur to us. One of these is the way in which the consequent may aect the times included in the pairs picked by a class selection function. Consider the sentences 44. If he had broken his leg, he would have missed the game. 45. If he had broken his leg, the mend would have shown on his X- ray. The times at which the leg might have been broken varies in the truth conditions for these two conditionals. This suggests that a semantics like Gabbay's which makes both antecedent and consequent arguments for the class selection function might after all be the preferred semantics. Another possibility is that despite its awkwardness we must introduce some sort of context-dependent tense operator like the operator H discussed earlier. When we represent (44) as H ( > F ), H has the whole of the conditional within its scope and can consider the consequent in determining which times are appropriate. A third possibility is that the consequent does not gure as an argument for the selection function but it does gure as part of the context which determines the selection function which is, in fact, used during a particular piece of discourse. This sort of approach utilizes the concept of conversational score discussed in Section 1.7 of this paper. One piece of evidence in favor of this approach is the fact that it would be unusual to assert both (44) and (45) in the same conversation. Whichever of these two sentences was asserted rst, the antecedent of the other would likely be modi ed in some appropriate way to indicate that a change in the times to be considered was required. Besides these interesting puzzles, we need
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also to explain the fact that we maintain the distinction between indicative and subjunctive conditionals involving present and past tense much more carefully than we do where the future tense is concerned. These topics are considered in more detail in [Nute, 1982 and 1991] and [Nute, 1991].
1.10 Other Conditionals Besides the subjunctive conditionals we have been considering, we also want an analysis for the might conditionals, the even-if conditionals, and the indicative conditionals mentioned in Section 1.1. It is time we took another look at these important classes of conditionals. Most authors who discuss the might and the even-if conditional constructions propose that their logical structure can be de ned by reference to subjunctive conditionals. Lewis [1973b] and Pollock [1976] suggest that English sentences having the form `If were the case, then might be the case' should be symbolized as :( > : ). Stalnaker [1981a] presents strong linguistic evidence against this suggestion, but the suggestion has achieved wide acceptance nonetheless. Pollock [1976] also oers a symbolisation of even-if conditionals. English sentences of the form ` even if ', he suggests, should be symbolized as ^ ( > ). The adequacy of this suggestion may depend upon our choice of conditional logic and particularly upon whether we accept the thesis CS. If we accept both CS and Pollock's proposal, then ` even if ' will be true whenever both and are true. An alternative analysis of even-if conditionals is developed in [Gardenfors, 1979]. Gardenfors's objection to Pollock's proposal seems to be that a person who knows that both and are true might still reject an assertion of the sentence ` even if '. Normally, says Gardenfors, one does not assert ` even if ' when one knows that is true; an assertion of ` even if ' presupposes that is true and is false. Even when the presupposition that is false truth turns out to be incorrect, Gardenfors argues that there is a presumption that the falsity of would not interfere with the truth of . Consequently, Gardenfors suggests that ` even if ' has the same truth conditions as ( > ) ^ (: > ). Another suggestion comes from Jonathan Bennett [1982]. Bennett gives a comprehensive account of even-if conditionals, tting them into the context of uses of `even' that don't involve `if', and uses of `if' that don't involve `even'. That is, Bennett rejects the treatment of `even if' as an idiom with no internal structure. The rst of three proposals we will consider concerning the analysis of indicative conditionals, which can be found in [Lewis, 1973b; Jackson, 1987] and elsewhere, is that indicative conditionals have the same truth conditions as do material conditionals, paradoxes of implication and problems with Transitivity, Contraposition, and Strengthening Antecedents notwithstanding. It is diÆcult and perhaps impossible to nd really persuasive
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counterexamples to Transitivity and Strengthening Antecedents using only indicative conditionals, but apparent counterexamples to Contraposition are easy to construct. Consider, for example, the following two sentences: 46. If it is after 3 o'clock, it is not much after 3 o'clock. 47. If it is much after 3 o'clock, it is not after 3 o'clock. It is easy to imagine situations in which (46) would be true or appropriate, but are there any situations in which (47) would be true or appropriate? Another problem with this analysis concerns denials of indicative conditionals. Stalnaker [1975] oers an interesting example: 48. If the butler didn't do it, then Fred did it. Being quite sure that Fred didn't do it, we would deny this conditional. At the same time, we may believe that the butler did it, and therefore when we hear someone say what we would express by 49. Either the butler did it or Fred did it. We might respond, \Yes, one of them did it, but it wasn't Fred". Yet (48) and (49) are equivalent if (48) has the same truth conditions as the corresponding material conditional. One possible response to these criticisms is that we must distinguish between the truth conditions for an indicative conditional and the assertion conditions for that conditional. It may be that a conditional is true even though certain conventions make it inappropriate to assert the conditional. This might lead us to say that (47) is true even though it would be inappropriate to assert it. We might also attempt to explain away the paradoxes of implication in this way, relying on the assumed convention that it is misleading and therefore inappropriate to assert a weaker sentence when we are in a position to assert a stronger sentence which entails . For example, it is inappropriate to assert _ when one knows that is true. Just so, the argument goes, it is inappropriate to assert ) when one is in a position to assert either : or . and in general we may reject other putative counterexamples to the proposal that indicative conditionals have the same truth conditions as material conditionals by saying that in these cases not all the assertion conditions are met for some conditional rather than admit that the truth conditions for the conditional are not met. This line of defence is suggested, for example, by [Grice, 1967; Lewis, 1973b; Lewis, 1976] and by [Clark, 1971]. A second proposal is that indicative conditionals are Stalnaker conditionals, i.e. that Stalnaker's world selection function semantics is the correct semantics for indicative conditionals and Stalnaker's conditional logic C2 is the proper logic for these conditionals. This suggestion is found in [Stalnaker, 1975] and in [Davis, 1979]. While both Stalnaker and Davis propose
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the same model theory for indicative and subjunctive conditionals, both also suggest that the properties of the world selection function appropriate to indicative conditionals are dierent from those of the world selection function appropriate to subjunctive conditionals. The dierence for Stalnaker has to do with the presuppositions involved in the utterance of the conditional. During the course of a conversation, the participants come to share certain presuppositions. In evaluating an indicative conditional ) , Stalnaker says that we look for the closest -world at which all of these presuppositions are true. In the case of a subjunctive conditional, on the other hand, we may look outside this `context set' for the closest -world. Of course the overall closest -world may not be a world at which all of the presuppositions are true since making true could tend to make one of the presuppositions false. This means that dierent worlds may be chosen by the selection function used to evaluate indicative conditionals and the selection function used to evaluate subjunctive conditionals. While accepting Stalnaker's model theory for both indicative and subjunctive conditionals, Davis oers a dierent distinction between the world selection function appropriate to indicative conditionals and that the appropriate to subjunctive conditionals. In fact, Davis claims that Stalnaker's analysis of subjunctive conditionals is actually the correct analysis of indicative conditionals. To evaluate an indicative conditional ! , Davis says we look at the -world which bears the greatest overall similarity to the actual world to see if it is a -world. For a subjunctive conditional > , we look at the - world which most resembles the actual world up until just before what Davis calls the time of reference of . Apparently, the time of reference of is the time at which events reported by occur, or states of aairs described by obtain, or etc. A third proposal, due to Adams [1966; 1975b; 1975a; 1981], holds that indicative conditionals lack truth conditions altogether. They do, however, have probabilities and these probabilities are just the corresponding standard conditional probabilities. Thus pr( ) ) = pr( ^ )=pr(), at least in those cases where pr() is non-zero. We must remember that Adams does not identify the probability of a conditional with the probability that that conditional is true since he rejects the very notion of truth values for conditionals. Adams proposes that an argument involving indicative conditionals is valid just in case its structure makes it possible to ensure that the probability of the conclusion exceeds any arbitrarily chosen value less than 1 by ensuring that the probabilities of each of the premises exceeds some appropriate value less than 1. In other words, we can push the probability of the conclusion arbitrarily high by pushing the probabilities of the premises suitably high. When an argument is valid in this sense, Adams says that the conclusion of the argument is `p-entailed' by its premises and the argument itself is `p-sound'. Since Adams rejects truth values for conditionals, conditionals can cer-
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tainly not occur as arguments for truth functions. Given his identi cation of the probability of a conditional with the corresponding standard conditional probability, this further entails that conditionals may not occur within the scope of the conditional operator. Adams attempts to justify this consequence of this theory by suggesting that we don't really understand sentences which involve the embedding of one conditional within another in any case. This claim, though is far from obvious. Such sentences as 50. If this glass will break if it is dropped on the carpet, then it will break if it is dropped on the bare wooden oor. seem absolutely ordinary and at least as comprehensible as most other indicative conditionals. the inability to handle such conditionals must count as a disadvantage of Adams's theory. In [Adams, 1977] it is shown that p-soundness is equivalent to soundness in Lewis's system-of-spheres semantics. This implies that the proper logic for indicative conditionals is the ` rst-degree fragment' of Lewis's VC. By the rst degree fragment of VC We mean the set of all those sentences in VC within which no conditional operator occurs within the scope of any other operator. Since the logic Adams proposes for indicative conditionals can be supported by a semantics which also allows us to interpret sentences involving iterated conditional operators, we will need very strong reasons to accept Adam's account with its restrictions rather than some possible worlds account like Lewis's. In fact it may be possible to reconcile Lewis's view that the truth conditions for indicative conditionals are the same as those for the corresponding material conditionals with Adams work on the probabilities of conditionals and p-entailment. Lewis [1973b] suggests that the truth conditions for ) are given by ! while the assertion conditions for ) are given by the corresponding standard conditional probability. Jackson [1987] also entertains such a possibility. If we accept this, then we might accept Adams's theory as a basis for an adequate account of the logic of assertion conditions for indicative conditionals. Since we would be assuming that conditionals have truth values as well as probabilities, we could also overcome the restrictions of Adams's theory and assign probabilities to conditionals which have other conditionals embedded in them. One problem with this approach, though, is that it would seem to require that we identify the probability of a conditional with the probability that the conditional is true. When we do this and also take the probability of a conditional to be the corresponding standard conditional probability, serious problems arise as is shown in [Lewis, 1976] and in [Stalnaker, 1976]. These diÆculties will be discussed brie y in Section 3.
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2 EPISTEMIC CONDITIONALS The idea that there is an important connection between conditionals and belief change seems to have been inspired by this suggestion of Frank Ramsey's: If two people are arguing \if p will q?" and are both in doubt as to p, they are adding p hypothetically to their stock of knowledge and arguing on that basis about q.12 The issue of how, precisely, to formalize Ramsey's suggestion and extend it from the case where p is in doubt to the general case has received a great deal of attention|too much attention to permit an exhaustive survey here. We will focus here on the Gardenfors triviality result for the Ramsey test (see [Gardenfors, 1986]) and related results, and the implications of these results for the project of formalizing the Ramsey test for conditionals. Despite the narrowness of this topic our discussion will not mention all worthy contributions to the subject. Sections 2.1 and 2.2 provide a general framework for formalizing belief change and the Ramsey test. Section 2.3 makes connections between this framework and the literature on belief change and the Ramsey test. Section 2.4 presents the Ramsey test itself, and Section 2.5 presents versions of several triviality results found in the literature, including a version of Gardenfors' 1986 result that subsumes several of the de nitions of triviality found in the literature. Section 2.6 examines how triviality can be avoided, and section 2.7 examines systems of conditional logic associated with the Ramsey test. We will provide proofs for some of the results stated below and in other cases refer the reader to the literature.
2.1 Languages By a Boolean language we will mean any logical language containing at least the propositional constant `?', the binary operator `^', and the unary operator `:'. We will assume that `>' is de ned as `:?' and that any other needed Boolean operators are de ned. We do not assume anything at this stage about how the operators and propositional constant of a Boolean language are interpreted, but it will turn out that in most cases `^', `:', and `?' will receive classical truth-functional interpretations. We will use the symbol ``' as a variable ranging over logical inference relations. Eective immediately we will cease using quotes when mentioning formulas and logical symbols. We de ne a language (whether Boolean or nonBoolean) to be of type L0 i it contains the propositional constants > and ? but does not contain the 12 [Ramsey,
1990], p. 155.
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binary conditional operator >. We next de ne two language-types for doing conditional logic. A language, whether Boolean or nonBoolean, is of type L1 i it contains > and ? and the only >-conditionals allowed as formulas are rst-degree or \ at" conditionals, i.e. conditionals > where ; are conditional-free. We de ne a language, whether Boolean or nonBoolean, to be of type L2 (a \full" conditional language) i it contains > and ? and allows arbitrary nesting of conditionals in formulas.
2.2 A general framework for belief change We will describe belief change using a framework that is related to the AGM (Alchourron, Gardenfors, Makinson) framework for belief revision.13 Our framework extends that of AGM and is adapted (with further enrichment) from the notion of an enriched belief revision model introduced in [Cross, 1990a]. For a given language L containing >; ? as formulas, let wffL be the set of all formulas of L and let KL be P (wffL ) f;g (where P (wffL ) is the powerset of wffL). For a given inference relation ` and set of formulas of L, de ne Cn` ( ) (the `-consequence set for ) to be f : ` g, and let TL;` = f : wffL and Cn` ( ) = g be the set of all theories in L with respect to `. A set is `-consistent i 6` ?. We next de ne the notion of a belief change model : (DefBCM) A belief change model on a language L containing >; ? as formulas is an ordered septuple
hK; I ; `; K? ; ; ; si whose components are as follows:
13 See
1. K KL and ` is a subset of P (wffL ) wffL ; 2. I and K? are sets of formulas meeting the following requirements: (a) K? 2 K; (b) >; ? 2 I ; (c) K? is the set of all formulas of L or a fragment of L; (d) I is the set of all formulas of L or a fragment of L, and I K?. (e) K K? for all K 2 K. 3. and are binary functions mapping each K 2 K and each 2 I to sets K and K , respectively, where K K? and K K?;
[Alchourron et al., 1985] and [Gardenfors, 1988].
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DONALD NUTE AND CHARLES B. CROSS
4. s is a function taking values in P (wffL ), where K dom(s) P (wffL ). A classical belief change model is a belief change model de ned on a Boolean language whose logical consequence relation ` includes all classical truthfunctional entailments and respects the deduction theorem for the material conditional. A deductively closed belief change model is a belief change model for which K = Cn` (K ) \ K? and K = Cn` (K ) \ K? and K = Cn` (K ) \ K? for all K 2 K and all 2 I . Note that in a deductively closed belief change model on language L, belief sets are theories in the fragment of L represented by K? and not necessarily theories in L itself. Informally, the items in a belief change model can be described as follows. K represents the set of all possible belief states recognized by the model; often K will be a subset of TL;` but not always. I represents the set of all formulas eligible to serve as inputs for contraction and revision. ` is an inference relation de ned on L and will in most cases be an extension of truth-functional propositional logic. K? contains all of the formulas of that fragment of L from which the belief sets in K are constructed and represents the absurd belief state; thus every belief set in K is a subset of K? . For each K 2 K and each 2 I , K represents the result of contracting K to remove (if possible), whereas K represents the result of revising K to include as a new belief. Revision is normally assumed to involve not only adding the given formula to the given belief set but also resolving any inconsistencies thereby created. For the sake of generality, we have not stipulated that K ; K 2 K, though this will usually be the case. Finally, s is the support function for the model, which determines for each belief state K (and perhaps for other sets, as well) the set of formulas of L supported by K . For belief sets K in belief change models for which the Ramsey test holds, s(K ) will contain Ramsey test conditionals even if K does not.
2.3 Comparisons With an eye toward our presentation of the basic triviality result for the Ramsey test we will brie y review diering positions about the elements making up a belief change model. The list of authors we mention here is not exhaustive but constitutes a representative sample of the diversity of positions taken with respect to belief change models and their elements in discussions of the Ramsey test. Belief states: the language of the model and the set K
Segerberg places no restrictions on the language in his discussion of the triviality result in [Segerberg, 1989]. Gardenfors ([1988] and elsewhere),
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49
Rott [1989], and Cross [1990a] all adopt a type-L2 language for their Ramsey test belief change models, whereas Makinson [1990], Morreau [1992], Hansson ([1992], section III), Arlo-Costa [1995], and Levi ([1996] and elsewhere) restrict themselves to a type-L1 language. Hansson, Arlo-Costa, and Levi allow only the type-L0 formulas of a type-L1 language to belong to the sets that individuate belief states. Makinson and Morreau, like Gardenfors, Rott, Segerberg, and Cross, do not restrict the membership of belief-stateindividuating sets to type-L0 formulas. Most authors on the Ramsey test follow Gardenfors in representing the set of all possible belief states as a set of theories. One exception is Hansson [1992], who takes each possible belief state to be represented by a pair consisting of a set of formulas and a revision operator that de nes the dynamic properties of the belief state. For Hansson, the set of formulas in question is a belief base , a set of conditional-free formulas that need not be deductively closed. The belief base of a belief state is a (not necessarily nite) axiom set for the belief state, the idea being to allow dierent belief states to be associated with the same deductively closed theory. A belief state in Hansson's model can still be individuated by means of its belief base, however, because the revision operator of a belief state is a function of that belief state's belief base. Morreau [1992] also gives a two-component analysis of belief states, but in Morreau's analysis the two components are a set of \worlds" (truth-value assignments to atomic formulas) and a selection function (of the Stalnaker-Lewis variety) that determines which conditionals are believed in the belief state. A third exception is Rott [1991], who identi es belief states with epistemic entrenchment relations and notes that a nonabsurd belief set can be recovered from an epistemic entrenchment relation that supports at least one strict entrenchment: the belief set will be the set of all formulas strictly more entrenched than ?. Among those authors who take belief states to be deductively closed theories, most follow Gardenfors in assuming that not every theory corresponds to a possible belief state. On this issue Segerberg and Makinson are exceptions. In their respective extensions of Gardenfors' basic triviality result Segerberg and Makinson assume that revision is de ned on all theories in a given language rather than on a nonempty subset of the set of all theories for that language.14 We note above that Hansson, Arlo-Costa, and Levi allow only conditionalfree formulas into the sets that individuate belief states.15 Why exclude conditionals from these sets? In Levi's view, the formulas eligible for membership in the theories that individuate belief states are precisely those statements about which agents can be concerned to avoid error. Levi argues against including conditionals in the theories that individuate belief 14 See [Segerberg, 1989] and [Makinson, 1990]. 15 See, for example, [Hansson, 1992], [Arl o-Costa,
and Levi, 1996].
