Handbook of Philosophical Logic Vol 8

Handbook of Philosophical Logic 2nd Edition Volume 8

edited by Dov M. Gabbay and F. Guenthner

CONTENTS

Editorial Preface

Dov M. Gabbay The Logic of Questions

David Harrah

Sequent Systems for Modal Logics

Heinrich Wansing

vii 1 61

Deontic Logic

147

Deontic Logic and Contrary-to-duties

265

Index

345

Lennart Aqvist Jose Carmo and Andrew J. I. Jones

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 D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, Volume 8, vii{ix.

c 2002, Kluwer Academic Publishers. Printed in the Netherlands.

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!


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 extension to logic programs

Negation by failure and modality

Horn clause logic is really intuitionistic. Extension of logic programming languages


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


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

DAVID HARRAH

THE LOGIC OF QUESTIONS

1 INTRODUCTION

1.1 Basic Notions Most theorists use ìnterrogative' to refer to a type of sentence. Some theorists posit questions as distinct entities that may be asked, or put, or expressed by interrogatives, just as propositions may be expressed by declaratives and commands may be expressed by imperatives. Intuitively it seems that some questions may be expressed by sentences other than interrogatives, and some interrogatives can be used to do other things besides ask questions. Thus it is reasonable to say that there are two overlapping subject matters: the logic of interrogatives, and the logic of questions. Most theorists use `reply' to refer to any verbal response that can be given to a question, and use ànswer' to refer to a distinguished kind of reply. Many kinds of reply may be appropriate from the respondent's point of view, but the replies that are appropriate from the questioner's point of view, the replies that the question calls for, are the answers. Most theorists de ne various types of answer. The most important distinction is between direct answers, each of which gives exactly what the question calls for, and partial answers, each of which may give some (but perhaps not all) of what the question calls for. The label `direct' was introduced in Harrah [1961] because it connotes both logical suÆciency and immediacy, as in the request: `Please give me a direct answer?' Just as statements and commands can be `good' or `bad' in various ways (valid or not, true or not, possible or not, and the like), so too with questions. The details, however, vary from theory to theory. Most theorists agree on labels for question-types approximately as follows: Label

whether yes-no which what who why deliberative

Example Ìs two even or odd?' Ìs two a prime number?' `Which even numbers are prime?' `What is Church's Thesis?' `Who is Bourbaki?' `Why does two divide zero?' `What shall I do now?'

D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, Volume 8, 1{60.


2

DAVID HARRAH

disjunctive hypothetical conditional given-that

`How long is your new proof, or do you have a shorter one?' Ìf you had a proof, how long would it be?' Ìf you now have a proof, how long is it?' `Given that Turing's Conjecture is provable, is Church's Thesis provable?'

No theorist holds that this is a complete list of question-types, and most would divide each of the types listed here into several sub-types. For discussions, see Hamblin [1967], Prior and Prior [1955], Belnap and Steel [1976], and Wisniewski [1995]. To see how problematic these basic notions can be, and how theorists' intuitions may dier, consider the English sentence: `Where is Jane or Ann?' Harrah [1975] formalizes this as a disjunction of two interrogatives (one about Jane, one about Ann); each of these interrogatives expresses one question that has its own set of answers. Belnap and Steel [1976] formalizes this as one interrogative that expresses one disjunctive question whose answers are the answers about Jane plus the answers about Ann. Groenendijk and Stokhof [1984] construes it as one interrogative that expresses two questions; the semantics of the interrogative yields one set of answers that contains the Jane answers and the Ann answers. There is another subject matter that is a generalization on the rst two: erotetic logic. In the narrow sense, èrotetic' can be paraphrased as `pertaining to questioning'. In this sense erotetic logic is the theory of questions, interrogatives, and the use of interrogatives. Generalizing on this notion, we may use èrotetic' with the sense of `pertaining to calling-for-reply'. Under such an interpretation, erotetic logic is the theory of all the sentences (interrogative, imperative, declarative, or whatever) that call for reply, or that are vulnerable to replies of certain sorts, and the theory of all the entities that can thus be called for. Erotetic logic in this general sense has not yet been developed to any signi cant depth, and we discuss it only brie y in Sections 7.7 { 7.8. For this reason, and because most of the theories discussed in this chapter are logics of questions, this chapter is most appropriately titled `The Logic of Questions'. (Helpful comments and advice concerning this chapter were supplied by Andrzej Wisniewski.)

1.2 Motivations The theories that have been developed up to now dier not only in superstructure and points of detail but also in foundation and basic conception. Many of these dierences are due to dierences in motivation. For this


3

reason we take note of motivation in our survey below. The following descriptive labels are used in our exposition: Empirical. This is the motivation of linguists and psychologists, for example, who wish to describe the sentences of a natural language and describe how those sentences are understood and used by the speakers of the language. Platonic. This is the motivation of some philosophers and logicians who wish to describe linguistic or semantic entities considered as mathematical objects, objects that exist in their own right, so to speak. Normative. This is the motivation of some philosophers and logicians who wish to describe how one ought to ask and answer question, or how a rational person asks and answers. Engineering. This is the motivation of those whose purpose is simply to construct a system that will be usable for certain practical purposes (e.g. computer-assisted information-retrieval) and satisfy certain criteria (e.g. eÆciency, eectiveness). Metalogical. This is the motivation of one who wishes to study how far, within a given logical system, a system of question-and-answer can be developed, or, more generally, to study what sorts of question-and answer system can be developed within a given logical system, where the given system might or might not have been intended by its creators to provide a question-and-answer system. Technical and aesthetic. These are additional motivations felt by the theorist in the course of constructing the theory. The most important of course are the desire to facilitate construction of theory and the desire to achieve simplicity or elegance of other kinds. Two cautions: First, the foregoing characterizations are rough and usually require some quali cation when applied to particular cases. Second, in any particular case there may be more than one motivation present; we note some examples below.

1.3 History and Bibliography Discussion of questions is at least as old as Aristotle; Kubinski [1980], pp. 118 { 119, says that Cohen [1929] \is the earliest known to me in which the logic of questions is treated by means of formal logic". Cohen suggested identifying questions with sentential functions having free variables. Closely related ideas were considered by Carnap and Reichenbach. Gornstein [1967], p. v { 1, says that Carnap seems to be the rst author who wrote a question in a formalized manner. Current activity in the eld began in the 1950s, stimulated to a large extent by Prior and Prior [1955], Stahl [1956], Hamblin [1958], and Kubinski [1960]. Since then many approaches have been suggested, and several have been developed in detail.

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DAVID HARRAH

For a general discussion of the history of the eld to about 1965, see Hamblin [1967]. Gornstein [1967] presents a detailed summary and discussion of many theories and logics from Aristotle to the 1960s. Kubinski [1980] oers many historical remarks and useful bibliography. Probably the most comprehensive and complete bibliography published to 1976 is the one by Egli and Schleichert that is included in Belnap and Steel [1976]. It contains sections on Logic and Philosophy of Language; Linguistics; Automatic Question-Answering; and Psychology and Pedagogy. Abstracts are included for many of the items listed. Ficht [1978] presents an updated version of this bibliography with an additional section on dialogue. Other bibliographies, more limited in scope but quite useful, are found in Hi_z (ed.) [1978], Lehnert [1978], Belnap [1981] and [1983], and Higginbotham [1993]. The bibliography presented in Wisniewski [1995] is valuable in many ways and has good coverage of the important work done up to 1994. Note: The list of References given at the end of this chapter is very selective. In general it concentrates on logic and ignores psychology, pedagogy, and heuristics. In particular it concentrates on the topics and authors discussed in this chapter.

1.4 Scope of this Chapter The main aims of this chapter are to indicate the variety of motivations, basic conceptions, and approaches to theorizing that are now evident in the eld, and to outline some topics and aspects that invite further study. This chapter does not aim at a complete historical account, or a complete catalogue of possible theories. We concentrate on some work, by a few logicians, that is especially signi cant and fruitful for further developments in various directions.

1.5 Abbreviations and Notation In most cases, when summarizing the work of another author, we use that author's terminology and notation. In a few cases we depart for the sake of clarity or simplicity. Occasionally, where there is no danger of ambiguity, we use symbols as names of themselves. Except where noted otherwise, we use wff for well-formed formula iff if and only if d(I ) direct answer to I (or, in Sections 7.3 { 7.4, the set of these answers) D(I ) the set of direct answers to I :X the negation of X the empty set


5

2 SET-OF-ANSWERS METHODOLOGY

2.1 Hamblin's Postulates In an informal paper Hamblin [1958] proposed three postulates: 1. An answer to a question is a statement. 2. Knowing what counts as an answer is equivalent to knowing the question. 3. The possible answers to a question are an exhaustive set of mutually exclusive possibilities. Hamblin also suggested a calculus of questions to formalize such ideas as containment and equivalence. For example, one question contains another when from every answer to the rst we can deduce an answer to the second; and the two questions are equivalent if they contain each other. This paper stimulated much formal work by others (see, e.g. Belnap and Steel [1976], p. 35). Some linguists and logicians have argued against adopting Postulate (1) (Hintikka [1976], Tichy [1978]). For those who adopt it, however, it eects an important simpli cation. Replies that are not statements (e.g. noun phrases, nods, grunts) can be treated as coded answers that are abbreviations of statements. Thus the logic of answers is concerned only with statements. Some logicians have argued against adopting Postulate (2) ( Aqvist [1965], Hintikka [1976]). For those who adopt it, however, it represents another giant step toward formalization. The techniques inspired by it are perhaps best thought of under the label `set-of-answers methodology' (or `SA methodology' for short). Hamblin's own technique for de ning containment and equivalence is one example; others appear below. Postulate (2), and SA methodology in general, are compatible with several dierent theories about the logical nature of questions, and about the connection between a question and its answers. In fact, most of the approaches to theorizing surveyed in this chapter exemplify SA methodology at various points. (Note: Postulate (3) is controversial and there is much un nished business connected with it, but we do not discuss it in this chapter beyond the brief mentions in 6.6 and 7.2.)

2.2 The SA Reduction of Questions One idea, not required by SA methodology but obviously compatible with it, is to identify a question with its set of answers. Let us call this the SA reduction of questions. In radical versions of SA reduction one allows

6

DAVID HARRAH

arbitrary sets of sentences to count as questions, regardless of whether these sets have de ning characteristics that can be expressed in a given language. In conservative versions one considers only certain kinds of sets | e.g. sets that are de nable in terms of the syntactical form of the sentences that are their members. In several papers in the 1960s Stahl developed an SA reduction. The following is a summary of Stahl [1962]. We assume a higher-order function calculus, and then distinguish three types of questions: (1) individual questions (e.g. [Hx?], read `Which things satisfy H ?'), (2) function questions (e.g. [F ?a], read `Which functions are satis ed by a?'), and (3) truth questions (e.g. [Af ?B ], read `Which truthfunctions hold between A and B ?'). To (1), simple answers are Ha, Hb, etc.; direct answers are simple answers that are not negations of theorems. We can form nite conjunctions [Ha ^ Hb ^ Hd], (:Ha ^ :Hc), etc., and also in nite conjunctions (x):Hx, (x)(H 0 x ! Hx), etc. A perfect answer is such a conjunction which is not the negation of a theorem. A suÆcient answer is a wff F such that F is not the negation of a theorem, and either F implies a perfect answer which is not a theorem or else F is a theorem and some perfect answer is a theorem. We now de ne the question [Hx?] as the class of its suÆcient answers. To (2) the simple answers are Ha, H 0 a, H 00 a, etc.; as before we then form direct, perfect, and suÆcient answers, and the question is de ned as the class of its suÆcient answers. For (3) we proceed likewise, except that there are no in nite conjunctions; e.g. ((A _ B ) ^ (A ! B )) is in [Af ?B ]. The initial de nitions can be generalized and made relative to a system X and a set of premises S ; we write [P ]X S to mean that P is a question in X relative to S . Here the suÆcient answers are required to be consistent with consequences of S . For discussion and criticism of Stahl [1962], see Harrah [1963b]. See also Section 7.5 below.