1995], [Levi, 1996], and [Arlo-Costa
50
DONALD NUTE AND CHARLES B. CROSS
states because in his view conditionals do not have truth conditions or truth values and so are not sentences about which agents can be concerned to avoid error.16 On Levi's view, a conditional > in a type-L1 language is acceptable relative to a belief set K in a type-L0 language i : is not epistemically possible relative to the result of revising K to include , and the negated conditional :( > ) is acceptable relative to K i : is epistemically possible relative to the revision of K to include . The part of this view governing negated conditionals, the negative Ramsey test, will be discussed later.17 An important consequence of Levi's view is the thesis that conditionals are \parasitic" on conditional-free statements in the following sense: the set of conditionals supported by a given belief state is determined by the conditional-free formulas accepted in that belief state or a subset thereof. Hansson [1992] shows, however, that it is possible to motivate a parasitic account of conditionals without taking a position on whether conditionals have truth conditions or truth values. Gardenfors [1988] criticizes Levi's view of conditionals on the grounds that it fails to account for iterated conditionals, a species of conditional about which Levi has expressed skepticism, but Levi [1996] and Hansson [1992] show that iterated conditionals can be accounted for (if necessary) even if conditionals do not have truth conditions or truth values. Levi [1996] points out, however, that axiom schema (MP) fails to be valid in the sense he favors if iterated conditionals are allowed.18 In this connection Levi exploits examples like the following, which was described by McGee [1985] as a counterexample to modus ponens : Opinion polls taken just before the 1980 election showed the Republican Ronald Reagan decisively ahead of the Democrat Jimmy Carter, with the other Republican in the race, John Anderson, a distant third. Those apprised of the poll results believed, with good reason: If a Republican wins the election, then if it's not Reagan who wins it will be Anderson. A Republican will win the election. Yet they will not have good reason to believe If it's not Reagan who wins, it will be Anderson.19 16 See [Levi, 1988] and [Levi, 1996], for example. As Arl o-Costa and Levi [1996] point out, Ramsey agreed that conditionals lack truth conditions and truth values: this is clear from the context of the quote from Ramsey with which we began Section 2. 17 For the most recent account of Levi's views on this topic, see [Levi, 1996]. 18 See [Levi, 1996], pp. 105-112. 19 [McGee, 1985], p. 462.
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Arlo-Costa [1998] embraces iterated conditionals and uses McGee's example to argue, via the Ramsey test, against the following principle of invariance for iterated supposition . (K INV) If 2 K 6= K? , then (K ) = K .
Supposition, i.e. hypothetical revision of belief \for the sake of argument," is the notion of revision that Arlo-Costa [1998] and Levi [1996] both associate with Ramsey test conditionals. Since (K INV) holds in any deductively closed classical belief change model that satis es (K 3) and (K 4), Arlo-Costa takes McGee's example as evidence against (K 4) as a principle governing supposition. Contraction and revision inputs: the set I
Gardenfors does not exclude conditionals from the class of formulas eligible to be inputs for belief change in the models he formulates, but Morreau, Arlo-Costa, and Levi do. In Levi's case this restriction clearly follows from his view that conditionals have neither truth conditions nor truth values, and Arlo-Costa appears to agree with this view. Morreau's exclusion of conditionals as revision inputs appears to be an artifact of the nontriviality theorem he proves for the Ramsey test ([Morreau, 1992], THEOREM 14, p. 48) rather than indicative of a philosophical position about the status of conditionals. Logical consequence and support:
` and s
Most authors on the Ramsey test follow Gardenfors in assuming a compact background logic ` that includes all truth functional propositional entailments while respecting the deduction theorem for the material conditional, but there has been research on the Ramsey test in frameworks where the background logic is nonclassical or not necessarily classical. For example, Segerberg's triviality result in [Segerberg, 1989] assumes only the minimal constraints of Re exiveness, Transitivity, and Monotony for `,20 and in [Gardenfors, 1987] Gardenfors credits Peter Lavers with having established in an unpublished note a triviality result for the Ramsey test in which ` is de ned to be minimal logic 21 instead of an extension of classical truth-functional logic. Also, Cross and Thomason [1987; 1992] investigate a four-valued system of conditional logic that is motivated by an application of the Ramsey test in the context of the nonmonotonic logic of multiple inheritance with exceptions in semantic networks. 20 See the de nition of a Segerberg belief change model near 21 Minimal logic has modus ponens as its only inference rule
following schemata as axioms:
the end of section 2.3. and every instance of the
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DONALD NUTE AND CHARLES B. CROSS
Levi [1988] introduces the function RL, which maps a conditional-free belief set to a conditional-laden belief set via the Positive and Negative Ramsey tests. Cross [1990a] formulates a version of the triviality result proved in [Gardenfors, 1986] in a framework where an extension ` of classical logic is coupled with a not-necessarily-monotonic consequence operation cl . Makinson [1990] does the same, calling his not-necessarily-monotonic consequence operation C . Hansson [1992] makes use of a function s which maps each belief state to the set of all formulas the belief state \supports." Support functions are also adopted by Arlo-Costa [1995] and by Arlo-Costa and Levi [1996]. Our view is that Levi's RL, Cross' cl , Makinson's C , and Hansson's s should be regarded as variations on the same theoretical construct, and we will follow Hansson in calling this construct a support function and in using s to represent it. More on this in Section 2.6 below. The following postulates are examples of requirements that might be imposed on s. Assume a belief change model on a language L, and assume that ranges over dom(s), which always includes K as a subset: (Identity over K) s(K ) = K for all K 2 K. (Monotonicity over K) For all H; K 2 K, if H K then s(H ) s(K ). (Re exivity)
s(
).
(Closure) Cn` [s( )] = s( ). (Consistency) If is `-consistent then s( ) is `-consistent. (Superclassicality) Cn` ( ) s( ). (Transitivity) s( ) = s[s( )].
(Reasoning by Cases) s( [fg) \ s( [f:g) s( ) for all ; such that [ fg 2 dom(s) and [ f:g 2 dom(s).
(Conservativeness) L has type-L0 fragment L0 and for all 2 wffL0 , 2 s( ) i 2 Cn` ( ). None of Gardenfors, Morreau, or Segerberg uses the notion of a support function: they assume, in eect, that s(K ) = K for all K 2 K. 1: ( ^ ) ! : 2: ( ^ ) ! : 3: ! ( _ ): 4: ! ( _ ): 5: ( ! ) ! [( ! ) ! (( _ ) ! )]: 6: ( ! ) ! [( ! ) ! (( ! ( ^ )]: 7: [ ! ( ! )] ! [( ! ) ! ( ! )]: 8: ! ( ! ): The formula : is de ned to be ! ?. This axiomatization is found in [Segerberg, 1968].
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Contraction and revision: postulates for belief change
53
Since and are to represent functions legitimately describable as contraction and revision, respectively, it is appropriate to consider additional conditions on these functions. Which additional conditions should be imposed is a matter of dispute, and some of the additional postulates that will be under consideration are listed below. In the case of postulates (K+ 1), (K 1), (K 2), (K 3), (K 4), (K 5), (K 6), (K 7), (K 8), (K 1), (K 2), (K 3), (K 4), (K 5), (K 6), (K 7), (K 8), (K L), (K M), and (K P) we follow the labeling used in [Gardenfors, 1988]. Please note that we have not adopted any of the postulates given below in the de nition of belief change model . In each postulate, the variable K is understood to range over K; also ; are understood to range over I . We begin with a de nition of a third important belief change operation: expansion. De nition of and postulate for expansion
2 K. (K is a belief set.) K K . If 62 K , then K = K . If 6` , then 62 K . If 6` and K = 6 K? , then 62 K . If 2 K , then K (K )+ . If ` $ , then K = K . K \ K K^ . If 62 K^ , then K^ K . K
Postulates for revision (K 1) K 2 K. (K is a belief set.) (K 2) 2 K .
54
(K 3) (K 4)
DONALD NUTE AND CHARLES B. CROSS
K K+ .
If : 62 K , then K+ K .
(K 4s) If : 62 K , then K+ = K .
(K 4ss) If K+ 6= K? , then K+ = K . (K 4w) If 2 K 6= K? , then K K. (K 5) K = K? i ` :.
(K 5w) If K = K? , then ` :. (K 5ws) If K = K? , then Cn` (fg) = K?. (K C) (K 6)
If K 6= K? and K = K? , then ` :. If ` $ , then K = K . (K 6s) If 2 K and 2 K , then K = K . (K 7) K^ (K )+ . (K 7 0 ) K \ K K_ . (K 8) If : 62 K , then (K )+ K^ . (K L) If :( > : ) 2 K , then (K )+ K^ . (K M) If s(K ) s(K 0 ), then K K0 . (K IM) If K 6= K? 6= K 0 and s(K ) s(K 0 ), then K 0 K . (K T) (K P)
If K 6= K? , then K> = K . If : 62 K , then K K.
(K PI ) If : 62 K then K \ I K \ I . (LI) K = (K: )+ . A few other postulates will be identi ed as needed. Our treatment of contraction and revision is not general enough to include every treatment of contraction and revision as a special case. For example, in the formalization of belief revision in [Morreau, 1992], the revision operation is nondeterministic, i.e. its value for a given belief set K and proposition is a set of belief sets rather than a belief set. We will not attempt to formalize nondeterministic contraction or revision. Also, for
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55
Levi, contraction and revision are to be evaluated by means of a measure of informational value , which we do not explicitly formalize. The contraction operation, which we include in every belief change model, is not often used in the presentation of triviality results for the Ramsey test. Exceptions to this pattern include Cross [1990a] and Makinson [1990], who involve contraction explicitly in their respective formulations of triviality results for the Ramsey test. A catalog of belief change models
We conclude our discussion of comparisons by de ning several categories of belief change model that illustrate how the framework de ned above can be made to re ect the diering assumptions of a subset of authors who have written on belief revision and the Ramsey test. In associating a name with a class of belief change models we do not claim that the person named de ned this class of models; rather, we claim that the belief change models associated with this name are the appropriate counterpart in our framework of models that the named person did de ne in the context of work on the Ramsey test. Note that postulates on contraction and revision are not part of these de nitions. 1. By a Gardenfors belief change model (see, for example, [Gardenfors, 1986], [Gardenfors, 1987], and [Gardenfors, 1988]) we will mean a deductively closed classical belief change model hK; I ; `; K? ; ; ; si de ned on a type-L2 language L where I = wffL = K?, and dom(s) = K, and s satis es Identity over K. 2. By a Segerberg belief change model (see [Segerberg, 1989]) we will mean a belief change model hK; I ; `; K? ; ; ; si, de ned on any language, such that the following hold: K = TL;` ; I = wffL = K? ; dom(s) = K; s satis es identity over K; and Cn` meets the following requirements, for all ; wffL : (Re exivity for `) Cn` ( ). (Monotonicity for `) If , then Cn` () Cn` ( ). (Transitivity for `) Cn` ( ) = Cn` [Cn` ( )]. 3. By a Makinson belief change model (see [Makinson, 1990]) we will mean a belief change model hK; I ; `; K?; ; ; si de ned on a typeL1 language L and satisfying the following: K = f : wffL and s( ) = g; I = wffL = K? , ` is classical propositional consequence; dom(s) = P (wffL ); and s satis es Superclassicality, Transitivity, and Reasoning by Cases.22 22 Note that in a Makinson belief change model s satis es both Re exivity and Closure. Closure holds since Superclassicality and Transitivity for s imply that for each wffL , we have s( ) Cn` [s( )] s[s( )] = s( ).
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DONALD NUTE AND CHARLES B. CROSS
4. By a Morreau belief change model (see [Morreau, 1992]) we will mean a deductively closed classical belief change model hK; I ; `; K?; ; ; si de ned on a type-L1 language L whose type-L0 fragment is L0 and where the following hold: I = wffL0 ; K? = wffL; dom(s) = K; and s satis es Identity over K. 5. By a Hansson belief change model (see [Hansson, 1992], section 3) we will mean a classical belief change model hK; I ; `; K?; ; ; si de ned on a type-L1 language L whose type-L0 fragment is L0 and where the following hold: K P (wffL0 ); I = wffL0 = K? ; dom(s) = K; and s satis es Re exivity, Conservativeness, and Closure. 6. By an Arlo-Costa/Levi belief change model (see [Arlo-Costa, 1990], [Arlo-Costa, 1995], [Arlo-Costa and Levi, 1996], and [Levi, 1996]) we will mean a deductively closed classical belief change model hK; I ; ` ; K? ; ; ; si de ned on a type-L1 language L whose type-L0 fragment is L0 and where the following hold: I = wffL0 = K?; dom(s) = K; and s satis es Re exivity, Conservativeness, and Closure. As we have already noted, our belief change models do not capture every feature of every belief revision model appearing in the literature on the Ramsey test, and the models we associate with the names of authors in some cases omit some of the structure that these authors include in their own respective accounts of what constitutes a belief revision model. On the other hand, we have stipulated more detail for the models we associate with certain authors than do the authors themselves. For example, none of Gardenfors, Morreau, or Segerberg uses the notion of a support function s in the sources cited above, and neither Gardenfors, nor Makinson, nor Segerberg restricts the applicability of contraction and revision to a subset I of the set of formulas of the language on which the model is de ned. Finally, as was pointed out earlier, the contraction operation, which we include in every belief change model, is not often discussed in connection with the Ramsey test. In general, the stipulation of extra detail will serve to highlight tacit assumptions and make comparisons easier.
2.4 The Ramsey test for conditionals Ramsey's original suggestion can be put as follows: if an agent's beliefs entail neither nor :, then the agent's beliefs support > i his or her initial beliefs together with entail , i.e. (RTR) For all K 2 K and all 2 I such that ; : 62 K and all > 2 s(K ) i 2 K+ .
2 K?,
This suggestion covers only the case in which the epistemic status of is undetermined. What about the case in which the agent's initial beliefs
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57
entail and the case in which the agent's initial beliefs entail :? Stalnaker [1968] suggests the following rule for evaluating a conditional in the general case: First, add the antecedent (hypothetically) to your stock of beliefs; second, make whatever adjustments are required to maintain consistency (without modifying the hypothetical belief in the antecedent); nally, consider whether or not the consequent is true.23 Stalnaker's proposal handles the general case by substituting the operation of revision for that of expansion in Ramsey's original proposal. In our framework Stalnaker's suggestion amounts to the following: (RT)
For all K 2 K .
2 K and all 2 I and all 2 K?, > 2 s(K ) i
Revision postulates (K 3) and (K 4) jointly entail (K 4s) If : 62 K then K + = K .
Hence, if (K 3), (K 4) are assumed, then (RT) agrees with (RTR) in the case where neither nor : belongs to K . That is, if (K 3) and (K 4) hold, then (RT) can be considered an extension of Ramsey's original proposal. In [Gardenfors, 1978] and in later writings Gardenfors adopts Stalnaker's version of the Ramsey test for type-L2 languages and assumes, in addition, the following: every formula of a type-L2 language L is an eligible input for revision and an eligible member of a belief set, i.e. I = wffL = K?, and a conditional, like any other formula, is accepted with respect to (supported by) a belief set K i it belongs to K , i.e. s(K ) = K for all K 2 K. We have already noted that Levi, in contrast to Gardenfors, excludes conditionals as revision inputs and as members of belief sets. Levi's view is that the conditional > in a type-L1 language expresses the attitude of an agent for whom : is not epistemically possible relative to K, and the negated conditional :( > ) expresses the attitude of an agent for whom : is epistemically possible relative to K. Assuming a type-L1 language L with type-L0 fragment L0 , and assuming that K TL0 ;` and I = wffL0 = K?, Levi's view amounts in our framework to the conjunction of the following: (PRTL) For all 2 I and all 2 K? and all K > 2 s(K ) i 2 K. (NRTL) For all 2 I and all :( > ) 2 s(K ) i
2 K such that K 6= K?,
2 K? and all K 2 K such that K 6= K? , 62 K.
23 [Stalnaker, 1968], p. 44. (The page reference is to [Harper et al., 1981], where [Stalnaker, 1968] is reprinted.)
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DONALD NUTE AND CHARLES B. CROSS
Note that in both (PRTL) and (NRTL), unlike in (RT), K is restricted to `-consistent members of K. Note also that the adoption of (RT) (or of (PRTL) without (NRTL)) places no constraints on how negated conditionals are related to belief change. Other versions of the Ramsey test appearing in the literature include the following, due to Hans Rott, who, like Gardenfors, assumes a language L of type L2 and no restrictions on which formulas can appear as members of belief sets or as revision inputs (i.e. I = wffL = K?): (R1) (R2) (R3)
For all K 2 K and all 2 K and 62 K . For all K 2 K and all 2 K and 62 K: . For all K 2 K and all 2 (K ) .
2I
and all
2 K? , >
2K
i
2I
and all
2 K? , >
2K
i
2I
and all
2 K? , >
2K
i
Here we follow the labeling in [Gardenfors, 1987]. The interest of (R1)-(R3) stems in part from the fact that whereas (RT) can be used with (K 3) and (K 4) to derive the following thesis (U), none of (R1)-(R3) can be so used: (U)
If 2 K and
2 K , then > 2 K .
Thesis (U) is related to the strong centering axiom CS of VC, and Rott [1986] suggests that (U) should be rejected. Since none of (R1)-(R3) entails (K M), one of the assumptions of Gardenfors' 1986 triviality result for the Ramsey test, (R1)-(R3) might seem worth investigating as alternatives to (RT), but Gardenfors [1987] shows that (R1)-(R3) do not avoid the problem faced by (RT). Consider the Weak Ramsey Test: (WRT) For all K 2 K and all 2 I and all > 2 K i 2 K.
2 K? such that _ 62 K ,
Each of (R1)-(R3) entails (WRT), and Gardenfors [1987] proves a triviality result that holds for any version of the Ramsey test which entails (WRT), including (R1)-(R3) and (RT).24
2.5 Triviality results for the Ramsey test The basic result
Many versions of the basic triviality result for the Ramsey test have appeared in the literature, all of them variations on the result proved by Gardenfors [1986]. All proofs of the basic triviality result we know of exploit the same maneuver, however, one which Hansson [1992] makes explicit: 24 See
also [Gardenfors, 1988], Chapter 7, Corollary 7.15
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59
in terms of our framework, the nding of forking support sets within a belief change model. (DefFORK) A belief change model hK; I ; `; K?; ; ; si will be said to contain forking support sets i there exist H; J; K 2 K, such that H = Cn` (H ) \ K?, and J = Cn` (J ) \ K?, and K = Cn` (K ) \ K? 6= K? , and H \ I 6 J , and J \ I 6 H , and s(H ) s(K ), and s(J ) s(K ). For Gardenfors and Segerberg belief change models this condition can be stated in the form in which Hansson originally formulated it: PROPOSITION 1. A Gardenfors or Segerberg belief change model hK; I ; `; K?; ; ; si contains forking support sets i there exist H; J; K 2 K, where H; J K 6= K? , H 6 J , and J 6 H . This proposition follows from the fact that in Gardenfors and Segerberg belief change models (i) s(K ) = K = Cn` (K ) for all K 2 K, and (ii) I and K? both exhaust the formulas of the language of the model. Next we present the main lemmas for the basic triviality result: LEMMA 2. If (RT) holds in a belief change model, then so does (K M).
Proof. Trivial; left to reader.
Postulate (K M) is a postulate of monotonicity for belief revision. We discuss Gardenfors' argument against (K M) in Section 2.6 below. LEMMA 3. No classical belief change model containing forking support sets satis es (K 2), (K C), (K P), and (K M).
Proof. Assume for reductio that hK; I ; `; K?; ; ; si is a classical belief change model that contains forking support sets and satis es (K 2), (K C), (K P), and (K M). For clarity, we follow the example of [Rott, 1989] in numbering the steps in the reductio argument. (1) H = Cn` (H )\K? , J = Cn` (J )\K? , K = Cn` (K ) \ K? 6= K? , H \I 6 J , J \ I 6 H , and s(H ); s(J ) s(K ), for some H; J; K 2 K (2) 2 (H \ I ) J , for some (3) 2 (J \ I ) H , for some (4) :( ^ ) 2 I (5) ::( ^ ) 62 H (6) H H: (^ )
(DefFORK)
(1) (1) (2), (3), (DefBCM) (3), classicality of `, fact that H = Cn` (H ) \ K? (4), (5), (K P)
(2), (6) (2), classicality of `, fact that J = Cn` (J ) \ K? (4), (8), (K P) (3), (9) (1), (4), (K M) (7), (10), (11) (4), (K 2) (12), (13), classicality of ` classicality of `, fact that K = Cn` (K ) \ K? 6= K? (14), (15), (K C) (5), classicality of `, fact that H = Cn` (H ) \ K?
Since (17) contradicts (16), this completes the proof.
Lemmas 2 and 3 suÆce to prove the following: THEOREM 4. No classical belief change model de ned on a language of type L1 or type L2 and containing forking support sets satis es (K 2), (K C), (K P), and (RT). Note that we have not assumed that K is a set of theories either in the language of the model or in the fragment thereof represented by K?. We have not even assumed (K 1): that the sets produced by revision always belong to K. It is however required that the belief sets H , J , and K used in the proof be theories in the fragment of the language represented by K?. Do we have a triviality result? Not yet: we do not yet have a criterion of triviality. The following criteria have appeared in the literature: 1. A belief change model is Gardenfors nontrivial i there is a K 0 2 K and ; ; 2 I such that :; : ; : 62 Cn` (K 0 ) and ` :( ^ ) and ` :( ^ ) and ` :( ^ ). 2. A belief change model is Rott nontrivial i there is a K 0 2 K and ; 2 I such that 6` and 6` and _ ; :_ ; _: ; : _: 62 Cn` (K 0 ). 3. A belief change model is Segerberg nontrivial i there exist ; ; 2 I such that 6` and 6` and Cn` f; ; g = K? and Cn` (fg), Cn` (f g), Cn` (f; g) 2 K.
Recall that a support function s is monotone over K i s(H ) s(K ) for all H; K 2 K such that H K . A support function can be monotone
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61
over K even if it is a nonmonotonic consequence operation provided that K does not exhaust dom(s). For example, s is monotone over K (but not necessarily over dom(s)) in all Makinson belief change models, since in a Makinson belief change model s(K ) = K for all K 2 K. Recall that the operation of expansion is de ned by (Def+); it turns out that if (K+ 1) and the monotonicity of s over K are assumed, then nontriviality by any of the above criteria will imply the existence of forking support sets: LEMMA 5. A classical belief change model de ned on a language of type L1 or L2 contains forking support sets if it satis es (K+ 1) and its support function is monotone over K and it is Gardenfors nontrivial.25
Proof. Suppose that the model is Gardenfors nontrivial; we will show that it contains forking support sets. Let K 0 , , , and be as in the de nition of Gardenfors nontriviality; also, let H = K0+_ ; let J = K0+_ ; and let K = K0+ . Then by (Def+) and the classicality of `, H = Cn` (H ) \ K? , J = Cn` (J )\K? , and K = Cn` (K )\K? 6= K?. (Def+) and the classicality of ` also imply that H; J K , hence by the monotonicity of s we have that s(H ); s(J ) s(K ). H \I 6 J holds because _ 2 (H \I ) J ; J \I 6 H holds because _ 2 (J \ I ) H . LEMMA 6. A classical belief change model de ned on a language of type L1 or L2 contains forking support sets if it satis es (K+ 1) and its support function is monotone over K and it is Rott nontrivial.26
Proof. Like the proof of Lemma 5, but let H = K0+_ ; let J = K0+_: , let K = K:0+, where K 0 , , and are as in the de nition of Rott nontriviality. H \ I 6 J holds because _ 2 (H \ I ) J ; J \ I 6 H holds because _ : 2 (J \ I ) H . LEMMA 7. A classical belief change model de ned on a language of type L1 or L2 contains forking support sets if it satis es (K+ 1) and its support function is monotone over K and it is Segerberg nontrivial.
Proof. Like the proof of Lemma 5, but let H = Cn` (fg), J = Cn` (f g), K = Cn` (f; g), where , , and are as in the de nition of Segerberg nontriviality. Theorem 4 and Lemmas 5, 6, and 7 immediately imply Theorem 8, the basic triviality result for the Ramsey test: THEOREM 8. No classical belief change model de ned on a language of type L1 or L2 that satis es (K+ 1), (K 2), (KC), (K P), and (RT) and 25 See 26 See
[Gardenfors, 1986]. [Rott, 1989].
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whose support function is monotonic over K is Gardenfors nontrivial or Rott nontrivial or Segerberg nontrivial. The basic result of [Gardenfors, 1986] can be derived by applying Theorem 8 to Gardenfors belief change models. Gardenfors [1987; 1988] notes that (K P) and (K 2) can be replaced by (K 4) in the triviality result he proves there, and this same replacement can be made in Theorem 8, with a corresponding change in Lemma 3 and its proof.27 As was mentioned in Section 2.3 above, Segerberg has proved a version of the Gardenfors result in which the constraints on ` are limited to Re exivity, Transitivity, and Monotonicity. The counterpart in our framework of Segerberg's result is the following: THEOREM 9. No Segerberg nontrivial Segerberg belief change model satis es (K M), (K 4ss), and (K5ws).28 If contraction and revision are assumed to be related by the Levi Identity (LI) in a deductively closed belief change model, then triviality results for the Ramsey test can be formulated in terms of contraction rather than in terms of revision. In particular, we have the following as a corollary of Theorem 8:29 THEOREM 10. No deductively closed classical belief change model de ned on a language of type L1 or L2 that satis es (K+ 1), (K 3), (K 4w), (LI), and (RT) and whose support function is monotonic over K is Gardenfors nontrivial or Rott nontrivial or Segerberg nontrivial.
Proof. It suÆces to note that where ` is classical, we have the following: (Def+) and (LI) jointly imply (K 2); (LI) and (K 4w) jointly imply (K C); (Def+), (K 3), and (LI) jointly imply (K P). Makinson [1990] proves a variant of Theorem 10 for type-L1 languages that replaces (K 4w) and weakens both (RT) and (LI) while making stronger assumptions about s than merely that it is monotone over K: THEOREM 11. Let hK; I ; `; K?; ; ; si be a Makinson belief change model de ned on a language L of type L1 . De ne postulates (RTM), (K 4c), and (MI) as follows: (RTM) For all ; 2 wffL0 , where L0 is the type-L0 fragment of L, > 2 s(K ) i 2 K . 27 Steps (6), (9), and (13) must be dierently justi ed. 28 See [Segerberg, 1989]. Segerberg's version of the G ardenfors
triviality result makes no assumption about which operators are available in the language, hence is used in the de nition of Segerberg nontriviality to play the role that :( ^ ) plays in the proof of Lemma 3. Also, Segerberg's result does not assume that the language contains both > and ?, which we assume here in (DefBCM). 29 A similar result is proved in [Cross, 1990a].
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(K 4c) If 2 s(K ), then 2 s(;). (MI)
K: K s(K: [ fg).
Then we have the following: (1) Limiting Case. If (RTM), (K 4c), and (MI) hold for K = K?, then the model is trivial in the sense that s(;) = K? . (2) Principal Case. If (RTM), (K 3), (K 4c), and (MI) hold for all K 2 K such that K 6= K?, then the model is trivial in the sense that there are no conditional-free formulas and of L such that ^ 62 s(;) and 62 s(f g) and 62 s(fg) and s[s(fg) [ s(f g)] 6= K? .
The Limiting Case generalizes Theorem 12 discussed below. It is the Principal Case that more closely corresponds to Theorem 10. (K 4c) neither entails nor is entailed by (K 4), its AGM counterpart, but (MI), which we will refer to as Makinson's Inequality, is the result of weakening (LI), the Levi Identity, to say that a revision of K to include must lie \between" K: and s(K: [ fg). Since, as we have seen, (LI), (Def+), and (K 3) entail (K P), one might expect that replacing (LI) with (MI) would leave (K P) unsupported, but this is not the case: (MI), (Def+), and (K 3) already entail (K P). Making up for the fact that (MI) is weaker than (LI) are Makinson's strengthened assumptions about s: that it satis es Superclassicality, Transitivity, and Reasoning by Cases. Makinson [1990] points out that these conditions are known not to imply that s is monotone over its entire domain (P (wffL )), but since contraction and revision in a Makinson belief change model are de ned only on K such that s(K ) = K , s is nevertheless monotone \where it counts", namely over the set K of belief sets on which contraction and revision are de ned. Several authors (e.g. Grahne [1991], Hansson [1992], and Morreau [1992]) have concluded from Makinson's result that nonmonotonic consequence does not provide a way out of the Gardenfors triviality result. In fact, adopting a nonmonotonic consequence operation does provide a way out, provided that this consequence relation plays the role of a support function s that is nonmonotonic over the belief sets to which contraction and revision are applied. Indeed, it is by adopting such support functions that Hansson, Arlo-Costa and Levi are able to make the Ramsey test nontrivial, though these authors do not describe the support function as a consequence operation. (See also Section 2.6 below.) Theorem 8 and its variants pose a dilemma: which of an inconsistent set of constraints on belief change models should be rejected? We return to this later in Section 2.6 below.
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The problem with (K 5w)
In the version of Theorem 8 that Gardenfors proves in [Gardenfors, 1988], postulate (K C) is replaced by the stronger (K 5w), but Arlo-Costa [1990] proves that (K 5w) faces problems that have nothing to do with (K P). Arlo-Costa's result, which is not so much a triviality result as an impossibility result, can be formulated as follows in our framework: THEOREM 12. There is no Gardenfors belief change model de ned on a language of type L1 or L2 in which 6` ?, (K 5w), and (RT) hold. Whereas Gardenfors' results against the Ramsey test exploit the fact that (RT) entails (K M), Arlo-Costa's result exploits the fact that (RT) entails the following, which Arlo-Costa calls \Unsuccess": (US) If K = K? , then K = K? .
In other words, the Ramsey test requires revision into inconsistency if the initial belief state is already inconsistent, regardless whether the revision input is a consistent proposition. Contrary to this, (K 5w) prohibits revision into inconsistency when the revision input is a consistent proposition, regardless whether the initial belief state is consistent. The labeling of (US) as a postulate of \unsuccess" is appropriate since (K 5w), which (US) contradicts, follows from (Def+), (LI), and the Postulate of Success for contraction (K 4). Arlo-Costa's result can be strengthened to include belief change models in which s(K ) = K does not always hold: THEOREM 13. There is no deductively closed classical belief change model de ned on a language of type L1 or L2 whose support function satis es Re exivity and Closure and in which 6` ?, (K 5w), and (RT) hold.
Proof. Let a deductively closed classical belief change model on a language of type L1 or L2 be given and suppose for reductio that 6` ?, that s satis es Re exivity and Closure, and that the model satis es (K 5w) and (RT). First we prove (US). Let K = K?; then we have K
s(K )
Re exivity of s = Cn` [s(K )] Closure of s
Since ? 2 K? = K we have s(K ) = wffL , by the classicality of `. Next, let 2 K? , and let 2 I . Since s(K ) = wffL , we have > 2 s(K ). Hence by (RT) we have B 2 K . Thus K? K ; the converse inclusion holds by (DefBCM), so K = K?, as required to show (US). By (DefBCM) K? 2 K and :? 2 I ; by (US) we have (K? ):? = K? . By hypothesis 6` ?, hence by the classicality of ` we have not only (K? ):? = K? but also 6` ::?, which contradicts (K 5w).
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The Limiting Case of Theorem 11, like Theorem 13, is a strengthening of Theorem 12. The problem posed by Theorems 12 and 13 can be solved by retreating from (K 5w) to something weaker, such as the postulate (K C) mentioned in Theorem 8, or by restricting the applicability of the Ramsey test, as Aro-Costa and Levi both do by adopting (PRTL) (see Section 2.4 above), which eliminates K? from the domain of belief sets to which the Ramsey test can be applied. The negative Ramsey test Levi has argued (see, for example, [Levi, 1988] and [Levi, 1996]) that a negated conditional :( > ) expresses the propositional attitude of an agent for whom : is a serious (i.e. epistemic) possibility relative to K . Abstracting from Levi's requirements on what is allowed to be a revision input, the result is this thesis, the negative Ramsey test: (NRTL) For all 2 I and all 2 K? and all K 2 K such that K 6= K? , :( > ) 2 s(K ) i 62 K. Rott [1989] takes the view that adopting both the negative Ramsey test and the Ramsey test amounts to an assumption of autoepistemic omniscience. Given the view of Gardenfors, Rott, and others that conditionals and negated conditionals belong in belief sets along with other beliefs (so that s satis es Identity over K), the conjunction of (RT) and (NRTL) does amount to a kind of epistemic omniscience. That is, if s satis es Identity over K, then \closing" each belief set under (RT) and (NRTL) amounts to an idealization that parallels the idealization represented by \closing" each belief set under `. On Levi's view, conditionals do not express propositions and so are not objects of belief, thus on Levi's view the positive and negative Ramsey tests cannot be said to represent an idealization concerning what beliefs an agent holds. For Levi, what the positive and negative Ramsey tests represent is not a pair of closure conditions on the unary propositional attitude of belief but rather a de nition of a binary propositional attitude toward the antecedent and consequent of a conditional that an agent is said to `accept'. Regardless how the issue of autoepistemic omniscience is resolved, the adoption of (NRTL) has consequences. Gardenfors, Lindstrom, Morreau, and Rabinowicz [1991] prove what they consider to be a triviality result for (NRTL) with assumptions weaker than those needed for Gardenfors' 1986 triviality result for (RT); in particular, (K P) is not needed. In our framework their result is equivalent to the following: THEOREM 14. If hK; I ; `; K? ; ; ; si is a belief change model de ned on a language of type L1 or L2 for which both (NRTL) and (K T) hold and for which s satis es Identity over K, then there are no K; K 0 2 K such that K 6= K? 6= K 0 and K 0 6= K K 0 .
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The latter can be derived as a corollary of the following stronger result: THEOREM 15. If hK; I ; `; K? ; ; ; si is a belief change model de ned on a language of type L1 or L2 for which both (NRTL) and (K T) hold and for which s is monotone over K, then there are no K; K 0 2 K such that K 6= K? 6= K 0 and K 0 6= K K 0.
Proof. Suppose that hK; I ; `; K? ; ; ; si is a belief change model de ned on a language of type L1 or L2 such that s is monotone over K. First we prove that (NRTL) implies (K IM): assume (NRTL) and suppose that K; K 0 2 K and 2 I and K 6= K? 6= K 0 and s(K ) s(K 0 ), and let 62 K. Then by (NRTL) :( > ) 2 s(K ), hence :( > ) 2 s(K 0 ). By (NRTL) it follows that 62 K0 , as required to establish (K IM). Now suppose for reductio that hK; I ; `; K?; ; ; si satis es both (NRTL) and (K T), that s is monotone over K, and there are K; K 0 2 K such that K 6= K? 6= K 0 and K 0 6= K K 0 . Since K K 0 we have s(K ) s(K 0 ) by the monotonicity of s. By (DefBCM) we know that > 2 I , so we have K>0 K> by (K IM). But K> = K and K>0 = K 0 by (K T), hence K 0 K , which contradicts K 0 6= K K 0 , completing the reductio. Note that neither theorem assumes that all members of K must be deductively closed, nor does either result include any assumption about `. In [Gardenfors et al., 1991] Theorem 14 is presented as a triviality result because the authors maintain that a model of belief change is trivial if it contains no consistent, conditional-laden belief sets K; K 0 such that K is a proper subset of K 0 . As Rott [1989], Morreau [1992], and Hansson [1992] point out, however, it is a substantive (and, they argue, mistaken) assumption to hold that principles of belief revision that are justi ed in the context of conditional-free belief sets (e.g. the closure of K under expansions) can be carried over without modi cation to conditional-laden belief sets. More on this in Section 2.6 below. One might therefore respond to Theorem 14 by questioning the criterion of triviality: perhaps a model of belief change whose belief sets are conditional-laden should not be classi ed as trivial simply because it contains no consistent K; K 0 such that K is a proper subset of K 0 , even though a belief change model with conditional-free belief sets would be trivial in that case. But if the criterion of triviality espoused by Gardenfors, et al [1991] is appropriate for conditional-free belief sets, then what about Theorem 15, which does cover belief change models with conditional-free belief sets? Our discussion in Section 2.6 below may appear to suggest that the problem raised by Theorems 14 and 15 might ultimately
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be solved by giving up the monotonicity of s over K, but even that is not guaranteed to be enough. Consider this result:30 THEOREM 16. No classical belief change model de ned on a language of type-L1 or type-L2 that satis es (K T), (PRTL), and (NRTL), and whose support function satis es Conservativeness and Closure, is Gardenfors nontrivial or Rott nontrivial or Segerberg nontrivial.
Proof. By Lemmas 5, 6, and 7, it suÆces to show that no classical belief change model de ned on a language of type-L1 or type-L2 that satis es (K T), (PRTL), and (NRTL), and whose support function satis es Conservativeness and Closure, contains forking support sets. Consider a classical belief change model hK; I ; `; K? ; ; ; si de ned on a language of type-L1 or type-L2 that satis es (K T), (PRTL), and (NRTL), and whose support function satis es Conservativeness and Closure. Note rst that since s satis es Conservativeness and Closure, s must also satisfy Consistency. Suppose the model contains forking support sets. Then there exist H; J; K 2 K such that H = Cn` (H ) \ K? , and J = Cn` (J ) \ K? , and K = Cn` (K ) \ K? 6= K? , and H \ I 6 J , and J \ I 6 H , and s(H ) s(K ), and s(J ) s(K ). Since J \ I 6 H we have 2 J but 62 H for some conditional-free . By (K T) we have H = H> and J = J> and K = K> , hence 2 J> and 62 H> . By (PRTL) we have > > 2 s(J ), and by (NRTL) we have :(> > ) 2 s(H ). We also have > > 2 s(K ), since s(J ) s(K ); hence s(K ) is not `-consistent. This contradicts the `-consistency of K , since s satis es Consistency. As Theorem 16 shows, (NRTL), (K T) and (PRTL) cannot be nontrivially combined, even in a broad category of models where s is not monotone over K, unless we abandon the Rott, Gardenfors, and Segerberg criteria of nontriviality.