2.3 Motivation There are several motivations for adopting an SA reduction. Besides the metalogical one (to see what can be done within set theory) there are technical and aesthetic motivations. All the operations on sets and relations between sets are directly available as operations on questions and relations between questions, and theorems about sets become theorems about questions. For some researchers there are also Platonic, normative, and engineering motivations. It has been argued that for a rational decision-maker in a choice situation the only essential thing about a question is its set of answers (e.g. Szaniawski [1973] and Dacey [1981]). Another motivation (a technical one?) appears in connection with Fregean principles of language construction. If every declarative sentence is to be


7

assigned a sense and a denotation, uniformity would suggest a similar treatment for interrogatives. The natural technique is to let each interrogative denote the set of its direct answers. Some logicians and linguists have adopted this technique, though others have argued that an interrogative denotes just the set of its true direct answers. For an entry into the literature of this topic, see Karttunen [1978], Belnap [1981], Kiefer [1983], Groenendijk and Stokhof [1984], and Higginbotham [1993].

2.4 Some Problems Some logicians have argued that the SA reduction is not intuitively or empirically plausible (e.g. Tichy [1978], pp. 279 { 280). Even for those who favor it, however, it presents the problem of deciding what kind of answers are to be in the sets under consideration. It might seem that all the direct answers must be included; but, as noted above, some theorists in the tradition of Montague Grammar have equated questions with the sets of just their true direct answers. Regardless of whether a question includes all of its direct answers, should it be allowed to include partial answers? and replies like Ì don't know'? Some linguists have argued for a view that in eect obliterates the distinction between direct and partial, and that would (on the SA reduction) treat a question as the set of its direct and partial answers together (e.g. Bolinger [1978], p. 104). For logicians who wish to preserve a sharp distinction between the direct and partial the simplest course (with the SA reduction) is to identify a question with the direct (or, true direct) answers, and then to de ne the partial answers separately. As noted above, this is the technique used in Stahl [1962]. On some motivations one who adopts SA reduction cares about questions but not about interrogatives, and speci cally does not care whether there are enough interrogatives to express all questions. On other motivations one is not so indierent, and decisions about what sets are to count as questions depend on what sentences are available as interrogatives for expressing questions. Among many possible policies of question-de nition the following are the natural ones: 1. Choose any arbitrary collection C of sets S of sentences. 2. Choose any C such that every S in C is describable in the assumed metalanguage. 3. Choose any C such that there is a set S 0 of sentences of L (the ìnterrogatives') with this property: There is a many-one mapping from S 0 onto C (so that every question S in C is expressible by at least one interrogative in S 0 ).

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DAVID HARRAH

4. As in (3), with the additional requirement that there is an eective procedure for recognizing interrogatives, and an eective procedure whereby, given an interrogative, we can recognize the question that it expresses. 5. Choose a C only if it, in eect, represents the set of questions belonging to some given natural language. In fact, most logicians in the eld thus far have followed (5) or (4). An empirical motivation of course leads to (5). A Platonic motivation might lead to any of (1){(5), and would lead also to metatheoretic questions about completeness. The topics of eectiveness and completeness are of such importance that we discuss them in a separate section below. 3 EFFECTIVENESS AND COMPLETENESS

3.1 Introduction It is usually of interest to investigate whether certain aspects of a questionand-answer system are eective, and whether the system is complete in certain respects. On some motivations it is required that certain kinds of completeness and eectiveness do indeed obtain. Probably the rst theorist to note the importance of these properties and to study them in a systematic way was Belnap [1963]; we outline his ideas and results in Section 4. In this section we give a more general discussion, because the topic is important for many theories of questions.

3.2 The Problem of Eectiveness Should the notion of interrogative, or question-expresser, be eective? Sup-

pose it is not. Then, when questioner Q utters X , respondent R might have to ask Àre you asking a question?' and then Q might have to ask Ìs the latter a question?' and so on back and forth inde nitely. This argument does not prove that all interrogatives must be recognizable as such, but it does suggest that some must be. Either all must be, or else there must be in the language some interrogatives with the force of `Does the expression X express a question?', where these are recognizable as such. A similar argument applies to answers. Suppose that Q utters X , and R recognizes that X expresses a question, but that answer-to-X is not eective. Then, when R gives a reply, Q must ask `Was that an answer?' As before, this argument does not prove that every question must have an eective set of answers, but it does suggest that some must. Either all must, or there must be some question with the force of Ìs W an answer to X ?' and with an eective set of answers.


9

On the other hand one might want to allow noneectiveness for certain types of question and answer. There seems to be no eective method for determining whether English sentences of the form Ì wonder whether . . . ' express questions as distinct from statements. Also it seems that many who-, what-, and why- questions do not have eective answer sets.

3.3 Concepts of Expressive Completeness A question system might be expressively complete (or fail to be complete) in any of several senses. First, it might be empirically complete, in that it provides for all the questions that natural languages do. Second, it might be complete in a Platonic sense, in that it provides for all the questions that `really exist'. The Platonic conception may be made precise via model theory without reference to a particular language. One theory for which this would be possible is that of Tichy [1978]; see below in Section 6.6. Alternatively, we may speak of àll the questions that really exist, relative to a given language'. This is Belnap's conception; see Section 4. With this conception the problem of completeness is to determine whether everything that counts as a real question is expressible by an interrogative in the given language. Further senses of completeness are generated by semantic and pragmatic considerations. For example, a system that fails to provide for all the real questions may nevertheless provide for all the questions that have a true direct answer, or all that are truly answerable by a human being. With respect to answers, a system may be complete if, for each of its questions, all the `real answers' (again, see Belnap) are expressible. A more reasonable alternative is to require merely that, for each question that has true real answers, at least one true real answer is expressible. For further discussion of concepts of completeness, see Belnap and Steel [1976] and Harrah [1969].

3.4 Diagonalization and Expressive Incompleteness In the logic of questions generally, and in SA methodology especially, we deal with sets of linguistic expressions, or sets of sets of them. Sometimes there is enough structure in the situation to permit Cantorian diagonalization. We give one example here, from Harrah [1969]. We assume that we have a language L with denumerably many expressions, and that an eective alphabetical ordering of them has been established. We assume that there is a set S of questions, and that S is recursively enumerable. Suppose that each question has denumerably many direct answers, or can be assigned denumerably many in a harmless way (e.g. by adding instances of (P ^ :P )). Suppose next that, given a question q, the

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DAVID HARRAH

set of direct answers to q is recursively enumerable. Finally suppose that direct answers are sentences, and that sentence is eective. Now we can diagonalize to construct new sets of sentences each of which is not the set of direct answers to any question in the assumed enumeration of questions. Viz: Choose any positive integer j . Then for the alphabetically rst member of the new set D we choose the alphabetically j th sentence after the rst direct answer of the rst question, and for the alphabetically (i + 1)th member of D we choose the alphabetically j th sentence after the rst direct answer d of the (i + 1)th question such that d is alphabetically later than the ith member of D. To make D more interesting we can specify in advance some recursive property P and stipulate that the construction is to move at least j sentences and keep going until it nds a sentence with the property P . One consequence that is relevant to many theories but to the SA reduction in particular is this: If question and direct answer are eective (or merely enumerable, as assumed above), then the system is incomplete in the sense that not every set of sentences can be the set of direct answers to a question. For further discussion, see Harrah [1969].

3.5 Deductive Completeness There is another family of completeness concepts that has been suggested by work of Kubinski and Wisniewski (see 8.5 in Wisniewski [1995]). The basic idea is that relations like implication can hold between (1) wffs and questions and (2) questions, and the particular relationships that do hold can be described or expressed via statements of a certain form F in a metalanguage ML of the given language L. Then, in analogy with the development of systems of logic for declarative sentences, we may choose some set S of statements of ML and ask whether S is a complete axiom set for all of the true statements of the form F . This area invites and awaits exploration. 4 BELNAP'S ANALYSIS In this section we summarize a part of Belnap and Steel [1976]. That book presents (1) a formal system for question-and-answer, (2) a rich metatheory for the system, (3) discussion of application to English, (4) discussion of application to data processing and information retrieval, and (5) an extensive bibliography by U. Egli and H. Schleichert. Here we summarize just (1) and (2). Because they were the work of Belnap (completed in 1968), we shall call them Belnap's system and Belnap's theory. We present Belnap's theory in detail because many of his concepts apply to systems other than his own, and many of his concepts deserve to become standard.


11

4.1 Motivation The main motivation is a normative one: to construct a rational system applicable in situations of a certain kind | namely, where questioner Q and respondent R are motivated to help each other, and R has access to a well structured information source. In particular R may be a machine, and the information source may be a data bank. For Belnap, a system is adequate for these situations only if it meets certain conditions of eectiveness. First, the interrogatives must be eectively recognizable as such. Second, given any question, its direct answers must be eectively recognizable as such. The latter is the fundamental criterion, emphasized in both Belnap [1963] and Belnap and Steel [1976].

4.2 The Assertoric Basis The language L is an applied rst-order functional calculus with identity. There are denumerably many individual variables w, x, y, z , . . . , and countably many individual constants, n-ary function constants f , g, . . . , and nary predicate constants F , G, . . . . There are signs = for identity, ^ for conjunction, _ for disjunction, ! for the material conditional, $ for the material biconditional, 9 and 8 for the existential and universal quanti ers. Term and wff are de ned as usual, except that n-ary conjunctions (A1 ^ : : : ^ An ) and disjunctions (A1 _ : : : _ An ) are permitted for each n. We use a, b, c, . . . for terms and A, B , . . . for wffs. An n-place condition is a wff with exactly n free variables. A statement is a wff with no free variables. A name is a term with no free variables. Given Ax1 : : : xn , we understand that Ab1 : : : bn comes from Ax1 : : : xn by proper substitution of bi for xi . Some one-place conditions are designated as elementary category conditions (including (x = a) for each name a). The set of category conditions is de ned recursively by: 1. Every elementary category condition is a category condition. 2. If Ax and Bx are category conditions with x as the only free variable, then (Ax ^ Bx) and (Ax _ Bx) are category conditions, and so too is any result of changing variables (free or bound) in Ax. With each elementary category condition Ax there is associated a decidable set of names, called the nominal category determined by Ax. If Ax is (x = a), the nominal category must be fag. If Ax and By dier only in their variables (free or bound), they determine the same nominal category. If Ax is (Bx ^ Cx) or (Bx _ Cx), then its nominal category is the intersection or union of the nominal categories determined by Bx and Cx. For the semantics of L, a candidate interpretation consists of a nonempty domain of individuals D and an interpretation function of the usual extensional kind. Denotation and truth are de ned as usual. The range of a

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DAVID HARRAH

one-place condition Ax in a candidate interpretation M is the set of individuals i in the domain of M such that Ax is true in M 0 , where M 0 is like M except in assigning i to the free variable x in Ax. The range in M of a category condition is also called the real range, or the real category determined by that condition in M . Category conditions diering only in their free or bound variables are equivalent; we write Cx for the set of conditions that are equivalent to Cx. An interpretation is a candidate interpretation M in which, for every category condition Ax, every name in the nominal category determined by Ax denotes in M some individual in the real category determined by Ax in M . Consistency, validity, logical implication and logical equivalence are de ned as usual. Where there is no explicit reference to an interpretation, it is understood that there is implicit reference to some principal interpretation.

4.3 Elementary Questions The elementary questions are whether-questions and which-questions. An elementary question is expressed by an elementary interrogative. These have the form ?, where denotes a request, denotes a subject and ? denotes the function which takes a request and a subject as arguments and produces a question as value. An abstract whether-subject is a nite set of wffs. The range determined by this subject is the set itself; likewise, the set of alternatives presented by this subject is the set itself. A lexical whether-subject is a nite list of wffs enclosed in parentheses, as: (A1 ; : : : ; An ). To simplify matters it is required of both abstract whether-subjects fA1 ; : : : ; An g and lexical whethersubjects (A1 ; : : : ; An ) that there be no repetitions among A1 ; : : : ; An , and that no Ai be a conjunction of other statements in the list A1 ; : : : ; An . The lexical whether-subject (A1 ; : : : ; An ) signi es the abstract whether-subject fA1 ; : : : ; An g. An abstract which-subject is a triple hX; g; Ai such that X is a nonempty set of variables (the queriables ), g is a category mapping in X (i.e. a mapping from a subset of X into the set of equivalence classes of category conditions), and A is a matrix (a wff whose free variables include the queriables). Let hX; g; Ax1 : : : xn i be an abstract which-subject, where X = fx1 ; . . . , xn g, and g is a category mapping in X . Then the nominal alternatives making up the nominal range determined by this subject and presented by any question with this subject are the results Aa1 : : : an of substituting a name ai for a queriable xi (for each i) in the matrix Ax1 : : : xn , under the restriction that, if g(xi ) is de ned and is Cx, then ai must be in the nominal category determined by Cx. For any interpretation M , the real M -alternatives, or alternatives in M , which make up the real M -range or range in M , determined by hX; g; Ax1 : : : xn i, and presented by any question with this subject are all the


13

pairs hf; Ax1 : : : xn i, where f is a function from X into the domain in M , under the restriction that, if g(xi ) is de ned and is Cx, then f (xi ) is in the real range in M of Cx. A real M -alternative hf; Ax1 : : : xn i is true in M just in case Ax1 : : : xn is true in that M 0 which is like M except in assigning f (xi ) to xi for each i. Roughly, a nominal alternative Aa1 : : : an signi es in M the real alternative hf; Ax1 : : : xn i, provided that ai denotes in M the individual f (xi ). A lexical which-subject is an expression of the form (C1 x1 ; : : : ; Cr xr ; xr+1 ; : : : ; xn kAx1 : : : xn ); where x1 , . . . , xn is a nonempty nonrepeating sequence of variables, and C1 x1 , . . . , Cr xr is a possibly empty sequence of category conditions, each Ci xi being a category condition with xi as its one free variable. Here each xi is governed by the category condition Ci xi , while xr+1 , . . . , xn are categoryfree. Given such a lexical which-subject, we can recover the abstract whichsubject that it signi es, in the obvious way. The queriables are all of x1 , . . . , xn , but g is de ned only for x1 , . . . , xr . In a footnote added late (p. 26), Belnap and Steel say that it was a mistake to de ne an abstract which-subject as the triple hX; g; Ai, because then (xkF x) and (ykF y) signify distinct abstract which-subjects. They suggest de ning an abstract which-subject not as hX; g; Ai but rather as \something amounting to the equivalence-class generated from this by means of uniform substitution for queriables". To avoid complicating our exposition here we shall not make this change but will continue as in Belnap's original development. A which-interrogative has the form ?(C1 x1 ; : : : ; Cr xr ; xr+1 ; : : : ; x1 kAx1 : : : xn ): The variables x1 ; : : : ; xn occur free in Ax1 : : : xn , and are said to be free in the list C1 x1 ; : : : ; Cr xr ; xr+1 ; : : : ; xn , but are considered to be bound in the interrogative as a whole. Roughly: Every direct answer to an elementary question is a conjunction (S ^ C ^ D), where S is a selection drawn from among the presented alternatives, C is a completeness-claim, and D is a distinctness-claim. The selection S is itself a conjunction (S1 ^ : : : ^ Sp ) without repetitions. (S1 ^ : : : ^ Sp ) is a lexical selection, and the corresponding set fS1 ; : : : ; Sp g is an abstract selection. In the case of which-questions, the nominal selection signi ed by (S1 ^ : : : ^ Sp ) is fS1 ; : : : ; Sp g; if each of the Si is in the range of a whichsubject , then the real selection signi ed by (S1 ^ : : : ^ Sp ) in M relative to is the set of real alternatives signi ed by the Si in M relative to . Because direct answers have (S ^ C ^ D) structure, the request in ? has a structure of the form (s c d). Here s is a selection-size speci cation,

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which is a pair of numerals , where is a positive numeral representing a lower bound on the selection size, and is either a positive numeral ( ) representing an upper bound, or a dash signifying the absence of an upper bound. The notation is a lexical selection-size speci cation, and it signi es the corresponding abstract selection-size speci cation, which is the corresponding ordered pair of cardinals (or the dash). A subject sanctions a selection (S1 ^ : : : ^ Sp ) if each Si is in the range determined by . A request sanctions a selection if the length of the selection falls within the limits speci ed by the selection-size speci cation of . An interrogative ? sanctions a selection if both and do. Roughly: The completeness-claim made by a direct answer is a claim as to how complete the selection is when measured against the totality of true alternatives presented by the question. Completeness-claims may be analyzed in terms of quanti ers, where a quanti er Q is de ned as a binary relation between classes T and S (here T would be the set of true alternatives, and S would be the selection) such that whether or not Q(T; S ) holds depends on the cardinalities of T \ S and T S (e.g. the universal quanti er is the quanti er in which T S = 0). Belnap mentions various examples of quanti ers that might be used, and the possibility of letting the speci cation indicate a range of completeness-claims. To develop these possibilities would require an enrichment of the assertoric basis, so Belnap con nes attention to just the universal quanti er and the claim it represents, the maximal completeness-claim. Notation: Max(; S ). For whether-questions: Given a lexical whether-subject and a selection S (S1 ^ . . . ^Sp ) sanctioned by that subject, we de ne Max(; S ), the maximum completeness-claim in and S , as (:B1 ^ . . . ^:Br ), where B1 , . . . , Br are (in order) all the members of the subject that are not in the selection S. For which-questions: Assume as given a lexical which-subject

= (C1 x1 ; : : : ; Cr xr ; xr+1 ; : : : ; xn kAx1 : : : xn ) and a selection

S = (Aa1 : : : a1

n

1

^ : : : ^ Aap : : : ap 1

n

)

sanctioned by . Then we de ne Max(; S ) as

8x1 : : : 8xn [C1 x1 ^ : : : ^ Cr xr ! [Ax1 : : : xn ! [(x1;n = = a1 ) _ : : : _ (x1;n = ap )]]]; where (x1;n = ak ) = [(x1 = ak ) ^ : : : ^ (xn = ak )]. A dash is used for the lexical empty completeness-claim speci cation. Then ?(s d) speci es no completeness-claim, and ?(s8d) speci es the maximum completeness-claim. Given an interrogative I and a selection S 1;n

1;n

1

1;n

n


15

sanctioned by I , the completeness-claim sanctioned by I relative to S , or Comp(I; S ), is not de ned if I speci es no completeness-claim, and is de ned as Max(; S ) if I speci es the maximum completeness-claim. 1

d) .

Single-example questions have the form ?(1 Some-examples questions have the form ?( 1

d) .

1

Unique-alternative questions have the form ?(1 8d). Complete-list questions have the form ?( 1 8d). For the following theorem we say that I1 is erotetically equivalent to I2 iff, for every d(I1 ) there is an equivalent d(I2 ), and for every d(I2 ) there is an equivalent d(I1 ). THEOREM 1. The completeness-claim speci cation is dispensable in some

cases but not in all. Viz: (A) For each whether-question interrogative, there is an erotetically equivalent single-example whether-interrogative. (B) Unique-alternative which-interrogatives are erotetically equivalent to certain single-example which-interrogatives. (C) There are certain complete-list which-interrogatives that are not erotetically equivalent to any some-examples which-interrogative.

Result (B) can be generalized from exactly-one to exactly-n. The trick is to add to the subject the appropriate completeness-clause 8y(Ay ! (y = x1 _ : : : _ y = xn )). Compare Aqvist [1965], pp. 123. Further results on equivalence of this sort are noted in Aqvist [1965] and Kubinski [1980], pp. 61{68. Concerning distinctness-claims, Belnap says that only two kinds have a systematic use in question logic: the empty and the nonempty. Let and S be as in the de nition of Max. Then Dist(; S ) is de ned as a conjunction of disjunctions of the form ((ai 6= aj ) _ : : : _ (ai 6= aj )) which says that the conjuncts of S signify, relative to , distinct real alternatives. Notation. for the lexical empty distinctness-claim speci cation, and 6= for the lexical nonempty distinctness-claim speci cation. Given an interrogative I and a selection sanctioned by I , the distinctness-claim sanctioned by I relative to S , or Dist(I; S ), is not de ned if I speci es no distinctnessclaim, and is de ned as Dist(; S ) if I does specify a distinctness-claim. 1

1

n

n

4.4 Answers Let I be an elementary interrogative and S be a selection sanctioned by I . Then A is a direct answer to I iff I and A have the corresponding forms indicated:

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DAVID HARRAH

I ?(s ) ?(s8 ) ?(s = 6 ) ?(s8 = 6 )

A S S ^ Comp(I; S ) S ^ Dist(I; S ) S ^ Comp(I; S ) ^ Dist(I; S )

4.5 Eectivity, Univocity, Completeness Belnap says that his elementary interrogatives and their answers as de ned above satisfy criteria of eectivity and univocity. One can eectively tell whether an expression is an elementary interrogative (eectivity) and, given that it is, what question it puts (univocity). Given a wff A and interrogative I , one can eectively tell whether A is a d(I ) (eectivity) and, if it is, how it answers I (univocity). It is for the sake of eectivity that we require the nominal ranges of category conditions to be eective, allow n-ary conjunction, and bar conjunctions of alternatives from counting as alternatives. Roughly, a real answer to a which-question is a sequence of real alternatives presented by that question, and the system is complete only if every real answer is expressible by some nominal answer. Completeness can fail if some entities do not have names, or do not have names in the proper nominal categories, and also if some true real answers are in nite. Also, as Belnap points out, no category system can be complete; for there are only denumerably many one-place wffs available to serve as category conditions, while there are nondenumerably many sets of names that might be wanted as categories. Roughly speaking, the positive result is that, for any given category system, Belnap's answer system is complete up to these limitations: the real answers must be nite, the entities involved must have names, and the names must be in the right nominal categories. (For the precise account, see Belnap and Steel [1976], Section 1.34.)

4.6 Useful Abbreviations Belnap suggests the following rules for abbreviating requests: (1) Drop parentheses around the request. (2) Omit dashes. (3) Omit `1' as a lower limit. Then, where is a subject, these rules allow ?1 for ? ?= 6 ?1 8 ?8 ?8 6=

single-example questions, some-examples questions, some-distinct-examples questions, unique alternative questions, complete-list questions, complete-and-distinct-list questions.


17

For abbreviating subjects, where A has no free variables, A abbreviates (A; :A); and, where A contains exactly x1 ; : : : ; xn free (given in order), then A abbreviates (x1 ; : : : ; xn kA). Then Ìs it the case that A?' can be expressed by either ?1 A or ?A.

4.7 Elementary-like Questions In rst-order functional calculi there are six parts of speech: open and closed wffs, open and closed terms, connectives, and quanti ers. These generate 36 types of elementary-like questions, each type positing an entity of some part of speech and having as desiderata entities of some parts of speech; e.g. a whether-question can be analyzed as positing a sequence of statements and asking for a truth-functional connection between them. Belnap credits this idea to Stahl [1962]. As noted above, Stahl assumed a higher-order function calculus and analyzed just individual-questions, function-questions, and truth-questions. Kubinski has considered enriched languages in which one can ask these and related questions (Kubinski [1980], Section I.12). It does not appear that anyone has yet made a full study of all 36 of the possibilities noted above, or of the analogous set for higher-order languages. The possibilities that Belnap discusses are six: 1. Whether-questions. Each of these posits a sequence (conjunction?) of statements and has truth-functional connectives as desiderata. The presented alternatives are formed by constructing truth-functional compounds of the posited wffs. 2. Which-questions. Each of these posits an open wff and has closed terms as desiderata. The presented alternatives are formed by substituting one of the closed terms for one of the free variables in the posited wff. 3. Description-questions. Each of these posits a closed term and has descriptors (i.e. open wffs) as desiderata. One way to formalize is this: Let L have a list of determinables, which are open wffs, and with each determinable let there be associated a list of descriptors. (E.g. with `x is a color' we associate `x is red', `x is green', etc.) For semantic coherence each candidate interpretation must be such that an individual satis es a determinable only if also satisfying some descriptor associated with it. Where Hx is a determinable with H1 x; : : : ; Hi x; : : : with it, to posit a term b and call for H1 x; : : : ; Hi x; : : : as desiderata we would use a new sort of subject Des(Hxkb) whose range is the set of presented alternatives H1 b; : : : ; Hi b; : : :. For this sort of question distinctness-claims can be expressed via wffs of the form

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DAVID HARRAH

:8x(Hi x $ Hj x).