2.6 Resolving the con ict On giving up (RT) Gardenfors interprets Theorem 8 as forcing a choice between (K P) and the Ramsey test (RT), and he has argued (see, e.g., [Gardenfors, 1986], pp. 8687 and [Gardenfors, 1988], p. 59 and p. 159) that (K M) and with it (RT) should be rejected. In this connection he oers the following example:
Let us assume that Miss Julie, in her present state of belief K , believes that her own blood group is O and that Johan is her 30 The authors thank Horacio Arl o-Costa for showing us the proof of this result in correspondence.
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father, but she does not know anything about Johan's blood group. Let A be the proposition that Johan's blood group is AB and C the proposition that Johan is Miss Julie's father. If she were to revise her beliefs by adding the proposition A, she would still believe that C , that is, C 2 KA . But in fact she now learns that a person with blood group AB can never have a child with blood group O. This information, which entails C ! :A, is consistent with her present state of belief K , and thus her new state of belief, call it K 0 , is an expansion of K . If she then revises K 0 by adding the information that Johan's blood group is AB, she will no longer believe that Johan is her father, that is C 62 KA0 . Thus (K M) is violated. ([Gardenfors, 1986], pp. 86-87) The example assumes that s satis es Identity over K, so let us assume that as well. In reply to Gardenfors one might say that if (RT) and Identity over K are assumed, then the presence of conditionals in belief sets prevents this example from being a counterexample to (K M): if (RT) and Identity over K are assumed, then since we have C 2 KA and C 62 KA0 it follows that A > C 2 K and A > C 62 K 0 , in which case K 6 K 0 , i.e. K 0 is not an expansion of K . But to accept this, Gardenfors argues, would violate certain intuitions: [I]f we assume (RT) and not only (K M), then Miss Julie would have believed A > C in K . But then the information that a person with blood group AB can never have a child with blood group O, would contradict her beliefs in K , which violates our intuitions that this information is indeed consistent with her beliefs in K . ([Gardenfors, 1986], p. 87) Let B stand for the statement that a person with blood group AB can never have a child with blood group O. Gardenfors has claimed in a context where s satis es Identity over K that if (RT) holds, then Miss Julie's beliefs in K contradict B , but this claim requires further justi cation: how exactly does K contradict B ? We might suppose that B entails L(C ! :A), where L is an alethic nomological necessity operator expressing the modal force of B . Assuming (RT) and Identity over K, the question whether K contradicts B depends on whether the set fC; B; A > C g is consistent, and this can be assumed to depend on whether the set fC; L(C ! :A); A > C g is consistent. But the latter set is consistent if the semantics for > is not tied to nomological necessity. For example, given a selection function semantics for > and an accessibility relation semantics for L, all of C , A > C , and L(C ! :A) can be true at possible world w if none of the A-worlds selected relative to w happen to be nomologically accessible at w.31 So in 31 For
a less abstract version of essentially this point, see [Cross, 1990a], pp. 229-232.
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order to sustain Gardenfors' claim in the passage cited above, the claim that K contradicts B if (RT) holds (and if s satis es Identity), we would have to assume the right sort of semantic connection between Ramsey test conditionals and the nomological modality in B , but the case for assuming that connection is not at all obvious: Ramsey test conditionals, after all, are epistemic . And if K indeed does not contradict B , then the same conditional that prevents the example from being a counterexample to (K M) makes the example a counterexample to (K P): since K 0 = KB , and since C 2 KA but C 62 KA0 , it follows that if (RT) holds, then A > C 2 K 6 KB 63 A > C even though :B 62 K . So it might be argued that the presence of Ramsey test conditionals in belief sets will render (K M) intuitively innocuous while providing perfectly reasonable counterexamples to (K P). Still, the arguments in favor of (K P) seem strong. One argument appeals to the Bayesian model of rationality. Suppose that an agent's belief state is represented as a probability function P . According to Bayesian doctrine, upon becoming certain of a rational agent in belief state P revises her belief state by conditionalizing on , assuming P () > 0. If this doctrine is correct and if an agent's belief set consists of those statements to which she assigns unit probability, then (K P) reduces to a theorem of probability theory: if P (:) 6= 1 and P ( ) = 1, then P ( j) = 1. A second argument appeals to the doctrine that revision can be de ned in terms of contraction and expansion via the Levi Identity (LI), which prescribes the following: to revise with , rst contract relative to : and then expand with . If ` is classical and if (LI), (Def+), and (K 3) hold, then (K P) follows, the role of (K 3) being to require any contraction of K to be vacuous if the proposition contracted does not belong to K : if one does not believe a given proposition then no prior belief need be discarded when one contracts one's beliefs to exclude that proposition|it is already excluded. As Theorem 10 shows, we can incorporate the second of these arguments for (K P) directly into the triviality result by recasting Theorem 8 in terms of an inconsistency between (RT), (LI), and postulates (K+ 1), (K 3), and (K 4w). (K 3) deals with what is in some sense the degenerate case of contraction: contraction with respect to an absent proposition. Postulate (K 4w) is similarly weak: it requires only that a contraction should really be a contraction in any case where a logically contingent proposition is contracted from a logically consistent belief set. Both postulates are very weak constraints on contraction, and their weakness makes the case against (RT) seem strong, as long as we assume that s satis es Identity over K or at least Monotonicity over K. Is there a weakened version of (RT) that is compatible with the other postulates mentioned in Theorem 8? Lindstrom and Rabinowicz [1992] show that there is. They suggest replacing (RT) with a condition whose counterpart in our framework is the following:
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(SRT)
For all K 2 K and all 2 I and all 2 K? , > 2 K0 for all K 0 2 K such that K K 0 .
2 s(K ) i
Like Gardenfors, Lindstrom and Rabinowicz do not distinguish between K and s(K ), and in a context where this distinction is not made (i.e where s satis es Identity over K), replacing (RT) with (SRT) has the eect of excluding from belief sets many of the conditionals that must be present in them if (RT) and Identity over K are assumed. For example, in Gardenfors' Miss Julie case, (RT) and Identity over K force the conclusion that A > C 2 K and A > C 62 KA0 , since C 2 KA and C 62 KA0 , giving us a counterexample to (K P) since :B 62 K and K 0 = KB . If (SRT) and Identity over K are assumed instead, then the falsity of (K P) no longer follows. The question whether A > C belongs to K depends not simply on KA but on the revision behaviour of all belief sets that include K as a subset, and similarly for the question whether A > C belongs to K 0 . On giving up (K P)
Should the Ramsey test be preserved at the expense of (K P)? The answer is certainly yes if the Ramsey test is applied to the notion of theory change to which Katsuno and Mendelzon in [Katsuno and Mendelzon, 1992] attach the label update . They write:
: : : [U]pdate , consists of bringing the knowledge base up to date when the world described by it changes. For example, most database updates are of this variety, e.g. \increase Joe's salary by 5%". Another example is the incorporation into the knowledge base of changes caused in the world by the actions of a robot.32 Update, according to Katsuno and Mendelzon, contrasts with revision :33
: : : [R]evision, is used when we are obtaining new information about a static world. For example, we may be trying to diagnose a faulty circuit and want to incorporate into the knowledge base the results of successive tests, where newer results may contradict old ones. We claim the AGM postulates describe only revision.34 Katsuno and Mendelzon represent knowledge bases as formulas and introduce a binary modal connective to represent the update operation. Following [Grahne, 1991] we will use the symbol `Æ' for this operation; then the 32 [Katsuno and Mendelzon, 33 See also [Winslett, 1990]. 34 Ibid.
1992], p. 183.
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formula Æ is the knowledge base that results from updating knowledge base with new information . Grahne [1991] provides an interpretation of the Ramsey test in terms of update. Given a type-L2 language that includes the binary connective `Æ' Grahne simply adds to Lewis' system VCU the following (validity preserving) rule of inference: RR:
From ! ( > ) infer ( Æ ) ! , and from ( Æ ) ! ! ( > ).
infer
Grahne calls the resulting logical system VCU2 . In Grahne's framework, the formula Æ is true in possible world w i w belongs to the set of closest worlds to w0 in which is true for at least one world w0 in which is true. Grahne proves soundness, completeness, decidability, and nontriviality results for VCU2 , and he notes that VCU2 fails to satisfy the following principle: (U 4s) If 6` :( ^ ), then ` ( Æ ) $ ( ^ ). (U 4s) states that if is consistent with knowledge base , then the result of updating with is a formula logically equivalent to ^ . Grahne cites the following example to illustrate the failure of (U 4s), which is the update counterpart of revision postulate (K 4s): A room has two objects in it, a book and a magazine. Suppose p1 means that the book is on the oor, and p2 means that the magazine is on the oor. Let the knowledge base be (p1 _ p2 ) ^ :(p1 ^ p2 ), i.e. either the book or the magazine is on the oor, but not both. Now we order a robot to put the book on the oor, that is, our new piece of knowledge is p1 . If this change is taken as a revision [so that (K 4s) is assumed], then we nd that since the knowledge base is consistent with p1 , our new knowlege base will be equivalent to p1 ^ :p2 , i.e. the book is on the oor and the magazine is not. But the above change is inadequate. After the robot moves the book to the oor, all we know is that the book is on the oor; why should we conclude that the magazine is not on the oor?35 That is, upon updating to include p1 , we should give up something we believed in our initial epistemic state, namely :(p1 ^ p2 ), even though the new information p1 is consistent with our initial epistemic state. Apparently, then, we have made a belief change using a method that does not satisfy an appropriate counterpart of (K P). Isaac Levi disagrees. The mechanism which underlies update is imaging : the \image" of a set S of possible worlds 35 [Grahne,
1991], pp. 274{275.
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under is the set of worlds each of which is one of the closest -worlds to some world belonging to S . Levi [Levi, 1996] argues that while imaging may be useful for describing how changes over time in the state of a system (such as the room in Grahne's example) are regulated, such changes are not an example of belief change. We may, of course, have beliefs about how changes in a system over time are regulated, but an analysis of Grahne's example along the lines recommended by Levi would show it to be a straightforward case in which belief revision took place via expansion : if t is a time before the book was moved and t0 is a time just after the book is moved and if propositional variables p1 and p2 are replaced by formulas containing predicates P1 and P2 , where Pi u means that pi is true at time u, then our initial epistemic state can be represented as (P1 t _ P2 t) ^ :(P1 t ^ P2 t), and upon learning of the change in the position of the book our new epistemic state is (P1 t _ P2 t) ^ :(P1 t ^ P2 t) ^ P1 t0 . On giving up (K+ 1)
Gardenfors interprets Theorem 8 as forcing a choice between (RT) and (K P), but Rott [1989], Morreau [1992], and Hansson [1992] have argued that (K+ 1) is the real culprit. Postulates (K 3) and (K 4) entail (K 4s): if is consistent with K , then a revision to accept should be the result of expanding K with . Rott [1989] argues that (K 4s), while ne for belief revision in a type-L0 language, is an inappropriate requirement on belief revision in a language with Ramsey test conditionals. Once (K 4s) is rejected in the context of Ramsey test conditionals, Rott argues, (K+ 1) is robbed of any intuitive basis: the only reason for thinking that belief change models should be closed under expansion would be the assumption that expansion is a species of revision. Why think that expansion is a species of revision in the rst place? One could justify (K 4s) as the qualitative analog of the Bayesian doctrine that upon becoming certain of a rational agent whose belief state is represented by probability function P revises her belief state by conditionalizing on if P () > 0. This doctrine supports (K 4s) because if P () > 0, then the set f : P ( j) = 1g is precisely the result of expanding the set f : P ( ) = 1g with . But, Morreau [1992] counters, Bayesian doctrine supports (K 4s) in this way only for belief sets over a type-L0 language. Still, one might argue, regardless whether revision ever leads from a belief set to one of its expansions, should not the expansion of every belief set in a belief change model be available in the model as a possible starting point for revision? Not so, argues Morreau [1992]: a belief change model over which (RT) holds and in which belief sets contain conditionals incorporates the idealizing assumption that the conditionals an agent believes form a complete and correct record of how the agent would revise his or her beliefs.
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Not just any collection of theories in a conditional language can be the belief sets of a Ramsey test respecting belief change model because not just any theory will conform to the idealization. Morreau interprets the Gardenfors triviality result as showing in particular that the idealization required by the Ramsey test cannot be achieved in a nontrivial belief change model that respects (K+ 1) while incorporating conditionals in belief sets. But are there in fact nontrivial belief change models containing conditionalladen belief sets in which (RT) holds but (K+ 1) does not? Morreau's Example 6 ([Morreau, 1992], p. 41), which we adapt to our framework, con rms that there are. Let L be a type-L1 language and let L0 be its type-L0 fragment. Assume that L0 contains at least two distinct atomic formulas. Let `0 be truth-functional consequence, and assume (Def+), and let K0 be the set of all `0 -theories in L0 . De ne a belief revision operation ? as follows for all 2 K0 and all formulas of L0 : 8 if ? 2 ; < if ? 62 and : 62 ; ? = + : Cn`0 (fg) otherwise. Let I0 = wffL0 = K?0 ; let dom(s0 ) = K0 ; and let s0 (K ) = K for all K 2 K0 . Letting the contraction operation ( 0 ) be arbitrary, note that hK0 ; I0 ; `0 ; K?0 ; 0 ; ?; s0 i satis es (K+ 1), (K 2), (K C), and (K P), but not (RT). Using K0 and ? we construct a second, Ramsey test supporting belief change model with conditional-laden belief sets as follows: for each 2 K0 , let K = Cn`0 ( [f > : 2 ? g). Let K = fK : 2 K0 g; let (K ) = K? ; let I = wffL0 (as before); let K? = wffL; let dom(s) = K; and let s(K ) = K for all K 2 K. Letting contraction ( ) again be arbitrary, hK; I ; `0; K? ; ; ; si satis es (K 2), (K C), and (RT), but this model, unlike the rst, does not satisfy (K+ 1) or (K P).36 For example, let A; B be distinct atomic formulas of L0, and let 0 = Cn`0 (fB g); thus 0 2 K0 . Since :A 62 0 , we have that 0A? = 0A+ = Cn`0 (fA; B g). Accordingly, A > B 2 K 2 K, but note that (K )+:A does not belong to K, for there is no 2 K0 such that both :A and A > B belong to K . The second belief change model constructed above is the result of closing the rst model under the Ramsey test (restricted to non-nested conditionals), and both models are Gardenfors nontrivial. The Gardenfors nontriviality of the second model is established by KCn 0 (;) , which belongs to K, and A ^ :B , B ^ :A, and A ^ B which belong to L. In addition to this example Morreau provides a general recipe for constructing nontrivial models of belief revision in a type-L1 language L whose type-L0 fragment is L0 and where (K 1), (K 2), (K C), (RT), and (K PI ) hold and I = wffL0 . 0
0
`
36 The model does satisfy a weakened version of (K P), however, as Morreau points out: (K PI ) For all 2 I , if : 62 K then K \ I K \ I .
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Where does the triviality proof break down when applied to Morreau's example? Hansson [1992] proves a theorem that provides the answer: Morreau's example is one of a set of belief change models in which forking support sets cannot be constructed. The counterpart of Hansson's theorem in our framework is the following: THEOREM 17. Suppose that hK; I ; `; K?; ; ; si is a belief change model de ned on a language L of type L1 or L2 and that ` includes all truthfunctional entailments and respects the deduction theorem for the material conditional. Suppose also that dom(s) = K, that s(K ) wffL for all K 2 K, and that s satis es the following for all K 2 K, all 2 I and all ; 2 K? :
s(K ) and ` then 2 s(K ). 2. If K is `-consistent and 2 s(K ), then : 62 s(K ). 3. If K is `-consistent and 6` : and > ; > 2 s(K ), then 6` :( ^ ). 4. If > ( ^ ) 2 wffL and 2 s(K ) and 62 s(K ) and : 62 s(K ), then > ( ^ ) 2 s(K ). Suppose that K1 ; K2 ; K 2 K fK?g and that s(K1 ) and s(K2 ) are both subsets of s(K ). Then either s(K1 ) s(K2 ) or s(K2 ) s(K1 ). 1. If
Conditions 1 and 2 are equivalent to Closure and Consistency for s, respectively. Note that the Ramsey test itself is not assumed: the point is that simply having conditionals in a belief change model that meets these four conditions ensures that forking support sets cannot be constructed. On giving up the monotonicity of s over K Rott [1989; 1991] suggests that nonmonotonic reasoning may provide a solution to the dilemma posed by the Gardenfors triviality result, and Cross [1990a] argues that Gardenfors' triviality result should be interpreted as showing not that the Ramsey test should be abandoned but that, given the Ramsey test, s must be nonmonotonic over K, i.e. for some H; K 2 K H K but s(H ) 6 s(K ).37 Makinson counters in [Makinson, 1990] with a triviality result for models in which s is permitted to be nonmonotonic, but Makinson's result does not bear on the suggestion endorsed by Cross and by Rott. More on this below. Other authors have brought nonmonotonic reasoning into the discussion of the Ramsey test without advertising it as such. For example, in [Hansson, 1992] Hansson writes: 37 Since the monotonicity of s is not assumed in Theorem 13, however, it is clear that the problem for (RT) posed by (K 5) cannot be solved by making s nonmonotonic.
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: : : the addition of an indicative sentence that is compatible with all previously supported indicative sentences typically withdraws the support of conditional sentences that were previously supported.38 The type-L1 statements that are in Hansson's sense supported by a given \indicative" (i.e. conditional-free) belief base K represent what Cross (and possibly Rott) would classify as the nonmonotonic consequences of K . Hansson and Cross both think of the sets that individuate belief states as belief bases and de ne contraction and revision as operations on these sets, but Cross' belief bases dier from Hansson's in two respects: rst, whereas Hansson's belief bases are conditional-free, Cross' are not; secondly, whereas Hansson's belief bases are not closed under ` or closed under s, in Cross' enriched belief revision models belief bases are closed under `, though not under s. That is, for Hannson, belief states are individuated in terms of sets that function as belief bases with respect to both ` and s, whereas for Cross, belief states are individuated in terms of sets that function as belief bases only with respect to s. For Hansson, belief bases need not be closed under ` and are never closed under s. Makinson [1990], like Cross [1990a], supplements the classical ` with a not-necessarily-monotonic s,39 and like Cross, Makinson explicitly advertises s as a consequence operation. But in Makinson's discussion revision and contraction are de ned only on K that are closed under s, and the proof of Makinson's triviality theorem, whose counterpart here is Theorem 11 above, requires a belief change model containing three belief sets closed under s. Makinson's triviality result does not apply to belief change models in which K contains no K such that s(K ) = K , and such authors as Arlo-Costa and Levi (see [Levi, 1988], [Arlo-Costa, 1995], and [Arlo-Costa and Levi, 1996]) avoid Makinson's triviality result precisely by requiring s(K ) 6= K for all K 2 K. As we noted above, Hansson does not explicitly speak of the support function as a consequence operation, nor does Arlo-Costa or Levi. Yet, if one looks at the conditions that Hansson, Arlo-Costa, and Levi place on the support function, mirrored here in the de nitions of Hansson and Arlo-Costa/Levi belief change models as the requirements of Re exivity, Conservativeness, and Closure, it seems natural to think of s as a nonmonotonic consequence operation. But if we do think of s in a Hansson or Arlo-Costa/Levi belief change model as a nonmonotonic consequence operation, what sort of nonmonotonic reasoning does it represent? In [Moore, 1983] Robert Moore distinguishes two types of nonmonotonic reasoning: By default reasoning, we mean drawing plausible inferences from less than conclusive evidence in the absence of any information 38 [Hansson, 39 Makinson
1992], p. 526. uses the symbol C and Cross the symbol cl for s.