Completeness-claims can be expressed in a rstorder way only in special cases, however | e.g. in the case where the set of the associated descriptors is nite. 4. Identity-questions. Each of these posits a closed term and has closed terms as desiderata. We de ne the subject as Ident(Cxkb), where b is the posit and Cx is a category condition with associated names a1 ; : : : ; ai ; : : :. The presented alternatives are (b = a1 ), (b = a2 ); : : :. 5. What-questions. Belnap distinguishes four sub-types: equivalence questions, with subject Equiv(HxkAx) and alternatives 8x(Ax $ Hi x); necessity-questions, with subject Nec(HxkAx) and alternatives 8x(Ax ! Hi x); suf ciency-questions, with subject Suf(HxkAx) and alternatives 8x(Hi x ! Ax); and intersection-questions, with subject Inter(HxkAx) and alternatives 9x(Hi x ^ Ax). In all cases here the Hi are descriptors associated with Hx. As with description-questions, completeness-claims can be expressed in a rst-order language only in the nite case. 6. How-many Questions. Each of these posits an open wff and has quanti ers as desiderata. The alternatives are formed by pre xing the quanti er to the wff. Full study of this awaits further work on the logic of quanti ers.

4.8 Compounding

Given interrogatives I1 ; : : : ; In , we write (I1 [ : : : [ In ) for the unionized interrogative of I1 ; : : : ; In . For this interrogative the concepts of subject and request are not de ned. We say that A is a direct answer to it iff A is a direct answer to at least one of the Ii . In a similar way we can form intersection, complement, and set-dierence questions. Belnap says that intersections and complements of questions do not appear to be useful, but set-dierence might be (cf. `Tell me about . . . , without telling me about | '). The above-mentioned operations are boolean. In contrast there are other operations best thought of as logical or syntactical. Viz: We de ne (I1 ^ : : : ^ In ) as the conjunction of I1 ; : : : ; In . Its direct answers are conjunctions (A1 ^ : : : ^ An ), where each Ai is a direct answer to Ii . Belnap says that negation, disjunction, implication, and equivalence, conceived as logical operations on questions, do not seem to be of much interest. Incidentally, the connective òr' in English interrogatives is ambiguous.


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Compare: `Who or what killed the dog?' [exclusive or ] `Have you been to Sweden, or have you been to Germany?' [inclusive or ] `What day have you chosen, or what week?' [nonsymmetric or ] Ìs it a bird or is it a plane?' [simple whether-question] For discussion see Stahl [1962], and Belnap and Steel [1976], p. 91. Given a list of whether- and which-subjects 1 ; : : : ; n , we can form the unionized subject (1 [ : : : [ n ). The set of presented alternatives is the union of the sets presented by 1 ; : : : ; n , but the selections sanctioned by the unionized subject are conjunctions of alternatives presented by it. Expression of the appropriate completeness- and distinctness-claims is tedious. Given elementary requests 1 ; : : : ; n , we let (1 [ : : : [ n ) be a unionized request. The interrogative ?(1 [ : : : [ n ) is to be treated like the union of the interrogatives ?1 ; : : : ; ?n . To formalize hypothetical questions, Belnap introduces hypothetical interrogatives (P j!j I ), where P is a wff and I is an interrogative. The direct answers are wffs (P ! A), where A is a d(I ). To formalize given-that questions, Belnap introduces interrogatives (P j ^ j I ), with direct answers (P ^ A), where A is a d(I ). Belnap says that (P j!j I ) could be called an added-condition question, and (P j ^ j I ) an added-conjunct question. \We leave it to the reader to determine whether there is any point in introducing àdded disjunction' or àdded equivalence' questions" (pp. 98 { 99). Let I be an interrogative with x free (hence, not a queriable). Then we can allow 8xIx to be an interrogative, and de ne the direct answers as conjunctions (A1 a1 ^ : : : ^ An an ^ 8x(x 6= a1 ^ : : : ^ x 6= an ! Bx)); where A1 x; : : : ; An x; Bx are all direct answers to I . To allow for quali cation in terms of a category condition Cx, we use 8[Cx]I and de ne its direct answers as conjunctions (A1 a1 ^ : : : ^ An an ^ 8x(Cx ! (x 6= a1 ^ : : : ^ x 6= an ! Bx)))

as above, but requiring also that each ai is in the nominal category determined by Cx. To formalize Ànswer I for some x in the domain' (where x is not free in I ), Belnap uses not 9xI but [xI , whose direct answers are the wffs Aa such that Ax is a direct answer to Ix. Thus [x?1 (P x) has the same answers as ?1 (x k P x). Belnap says that 9xI might have a use if the operation of disjunction on questions can be shown to have a use.

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DAVID HARRAH

As emphasized by many writers, conditional questions call for an answer only if a given condition obtains. A general theory may use these basic notions: I calls for an answer in an interpretation M , and A is a direct answer to I in M . The latter is unde ned unless I is operative (i.e. calls for an answer) in M . Absolute interrogatives are those for which direct answer is de ned without relativization to M . Relativized interrogatives are those for which direct answerhood is relativized to interpretations. Categorical interrogatives call for an answer in every M and have the same direct answers in every M . Belnap writes (P=I ) for a conditional interrogative with condition P and conditioned interrogative I . Then (P=I ) calls for an answer in an interpretation M iff P is true in M and I calls for an answer in M . If (P=I ) calls for an answer in M , then A is a direct answer to (P=I ) iff A is a direct answer to I in M . Belnap also discusses conjunction and union of relativized interrogatives, and points out the natural generalization via universal quanti cation.

4.9 Presupposition and Truth Belnap's intention is, roughly, that every question presupposes precisely that at least one of its direct answers is true. (For relativized questions we should pre x ìf the question is operative, then . . . '.) Presuppositions can be attached to interrogatives in a way that parallels their attachment to questions. For simplicity we formulate the discussion here in terms of interrogatives. We assume throughout that A is any wff, and M is any interpretation. As before, d(I ) abbreviates `direct answer to I ', and D(I ) denotes the set of direct answers to I . Let I be a whether-interrogative. Then: I is true in M iff some d(I ) is true in M , and I is false in M otherwise. I presupposes A if A is true in every M in which I is true. A expresses-the-presupposition-of I if A is true in exactly those M in which I is true. Let I be a which-interrogative. Then: I is really true [really false ] in M iff some real answer to I is true in M [every real answer to I is false in M ]. I is nominally true [nominally false ] in M iff some (nominal) d(I ) is true in M [every d(I ) is false in M ]. I really [nominally ] presupposes A iff A is true in every M in which I is really [nominally] true. A expressesthe-real- [nominal-] presupposition of I iff A is true in exactly those M in which I is really [nominally] true. For uniformity we use unquali ed Ì is true in M ', Ì presupposes A', and À expresses-the-presupposition-of I ' for both which- and whether-interrogatives, meaning for which-interrogatives the `real' variety. If there is an A that expresses-the-presupposition-of I , then there are inde nitely many such A, so it is convenient to pick one such and call it the presupposition of I ; notation: Pres(I ). For whether-interrogatives we


21

choose some disjunction of the direct answers. For which-interrogatives the construction is straightforward but tedious. Belnap's example: Let I be ?(Cx k F x). Then Pres(I ) will be a conjunction (P1 ^ P2 ^ P3 ) of at most three conjuncts. P1 is always present, and says that at least one C is an F . P2 is present just in case is ( 8d), where is an integer; P2 says that at most C 's are F 's. P3 is present just in case has the form ( c 6=); P3 says that at least C 's are F 's. It is possible to de ne Pres(I ) in such a way that, for each elementary interrogative I , Pres(I ) is an eectively speci ed wff that expresses-thepresupposition-of I . In the case of nominal presuppositions it is not in general possible to nd for each which-interrogative I a wff that expressesthe-presupposition-of I . See Belnap [1963], Section 7.5. Let us say that X is a quasi-wff iff X is a wff or an interrogative. Then, using X for quasi-wffs and H for sets of quasi-wffs, we de ne: M is an H -interpretation iff every member of H is true in M . X is logically H-true, H-consistent, or H-inconsistent according as X is true in all, some, or no H -interpretations. H 0 propositionally H -implies X iff X is true in every H -interpretation in which every member of H 0 is true. Similarly with other semantic concepts. For brevity the `propositionally' may be omitted before `H -implies' and `H -equivalent'.

4.10 Types of Answer Let us say that a wff, considered as a reply to I , is H -uninformative if it is H -implied by I , and otherwise H -informative. Such a wff is H -foolish if it is H -inconsistent, and otherwise H -possible. It is relatively H -foolish if it is H -inconsistent with I , and otherwise relatively H -possible. Such a wff is an H -complete answer to I iff it H -implies some d(I ), and an H -just-complete answer if it is H -equivalent to some d(I ). It is an H -partial answer iff it is H -implied by some d(I ). It is an H -eliminative answer iff it H -implies the negation of some d(I ), and H -quasi-eliminative answer iff it is H -implied by the negation of some d(I ). Belnap says that A is a Harrah-H-complete answer to I iff A is a (H [ fI g)-complete answer to I . The essential idea, which was suggested by Harrah [1961], is that in normal cases the questioner believes that I is true and thus includes Pres(I ) in his background knowledge. Example: Let I be ?1 ((A _ B ), (C ^ D)). Then :C is not a complete answer, but it is a Harrah-complete answer. The same idea can be used to generate `Harrah' variants on many other concepts, some of which are noted below. Suggestion: Find a substitute for the label `Harrah', preferably a short word meaning `presupposition-aided'. A proper H -partial answer is a partial answer that is implied by some H -consistent d(I ). A is a highly proper H -partial answer iff A is both H -

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DAVID HARRAH

informative and a proper H -partial answer. (In contrast, the safe answers are the uninformative partial answers.) A is a nominal H -corrective answer to I iff A implies the negation of every d(I ). A is an H -corrective answer to I iff I is (really) false in every M in which A is true. For a standard corrective answer we may choose :Pres(I ) and abbreviate it (following Aqvist [1965]) by Corr(I ). Any wff that counts as an answer to I relative to H , in any of the senses of answer de ned above, may be said to be erotetically H -relevant to I .

4.11 Properties of Interrogatives An interrogative I is H -safe iff I is logically H -true (i.e., H implies I ); otherwise I is H -risky. I is H -foolish iff I is H -inconsistent; otherwise I is H -possible. (Thus I is H -safe if H implies Pres(I ), and H -foolish if H

implies Corr(I ).) Parallel to these (real) concepts for interrogatives there are nominal variants, and there are concepts for questions. THEOREM 2. [Belnap's Hauptsatz, or, the Theorem of the Fifth Gym-

nosophist (see Plutarch's Alexander)] Ask a foolish question and you get a foolish answer.

Belnap gives a proof of the corresponding result for interrogatives. The proof is straightforward. (See Belnap and Steel [1976], pp. 131 { 133.) Continuing in terms of interrogatives: I is dumb iff I has no direct answers. I is H -exclusive if in each H -interpretation there is at most one true real answer (for which-interrogatives) or abstract answer (for whetherinterrogatives). (Thus ?1 8(: : :) and ?8(A1 ; : : : ; An ) turn out to be exclusive.) I is a Hobson's Choice if I has exactly one direct answer (e.g. ?1 (A)). I is answerable by H iff H implies some d(I ); otherwise I is H -unanswerable. I is Harrah-answerable by H if (H [ fI g) implies some d(I ). Here, if H is the questioner's beliefs, we can say that I is H -rhetorical. Similarly, I may be Harrah-unanswerable by H , and, if H is the questioner's beliefs, we say that I is moot, or open. I is hyper-H -moot, or H -wide-open, if H provides neither a Harrah-complete answer nor an eliminative answer. Following the suggestion of Aqvist (who used the label `normal'), I is H -independent if no d(I ) H -implies any other d(I ). I is H -minimal if, for every A in D(I ), there is an H -interpretation M in which A is the one and only true d(I ). Minimality implies independence, but not conversely. I1 H -contains I2 just in case every H -consistent d(I1 ) is an H -complete answer to I2 . I1 Harrah-H -contains I2 iff every d(I1 ) is a Harrah-H complete answer to I2 .