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to the contrary. The examples about birds being able to y are of this type.40 He continues: Default reasoning is nonmonotonic because, to use a term from philosophy, it is defeasible . Its conclusions are tentative, so, given better information, they may be withdrawn.41 Default reasoning, according to Moore, contrasts with autoepistemic reasoning , or reasoning about one's state of belief. Moore writes: Autoepistemic reasoning is nonmonotonic because the meaning of an autoepistemic statement is context-sensitive; it depends on the theory in which the statement is embedded.42 For example, if } is de ned as being accepted in belief state K just in case : is not accepted in K , then } is an autoepistemic statement in Moore's sense. If the support function s in a belief change model is thought of as a nonmonotonic consequence operation, then how should s be classi ed with respect to Moore's distinction? It depends on the properties s is assumed to have. If a belief change model satis es some version of the Ramsey test (e.g. (RT), (PRTL), or (NRTL)), then the support function of that model is at least a form of autoepistemic reasoning. This is clear since the acceptability of a Ramsey test conditional for a given agent is in part a function of the agent's current epistemic state, and this holds true regardless whether conditionals themselves are objects of belief.43 Moreover, the context sensitivity to which Moore refers in the passage just quoted is clearly present in the support function of any belief change model that satis es (RT), and indeed this context sensitivity was exploited by Morreau [1992] in his construction of a nontrivial Ramsey test, by Lindstrom and Rabinowicz [1995] and Lindstrom [1996] in a proposed indexical interpretation of conditionals,44 by Hansson [1992] in his accounts of type-L1 conditionals and iterated conditionals, respectively, and by Boutilier and Goldszmidt [1995] in their account of the revision of conditional belief sets. Given a support function s for a belief change model that satis es a version of the Ramsey test, can s be not only a mechanism for autoepistemic reasoning but a mechanism for default reasoning, too? This depends on whether s can be used to make ampliative inferences to conclusions that 40 [Moore, 1983], p. 273. 41 [Moore, 1983], p. 274. 42 [Moore, 1983], p. 274. 43 Rott [1989] and Morreau
tionals are autoepistemic. 44 See also [D oring, 1997].
[1992] explicitly adopt the view that Ramsey test condi-
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are not epistemically context-sensitive from premises that are not epistemically context-sensitive. Since Hansson, Arlo-Costa, and Levi assume that s satis es Conservativeness, it is clear that for them s is an operation of autoepistemic reasoning but not an operation of default reasoning: s(K ) will contain conditionals, i.e. autoepistemic statements, that are not logical consequences of K , but no conditional-free formula gets into s(K ) without being a logical consequence of K , which is itself conditional-free for any K 2 dom(s) according to Hansson, Arlo-Costa, and Levi. Cross and Makinson, on the other hand, do not require the support function to satisfy Conservativeness; accordingly, they allow belief change models in which s supports default reasoning. No distinction between s(K ) and K exists for Morreau, Gardenfors, and Segerberg, hence the issue of the status of s does not arise in their respective cases.
2.7 Logics for Ramsey test conditionals Gardenfors [1978] proves the soundness and completeness of David Lewis' system of conditional logic VC with respect to an epistemic, Ramsey test semantics for the conditional. Several other authors have proposed variants of Gardenfors' Ramsey test semantics, including variants that generalize Gardenfors' semantics, but it will be convenient for our purposes to adopt formalisms similar to those of [Arlo-Costa, 1995] and [Arlo-Costa and Levi, 1996]. Primitive belief revision models
Since the conditional is to be given a semantics in terms of belief revision, the notion of a belief set must be de ned in terms that do not assume a logic for the conditional. To this end we de ne primitive belief sets , primitive expansion , and primitive belief revision models . For a Boolean language L of type L1 or L2 let a primitive belief set de ned on this language be any set K of formulas of L meeting three requirements: (pBS1) K 6= ;; (pBS2) if 2 K and (pBS3)
2 K , then ^ 2 K ; if 2 K and ! is a truth-functional tautology, then 2 K .
If K K 0 and K , K 0 are both primitive belief sets, then K 0 is a primitive expansion of K . The operation of primitive expansion is de ned as follows: (DEF+) K+ = f : !
2 K g.
It is easy to see that if K is a primitive belief set, then so is K+. Finally, let us de ne the notion of a primitive belief revision model on L:
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(DefpBRM) A primitive belief revision model (or pBRM) on a Boolean language L is an ordered quadruple hK; ; K? ; si whose components are as follows: 1. K? = wffL , where L0 is L or a fragment of L; 2. K is a nonempty set of primitive belief sets de ned on L0, and if K 2 K then K contains every primitive expansion of K on L0 ; 3. is a function mapping each K 2 K and each formula 2 K? to a primitive belief set K belonging to K; 4. s is a function mapping each K 2 K to a primitive belief set s(K ) of formulas of L, where s satis es the following: (a) if 2 K? and 2 s(K ) then 2 K ; (b) K s(K ) if K? 6= K 2 K. 0
Note that while K and s(K ) must both be primitive belief sets, they need not be primitive belief sets of the same language. When referring to the belief revision postulates (K 1), (K 2), etc., in the context of primitive belief revision models we will assume that and range over K? . A primitive belief revision model hK; ; K? ; si de ned on a Boolean language L is a Gardenfors pBRM i L is of type L2 and K? = wffL and s is the identity function on K and s satis es the following unrestricted version of the positive Ramsey test: (pRTG) For all K 2 K, if ; 2 K?, then ( > ) 2 s(K ) i 2 K .
A primitive belief revision model hK; ; K? ; si de ned on a Boolean language L is an Arlo-Costa/Levi pBRM i L is of type L1 and K? is the set of all formulas of the largest conditional-free fragment of L and s satis es the following versions of both the positive and negative Ramsey tests: (pPRT) For all K s(K ) i
2 K such that K 6= K? , if ; 2 K? , then ( > 2 K . For all K 2 K such that K = 6 K? , if ; 2 K? , then :( > s(K ) i 62 K .
)2
(pNRT)
)2
Positive and negative validity In [Arlo-Costa, 1995] (and in [Arlo-Costa and Levi, 1996], with Isaac Levi) Arlo-Costa distinguishes between positive and negative concepts of validity. The concepts are distinct in Arlo-Costa/Levi pBRMs, though not in Gardenfors pBRMs. A formula is positively valid (PV) relative to hK; ; K? ; si, where the latter is a primitive belief revision model, i 2 s(K ) for all K such that
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K? 6= K 2 K; and is positively valid relative to a set of belief revision models i is positively valid relative to each member of the set. is negatively valid (NV) relative to hK; ; K? ; si i : 62 s(K ) for each K such that K? 6= K 2 K; and is negatively valid relative to a set of belief revision models i is negatively valid relative to each member of the set. Notions of entailment can be associated with positive and negative validity, respectively. Given a set of formulas of a type-L1 or type-L2 language L, and a formula of L, positively entails ( j=+ ) with respect to a primitive belief revision model hK; ; K?; si i 2 s(K ) for every K such that s(K ) and K? 6= K 2 K. By contrast, negatively entails ( j= ) with respect to a primitive belief revision model hK; ; K? ; si i there is no K such that K? 6= K 2 K and [ f:g s(K ). In a Gardenfors pBRM, positive and negative validity coincide: PROPOSITION 18. Relative to any Gardenfors pBRM de ned on a language L of type L2 , a formula of L is positively valid i is negatively valid. Proof. Let hK; ; K?; si be a Gardenfors pBRM de ned on a language L of type L2 , and let be a formula of L. First, suppose that is positively valid relative to hK; ; K?; si and choose an arbitrary K such that K? 6= K 2 K. Then 2 s(K ), but since s(K ) = K 6= K? we have : 62 s(K ), as required. Conversely, assume that is negatively valid relative to hK; ; K? ; si and choose an arbitrary K such that K? 6= K 2 K. Assume for reductio that 62 s(K ). Since s is the identity function, we have that 62 K , in which case K:+ 6= K?. Since K is closed under primitive expansions, we have in addition that K:+ 2 K. Thus, : 2 s(K:+) and K? 6= K:+ 2 K, which is contrary to the negative validity of . Positive and negative validity do not coincide in Arlo-Costa/Levi pBRMs, however. The negative Ramsey test prevents it. Consider the following pair of lemmas regarding the thesis (CS): LEMMA 19. For any type-L1 language L, if , are conditional-free, then ( ^ ) ! ( > ) is negatively valid in an Arlo-Costa/Levi pBRM de ned on L i the model satis es (K 4w). The latter is equivalent to Observation 4.7 of [Arlo-Costa and Levi, 1996]. LEMMA 20. For any type-L1 language L containing at least one atomic formula other than > and ?, there are conditional-free and such that ( ^ ) ! ( > ) is not positively valid in any Arlo-Costa/Levi pBRM de ned on L that satis es (K 3) and contains a primitive belief set K where :; 62 K . Proof. Suppose L is a type-L1 language containing at least one atomic formula dierent from > and ?, and consider an Arlo-Costa/Levi pBRM
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hK; ; K? ; si de ned on L that satis es (K 3) and contains a primitive belief set K such that :; 62 K . Suppose for reductio that ( ^ ) ! ( >
) is positively valid for all conditional-free and . Then, in particular (> ^ ) ! (> > ) is positively valid relative to hK; ; K?; si. By (K 3) we have K> K>+ = K . Since, in addition, :; 62 K we have that :; 62 K> . Since 62 K> , it follows by the Negative Ramsey test that :(> > ) 2 s(K ), hence by the positive validity of (> ^ ) ! (> > ) relative to hK; ; K? ; si we have that :(> ^ ) 2 s(K ). Since :(> ^ ) is conditional-free, it follows that :(> ^ ) 2 K . Since primitive belief sets are deductively closed, we have : 2 K , contrary to assumption. The proof just given is derived from that given by Arlo-Costa for Observation 3.14 in [Arlo-Costa, 1995]. Finally, we state the following obvious but necessary lemma: LEMMA 21. For some type-L1 language L, there is an Arlo-Costa/Levi pBRM de ned on L that satis es (K 3) and (K 4w) and also contains a primitive belief set K where :; 62 K for some conditional-free formula of L. These three lemmas suÆce to show the following: THEOREM 22. There are Arlo-Costa/Levi pBRMs relative to which at least some formulas of the form ( ^ ) ! ( > ) are negatively valid but not positively valid. Interestingly, despite Theorem 22, f ^ g j=+ > holds relative to every Arlo-Costa/Levi pBRM that satis es (K 4w).45 Belief revision models for VC
Gardenfors provides an epistemic semantics for VC based on negative validity. He begins with a minimal conditional logic CM de ned as follows: Axiom schemata
Taut:
All truth-functional tautologies;
CC:
[( > ) ^ ( > )] ! [ > (
CN:
> >.
^ )];
Rules of inference
Modus Ponens From and !
RCM: From 45 See
! to infer ( >
to infer ; ) ! ( > ).
OBSERVATION 3.15 in [Arlo-Costa, 1995].
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Gardenfors [1978] proves a soundness/completeness theorem for CM that is equivalent to the following: THEOREM 23. A formula of any type L2 language is a theorem of CM i it is negatively valid in every Gardenfors pBRM. Gardenfors then proves the following: THEOREM 24. A formula is a theorem of VC i it is derivable from CM together with (ID), (CSO0 ), (CS), (MP), (CA), and (CV) as additional axiom schemata: ID:
An epistemic semantics for VC is obtained by restricting attention to Gardenfors pBRMs that satisfy constraints corresponding to axioms ID, CSO 0, CS, MP, CA, and CV. Gardenfors [1978] proves lemmas equivalent to the following: LEMMA 25. Where M is any Gardenfors pBRM, 1. all instances of ID are negatively valid in M i M satis es (K2); 2. all instances of CSO 0 are negatively valid in M i M satis es (K 6s); 3. all instances of CS are negatively valid in M i M satis es (K 4w);
4. all instances of MP are negatively valid in M i M satis es (K 3); 5. if M satis es (K 2), (K 6s), (K 4w), and (K 3), then all instances of CA are negatively valid in M if M satis es (K 7); 6. if all instances of ID, CSO 0 , CS, and MP are negatively valid in M, then M satis es (K 7) if all instances of CA are negatively valid in M; 7. if M satis es (K 2), (K 6s), (K 4w), and (K 3), then all instances of CV are negatively valid in M if M satis es (K L); 8. if all instances of ID, CSO 0 , CS, and MP are negatively valid in M, then M satis es (K L) if all instances of CV are negatively valid in M.
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Theorem 24 and Lemma 25 allow the soundness and completeness result of Theorem 23 to be extended to yield the following: THEOREM 26. A formula is a theorem of VC i it is negatively valid in all Gardenfors pBRMs that satisfy (K 2), (K3), (K 4w), (K 6s), (K 7), and (K L). This theorem shows that if VC is translated into a theory of belief revision on Gardenfors pBRMs using that version of the Ramsey test which is built into the notion of a Gardenfors pBRM, then the resulting theory of belief revision is de ned by (K 1) (which is built into the de nition of a pBRM), (K 2), (K 3), (K 4w), (K 6s), (K 7), and (K L). The absence of (K 5w) should not be surprising, given Theorem 13. Since in a Gardenfors pBRM (K 3), (K 4w), and (DEF+) imply (K T), and since (K T) together with (DEF+), (K 6s) and (K 8) imply (K P), and given Theorem 8, the absence of (K 8) should not be surprising. A conditional logic that approximates AGM belief revision
Whereas Gardenfors [1978] sets out to nd epistemic models for Lewis's system VC of conditional logic, Arlo-Costa [1995] sets out to nd a system of conditional logic whose primitive belief revision models are de ned at least approximately by the AGM belief revision postulates for transitive relational partial meet contraction (see [Gardenfors, 1988], Chapters 3{4). The result is the system EF , which is de ned only on languages of type L1 (languages of at conditionals). EF has the following axioms and rules, where ; ; and are conditional free: Axiom schemata
Taut:
All truth-functional tautologies
ID:
>
MP:
( > ) ! ( ! )
CC:
[( > ) ^ ( > )] ! [ > (
CA:
[( > ) ^ ( > )] ! [( _ ) > ]
CV:
[( > ) ^ :( > :)] ! [( ^ ) > ]
CN:
>>
CD:
:( > ?) for all non-tautologous .
^ )]
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Rules of inference
Modus Ponens: From and !
to infer .
! to infer ( > ) ! ( > ). From $ to infer ( > ) $ ( > ).
RCM: From RCEA:
One obvious dierence between VC and Arlo-Costa's EF is that EF is de ned for type L1 languages only whereas VC is de ned for type L2 languages. Another dierence is that CS, an axiom of VC, is not a theorem of EF . A third dierence is that CD, an axiom of EF , is not a theorem of VC. Arlo-Costa's epistemic semantics for EF is crucially dierent from Gardenfors' epistemic semantics for VC in that the semantics of EF is de ned in terms of positive validity over Arlo-Costa/Levi pBRMs rather than in terms of negative validity over Gardenfors pBRMs. Positive and negative validity coincide in Gardenfors pBRMs (see Proposition 18) but not in Arlo-Costa/Levi pBRMs (see Theorem 22). Which notion of validity should then be adopted? Arlo-Costa and Levi argue that positive validity should be adopted rather than negative validity both because positive validity is more intuitive and because in Arlo-Costa/Levi models, which satisfy the Negative Ramsey Test favored by Arlo-Costa and Levi, the inference rule modus ponens does not preserve negative validity.46 Consider a type-L1 language L; relative to L the logical system Flat CM is the smallest set of formulas of L that contains all instances of the axiom schemata of Gardenfors' CM and is closed under the rules of CM. Note that EF is an extension of Flat CM. Arlo-Costa [1995] proves the completeness of EF with respect to an epistemic semantics (based on positive validity) by proving a result equivalent to Theorem 31 below. We begin with a series of results to be used as lemmas for Theorem 31:47 THEOREM 27. A formula of any type L1 language L is a theorem of Flat CM i it is positively valid in every Arlo-Costa/Levi pBRM de ned on L. THEOREM 28. Let CM+ be the result of extending Flat CM by adding the rule (RCEA) (restricted to the conditionals of a type-L1 language). A formula is derivable in CM+ i it is positively valid in the class of all ArloCosta/Levi pBRMs that satisfy (K 6). THEOREM 29. Let CMU + be the result of extending CM+ by adding :( > ?) for +every non-tautologous conditional-free . A formula is derivable in CMU i it is positively valid in the class of all Arlo-Costa/Levi pBRMs that satisfy (K 6) and (K C). 46 See [Arl o-Costa and Levi, 1996], pp. 239-240. 47 Our formulation of these results re ects the organization
Levi, 1996].
found in [Arlo-Costa and
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LEMMA 30. Where M is any Arlo-Costa/Levi pBRM,
1. all instances of ID are positively valid in M i M satis es (K 2); 2. all instances of MP are positively valid in M i M satis es (K 3); 3. all instances of CA are positively valid in M i M satis es (K 70 ); 4. if all instances of ID are positively valid in M, then all instances of CV are positively valid in M if M satis es (K 8);
Theorems 27, 28, and 29, together with Lemma 30 and the fact that (K 7) and (K 70 ) are equivalent in any pBRM that satis es (K 2) and (K 6), yield the following completeness theorem for EF :48 THEOREM 31. A formula of any type L1 language L is a theorem of EF i it is positively valid in every Arlo-Costa/Levi pBRM de ned on L satisfying (K 2), (K 3), (K C ), (K 6), (K 7), and (K 8). Postulates (K 1), (K 2), (K 3), (K 4), (K 5), (K 6), (K 7), and (K 8) jointly capture that notion of revision that is derivable via the Levi Identity (LI) from the AGM notion of transitively relational partial meet contraction (AGM Revision , for short).49 Since (K 1) holds in all pBRMs, EF comes very close to capturing AGM Revision, but (K 1) and the postulates mentioned in Theorem 31 de ne a notion of revision (EF Revision, for short) that is strictly weaker than AGM revision in two respects. First, whereas AGM revision includes (K 4), EF revision does not. It turns out that (K 4) does not correspond to the positive validity of any typeL1 formula. Still, (K 4) does correspond to a certain positive entailment, as Arlo-Costa [1995] shows: PROPOSITION 32. If M is an Arlo-Costa/Levi pBRM de ned on a typeL1 language L, then M satis es (K 4) i
f ! ; :(> > :)g j=+ > holds in M for all conditional-free formulas and of L. This result is equivalent to OBSERVATION 3.16 of [Arlo-Costa, 1995]. Note that Proposition 32 does not establish conditions for the positive validity of ( ! ) ! [:(> > :) ! ( > )]: But if nesting of conditionals is allowed, then (K 4) can be associated with the positive validity of nested conditionals of the form [( ! ) ^ :(> > 48 Arl o-Costa [1995] notes that Theorems 27 and 29 and Lemma 30 suÆce to yield completeness theorem for the type-L1 fragment of David Lewis' system VW. 49 See, for example, [G ardenfors, 1988], Chapters 3 and 4.