Example : ?1 (A; B ) Harrah-contains ?1 8(A; B ). I1 is erotetically H -equivalent to I2 iff for every d(I1 ) there is an H equivalent d(I2 ), and, for every d(I2 ) there is an H -equivalent d(I1 ). I1 is


23

Harrah-erotetically H -equivalent to I2 iff I1 is erotetically (fI1 ; I2 g [ H )equivalent to I2 . Example: ?1 (A; B ) and ?1 8(A; B ). I1 is erotetically H -relevant to I2 iff some d(I1 ) is erotetically H -relevant to I2 .

Example : ?1 (A; :A) is (propositionally) equivalent to ?1 (B; :B ) but is not

erotetically relevant to it. I1 H -obviates I2 iff every d(I1 ) is either an H -corrective answer or an H -complete answer to I2 .

4.12 Extending Belnap's Analysis There are many possibilities for further development of Belnap's system. Some of these are suggested in an obvious way by the system itself. First: Belnap formalizes six types of elementary-like question (4.7 above), but, as Belnap and Steel point out, thirty types remain. How should they be formalized? Do they have a natural semantics? Do they have any interesting use? (For a relevant discussion, see Hi_z [1962].) Second: Belnap formalizes the maximum completeness-claim. As he points out, inde nitely many other types of completeness-claim remain to be studied. Third: Belnap and Steel say that only one type of distinctness-claim seems to have a systematic use in question logic. This may be doubted, however, because of examples like the police chief who says `Give me a (nominal) list of at least ten suspects, of whom at least seven are (really) distinct.' Surely the matter warrants further study. Fourth: Several possibilities are suggested by ideas of Kubinski. (For a summary in English, see Kubinski [1980].) Kubinski assumes an inde nitely large stock of interrogative operators. Simple interrogatives are formed by pre xing an interrogative operator to a sentential function that contains free variables (which are then bound by the queriables in the operator). One interesting dierence from Belnap concerns answer-size speci cations. Belnap's interrogatives contain a selection-size speci cation, but each of Kubinski's interrogative operators is in eect a string of numerical quanti ers, with each quanti er binding its corresponding queriable, so the quantity speci cations are attached to the queriables individually. This makes possible a straightforward formalization of interrogatives like: `Which two ministers voted against which three projects on which ve committees?' `Which at most three kings have ruled which at least two countries?'

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DAVID HARRAH

(Possible objection: English does not provide interrogatives like these. Reply: Polish shows that it ought to.) For more on interrogatives like these, and eÆcient ways of formalizing them, see Wisniewski [1995], x2.2.6.2. In connection with Belnap's system, what remains to be studied are (1) which of these are warranted in the system, and (2) how they can be accommodated in a smooth way. Fifth: As noted by many authors, the more one enriches the underlying assertoric logic, the more questions one can construct. Some possible enrichments mentioned by Belnap and Steel are: adding variables of higher type, adding modal operators, and allowing in nite conjunctions. Sixth: Although Belnap has argued for the importance of eectiveness at certain points in a logic of questions, and indeed one might take this as essential to his approach, we might study how the system could be extended if various of the eectivity requirements were relaxed. One example: If we drop the requirement that nominal categories be decidable, we can allow questions like `Which theorems of set theory should a number-theorist know?' 5 EPISTEMIC ANALYSIS OF QUESTIONS

5.1 Motivation

In this section we outline what is called the imperative-epistemic approach to questions. The object of study is said to be the `standard' situation, in which (1) the questioner does not know any direct answer, and (2) the interrogative is taken to be synonymous with an imperative of the form `Let it be the case that I know . . . ' or `Make it the case that I know . . . '. In our discussion below we may refer to this as the MMK (or `Make Me Know') approach. Kubinski mentions Bolzano and Loeser as precursors of this approach (Kubinski [1980], p. 131), but the rst to give a substantial formal analysis was Aqvist. Accordingly we begin here by summarizing, in 5.2 { 5.6, the work presented in Aqvist [1965]. In the next sections (5.7 { 6.3) we outline some re nements and further developments, and some other theories that analyze questions in terms of imperatives of various kinds. Many of these theories have an empirical (perhaps phenomenological) motivation, with an implied claim that most actual question situations are \standard" (see, e.g., Wachowicz [1978], p. 157). Complications may arise if one formalizes via a logical framework (e.g., epistemic logic) that has a normative motivation.


25

On all variants of this general approach one may use SA methodology, but it is not necessary. It is possible to develop a substantial part of the logic of questions within the logic of imperatives, without considering answers at all. To show this in detail is a contribution of Aqvist.

5.2 Foundations Aqvist assumes an applied rst-order predicate calculus with the quanti ers (Ux) and (Ex) and identity, supplemented with the operators ! i K P

(`Let it turn out to be the case that') (Ìt is permissible that') (Ì know that') (Ìt is compatible with everything that I know that')

To refer to free individual symbols (individual constants and free variables) Aqvist uses: a, b, c. To refer to bound individual variables: x, y, z . For one-place predicate constants: Ji . For n-place predicate constants: Fin . For the result of putting x for all free occurrences of the variable a in p we write: p(x=a). For QIE (quanti ed imperative-epistemic) logic: 1. Every propositional constant, predication (predicate followed by its arguments), and identity (a = b) is a QIE-wff. 2. If p and q are QIE-wffs, then so are :p, (p ^ q), (p _ q). 3. If p is a QIE-wff containing free occurrences of a, but not containing ! or i, then (Ux)p(x=a) and (Ex)p(x=a) are QIE-wffs. 4. If p is a QIE-wff not containing ! or i, then Kp and P p are QIE-wffs. 5. If p is a QIE-wff not containing ! or i or any free variables, then !p and ip are QIE-wffs. A sentence is a QIE-wff with no free variables. A statement is a sentence that does not contain ! or i. An ordinary statement is a sentence that does not contain !, i, K , or P . For the semantics Aqvist adopts (with some re nements) Hintikka's notions of model set and model system for the logic of K and P , and (following Kanger) adds a relation of imperative alternativeness and imposes appropriate conditions, e.g. if is a model set in a model system and + is an imperative alternative to in , then, if !p is in then p is in + . On the other hand, ! and i are to act as genuine imperative operators only when their arguments are epistemic statements; if p is an ordinary statement in and + is an imperative alternative to , then p is in + . To have

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DAVID HARRAH

a complete logic of imperatives one would have to add further imperative operators (with distinct properties) that apply to ordinary statements. If one wants to satisfy both normative and empirical motivations, one must impose a relatively complex set of conditions on model systems. We omit details. The culminating idea is the usual one: A set of sentences is consistent if it can be embedded in a model set that is a member of a model system. A sentence p is valid if f:pg is not consistent, and p entails q if fqg is embeddable wherever fpg is.

5.3 De nition of Questions

Aqvist de nes interrogatives by introducing various interrogative operators in abbreviations of sentences of the form !p. The following are examples. Whether-questions. (1) ?n (p1 ; : : : ; pn ) =df !(Kp1 _ : : : _ Kpn); where n 2 and each pi is an ordinary statement.

(2) ?n=m (p1 ; : : : ; pn j r1 ; : : : ; rm ) =df !(r1 ^ : : : ^ rm ! (Kp1 _ : : : _ Kpn)) where n 2, m 1, and each pi and ri is an ordinary statement. Monadic complete-list which-questions. Let p be a QIE-wff not containing !, i, K , or P and containing just one free variable a; let x, y, z be the alphabetically earliest variables not bound in p. Then we de ne: (3) (?A a)p =df !(Ux)(p(x=a) ! (Ey)(y = x ^ (Ez )K (z = y))) (4) (?B a)p =df !(Ux)(p(x=a) ! (Ey)(y = x ^ Kp(y=a))) (5) (?KB a)p =df !K (Ux)(p(x=a) ! (Ey)(y = x ^ Kp(y=a))) (6) (?EB a)p =df !((Ex)Kp(x=a) ^ (Ux)(p(x=a) ! (Ey)(y = x ^ Kp(y=a)))) (7) (?EKB a)p =df !((Ex)Kp(x=a) ^ K (Ux)(p(x=a) ! (Ey)(y = x ^ Kp(y=a)))) The latter two are equivalent to (8) (Ex)p(x=a) ^ (?B a)p (9) (Ex)p(x=a) ^ (?KB a)p Thus (?EKB a) and (?EB a) carry a nonemptiness claim; (?EKB a) and (?KB a) carry a completeness request. Aqvist says that (?A a) is too weak to be of interest, and that (?EKB a) best re ects ordinary use.


27

Monadic at-least-which and exactly-which. (10) (?C a)p =df !(Ex1 ) : : : (Exn )K (p(x1 =a) ^ : : : ^ p(xn =a) ^ xi 6= xj ) n

(11) (?D a)p =df !(Ex1 ) : : : (Exn )K (p(x1 =a) ^ : : : ^ p(xn =a) ^ xi 6= xj ^ ^ (Uy)(p(y=a) ! y = x1 _ : : : _ y = xn )) n

where x1 , . . . , xn , y are the earliest distinct variables not bound in p, and where `xi 6= xj ' abbreviates the obvious distinctness-clause. The interrogative (?D a)p is equivalent to (?C a)p0 , where p0 is n

n

(p ^ (Uy)(p(y=a) ! y = x1 _ : : : _ y = xn )): Cf. the comments on the Theorem in Section 4.3 above. Pure polyadic relational which-questions. The generalization to the case where monadic p is replaced by polyadic p is straightforward. Here, the de ned operators have the form (?m a1 ; : : : ; am), where is B , KB , EB , EKB , Cn , or Dn . Mixed polyadic which-questions. Let R be a binary relation. Aqvist shows how to de ne operators for: `Which [At least which n] [Exactly which n] objects bear R to which (or, at least which k, or, exactly which k) objects?' e.g., the interrogative (?2D B a; b) is introduced to abbreviate n

!(Ex)(Ey)K ((Ez )Rxz ^ (Ez )Ryz ^ x = 6 y^ ^(Ut)((Ez )Rtz ! t = x _ t = y) ^ (Uu)(Rxu ! ! (Ew)(w = u ^ KRxw) ^ ^(Uu)(Ryu ! (Ew)(w = u ^ KRyw))));

which expresses Èxactly which two objects bear R to which things?' Generalization from binary R to n-ary R is straightforward but tedious. Categoreally quali ed which-questions. One way to provide for categoreal quali cation is to use many sorts of variables x and then de ne interrogatives of the form (?m x1 : : : xm )p: Aqvist prefers to retain one-sorted theory and de ne interrogatives of the form m

1

;:::;J (?mJ a ;:::;a )p: Roughly, the latter are de ned by inserting [Ji xi appropriate places. 1

1

m m

!]

or [Ji xi ^] in the

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DAVID HARRAH

Mixed whether-which and conditional-which questions. Aqvist de nes CoreQ as the epistemic statement that is the scope of the ! operator in Q. Now let r1 ; : : : ; rk be ordinary QIE-statements. Then: (12)

(?m EKB a1 ; : : : ; am ; + k )(p;r1 ; : : : ; rk ) m =df !(Core(?EKB a1 ; : : : ; am )p _ Kr1 _ : : : _ Krk )

(13)

(?m EKB a1 ; : : : ; am j k )(p j r1 ; : : : ; rk ) =df !(r1 ^ : : : ^ rk ! Core(?m EKB a1 ; : : : ; am )p):

Similar operators can be de ned for EB, Cn , Dn . Also, categoreal quali cation can be added in the obvious way.

5.4 Presupposition, Riskiness, and Guarding Let us use Q as a variable over questions. Then: Q presupposes p iff p is an ordinary statement, and CoreQ entails p. [For Core, see above.] A statement p is a correction to Q iff p is the negation of some presupposition of Q. A safe question is one whose presuppositions are valid. A risky question is one that is not safe. PresQ is the result of dropping all occurrences of imperative and epistemic operators from Q. Claim: If Q entails an ordinary statement p, then PresQ entails p ( Aqvist [1965], p. 133). We interpret PresQ as being the presupposition of Q. The correction of Q = CorrecQ = :PresQ. Roughly, guarding a risky question consists of transforming it into a safe one. Three methods for guarding a risky whether-question of the form ?n (p1 ; : : : ; pn ) are: I. [Whether-Whether Method] Use ?n+1 (Correc(?n (p1 ; : : : ; pn )); p1 ; : : : ; pn ) II. [Whately-Prior Method] Use (?1 Pres(?n (p1 ; : : : ; pn ))^?n=1 (p1 ; : : : ; pn=Pres(?n (p1 ; : : : ; pn )))) III. [Whether-If Method] Use ?n=1 (p1 ; : : : ; pn =Pres(?n (p1 ; : : : ; pn ))) Here methods I and II are equivalent. Four methods for guarding a risky which-question of the form (?m EKB a1 , . . . , am )p are:


29

IV. [Mixed Whether-Which Method] Use m (?m EKB a1 ; : : : ; am ; +1)(p; Correc((?EKB a1 ; : : : ; am )p))

V. [Whately-Prior Method] Use

(?1 Pres((?m EKB a1 ; : : : ; am )p) ^ m (?EKB a1 ; : : : ; am =1)(p=Pres((?m EKB a1 ; : : : ; am )p)))

VI. [Which-If Method] Use m (?m EKB a1 ; : : : ; am =1)(p=Pres((?EKB a1 ; : : : ; am )p))

VII. [Weak Mixed Whether-Which Method] Use (?m KB a1 ; : : : ; am )p: Here methods IV and V are equivalent. Similar techniques can be used for the cases of EB , Cn , and Dn .

5.5 Direct Answers Roughly, a direct answer d to Q should be such that, if the questioner comes to know that d is true, then the epistemic request expressed by Q is satis ed. Speci c criteria that a theory should meet include the following. (These criteria will become clear in terms of the examples below.) First, if p is either a direct answer to Q [relative to r] or a direct pseudo-answer to Q, then p[p ^ r] should entail PresQ. Second, Q1 is to entail Q2 iff every direct proper answer to Q1, as well as every direct pseudo-answer (if any) to Q1 , entails some direct proper answer or direct pseudo-answer to Q2 . Aqvist does not yet have a complete theory of direct answers, but makes at least the following speci c proposals. The numbers refer to the questions de ned above in section 5.3. To (1) the direct answers are p1 ; : : : ; pn. To (2) the direct answers are p1 ; : : : ; pn ; and :(r1 ^ : : : ^ rm ) is a direct pseudo-answer. To (7) the direct answers are (p(c1 =a) ^ : : : ^ p(ck =a) ^ (Ux)(p(x=a) ! ! (x = c1 _ : : : _ x = ck )) ^ [: : : (Ey)(y = ck )]):

Here, and below, x and y are to be the earliest variables not bound in p, and [: : : (Ey)(y = ck )] abbreviates ((Ey)(y = c1 ) ^ : : : ^ (Ey)(y = ck )):

To (6) the statements

(p(c1 =a) ^ : : : ^ p(ck =a) ^ [: : : (Ey)(y = ck )])

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DAVID HARRAH

are direct answers relative to the ordinary statement (Ux)(p(x=a) ! (x = c1 _ : : : _ x = ck )): To (10) the direct answers are (p(c1 =a) ^ : : : ^ p(cn =a) ^ [(ci 6= cj )] ^ [: : : (Ey)(y = cn )]); where there are no repetitions among the c's and [(ci 6= cj )] is the obvious conjunction of nonidentities. To (11) the direct answers are (p(c1 =a) ^ : : : ^ p(cn =a) ^ [(ci 6= cj )] ^ (Ux)(p(x=a) ! ! (x = c1 _ : : : _ x = cn )) ^ [: : : (Ey)(y = cn )]): To (12) the direct answers are the direct answers to (7), plus r1 ; : : : ; rk . To (?EB a; +k)(p; r1 ; : : : ; rk ) the direct answers are the relativized direct answers to (6), plus r1 ; : : : ; rk . Each direct answer to (?C a)p [i.e. (10)] is a direct answer to (?C Ja)p relative to the statement (Jc1 ^ : : : ^ Jcn ). Each direct answer to (?D a)p [i.e. (11)] is a direct answer to (?D Ja)p relative to (Jc1 ^ : : : ^ Jcn ), except that in the direct answer we insert [Jx !] after (Ux). (?EKB Ja) is handled in the same way (?D Ja ) is. For (?EB Ja) we relativize to a complete-list statement. For all the cases given above the generalization from monadic p to polyadic p is straightforward. For details see Aqvist [1965], pp. 156 { 158. Aqvist indicates, but does not prove, that the foregoing proposals satisfy the criteria mentioned at the beginning of Section 5.5 ( Aqvist [1965], p. 156). On the other hand, he says that the theory is not yet complete. There is, e.g., an open problem concerning how to de ne direct answer for (?KB ) questions (Ibid., p. 158). Aqvist cites the following as an incoherence. Let Q be (?1EKB x)Nx, expressing `Which things are the natural numbers?'. Then PresQ is true, but no d(Q) is true. He conjectures that this is the only type of question that falsi es the Coherence Principle : For every Q and M , PresQ is true in M iff some d(Q) is true in M . (Ibid., p. 160) (Cf. Section 4.9 above.) n

n

n

n

n

5.6 Extending the System

Aqvist conjectures: We can accommodate all of Belnap's questions if we add set theory to QIE, construe questions as presenting sets of alternatives, construe Belnap's request-indicators as quanti ers over these sets, and interpret these quanti ers in terms of imperative, epistemic, and ordinary quanti ers (Ibid., p. 82). This program remains to be carried out.


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5.7 Revising the Foundations

Several motivations led to a revised proposal in Aqvist [1971]. Roughly: The 1965 analysis assumed a single context of use, a single knower, a nonindexical K , and monadic imperative operators ! and i. The revision assumes a set of possible contexts (each context being a tuple C = hs; r; t; : : :i, where s is a sender, r a receiver, t a time, . . . ). It also assumes a set of possible knowers, an indexical K , and imperative operators ! and i that are dyadic (to express conditional obligation and permission) and may be indexical as well. The new K carries a subscript for the knower and may carry other subscripts for other parameters such as time. Thereby we have not only standard questions (`Make me know') but also test questions (`Make me know that you know'). Because ! is conditional, the basic form of interrogative is now the conditional. However, unconditional forms can be de ned straightforwardly, e.g. ?n (p1 ; : : : ; pn ) =df !(Kp1 _ : : : _ Kpn =p1 _ : : : _ pn) ^ ^(p1 _ : : : _ pn ): To make the epistemic logic work more smoothly, Aqvist introduces several new kinds of existential quanti er. These might generate new types of interrogative, but this matter has not been studied in detail.

5.8 Hintikka's Development In general, on the MMK approach the point of a question is to express an epistemic request, and the point of an answer is to satisfy this request. Loosely speaking, in the 1960s Aqvist developed a theory of epistemic requests, and, since the mid 1970s, Hintikka has been articulating a theory of epistemic request-satisfaction. Consider `Bring it about that I know that A or I know that B .' In Hintikka's exposition this has a presupposition | namely, À or B ' and a desideratum | namely, Ì know that A or I know that B .' A reply that satis es the epistemic request of the questioner completely is a conclusive or full answer. A reply that is not conclusive but does contribute some information toward satisfying the request is a partial answer. One of Hintikka's aims is to develop the logic of conclusiveness.

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DAVID HARRAH

The conditions for conclusiveness may dier from one type of question to another. For example, the question `Who killed Julius Caesar?' has Ì know who killed Julius Caesar.' as desideratum, and the reply `Cassius killed Julius Caesar.' is a conclusive answer only if the questioner knows who Cassius is. Hintikka says that, for all simple questions (without intensional operators), if the desideratum has the form (9x)KS (x); then b is a conclusive answer to the question if and only if (9x)K (b = x) is established. Loosely speaking, conclusiveness requires that bound variables range over the questioner's domain of acquaintance, and singular terms denote things in that domain. One consequence is that conclusive answer is not eectively recognizable from the syntactical form of the question. For an introduction to Hintikka's theory, see Hintikka [1976], [1983], and [1992], and Hintikka (ed.) [1988]. For further discussion and criticism of Hintikka, see Harrah [1979] and [1987]. For further studies of the epistemics and pragmatics of questions, see Kiefer (ed.) [1983] and Groenendijk and Stokhof [1984]. Hintikka and others have developed what has come to be called an interrogative model of inquiry that is intended to correspond to rational inquiry in science and other elds. In general, using the basic concepts of a given theory of questions one can formulate rules for rational question-and-answer procedures, including question-and-answer dialogues. In the case of Hintikka's approach the idea is to choose rules whose rationale derives from his epistemic logic and game theoretic semantics. One promising way of developing the semantics for this approach uses the concept of interrogative tableau. Such tableaux are formed from Bethstyle deductive tableaux by adding interrogative rules | namely, rules that concern the formula that is being queried and the responses that might be given. Dierent sets of rules are possible, and correspondingly dierent systems of tableaux may be developed. For an introduction to this approach and the techniques involved, see Hintikka (ed.) [1988], Hintikka [1992], and Harris [1994].


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6 OTHER APPROACHES

6.1 MMB (`Make Me Believe') The MMK analysis of questions, and especially Hintikka's development of it, is really a family of dierent analyses. Family members share the reference to knowledge; family members may dier in their conception of knowledge. Consider the following (where B means Ì believe that'):

!P KP ! KKP KP ! BP (KP ^ [P implies P 0 ]) ! KP 0

1. KP 2. 3. 4.

This set of assertions articulates a conception of knowledge that is appropriate for some normative models of knowledge and belief. One might for various reasons hold that such a conception is too strong. One might decide to drop (4) and perhaps also (2). One might decide to drop knowledge altogether. Instead of using epistemic imperatives, one would use doxastic imperatives like `Bring it about that I believe that A or I believe that B .' Conclusive answers would bring suÆcient evidence to produce stable rm belief. Obviously there is a spectrum of possible systems, corresponding to possible conceptions of belief and evidence. This spectrum awaits detailed study.

6.2 TMT (`Tell Me Truly') D. and S. Lewis criticize Aqvist's MMK approach and argue that a more adequate theory results if one takes interrogatives to be synonymous with imperatives of the form `Tell me truly . . . ' (Lewis [1975]). Adopting some of the Lewises' ideas, Aqvist [1983] outlines a way of developing this approach. Aqvist assumes a three-place relation of presentation (sender X presents sentence S to receiver Y ). The language includes an èmpty' sentence representing silence, so that each X always presents some S to each Y . Then the question Ìs it the case that P ?' is to be analyzed as Let it be the case in the immediate future that either there is a sentence S such that (i) you present S to me, (ii) S is true iff P , and (iii) S is true, or there is a sentence S such that (i) you present S to me, (ii) S is true iff not-P , and (iii) S is true.

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The question `For which time x is it the case that F x?' is to be analyzed as: Let it be the case in the immediate future that, for some time x, there is a sentence S such that (i) you present S to me, (ii) S is true iff F x, and (iii) S is true. The locution `S is true' is to be analyzed as suggested in Kripke [1975]. A detailed working out of this approach remains to be undertaken.

6.3 GMA (`Give Me an Answer') In some situations the questioner doesn't need MMK or TMT. What will suÆce is simply `Give me an answer.' For arguments and examples, see Harrah [1987]. One way to develop this approach is to assume set theory and a predicate like Aqvist's presentation predicate. Suppose that b is a term denoting a set D of sentences. Then, to ask the question whose direct answers are the members of D, we use `Let it be the case that, for some x in b, you present me with x'. Suppose we want GMACT (Give me an answer and claim that it is true'). One method is to add syntax (including the theory of the concatenation operator^) and a sentential operator `, so that we have \. . . you present me with ``'^x." The GMA and GMACT approaches, like TMT, await detailed study. The general problem is to determine which systems model the GMA or GMACT idea at some level of abstraction (see Belnap [1969], p. 122, concerning his own system). The speci c problem is to develop systems in which the GMA or GMACT idea is directly expressed by interrogative-imperatives.