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:)] > ( >
) (see THEOREM 8.1 and OBSERVATION 8.3 in [ArloCosta, 1995]). In general, [ fg j=+ is not equivalent to j=+ > in an Arlo-Costa/Levi pBRM, but this equivalence does hold for certain and when = ; (see OBSERVATION 3.17 of [Arlo-Costa, 1995]). A second dierence between EF revision and AGM revision is this: where`as AGM Revision includes (K 5), which entails (K 5w), EF revision includes neither (K 5) nor (K 5w) but instead includes (K C). The only difference between (K 5w) and (K C) is that (K 5w) places a constraint on the revision of all belief sets that (K C) places just on the revision of consistent belief sets. In particular, where is nontautologous, (K 5w) requires (K?) to be distinct from K? (and therefore, actually, a contraction of K?), whereas (K C) implies no such requirement. Theorem 29 reveals that (K C) is secured in Arlo-Costa/Levi pBRMs via (pNRT) and the positive validity of negated conditionals of the form :( > ?), where is nontautologous. These negated conditionals also belong to K?, of course, but allowing K to take K? as a value in (pNRT) is not an option. Allowing K to take K? as a value in (pPRT) also does not help: Theorem 13 shows that (K 5w) and a consistent underlying logic cannot be combined with the positive Ramsey test in that case. Still, leaving aside the revision of K? , it is true, as Arlo-Costa [1995] has shown, that AGM revision of nonabsurd belief sets can be speci ed in terms of positive validity in a type-L2 language or in terms of positive validity and positive entailment in a type-L1 language. 3 OTHER TOPICS Our discussion of the major kinds of conditionals is far from exhaustive. We have looked at several dierent approaches to the problem of providing an adequate formal semantics and logic for various kinds of conditionals without being able to demonstrate that one approach is clearly superior to all the others. Furthermore, there are many problems involved in the analysis of conditionals which we either have not discussed at all or have only just mentioned in passing. In this section we will look at several of these, giving each the very briefest attention. One issue which has received much attention is the relationship between conditionals and probability. Stalnaker [1970] proposed that the probability that a conditional is true should be identical with the standard conditional probability. Lewis demonstrates in [Lewis, 1976], however, that this assumption can only be true if we restrict our probability functions to those which assign only a small nite number of distinct values to propositions. Stalnaker [1976] provides a dierent proof for a similar result, a proof which does not depend upon certain assumptions which Lewis used and which some investigators have questioned. Van Fraassen [1976] avoids
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these Triviality Results for a weakened, non-classical version of Stalnaker's conditional logic C2. Lewis [1976] shows, however, that we can embrace a result which resembles Stalnaker's while avoiding the Triviality Result. Lewis's suggestion depends upon a technique which he calls imaging. This technique, which provides an alternative method for determining conditional probabilities, requires that in conditionalising a probability assignment with respect to , i.e. in modifying the assignment in a way which produces a new assignment which assigns probability 1 to , all the probability which was originally assigned to each :-world i would be transferred to the -world closest to i. Lewis demonstrates that if we accept Stalnaker's semantics and if we assign conditional probabilities in this new, non-standard way, then the probability that a conditional is true turns out to be identical with the conditional probability even when the probabilities of truth for conditionals take on in nitely many dierent values. Lewis's imaging techniques can be adapted to semantics other than Stalnaker's. Nute [1980b] adapts Lewis's imaging technique to class selection function semantics, producing a notion of subjunctive probability which diers from both the standard conditional probability and the probability that the corresponding conditional is true. While promising in some ways, Nute's account is extremely cumbersome. Gardenfors [1982] presents a generalized form of imaging and shows that conditional probability cannot be described even in terms of generalized imaging. Other papers on conditionals and probability include [Doring, 1994; Fetzer and Nute, 1979; Fetzer and Nute, 1980; Hajek, 1994; Hall, 1994; Lance, 1991; Lewis, 1981b; Lewis, 1986; McGee, 1989; Nute, 1981a; Stalnaker and Jerey, 1994]. For a careful and comprehensive survey of results relating the probabilities of conditionals to conditional probabilities see [Hajek and Hall, 1994]. The relationship between causation and conditionals has certainly not been overlooked either. Many authors like Jackson [1977] and Kvart [1980; 1986] assign a special role to causation in their analyses of counterfactual conditionals. Others like Lewis [1973a] and Swain [1978] attempt to provide analyses of causation in terms of counterfactual dependence. Still others like Fetzer and Nute [1979; 1980] have tried to develop a semantics for a special kind of causal conditional. These special causal conditionals have then been employed in the formulation of a single-case propensity interpretation of law statements. Conditional logic also has applications in deontic logic (see, for example, [Hilpinen, 1981]), in decision theory (see, for example, [Gibbard and Harper, 1981; Stalnaker, 1981a]), and in nonmonotonic logic (for a summary of some of this work, see [Nute, 1994]). In addition, there has been signi cant attention in recent years to the issue of whether so-called future indicative conditionals (e.g. `If Oswald doesn't shoot President Kennedy, then someone else will') should be classi ed as indicative or as subjunctive (see, for example, [Bennett, 1988; Bennett, 1995; Dudman, 1984; Dudman, 1989;
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Dudman, 1994; Jackson, 1990]). For a careful and comprehensive review of this and other recent topics of discussion see [Edgington, 1995]. It is not possible in this essay to discuss or even to list all of the material that can be found in the literature on conditional logic and its applications. 4 LIST OF SOME IMPORTANT RULES, THESES, AND LOGICS In this section we collect some of the most important rules and theses of conditional logic together with de nitions for a few of the better known conditional logics. Rules RCEC: from $ , to infer ( > ) $ ( > ). RCK:
from (1 ^ : : : ^ n ) ! , to infer [( > 1 )(^ : : : ( > n )] ! ( > ); n 0.
RCEA: from $ , to infer ( > ) RCE:
from ! , to infer > .
RCM:
from
RR:
$ ( > ).
! , to infer ( > ) ! ( > ). from ! ( > ) infer ( Æ ) ! , and from ( Æ ) ! ! ( > ).
Recall that in Section 1 we de ned a conditional logic as any collection L of sentences formed in the usual way from the symbols of classical sentential logic together with a conditional operator >, such that L is closed under modus ponens and L contains every tautology. We now modify this de nition as follows, adopting the terminology of Section 2, to take dierent language types for conditional logic into account: let a conditional logic on a Boolean language L of type L1 or type L2 be any collection L of sentences of L such that L is closed under modus ponens and L contains every tautology. Logics for full conditional languages For a given Boolean language L of type L2 , each of the following is the smallest conditional logic on L closed under all the rules and containing all the theses associated with it below.
CM:
RCM, CC, CN
VW:
RCEC, RCK; ID, MOD, CSO, MP, CV
SS:
RCEC, RCK; ID, MOD, CSO, MP, CA, CS
VC:
RCEC, RCK; ID, MOD, CSO, MP, CV, CS
VCU: RCEC, RCK; ID, MOD, CSO, MP, CV, CS, CT, CU VCU2 : RCEC, RCK, RR; ID, MOD, CSO, MP, CV, CS, CT, CU (with Æ as an additional binary operator) C2:
RCEC, RCK; ID, MOD, CSO, MP, CV, CEM
Neither of VW and SS is an extension of the other, and neither of VCU and C2 is an extension of the other. VCU2 is an extension of VCU, and C2 and VCU are both extensions of VC, which is an extension of both VW and SS. VW and SS are both extensions of CM. For the de nitions
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of several weaker conditional logics, see [Lewis, 1973b; Chellas, 1975; Nute, 1980b]. Logics for languages of \ at" conditionals If L is a Boolean language of type L1 , then each of the following logics is the smallest conditional logic on L closed under all the rules and containing all the theses associated with it below.
Flat CM is contained in Flat VW, which is contained in both Flat VC and EF , but neither of Flat VC and EF is contained in the other. For a discussion of the logic of at conditionals aimed at being as true as possible to Ramsey's ideas, see [Levi, 1996], Chapter 4.
Acknowledgements Sections 1, 3 and 4 are primarily the work of the rst author, but revised from the rst edition of this Handbook with input from the second author. Section 2 is primarily the work of the second author. We are grateful to Lennart Aqvist, Horacio Arlo-Costa, Ermanno Bencivenga, John Burgess, David Butcher, Michael Dunn, Dov Gabbay, Christopher Gauker, Franz Guenthner, Hans Kamp, David Lewis and Christian Rohrer for their helpful comments and suggestions on material contained in this paper. We are also grateful to Richmond Thomason for help in assembling our list of references. Finally, we thank Kluwer and the editor of the Journal of Philosophical Logic for permission to use material from [Nute, 1981b] in this article. Donald Nute University of Georgia, USA. Charles B. Cross University of Georgia, USA. BIBLIOGRAPHY [Adams, 1966] E. Adams. Probability and the logic of conditionals. In J. Hintikka and P. Suppes, editors, Aspects of Inductive Logic. North Holland, Amsterdam, 1966. [Adams, 1975a] E. Adams. Counterfactual conditionals and prior probabilities. In A. Hooker and W. Harper, editors, Proceedings of International Congress on the Foundations of Statistics. Reidel, Dordrecht, 1975.
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[Kratzer, 1981] A. Kratzer. Partition and revision: the semantics of counterfactuals. Journal of Philosophical Logic, 10:201{216, 1981. [Kremer, 1987] M. Kremer. `if' is unambiguous. No^us, 21:199{217, 1987. [Kvart, 1980] I. Kvart. Formal semantics for temporal logic and counterfactuals. Logique et analyse, 23:35{62, 1980. [Kvart, 1986] I. Kvart. A Theory of Counterfactuals. Hackett, Indianapolis, 1986. [Kvart, 1987] I. Kvart. Putnam's counterexample to `A theory of counterfactuals'. Philosophical Papers, 16:235-239, 1987. [Kvart, 1991] I. Kvart. Counterfactuals and causal relevance. Paci c Philosophical Quarterly, 72:314-337, 1991. [Kvart, 1992] I. Kvart. Counterfactuals. Erkenntnis, 36:139-179, 1992. [Kvart, 1994] I. Kvart. Counterfactual ambiguities, true premises and knowledge. Synthese, 100:133-164, 1994. [Lance, 1991] M. Lance. Probabilistic dependence among conditionals. Philosophical Review, 100:269{276, 1991. [Lehmann and Magidor, 1992] D. Lehmann and M. Magidor. What does a conditional knowledge base entail? Arti cial intelligence, 55:1{60, 1992. [Levi, 1977] I. Levi. Subjunctives, dispositions and chances. Synthese, 34:423{455, 1977. [Levi, 1988] I. Levi. Iteration of conditionals and the Ramsey test. Synthese, 76:49{81, 1988. [Levi, 1996] I. Levi. For the Sake of the Argument: Ramsey Test Conditionals, Inductive Inference, and Nonmonotonic Reasoning. Cambridge University Press, Cambridge, England, 1996. [Lewis, 1971] D. Lewis. Completeness and decidability of three logics of counterfactual conditionals. Theoria, 37:74{85, 1971. [Lewis, 1973a] D. Lewis. Causation. Journal of Philosophy, 70:556{567, 1973. [Lewis, 1973b] D. Lewis. Counterfactuals. Harvard, Cambridge, MA, 1973. [Lewis, 1973c] D. Lewis. Counterfactuals and comparative possibility. Journal of Philosophical Logic, 2:418{446, 1973. [Lewis, 1976] D. Lewis. Probabilities of conditionals and conditional probabilities. Philosophical Review, 85:297{315, 1976. [Lewis, 1977] D. Lewis. Possible world semantics for counterfactuals logics: a rejoinder. Journal of Philosophical Logic, 6:359{363, 1977. [Lewis, 1979a] D. Lewis. Counterfactual dependence and time's arrow. No^us, 13:455{ 476, 1979. [Lewis, 1979b] D. Lewis. Scorekeeping in a language game. Journal of Philosophical Logic, 8:339{359, 1979. [Lewis, 1981a] D. Lewis. Ordering semantics and premise semantics for counterfactuals. Journal of Philosophical Logic, 10:217{234, 1981. [Lewis, 1981b] D. Lewis. A subjectivist's guide to objective change. In W. Harper, R. Stalnaker, and G. Pearce, editors, Ifs. Reidel, Dordrecht, 1981. [Lewis, 1986] D. Lewis. Probabilities of conditionals and conditional probabilities II. Philosophical Review, 95:581{589, 1986. [Lindstrom and Rabinowicz, 1992] S. Lindstrom and W. Rabinowicz. Belief revision, epistemic conditionals, and the Ramsey test. Synthese, 91:195{237, 1992. [Lindstrom and Rabinowicz, 1995] S. Lindstrom and W. Rabinowicz. The Ramsey test revisited. In G. Crocco, L. Fari~nas del Cerro, and A. Herzig, editors, Conditionals: From Philosophy to Computer Science. Oxford University Press, Oxford, 1995. [Lindstrom, 1996] S. Lindstrom. The Ramsey test and the indexicality of conditionals: A proposed resolution of Gardenfors' paradox. In A. Fuhrmann and H. Rott, editors, Logic, Action, and Information: Essays on Logic in Philosophy and Arti cial Intelligence. Walter de Gruyter, Berlin, 1996. [Loewer, 1976] B. Loewer. Counterfactuals with disjunctive antecedents. Journal of Philosophy, 73:531{536, 1976. [Loewer, 1978] B. Loewer. Cotenability and counterfactual logics. Journal of Philosophical Logic, 8:99{116, 1978. [Lowe, 1991] E.J. Lowe. Jackson on classifying conditionals. Analysis, 51:126-130, 1991.
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[Makinson, 1989] D. Makinson. General theory of cumulative inference. In M. Reinfrank, J. de Kleer, and M. Ginsberg, editors, Lecture Notes in Arti cial Intelligence: NonMonotonic Reasoning, volume 346. Springer-Verlag, Berlin, 1989. [Makinson, 1990] D. Makinson. The Gardenfors impossibility theorem in nonmonotonic contexts. Studia Logica, 49:1{6, 1990. [Mayer, 1981] J. C. Mayer. A misplaced thesis of conditional logic. Journal of Philosophical Logic, 10:235{238, 1981. [McDermott, 1996] M. McDermott. On the truth conditions of certain `if'-sentences. The Philosophical Review, 105:1{37, 1996. [Mcgee, 1981] V. Mcgee. Finite matrices and the logic of conditionals. Journal of Philosophical Logic, 10:349{351, 1981. [McGee, 1985] V. McGee. A counterexample to modus ponens. Journal of Philosophy, 82:462{471, 1985. [McGee, 1989] V. McGee. Conditional probabilities and compounds of conditionals. Philosophical Review, 98:485{541, 1989. [McGee, 2000] V. McGee. To tell the truth about conditionals. Analysis, 60:107-111, 2000. [McKay and Inwagen, 1977] T. McKay and P. Van Inwagen. Counterfactuals with disjunctive antecedents. Philosophical Studies, 31:353{356, 1977. [Mellor, 1993] D.H. Mellor. How to believe a conditional. Journal of Philosophy, 90:233248, 1993.` [Moore, 1983] R. Moore. Semantical considerations on nonmonotonic logic. In Proceedings of the Eighth International Joint Conference on Arti cial Intelligence, volume 1. Morgan Kaufman, San Mateo, 1983. [Morreau, 1992] M. Morreau. Epistemic semantics for counterfactuals. Journal of Philosophical Logic, 21:33{62, 1992. [Morreau, 1997] M. Morreau. Fainthearted conditionals. The Journal of Philosophy, 94:187{211, 1997. [Nute and Mitcheltree, 1982] D. Nute and W. Mitcheltree. Review of [Adams, 1975b]. No^us, 15:432{436, 1982. [Nute, 1975a] D. Nute. Counterfactuals. Notre Dame Journal of Formal Logic, 16:476{ 482, 1975. [Nute, 1975b] D. Nute. Counterfactuals and the similarity of worlds. Journal of Philosophy, 72:73{778, 1975. [Nute, 1977] D. Nute. Scienti c law and nomological conditionals. Technical report, National Science Foundation, 1977. [Nute, 1978a] D. Nute. An incompleteness theorem for conditional logic. Notre Dame Journal of Formal Logic, 19:634{636, 1978. [Nute, 1978b] D. Nute. Simpli cation and substitution of counterfactual antecedents. Philosophia, 7:317{326, 1978. [Nute, 1979] D. Nute. Algebraic semantics for conditional logics. Reports on Mathematical Logic, 10:79{101, 1979. [Nute, 1980a] D. Nute. Conversational scorekeeping and conditionals. Journal of Philosophical Logic, 9:153{166, 1980. [Nute, 1980b] D. Nute. Topics in Conditional Logic. Reidel, Dordrecht, 1980. [Nute, 1981a] D. Nute. Causes, laws and law statements. Synthese, 48:347{370, 1981. [Nute, 1981b] D. Nute. Introduction. Journal of Philosophical Logic, 10:127{147, 1981. [Nute, 1981c] D. Nute. Review of [Pollock, 1976]. No^us, 15:212{219, 1981. [Nute, 1982 and 1991] D. Nute. Tense and conditionals. Technical report, Deutsche Forschungsgemeinschaft and the University of Georgia, 1982 and 1991. [Nute, 1983] D. Nute. Review of [Harper et al., 1981]. Philosophy of Science, 50:518{520, 1983. [Nute, 1991] D. Nute. Historical necessity and conditionals. No^us, 25:161{175, 1991. [Nute, 1994] D. Nute. Defeasible logic. In D. Gabbay and C. Hogger, editors, Handbook of Logic for Arti cial Intelligence and Logic Programming, volume III. Oxford University Press, Oxford, 1994.
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[Pearl, 1994] J. Pearl. From Adams' conditionals to default expressions, causal conditionals, and counterfactuals. In E. Eells and B. Skyrms, editors, Probability and Conditionals: Belief Revision and Rational Decision. Cambridge University Press, Cambridge, England, 1994. [Pearl, 1995] J. Pearl. Causation, action, and counterfactuals. In A. Gammerman, editor, Computational Learning and Probabilistic Learning. John Wiley and Sons, New York, 1995. [Pollock, 1976] J. Pollock. Subjunctive Reasoning. Reidel, Dordrecht, 1976. [Pollock, 1981] J. Pollock. A re ned theory of counterfactuals. Journal of Philosophical Logic, 10:239{266, 1981. [Pollock, 1984] J. Pollock. Knowledge and Justi cation. Princeton University Press, Princeton, 1984. [Posch, 1980] G. Posch. Zur Semantik der Kontrafaktischen Konditionale. Narr, Tuebingen, 1980. [Post, 1981] J. Post. Review of [Pollock, 1976]. Philosophia, 9:405{420, 1981. [Ramsey, 1990] F. Ramsey. Philosophical Papers. Cambridge University Press, Cambridge, England, 1990. [Rescher, 1964] N. Rescher. Hypothetical Reasoning. Reidel, Dordrecht, 1964. [Rott, 1986] H. Rott. Ifs, though, and because. Erkenntnis, 25:345{370, 1986. [Rott, 1989] H. Rott. Conditionals and theory change: revisions, expansions, and additions. Synthese, 81:91{113, 1989. [Rott, 1991] H. Rott. A nonmonotonic conditional logic for belief revision. In A. Fuhrmann and M. Morreau, editors, The Logic of Theory Change. Cambridge University Press, Cambridge, England, 1991. [Sanford, 1992] D. Sanford. If P then Q. Routledge, London, 1992. [Schlechta and Makinson, 1994] K. Schlechta and D. Makinson. Local and global metrics for the semantics of counterfactual conditionals. Journal of applied Non-Classical Logics, 4:129{140, 1994. [Segerberg, 1968] K. Segerberg. Propositional logics related to Heyting's and Johansson's. Theoria, 34:26{61, 1968. [Segerberg, 1989] K. Segerberg. A note on an impossibility theorem of Gardenfors. No^us, 23:351{354, 1989. [Sellars, 1958] W. S. Sellars. Counterfactuals, dispositions and the causal modalities. In Feigl, Scriven, and Maxwell, editors, Minnesota Studies in the Philosophy of Science, volume 2. University of Minnesota, Minneapolis, 1958. [Slote, 1978] M. A. Slote. Time and counterfactuals. Philosophical Review, 87:3{27, 1978. [Stalnaker and Jerey, 1994] R. Stalnaker and R. Jerey. Conditionals as random variables. In E. Eells and B. Skyrms, editors, Probability and Conditionals. Cambridge University Press, Cambridge, England, 1994. [Stalnaker and Thomason, 1970] R. Stalnaker and R. Thomason. A semantical analysis of conditional logic. Theoria, 36:23{42, 1970. [Stalnaker, 1968] R. Stalnaker. A theory of conditionals. In N. Rescher, editor, Studies in Logical Theory. American Philosophical quarterly Monograph Series, No. 2, Blackwell, Oxford, 1968. Reprinted in [Harper et al., 1981]. [Stalnaker, 1970] R. Stalnaker. Probabilities and conditionals. Philosophy of Science, 28:64{80, 1970. [Stalnaker, 1975] R. Stalnaker. Indicative conditionals. Philosophia, 5:269{286, 1975. [Stalnaker, 1976] R. Stalnaker. Stalnaker to Van Fraassen. In C. Hooker and W. Harper, editors, Foundations of Probability Theory, Statistical Inference and Statistical Theories of Science. Reidel, Dordrecht, 1976. [Stalnaker, 1981a] R. Stalnaker. A defense of conditional excluded middle. In W. Harper, R. Stalnaker, and G. Pearce, editors, Ifs. Reidel, Dordrecht, 1981. [Stalnaker, 1981b] R. Stalnaker. Letter to David Lewis. In W. Harper, R. Stalnaker, and G. Pearce, editors, Ifs. Reidel, Dordrecht, 1981. [Stalnaker, 1984] R. Stalnaker. Inquiry. MIT Press, Cambridge, MA, 1984. [Swain, 1978] M. Swain. A counterfactual analysis of event causation. Philosophical Studies, 34:1{19, 1978.
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[Thomason and Gupta, 1981] R. Thomason and A. Gupta. A theory of conditionals in the context of branching time. In W. Harper, R. Stalnaker, and G. Pearce, editors, Ifs. Reidel, Dordrecht, 1981. [Thomason, 1985] R. Thomason. Note on tense and subjunctive conditionals. Philosophy of Science, pages 151{153, 1985. [Traugott et al., 1986] E. Traugott, A. ter Meulen, J. Reilly, and C. Ferguson, editors. On Conditionals. Cambridge University Press, Cambridge, England, 1986. [Turner, 1981] R. Turner. Counterfactuals without possible worlds. Journal of Philosophical Logic, 10:453{493, 1981. [van Benthem, 1984] J. van Benthem. Foundations of conditional logic. Journal of Philosophical Logic, 13:303{349, 1984. [Van Fraassen, 1974] B. C. Van Fraassen. Hidden variables in conditional logic. Theoria, 40:176{190, 1974. [Van Fraassen, 1976] B. C. Van Fraassen. Probabilities of conditionals. In C. Hooker and W. Harper, editors, Foundations of Probability Theory, Statistical Inference, and Statistical Theories of Science. Reidel, Dordrecht, 1976. [Van Fraassen, 1981] B. C. Van Fraassen. A temporal framework for conditionals and chance. In W. Harper, R. Stalnaker, and G. Pearce, editors, Ifs. Reidel, Dordrecht, 1981. [Veltman, 1976] F. Veltman. Prejudices, presuppositions and the theory of conditionals. In J. Groenendijk and M. Stokhof, editors, Amsterdam Papers in Formal Grammar. Vol. 1, Centrale Interfaculteit, Universiteit van Amsterdam, 1976. [Veltman, 1985] Frank Veltman. Logics for Conditionals. Ph.D. dissertation, University of Amsterdam, Amsterdam, 1985. [Warmbrod, 1981] Warmbrod. Counterfactuals and substitution of equivalent antecedents. Journal of Philosophical Logic, 10:267{289, 1981. [Winslett, 1990] M. Winslett. Updating Logical Databases. Cambridge University Press, Cambridge, England, 1990. [Woods, 1997] M. Woods. Conditionals. Oxford University Press, Oxford, 1997. Published posthumously. Edited by D. Wiggins, with a commentary by D. Edgington.
DAVID HAREL, DEXTER KOZEN, AND JERZY TIURYN
DYNAMIC LOGIC PREFACE Dynamic Logic (DL) is a formal system for reasoning about programs. Traditionally, this has meant formalizing correctness speci cations and proving rigorously that those speci cations are met by a particular program. Other activities fall into this category as well: determining the equivalence of programs, comparing the expressive power of various programming constructs, synthesizing programs from speci cations, etc. Formal systems too numerous to mention have been proposed for these purposes, each with its own peculiarities. DL can be described as a blend of three complementary classical ingredients: rst-order predicate logic, modal logic, and the algebra of regular events. These components merge to form a system of remarkable unity that is theoretically rich as well as practical. The name Dynamic Logic emphasizes the principal feature distinguishing it from classical predicate logic. In the latter, truth is static : the truth value of a formula ' is determined by a valuation of its free variables over some structure. The valuation and the truth value of ' it induces are regarded as immutable; there is no formalism relating them to any other valuations or truth values. In Dynamic Logic, there are explicit syntactic constructs called programs whose main role is to change the values of variables, thereby changing the truth values of formulas. For example, the program x := x + 1 over the natural numbers changes the truth value of the formula \x is even". Such changes occur on a metalogical level in classical predicate logic. For example, in Tarski's de nition of truth of a formula, if u : fx; y; : : : g ! N is a valuation of variables over the natural numbers N , then the formula 9x x2 = y is de ned to be true under the valuation u i there exists an a 2 N such that the formula x2 = y is true under the valuation u[x=a], where u[x=a] agrees with u everywhere except x, on which it takes the value a. This de nition involves a metalogical operation that produces u[x=a] from u for all possible values a 2 N . This operation becomes explicit in DL in the form of the program x := ?, called a nondeterministic or wildcard assignment . This is a rather unconventional program, since it is not eective; however, it is quite useful as a descriptive tool. A more conventional way to obtain a square root of y, if it exists, would be the program (1) x := 0 ; while x2 < y do x := x + 1:
In DL, such programs are rst-class objects on a par with formulas, complete with a collection of operators for forming compound programs inductively
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from a basis of primitive programs. To discuss the eect of the execution of a program on the truth of a formula ', DL uses a modal construct <>', which intuitively states, \It is possible to execute starting from the current state and halt in a state satisfying '." There is also the dual construct []', which intuitively states, \If halts when started in the current state, then it does so in a state satisfying '." For example, the rst-order formula 9x x2 = y is equivalent to the DL formula x2 = y. In order to instantiate the quanti er eectively, we might replace the nondeterministic assignment inside the < > with the while program (1); over N , the two formulas would be equivalent. Apart from the obvious heavy reliance on classical logic, computability theory and programming, the subject has its roots in the work of [Thiele, 1966] and [Engeler, 1967] in the late 1960's, who were the rst to advance the idea of formulating and investigating formal systems dealing with properties of programs in an abstract setting. Research in program veri cation
ourished thereafter with the work of many researchers, notably [Floyd, 1967], [Hoare, 1969], [Manna, 1974], and [Salwicki, 1970]. The rst precise development of a DL-like system was carried out by [Salwicki, 1970], following [Engeler, 1967]. This system was called Algorithmic Logic. A similar system, called Monadic Programming Logic, was developed by [Constable, 1977]. Dynamic Logic, which emphasizes the modal nature of the program/assertion interaction, was introduced by [Pratt, 1976]. Background material on mathematical logic, computability, formal languages and automata, and program veri cation can be found in [Shoen eld, 1967] (logic), [Rogers, 1967] (recursion theory), [Kozen, 1997a] (formal languages, automata, and computability), [Keisler, 1971] (in nitary logic), [Manna, 1974] (program veri cation), and [Harel, 1992; Lewis and Papadimitriou, 1981; Davis et al., 1994] (computability and complexity). Much of this introductory material as it pertains to DL can be found in the authors' text [Harel et al., 2000]. There are by now a number of books and survey papers treating logics of programs, program veri cation, and Dynamic Logic [Apt and Olderog, 1991; Backhouse, 1986; Harel, 1979; Harel, 1984; Parikh, 1981; Goldblatt, 1982; Goldblatt, 1987; Knijnenburg, 1988; Cousot, 1990; Emerson, 1990; Kozen and Tiuryn, 1990]. In particular, much of this chapter is an abbreviated summary of material from the authors' text [Harel et al., 2000], to which we refer the reader for a more complete treatment. Full proofs of many of the theorems cited in this chapter can be found there, as well as extensive introductory material on logic and complexity along with numerous examples and exercises.
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1 REASONING ABOUT PROGRAMS
1.1 Programs For us, a program is a recipe written in a formal language for computing desired output data from given input data. EXAMPLE 1. The following program implements the Euclidean algorithm for calculating the greatest common divisor (gcd) of two integers. It takes as input a pair of integers in variables x and y and outputs their gcd in variable x: while y 6= 0 do begin z := x mod y; x := y; y := z end The value of the expression x mod y is the (nonnegative) remainder obtained when dividing x by y using ordinary integer division. Programs normally use variables to hold input and output values and intermediate results. Each variable can assume values from a speci c domain of computation , which is a structure consisting of a set of data values along with certain distinguished constants, basic operations, and tests that can be performed on those values, as in classical rst-order logic. In the program above, the domain of x, y, and z might be the integers Z along with basic operations including integer division with remainder and tests including 6=. In contrast with the usual use of variables in mathematics, a variable in a program normally assumes dierent values during the course of the computation. The value of a variable x may change whenever an assignment x := t is performed with x on the left-hand side. In order to make these notions precise, we will have to specify the programming language and its semantics in a mathematically rigorous way. In this section we give a brief introduction to some of these languages and the role they play in program veri cation.
1.2 States and Executions As mentioned above, a program can change the values of variables as it runs. However, if we could freeze time at some instant during the execution of the program, we could presumably read the values of the variables at that instant, and that would give us an instantaneous snapshot of all information that we would need to determine how the computation would proceed from that point. This leads to the concept of a state |intuitively, an instantaneous description of reality.
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Formally, we will de ne a state to be a function that assigns a value to each program variable. The value for variable x must belong to the domain associated with x. In logic, such a function is called a valuation . At any given instant in time during its execution, the program is thought to be \in" some state, determined by the instantaneous values of all its variables. If an assignment statement is executed, say x := 2, then the state changes to a new state in which the new value of x is 2 and the values of all other variables are the same as they were before. We assume that this change takes place instantaneously; note that this is a mathematical abstraction, since in reality basic operations take some time to execute. A typical state for the gcd program above is (15; 27; 0; : : : ), where (say) the rst, second, and third components of the sequence denote the values assigned to x, y, and z respectively. The ellipsis \: : : " refers to the values of the other variables, which we do not care about, since they do not occur in the program. A program can be viewed as a transformation on states. Given an initial (input) state, the program will go through a series of intermediate states, perhaps eventually halting in a nal (output) state. A sequence of states that can occur from the execution of a program starting from a particular input state is called a trace . As a typical example of a trace for the program above, consider the initial state (15; 27; 0) (we suppress the ellipsis). The program goes through the following sequence of states: (15,27,0), (15,27,15), (27,27,15), (27,15,15), (27,15,12), (15,15,12), (15,12,12), (15,12,3), (12,12,3), (12,3,3), (12,3,0), (3,3,0), (3,0,0). The value of x in the last (output) state is 3, the gcd of 15 and 27. The binary relation consisting of the set of all pairs of the form (input state, output state) that can occur from the execution of a program , or in other words, the set of all rst and last states of traces of , is called the input/output relation of . For example, the pair ((15; 27; 0); (3; 0; 0)) is a member of the input/output relation of the gcd program above, as is the pair (( 6; 4; 303); (2; 0; 0)). The values of other variables besides x, y, and z are not changed by the program. These values are therefore the same in the output state as in the input state. In this example, we may think of the variables x and y as the input variables , x as the output variable , and z as a work variable , although formally there is no distinction between any of the variables, including the ones not occurring in the program.
1.3 Programming Constructs In subsequent sections we will consider a number of programming constructs. In this section we introduce some of these constructs and de ne a few general classes of languages built on them.
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In general, programs are built inductively from atomic programs and tests using various program operators .
While Programs A popular choice of programming language in the literature on DL is the family of deterministic while programs. This language is a natural abstraction of familiar imperative programming languages such as Pascal or C. Dierent versions can be de ned depending on the choice of tests allowed and whether or not nondeterminism is permitted. The language of while programs is de ned inductively. There are atomic programs and atomic tests, as well as program constructs for forming compound programs from simpler ones. In the propositional version of Dynamic Logic (PDL), atomic programs are simply letters a; b; : : : from some alphabet. Thus PDL abstracts away from the nature of the domain of computation and studies the pure interaction between programs and propositions. For the rst-order versions of DL, atomic programs are simple assignments x := t, where x is a variable and t is a term. In addition, a nondeterministic or wildcard assignment x := ? or nondeterministic choice construct may be allowed. Tests can be atomic tests , which for propositional versions are simply propositional letters p, and for rst-order versions are atomic formulas p(t1 ; : : : ; tn ), where t1 ; : : : ; tn are terms and p is an n-ary relation symbol in the vocabulary of the domain of computation. In addition, we include the constant tests 1 and 0. Boolean combinations of atomic tests are often allowed, although this adds no expressive power. These versions of DL are called poor test . More complicated tests can also be included. These versions of DL are sometimes called rich test . In rich test versions, the families of programs and tests are de ned by mutual induction. Compound programs are formed from the atomic programs and tests by induction, using the composition , conditional , and while operators. Formally, if ' is a test and and are programs, then the following are programs:
; if ' then else while ' do . We can also parenthesize with begin : : : end where necessary. The gcd program of Example 1 above is an example of a while program. The semantics of these constructs is de ned to correspond to the ordinary operational semantics familiar from common programming languages.
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Regular Programs Regular programs are more general than while programs, but not by much. The advantage of regular programs is that they reduce the relatively more complicated while program operators to much simpler constructs. The deductive system becomes comparatively simpler too. They also incorporate a simple form of nondeterminism. For a given set of atomic programs and tests, the set of regular programs is de ned as follows:
(i) any atomic program is a program (ii) if ' is a test, then '? is a program (iii) if and are programs, then ; is a program; (iv) if and are programs, then [ is a program; (v) if is a program, then is a program.
These constructs have the following intuitive meaning: (i) Atomic programs are basic and indivisible; they execute in a single step. They are called atomic because they cannot be decomposed further. (ii) The program '? tests whether the property ' holds in the current state. If so, it continues without changing state. If not, it blocks without halting. (iii) The operator ; is the sequential composition operator. The program ; means, \Do , then do ." (iv) The operator [ is the nondeterministic choice operator. The program [ means, \Nondeterministically choose one of or and execute it." (v) The operator is the iteration operator. The program means, \Execute some nondeterministically chosen nite number of times."
Keep in mind that these descriptions are meant only as intuitive aids. A formal semantics will be given in Section 2.2, in which programs will be interpreted as binary input/output relations and the programming constructs above as operators on binary relations. The operators [; ; ; may be familiar from automata and formal language theory (see [Kozen, 1997a]), where they are interpreted as operators on sets of strings over a nite alphabet. The language-theoretic and relationtheoretic semantics share much in common; in fact, they have the same equational theory, as shown in [Kozen, 1994a].
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The operators of deterministic while programs can be de ned in terms of the regular operators: (2) if ' then else (3) while ' do
def = '? ; [ :'? ; def = ('? ; ) ; :'?
The class of while programs is equivalent to the subclass of the regular programs in which the program operators [, ?, and are constrained to appear only in these forms. Recursion Recursion can appear in programming languages in several forms. Two such manifestations are recursive calls and stacks . Under certain very general conditions, the two constructs can simulate each other. It can also be shown that recursive programs and while programs are equally expressive over the natural numbers, whereas over arbitrary domains, while programs are strictly weaker. While programs correspond to what is often called tail recursion or iteration. R.E. Programs
A nite computation sequence of a program , or seq for short, is a nitelength string of atomic programs and tests representing a possible sequence of atomic steps that can occur in a halting execution of . Seqs are denoted ; ; : : : . The set of all seqs of a program is denoted CS (). We use the word \possible" loosely|CS () is determined by the syntax of alone. Because of tests that evaluate to false, CS () may contain seqs that are never executed under any interpretation. The set CS () is a subset of A , where A is the set of atomic programs and tests occurring in . For while programs, regular programs, or recursive programs, we can de ne the set CS () formally by induction on syntax. For example, for regular programs, CS (a) CS (skip) CS (fail) CS ( ; ) CS ( [ ) CS ( )
For example, if a is an atomic program and p an atomic formula, then the program
while p do a = (p? ; a) ; :p?
has as seqs all strings of the form (p? ; a)n ; :p? = p| ?; a; p?; a{z; ; p?; a}; :p? n
for all n 0. Note that each seq of a program is itself a program, and CS () = fg:
While programs and regular programs give rise to regular sets of seqs, and recursive programs give rise to context-free sets of seqs. Taking this a step further, we can de ne an r.e. program to be simply a recursively enumerable set of seqs. This is the most general programming language we will consider in the context of DL; it subsumes all the others in expressive power. Nondeterminism
We should say a few words about the concept of nondeterminism and its role in the study of logics and languages, since this concept often presents diÆculty the rst time it is encountered. In some programming languages we will consider, the traces of a program need not be uniquely determined by their start states. When this is possible, we say that the program is nondeterministic. A nondeterministic program can have both divergent and convergent traces starting from the same input state, and for such programs it does not make sense to say that the program halts on a certain input state or that it loops on a certain input state; there may be dierent computations starting from the same input state that do each. There are several concrete ways nondeterminism can enter into programs. One construct is the nondeterministic or wildcard assignment x := ?. Intuitively, this operation assigns an arbitrary element of the domain to the variable x, but it is not determined which one.1 Another source of nondeterminism is the unconstrained use of the choice operator [ in regular 1 This construct is often called random assignment in the literature. This terminology is misleading, because it has nothing at all to do with probability.