6.4 Questions as Context Descriptions Hamblin discusses analyses like that of Jereys, in which an interrogative is taken to be synonymous with Ì do not know . . . ; I want to know . . . ; and I think you know ...' He says that even if such analyses can be made precise in a noncircular way (avoiding the phrase `know whether'), they nevertheless confuse two things that should always be kept distinct: (1) the description of the situation, and (2) the content of the question (Hamblin [1958]).


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6.5 Questions as their Presuppositions Another kind of analysis takes an interrogative to be synonymous with the declarative sentence that is (roughly) the presupposition in the sense of Belnap or Aqvist (see Section 4.9 and 5.4 above). In Harrah [1961, 1963a] a whether-question is an exclusive disjunction; its direct answers are the disjuncts. A which-question is an existential generalization (the existentially quanti ed variables being the queriables); the direct answers are substitution-instances of the quanti ed matrix. In Harrah [1961] a question is required to be true. In Harrah [1963a] a question may be true or false; if false, it is said to commit the fallacy of many questions. The motivation for this sort of analysis is metalogical (to see what can be done within rst-order languages) and technical, rather than empirical. The analysis becomes plausible for application if the question-and-answer situation is interpreted as an information-matching game. The questioner begins by making an assertion (e.g. (A1 _ A2 )), and the respondent then replies by making another assertion (e.g. A1 ) that gives more information about the given subject matter. There is of course no provision for `tagging' sentences to indicate when they are being used to ask questions and when they are being used simply to make assertions. Thus the analysis is plausible where the communication situation can be interpreted as a question-and-answer situation and hence as an information-matching game, but is less plausible in wider contexts where this interpretation is not possible.

6.6 Questions as Intensional Entities According to Tichy [1978], an interrogative expresses a question, and a question is an oÆce | i.e. a function de ned on possible worlds. The commonest types of question are propositions, individual concepts, and properties. These are functions whose values for a given world are a truth value, an individual, and a set of individuals, respectively. To answer a question is to cite an entity of the right type (depending on the type of question); the answer is right if the entity is a value of the function at the actual world. A complete answer cites a single entity of the right type. An incomplete answer cites a class of entities of the right type. An incomplete answer is correct if the right complete answer is one of its members. The motivation here is partly empirical (to account for intuitions about the informativeness of answers to questions), partly metalogical (to see what can be done within the logic of propositional entities), and partly philosophical. It is based on the assumption that logic is the study of logical objects or topics, rather than speakers' concerns and attitudes. It assumes that the declarative-interrogative distinction is not one of logic, and that the logi-

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cian does not have to provide distinct syntactic forms corresponding to the various distinct speech acts such as asserting, asking, and the like. Higginbotham [1993] presents an intensional analysis that is motivated in large part by empirical-linguistic considerations. The aim is to discover the semantics of English interrogatives. The strategy is to develop a theory of questions as intensional entities, and then show how questions may be expressed by interrogatives. For Higginbotham, an elementary abstract question is a nonempty partition of the possible states of nature into cells P such that no more than one cell corresponds to the true state of nature. A partition is proper if at least one cell must correspond to the true state of nature. (The elements of a cell may be thought of as statements; the cell corresponds to the true state of nature if all the statements in the cell are true.) An answer to a question is a set S of sentences that is inconsistent with one or more cells in . An answer is proper if it is consistent with at least one cell. A partial answer is one that is inconsistent with some (but not with all but one) of the cells. X is a presupposition of if every cell in implies X . Complex abstract questions are constructed from elementary ones by quanti cation, conjunction, or disjunction; these form a hierarchy of orders, with abstract questions of order n being sets of abstract questions of order n 1. Elementary abstract questions may be expressed by simple interrogatives and referred to by indirect-question phrases. Complex abstract questions may be expressed by various syntactical means. Most abstract questions are not expressed by any interrogative (there are too many of them). Each interrogative may indicate partition of a limited universe, or partition of a limited part of a given universe. `Who did John see?' has the form [W H : person()]? John saw where the quanti cation is restricted to persons. This interrogative expresses a partition whose cells describe the possibilities of John's seeing (or not seeing) persons. To account for multiple questions (see below), Higginbotham generalizes as follows. Where Q is a restricted quanti er, ' is the restriction on Q, and is another interrogative, an interrogative of the form [Qv : '] expresses a question that is composed of sets of questions, one set for each way in which the quanti er, construed as a function from pairs of extensions to truth values, gives the value true. Consider


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`Where can I nd two screwdrivers?' This has the form [Two x : screwdriver(x)][What : place()]x at and the question it expresses is the class of all classes of partitions each of which, for at least two screwdrivers a and b as values of x (and for no objects other than screwdrivers as values of x), contains the partition for the interrogatives [What : place()]a at [What : place()]b at Classes of partitions are blocs, and classes of them are questions of order 1. To answer a question of order 1 is to answer every question in one of its blocs. On this approach the interrogative-question-answer relationship is not eective; one reason is that some English interrogatives are ambiguous. For example, where an interrogative seems to involve quanti ers, the semantics might not involve quanti ers and sets of questions but might instead involve a functional interpretation. E.g., to `Which of his poems does every poet like least?' the answers that are wanted are not lists of poet-poem pairs but are replies like `His earliest poems.' Concerning the various motivations for, and the rich potential of, the intensional approach, see Tichy [1978], Materna [1981], Belnap [1981], Higginbotham [1993], and Groenendijk and Stokhof [1984]. The latter work is rich in discussion and examples, and it argues that many theories can be articulated to provide a semantic equivalent of some Hintikka-type pragmatic dimensions. (Incidental note: On many approaches, Hamblin's Postulate (3) [recall 2.1 above] is false. On the intensional approach it might be true; see 7.2 below.)

6.7 Questions as Incomplete Entities Several linguists have proposed theories in which the semantically meaningful unit (or at least the unit of truth) is not the question but the questionanswer pair. One example is Keenan and Hull [1973]. Here the motivation is to account for our intuitions about the presuppositions of questions in

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natural language | in particular, the intuitions that (1) questions have no truth value, but (2) questions have presuppositions, and (3) presuppositions are to be analyzed in terms of truth-value. Keenan and Hull in eect identify questions with interrogatives. They de ne the class of L-sentences so that every declarative is an L-sentence, and, if Q is a question and A is a de nite noun phrase, then hQ; Ai is an L-sentence. For the case of wh-L-questions Q, which have the form (which, NP, S ), we say: Q is valid in a state of aairs iff NP speci es a nonempty set (the domain of the question) and S, which is an L-sentence expressing the question property, is true of some members of the domain. The answer set determined by hQ; Ai is the set of objects denoted by A that are also in the domain. The L-sentence hQ; Ai is true in i iff (1) Q is valid in i, (2) the answer set determined by hQ; Ai is nonempty in i, and (3) S is true in i of every member of the answer set. Similarly, hQ; Ai is false in i iff (1) and (2) hold but S is false of some member of the answer set. In addition, hQ; Ai is zero in i iff it is neither true nor false in i. On the basis of these de nitions we can in an obvious way de ne consequence and presupposition. This de nitional chain rests on (begins with) the de nition of valid, and the concept of validity adopted here is similar to Belnap's concept of real truth. Thus it remains to be seen whether there is any formal advantage in using this framework rather than Belnap's or Aqvist's. Hi_z [1978] presents a development of the question-answer pair approach that is like Keenan and Hull's in some of its basic conceptions but diers considerably in details of superstructure. It remains to be seen whether there are any advantages from a formal point of view.

6.8 Questions as Hyper-complete Entities According to an approach suggested by Finn [1974] (generalizing on Harrah [1963a]), a question may be treated as an ordered pair hA; B i such that (roughly) B is an interrogative (or ìnquiry') term, and A is a statement that expresses the presuppositions of the question. In general this approach is motivated by technical and engineering considerations; and, if the context of application is narrowly conceived, then A and B can be narrowly conceived. This idea leads, however, to the following generalization. If the context of application is a relatively general type of problem-solving or inquiry situation, then A and B can be conceived in a relatively general way. Among the interrogative terms B there may be noun phrases like àn x such that . . . '; this phrase indicates what sort of entity is to be found. Among the presupposition statements A there may be any statements giving background information or imposing constraints on the search.


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6.9 IQW (Ìt Is the Question Whether') Hoepelman [1983] proposes to read the propositional operator ? as Ìt is the question whether' Cf. the German Ès ist die Frage ob' and in English `[For me] there is the question whether' `[For me] it is an open question whether' The basic assumption is that Ìs it the case that p?' is a question for me not when p has a truth value for me but when its truth value is still undetermined for me. Hoepelman develops a truth value analysis of interrogatives to serve an empirical motivation | in general, to account for interrogatives in natural language, and in particular to account for distinctions that re ect dierences in the questioner's certainty, as in the following pairs: Ìs John ill?' Ìsn't John ill?' Inter alia, Hoepelman adopts the following truth tables: :p (p ! q) (p $ q) ?p q . . . 11 10 01 00 11 10 01 00 . . . p 11 00 11 10 01 00 11 10 01 00 00 10 01 11 11 01 01 10 11 00 01 00 01 10 11 10 11 10 01 00 11 10 10 00 11 11 11 11 11 00 01 10 11 00 The idea here is to articulate the questioner's certainty in terms of a comparison of two worlds | the world known by the questioner and the world known by the authority to whom the question is to be put. In the truth tables each pair of numbers represents a comparison; think of the rst number as the questioner's certainty about the given statement, and the second number as what the questioner believes is the authority's certainty. According to these truth tables, the following are valid: :?(p ! p) ?:p ! :?p ?p !?:?p ?(p ! q) ! (?p !?q) ?p ! (p ! q)

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The following are not valid: ?p !?:p (p $ q) ! (?p $?q) To accommodate predicate logic and wh-questions Hoepelman extends the propositional apparatus by assuming a pair of models (each with its domain of individuals and assignment of denotations), incorporating the propositional truth value conditions, and then adding truth value conditions for the two quanti ers and wffs with free variables. Loosely speaking: 1. ?'x = 10 if 'x = 01, and ?'x = 00 otherwise. 2. The value of 8x'x is the minimum of the values for 'x.

3. The value of 9x'x is the maximum of the values for 'x. If the two domains of individuals are identical, then the following are valid:

8x?' !?8x' ?9x' ! 9x?' If we add = as a predicate, then

8x8y8z (x = y^?(y = z ) !?(x = z )) is valid, but

8x8y(?(x = y) ^ P x !?P y) is not. For some readers it is an open question whether all of the validity results noted above accord well enough with intuition; but the assertion that this sort of approach is interesting and worth exploring further is not in question. 7 OTHER TOPICS AND FURTHER WORK

7.1 Other Types of Question Some logicians have been concerned to theorize in a global way about questions in general, setting aside or de-emphasizing the problem of distinguishing and analyzing particular types of question (e.g. Tichy [1978], and radical SA reduction). Most logicians, however, have concentrated on distinguishing and analyzing question types, one at a time; and most have concentrated on whether- and which-questions. Little progress was made on who, what, and why questions until the 1970s. See Belnap and Steel [1976], pp. 78 { 87. Hintikka, beginning with Hintikka [1976], has made an aggressive attack on who questions. See also,


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e.g., Grewendorf [1983]. Since the 1970s many logicians and linguists have studied the various types of wh-question found in natural languages, but everyone would agree that much more work needs to be done. In particular, why questions pose a special challenge. Bromberger [1992] contains illuminating and suggestive discussion of why questions, and proposals concerning one important type. This type is exempli ed by: `Why is it the case that X has property Y (instead of property Z )?' Bromberger's conception is that normally, for such questions, the questioner has had in mind a general rule R that represents some expectation E about X but the questioner fails to observe E and hence asks the question. The respondent answers the question by citing (1) an abnormic law L that speci es exceptions to the rule R, and (2) one or more exceptions that are speci ed by L. E.g., expecting the milk to taste good, the child asks `Why does this milk taste sour?' and is told Àll milk tastes good unless it is spoiled or adulterated, and this milk is spoiled.' To formulate this proposal in a way that is general, precise, and not subject to counterexamples is a task that is not yet complete; see Bromberger [1992], pp. 88 { 97. On the diÆculties of accommodating a Bromberger-style analysis within an extensional framework see Belnap and Steel [1976], pp. 84 { 87. Koura [1988] outlines a Hintikka-style analysis of one family of why questions. In these the explanandum is the occurrence of event e, and the question in eect is `Which event caused the event e?' Dierent subtypes correspond to dierent types of causation. For example, let E (x) =df 9y(y = x) and let N be a primitive signifying nomical necessity. Then N [E (x) ! E (y)] expresses that x is a possible (suÆcient) cause of y, and to say that x causes y we use E (x) ^ N [E (x) ! E (y)]: Following Hintikka's approach the desideratum of

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`What caused e?' is

9xK [E (x) ^ N [E (x) ! E (e)]]: Koura shows that a reply f is an adequate answer | i.e., brings about the desideratum | if and only if the conclusiveness conditions

KN [E (f ) ! E (e)] (relevance) 9xKN (x = f ) (uniqueness) both hold. Koura discusses other concepts of cause and suggests that some types of why question might involve pragmatic parameters. Hintikka and Halonen [1995] (hereafter `H&H') rejects this approach and says in particular that no modal element is needed for why questions per se. The gist of the H&H proposal is as follows. Consider `Why does b have property P ?' Suppose we have a sentence T (e.g., a general theory) and a sentence A (e.g., some additional ad hoc information supplied by an oracle) such that 1. (T ^ A) ` P (b)

` P (b) not A ` P (b)

2. not T 3.