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programs. A third source is the iteration operator in regular programs. A fourth source is r.e. programs, which are just r.e. sets of seqs; initially, the seq to execute is chosen nondeterministically. For example, over N , the r.e. program
fx := n j n 0g is equivalent to the regular program
x := 0 ; (x := x + 1) :
Nondeterministic programs provide no explicit mechanism for resolving the nondeterminism. That is, there is no way to determine which of many possible next steps will be taken from a given state. This is hardly realistic. So why study nondeterminism at all if it does not correspond to anything operational? One good answer is that nondeterminism is a valuable tool that helps us understand the expressiveness of programming language constructs. It is useful in situations in which we cannot necessarily predict the outcome of a particular choice, but we may know the range of possibilities. In reality, computations may depend on information that is out of the programmer's control, such as input from the user or actions of other processes in the system. Nondeterminism is useful in modeling such situations. The importance of nondeterminism is not limited to logics of programs. Indeed, the most important open problem in the eld of computational complexity theory, the P =NP problem, is formulated in terms of nondeterminism.
1.4 Program Veri cation Dynamic Logic and other program logics are meant to be useful tools for facilitating the process of producing correct programs. One need only look at the miasma of buggy software to understand the dire need for such tools. But before we can produce correct software, we need to know what it means for it to be correct. It is not good enough to have some vague idea of what is supposed to happen when a program is run or to observe it running on some collection of inputs. In order to apply formal veri cation tools, we must have a formal speci cation of correctness for the veri cation tools to work with. In general, a correctness speci cation is a formal description of how the program is supposed to behave. A given program is correct with respect to a correctness speci cation if its behavior ful lls that speci cation. For the gcd program of Example 1, the correctness might be speci ed informally by the assertion If the input values of x and y are positive integers c and d, respectively, then
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(i) the output value of x is the gcd of c and d, and (ii) the program halts. Of course, in order to work with a formal veri cation system, these properties must be expressed formally in a language such as rst-order logic. The assertion (ii) is part of the correctness speci cation because programs do not necessarily halt, but may produce in nite traces for certain inputs. A nite trace, as for example the one produced by the gcd program above on input state (15,27,0), is called halting, terminating, or convergent. In nite traces are called looping or divergent. For example, the program
while x > 7 do x := x + 3 loops on input state (8; : : : ), producing the in nite trace (8; : : : ); (11; : : : ); (14; : : : ); : : : Dynamic Logic can reason about the behavior of a program that is manifested in its input/output relation. It is not well suited to reasoning about program behavior manifested in intermediate states of a computation (although there are close relatives, such as Process Logic and Temporal Logic, that are). This is not to say that all interesting program behavior is captured by the input/output relation, and that other types of behavior are irrelevant or uninteresting. Indeed, the restriction to input/output relations is reasonable only when programs are supposed to halt after a nite time and yield output results. This approach will not be adequate for dealing with programs that normally are not supposed to halt, such as operating systems. For programs that are supposed to halt, correctness criteria are traditionally given in the form of an input/output speci cation consisting of a formal relation between the input and output states that the program is supposed to maintain, along with a description of the set of input states on which the program is supposed to halt. The input/output relation of a program carries all the information necessary to determine whether the program is correct relative to such a speci cation. Dynamic Logic is well suited to this type of veri cation. It is not always obvious what the correctness speci cation ought to be. Sometimes, producing a formal speci cation of correctness is as diÆcult as producing the program itself, since both must be written in a formal language. Moreover, speci cations are as prone to bugs as programs. Why bother then? Why not just implement the program with some vague speci cation in mind? There are several good reasons for taking the eort to produce formal speci cations:
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1. Often when implementing a large program from scratch, the programmer may have been given only a vague idea of what the nished product is supposed to do. This is especially true when producing software for a less technically inclined employer. There may be a rough informal description available, but the minor details are often left to the programmer. It is very often the case that a large part of the programming process consists of taking a vaguely speci ed problem and making it precise. The process of formulating the problem precisely can be considered a de nition of what the program is supposed to do. And it is just good programming practice to have a very clear idea of what we want to do before we start doing it. 2. In the process of formulating the speci cation, several unforeseen cases may become apparent, for which it is not clear what the appropriate action of the program should be. This is especially true with error handling and other exceptional situations. Formulating a speci cation can de ne the action of the program in such situations and thereby tie up loose ends. 3. The process of formulating a rigorous speci cation can sometimes suggest ideas for implementation, because it forces us to isolate the issues that drive design decisions. When we know all the ways our data are going to be accessed, we are in a better position to choose the right data structures that optimize the tradeos between eÆciency and generality. 4. The speci cation is often expressed in a language quite dierent from the programming language. The speci cation is functional |it tells what the program is supposed to do|as opposed to imperative |how to do it. It is often easier to specify the desired functionality independent of the details of how it will be implemented. For example, we can quite easily express what it means for a number x to be the gcd of y and z in rst-order logic without even knowing how to compute it. 5. Verifying that a program meets its speci cation is a kind of sanity check. It allows us to give two solutions to the problem|once as a functional speci cation, and once as an algorithmic implementation| and lets us verify that the two are compatible. Any incompatibilities between the program and the speci cation are either bugs in the program, bugs in the speci cation, or both. The cycle of re ning the speci cation, modifying the program to meet the speci cation, and reverifying until the process converges can lead to software in which we have much more con dence.
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Partial and Total Correctness
Typically, a program is designed to implement some functionality. As mentioned above, that functionality can often be expressed formally in the form of an input/output speci cation. Concretely, such a speci cation consists of an input condition or precondition ' and an output condition or postcondition . These are properties of the input state and the output state, respectively, expressed in some formal language such as the rst-order language of the domain of computation. The program is supposed to halt in a state satisfying the output condition whenever the input state satis es the input condition. We say that a program is partially correct with respect to a given input/output speci cation '; if, whenever the program is started in a state satisfying the input condition ', then if and when it ever halts, it does so in a state satisfying the output condition . The de nition of partial correctness does not stipulate that the program halts; this is what we mean by partial. A program is totally correct with respect to an input/output speci cation '; if
it is partially correct with respect to that speci cation; and
it halts whenever it is started in a state satisfying the input condition '.
The input/output speci cation imposes no requirements when the input state does not satisfy the input condition '|the program might as well loop in nitely or erase memory. This is the \garbage in, garbage out" philosophy. If we really do care what the program does on some of those input states, then we had better rewrite the input condition to include them and say formally what we want to happen in those cases. For example, in the gcd program of Example 1, the output condition might be the condition (i) stating that the output value of x is the gcd of the input values of x and y. We can express this completely formally in the language of rst-order number theory. We may try to start o with the input speci cation '0 = 1 (true ); that is, no restrictions on the input state at all. Unfortunately, if the initial value of y is 0 and x is negative, the nal value of x will be the same as the initial value, thus negative. If we expect all gcds to be positive, this would be wrong. Another problematic situation arises when the initial values of x and y are both 0; in this case the gcd is not de ned. Therefore, the program as written is not partially correct with respect to the speci cation '0 ; . We can remedy the situation by providing an input speci cation that rules out these troublesome input values. We can limit the input states to those in which x and y are both nonnegative and not both zero by taking
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the input speci cation
'1 = (x 0 ^ y > 0)
_ (x > 0 ^ y 0):
The gcd program of Example 1 above would be partially correct with respect to the speci cation '1 ; . It is also totally correct, since the program halts on all inputs satisfying '1 . Perhaps we want to allow any input in which not both x and y are zero. In that case, we should use the input speci cation '2 = :(x = 0 ^ y = 0). But then the program of Example 1 is not partially correct with respect to '2 ; ; we must amend the program to produce the correct (positive) gcd on negative inputs.
1.5 Exogenous and Endogenous Logics There are two main approaches to modal logics of programs: the exogenous approach, exempli ed by Dynamic Logic and its precursor Hoare Logic [Hoare, 1969], and the endogenous approach, exempli ed by Temporal Logic and its precursor, the invariant assertions method of [Floyd, 1967]. A logic is exogenous if its programs are explicit in the language. Syntactically, a Dynamic Logic program is a well-formed expression built inductively from primitive programs using a small set of program operators. Semantically, a program is interpreted as its input/output relation. The relation denoted by a compound program is determined by the relations denoted by its parts. This aspect of compositionality allows analysis by structural induction. The importance of compositionality is discussed in [van Emde Boas, 1978]. In Temporal Logic, the program is xed and is considered part of the structure over which the logic is interpreted. The current location in the program during execution is stored in a special variable for that purpose, called the program counter, and is part of the state along with the values of the program variables. Instead of program operators, there are temporal operators that describe how the program variables, including the program counter, change with time. Thus Temporal Logic sacri ces compositionality for a less restricted formalism. We discuss Temporal Logic further in Section 14.2. 2 PROPOSITIONAL DYNAMIC LOGIC (PDL) Propositional Dynamic Logic (PDL) plays the same role in Dynamic Logic that classical propositional logic plays in classical predicate logic. It describes the properties of the interaction between programs and propositions that are independent of the domain of computation. Since PDL is a subsystem of rst-order DL, we can be sure that all properties of PDL that we discuss in this section will also be valid in rst-order DL.
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Since there is no domain of computation in PDL, there can be no notion of assignment to a variable. Instead, primitive programs are interpreted as arbitrary binary relations on an abstract set of states K . Likewise, primitive assertions are just atomic propositions and are interpreted as arbitrary subsets of K . Other than this, no special structure is imposed. This level of abstraction may at rst appear too general to say anything of interest. On the contrary, it is a very natural level of abstraction at which many fundamental relationships between programs and propositions can be observed. For example, consider the PDL formula (4) [](' ^ )
$ []' ^ [] :
The left-hand side asserts that the formula ' ^ must hold after the execution of program , and the right-hand side asserts that ' must hold after execution of and so must . The formula (4) asserts that these two statements are equivalent. This implies that to verify a conjunction of two postconditions, it suÆces to verify each of them separately. The assertion (4) holds universally, regardless of the domain of computation and the nature of the particular , ', and . As another example, consider (5) [ ; ]'
$ [][ ]':
The left-hand side asserts that after execution of the composite program ; , ' must hold. The right-hand side asserts that after execution of the program , [ ]' must hold, which in turn says that after execution of , ' must hold. The formula (5) asserts the logical equivalence of these two statements. It holds regardless of the nature of , , and '. Like (4), (5) can be used to simplify the veri cation of complicated programs. As a nal example, consider the assertion (6) []p
$ [ ]p
where p is a primitive proposition symbol and and are programs. If this formula is true under all interpretations, then and are equivalent in the sense that they behave identically with respect to any property expressible in PDL or any formal system containing PDL as a subsystem. This is because the assertion will hold for any substitution instance of (6). For example, the two programs
= if ' then else Æ = if :' then Æ else are equivalent in the sense of (6).
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2.1 Syntax Syntactically, PDL is a blend of three classical ingredients: propositional logic, modal logic, and the algebra of regular expressions. There are several versions of PDL, depending on the choice of program operators allowed. In this section we will introduce the basic version, called regular PDL. Variations of this basic version will be considered in later sections. The language of regular PDL has expressions of two sorts: propositions or formulas '; ; : : : and programs ; ; ; : : : . There are countably many atomic symbols of each sort. Atomic programs are denoted a; b; c; : : : and the set of all atomic programs is denoted 0 . Atomic propositions are denoted p; q; r; : : : and the set of all atomic propositions is denoted 0 . The set of all programs is denoted and the set of all propositions is denoted . Programs and propositions are built inductively from the atomic ones using the following operators: Propositional operators:
! 0
implication falsity
Program operators: ;
[
composition choice iteration
Mixed operators: []
?
necessity test
The de nition of programs and propositions is by mutual induction. All atomic programs are programs and all atomic propositions are propositions. If '; are propositions and ; are programs, then
'! 0 []'
propositional implication propositional falsity program necessity
are propositions and
; [ '?
sequential composition nondeterministic choice iteration test
are programs. In more formal terms, we de ne the set of all programs and the set of all propositions to be the smallest sets such that
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0 if '; 2 , then ' ! 2 and 0 2 if ; 2 , then ; , [ , and 2 if 2 and ' 2 , then []' 2 if ' 2 then '? 2 : 0
Note that the inductive de nitions of programs and propositions are intertwined and cannot be separated. The de nition of propositions depends on the de nition of programs because of the construct []', and the de nition of programs depends on the de nition of propositions because of the construct '?. Note also that we have allowed all formulas as tests. This is the rich test version of PDL. Compound programs and propositions have the following intuitive meanings: []' \It is necessary that after executing ,
;
' is true."
\Execute , then execute ."
[ \Choose either or nondeterministically and execute it." '?
\Execute a nondeterministically chosen nite number of times (zero or more)." \Test '; proceed if true, fail if false."
We avoid parentheses by assigning precedence to the operators: unary operators, including [], bind tighter than binary ones, and ; binds tighter than [. Thus the expression [; [ ]' _
should be read
([(; ( )) [ ( )]') _ :
Of course, parentheses can always be used to enforce a particular parse of an expression or to enhance readability. Also, under the semantics to be given in the next section, the operators ; and [ will turn out to be associative, so we may write ; ; and [ [ without ambiguity. We often omit the symbol ; and write the composition ; as . The propositional operators ^, _, :, $, and 1 can be de ned from ! and 0 in the usual way.
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The possibility operator < > is the modal dual of the necessity operator
[ ]. It is de ned by
def = :[]:': The propositions []' and <>' are read \box '" and \diamond '," <>'
respectively. The latter has the intuitive meaning, \There is a computation of that terminates in a state satisfying '." One important dierence between < > and [ ] is that <>' implies that terminates, whereas []' does not. Indeed, the formula []0 asserts that no computation of terminates, and the formula []1 is always true, regardless of . In addition, we de ne
if '1 ! 1
skip fail 'n ! n
j j do '1 ! 1 j j 'n ! n od if ' then else while ' do repeat until '
f'g f g
def = 1? def = 0? def = '1 ?; 1 [ [ 'n ?; n n n [ ^ def = ( 'i ?; i ) ; ( :'i )? i=1 i=1 def = if ' ! j :' ! = '?; [ :'?; def = do ' ! od = ('?; ) ; :'? def = ; while :' do = ; (:'?; ) ; '? def = ' ! [] :
The programs skip and fail are the program that does nothing (noop) and the failing program, respectively. The ternary if-then-else operator and the binary while-do operator are the usual conditional and while loop constructs found in conventional programming languages. The constructs if-j- and do-j-od are the alternative guarded command and iterative guarded command constructs, respectively. The construct f'g f g is the Hoare partial correctness assertion. We will argue later that the formal de nitions of these operators given above correctly model their intuitive behavior.
2.2 Semantics The semantics of PDL comes from the semantics for modal logic. The structures over which programs and propositions of PDL are interpreted
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are called Kripke frames in honor of Saul Kripke, the inventor of the formal semantics of modal logic. A Kripke frame is a pair K = (K; mK ); where K is a set of elements u; v; w; : : : called states and mK is a meaning function assigning a subset of K to each atomic proposition and a binary relation on K to each atomic program. That is, mK (p) K; p 2 0 mK (a) K K; a 2 0 : We will extend the de nition of the function mK by induction below to give a meaning to all elements of and such that mK (') K; '2 mK () K K; 2 : Intuitively, we can think of the set mK (') as the set of states satisfying the proposition ' in the model K, and we can think of the binary relation mK () as the set of input/output pairs of states of the program . Formally, the meanings mK (') of ' 2 and mK () of 2 are de ned by mutual induction on the structure of ' and . The basis of the induction, which speci es the meanings of the atomic symbols p 2 0 and a 2 0 , is already given in the speci cation of K. The meanings of compound propositions and programs are de ned as follows.
The operator Æ in (7) is relational composition. In (8), the rst occurrence of is the iteration symbol of PDL, and the second is the re exive transitive closure operator on binary relations. Thus (8) says that the program is interpreted as the re exive transitive closure of mK (). We write K; u ' and u 2 mK (') interchangeably, and say that u satis es ' in K, or that ' is true at state u in K. We may omit the K and write u '
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when K is understood. The notation u 2 ' means that u does not satisfy ', or in other words that u 62 mK ('). In this notation, we can restate the de nition above equivalently as follows:
8v if (u; v) 2 mK () then v ' 9w (u; w) 2 mK () and (w; v) 2 mK ( ) (u; v) 2 mK () or (u; v) 2 mK ( ) 9n 0 9u0 ; : : : ; un u = u0; v = un ; and (ui ; ui+1 ) 2 mK (); 0 i n 1 def () u = v and u ':
The de ned operators inherit their meanings from these de nitions:
mK (' _ ) mK (' ^ ) mK (:') mK (<>')
def = def = def = def =
= mK (1) def = def mK (skip) = mK (fail) def =
mK (') [ mK ( ) mK (') \ mK ( ) K mK (')
fu j 9v 2 K (u; v) 2 mK () and v 2 mK (')g mK () Æ mK (') K
mK (1?) = ; the identity relation mK (0?) = ?:
In addition, the if-then-else, while-do, and guarded commands inherit their semantics from the above de nitions, and the input/output relations given by the formal semantics capture their intuitive operational meanings. For example, the relation associated with the program while ' do is the set of pairs (u; v) for which there exist states u0 ; u1; : : : ; un, n 0, such that u = u0 , v = un, ui 2 mK (') and (ui ; ui+1 ) 2 mK () for 0 i < n, and un 62 mK ('). This version of PDL is usually called regular PDL and the elements of are called regular programs because of the primitive operators [, ;, and , which are familiar from regular expressions. Programs can be viewed as regular expressions over the atomic programs and tests. In fact, it can be shown that if p is an atomic proposition symbol, then any two test-free programs ; are equivalent as regular expressions|that is, they represent the same regular set|if and only if the formula <>p $ < >p is valid.
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EXAMPLE 2. Let p be an atomic proposition, let a be an atomic program, and let K = (K; mK ) be a Kripke frame with