4. b does not occur in T 5. P does not occur in A where ``' indicates derivability in a Hintikka-style interrogation game based on rst-order logic. Then by Craig's Interpolation Theorem it follows that there is a formula H [b] such that 1. T

` (8x)(H [x] ! P (x))

2. All the constants in H , except for b, occur in both T and A. 3. A ` H [b]

Call H [b] the initial condition and (8x)(H [x] ! P (x)) the covering law. The proposal is that we answer the given why question by citing the initial condition or the covering law or both. The claim is that these answers are conclusive for anyone who believes that (T ^ A) is true and sees that P (b) is derivable from (T ^ A). (Comment: Those who are not committed to Hintikka-style interrogation games or criteria of conclusiveness may wish to say simply that to `Why


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P (b)?' the direct answers are all sentences of the form (H [b] ^ (8x)(H [x] ! P (x))), where H [x] is restricted in certain ways to exclude cases that are trivial or unacceptable for other reasons. The restrictions imposed by H&H are probably minimal; in practice, users of why interrogatives usually want to put further restrictions on the size or content of H [x]. This matter of restrictions on T , A, and H deserves much more study.) Concerning how questions: H&H suggests brie y that, if b is allowed to occur in T , then no covering law is obtainable in general and deriving P (b) from (T ^ A) seems more like answering a how question than answering a why question. This suggestion deserves to be clari ed and studied in detail. Another type of question awaiting further study is the deliberative question, exempli ed by, `What shall I do?' These seem to call for a decision or resolution (Wheatley [1955]). They might be analyzed as genuine questions, but of a type peculiar to oneperson decision-making situations. Alternatively they might be analyzed as having properties appropriate to the usual two-person question-and-answer situation, plus further pragmatic properties as well (so that they call for both an answer and a resolution). Alternatively, as suggested by Mayo [1956], they might be interpreted as calling for an imperative (`Do X !').

7.2 Other Types of Reply As noted in Section 2.1, Hamblin's Postulate (1) has not been universally accepted. In the rst place, as already noted, some logicians and many linguists have argued for allowing noun phrases to count as direct answers. There is much un nished business here, e.g. no one has yet provided a comprehensive formal system for connecting noun phrases with the system of completeness-claims and distinctness-claims. Natural language seems to have at least a rudimentary system; e.g. to questions like `Who were all the students who passed?' we give answers like Ònly two: Shane and Mark.' The problems for research are: What exactly is the system in natural language? and How can we formalize analogs for arti cial languages? In the second place, we might accept Hamblin's assumption that every direct answer is a sentence, but question whether it must be a statement. The suggestion of Mayo (see Section 7.1) that imperatives like `Do X !' should count as direct answers presumably applies to deliberative questions and their close relatives. In the case of most other kinds of question, however, it often seems appropriate to reply with certain kinds of imperative, as in

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Àsk someone who knows the Lexitron!' or interrogative, as in `How detailed an answer do you want?' Also, of course, there are nonanswer declaratives like Ì don't know' and `That's a long story' It might be argued that some of these nonanswer sentences are evasions (or possibly corrections) of the given question, but some should count as illuminations or at least helps. In any case the eld awaits formalization. For a beginning, on Ì don't know' replies at least, see Todt and SchmidtRadefeldt [1979], p. 15. There is much work to be done on the general topic of appropriate response, and in particular on the topic of nding or constructing an appropriate response. See, e.g., Lehnert [1978]. The notions of incomplete answer and partial answer invite further study. In the usual conception a partial answer is one that is implied by a direct answer. Belnap and Steel [1976] emphasize that this conception is relative to the type of implication assumed, and that we may re ne the type of partial answerhood by re ning the type of implication. For other problems and other perspectives on partial answerhood, see Cresswell [1965], Kubinski [1967], Groenendijk and Stokhof [1984], pp. 233 { 236, Higginbotham [1993], and Wisniewski [1995], pp. 114 { 115, 179 { 180. Finally, there is the challenge of Hamblin's Postulate (3). If it is interpreted as saying that, for every question, exactly one answer is true, then it seems obviously false. Is there some other plausible interpretation under which it is true? Consider Higginbotham's concept of proper partition (noted in 6.6 above), in which exactly one cell corresponds to the true state of nature. This might provide an intensional-analytic rationale for Postulate (3). On this and related matters, see Groenendijk and Stokhof [1984].

7.3 Implying, Raising, and Suppressing Hamblin's containment (recall 2.1 above) may be thought of as a kind of implication between questions. When that relation holds, each answer to the rst question implies an answer to the second, so the rst covers the second, so, if you ask the rst, you need not ask the second. Belnap's propositional implication (recall 4.9) is a simple generalization from implication between statements to, loosely speaking, implication between statements and questions. When that relation holds, truth of the implying statements and questions guarantees truth of the implied statements and questions; so,


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if you know that the rst question is good, you know that the second is good. Wisniewski [1994a and 1995] de nes a cluster of concepts that are less like the Hamblin and Belnap concepts and more like the informal notion of raising, as in `Your statement raises some hard questions.' `Your question raises a more basic question.' Loosely speaking, Wisniewski's concept of implication expresses the idea that, if you need to ask the rst question, then you would be well advised to ask the second also. The remainder of this section, except for the nal three paragraphs, summarizes part of Wisniewski's analysis. Wisniewski de nes his concepts for a very wide class of languages. This class includes more than the usual rst-order languages, but it will be helpful here for the reader to think of a rst-order language with the usual extensional semantics. Wisniewski uses `d-wff' for `declarative well-formed formula', Q for questions, and dQ for the set of direct answers to Q. Some interpretations are distinguished as normal. A set X entails a d-wff A iff A is true in every normal interpretation in which all the d-wffs in X are true. X logically entails A iff A is true in every interpretation in which all the d-wffs in X are true. X multiple-conclusion entails Y (or, X mc-entails Y ) iff, for each normal interpretation I where all the d-wffs in X are true, at least one d-wff in Y is true in I . Assumptions about questions: 1. Each question has at least two direct answers. 2. Direct answers are sentences (d-wffs without free variables). (Hence each dQ is at most denumerable.) 3. Each set of sentences that is nite and has at least two members is the dQ for some Q.

Q is sound in an interpretation I iff some A in dQ is true in I . Q is safe iff Q is sound in every normal I . Q is sound relative to X iff X mc-entails dQ. Q is risky iff Q is not safe. X evokes Q iff (i) X mc-entails dQ, (ii) for each A in dQ, X does not entail A. X generates Q iff X evokes Q and Q is risky. Q implies Q0 on the basis of a set X of d-wffs [or Im(Q; X; Q0)] iff (i) for each A in dQ: X + A mc-entails dQ0 , and (ii) for each B in dQ0 : there is a nonempty proper subset Y of dQ such that X + B mc-entails Y . (Roughly, the implying question Q raises the implied Q0 because Q0 helps to answer Q; Q0 helps because each B in dQ0 directs attention to a proper subset of dQ.) Q implies Q0 [or Im(Q; Q0)] iff Im(Q; ; Q0).

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It is not the case that every safe question is implied by every Q on the basis of every X ; safety guarantees that clause (i) holds but does not guarantee that clause (ii) holds. On the other hand, if Im(Q; Q0 ), then Q0 is safe iff Q is safe. THEOREM 3. If Q is sound relative to X , then there exists a sequence Z of simple yes-no questions (i.e., questions of the form ?fA; :Ag) such that: 1. each question in Z is implied by Q on the basis of X ,

2. each set consisting of direct answers to the questions in Z that contains exactly one direct answer to each question in Z entails along with X some A in dQ, and 3. each nonlogical constant that occurs in some direct answer to a question in Z occurs in some A in dQ. (Note: If dQ is nite, then Z can be nite.) Let X be a nite nonempty set of d-wffs. Then the pair hX; Qi is an e1 -argument, and is valid iff X evokes Q. The triple hQ; X; Q0i is an e2 argument, and is valid iff Q implies Q0 on the basis of X . For any given

language L having d-wffs and questions, we can construct a metalanguage ML that has statements asserting that evocation and implication hold between particular X 's and Q's. Then, for the given L, the set of all of these ML statements that are true can be regarded as the logic of questions of L. (The basic idea for this conception was suggested by Kubinski.) In addition to establishing his general framework as outlined above, Wisniewski studies several particular languages of the usual kinds (propositional, rst-order, etc.) and gives many results and examples for these. Many of these results and examples are for the case where everything has a name | i.e., where every entity in the universe of the assumed interpretation I is denoted by some closed term in the language (under I ). A problem for future research is to explore in detail the cases in which this is not so | i.e., where there are `real answers' (in Belnap's sense) that are not expressed by nominal answers. Another area that awaits development is the case of questions in logic and mathematics, questions whose direct answers are logical or normal truths. Wisniewski's de nition of evocation is tailored to t the case of factual questions, where no direct answer is normally or logically true. The challenge is to extend the analysis to capture the concept of evocation for the wider class of questions. Questions can be raised. Can questions be suppressed? Is suppression a dual of evocation? Is it suÆcient to say simply that X suppresses Q iff X entails :A for all A in dQ, and that Q0 suppresses Q iff every B in dQ0 suppresses Q? Or are there other conditions that are suÆcient for suppression? Why don't we know more about suppression? Don't ask.


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7.4 Reduction of Questions As noted in earlier sections, some theorists hold that questions are reducible to entities of other kinds. E.g., some hold that each interrogative is semantically or pragmatically equivalent to an imperative sentence. We might say that this sort of reduction is inter-categorial. In contrast, as common speech has long recognized, there is what we might call intra-categorial reduction, where one interrogative is construed as equivalent to some other interrogative, as indicated by locutions like `Let me rephrase my question; what I am really asking is . . . ' There are several problems of interest to logicians. One is to develop precise concepts for the general notions of equivalence and reduction, another is to nd techniques for demonstrating reducibility, and another is to establish particular results. Wisniewski [1994b and 1995] has suggested some general concepts and established some results. The basic de nition is this: [Concerning notation, see 7.3 above.] A question Q is reducible to a nonempty set S of questions iff: 1. for each A in dQ, for each question Q0 in S , A mc-entails dQ0 , 2. each set consisting of direct answers to questions in S that contains exactly one direct answer to each question in S entails some A in dQ, and 3. no question in S has more direct answers than Q. Some of Wisniewski's theorems are these: [Concerning terminology see 7.3.] 1. A question Q is safe iff Q is reducible to some set of simple yes-no questions that are implied by Q. 2. If dQ is nite, then Q is safe iff Q is reducible to some nite set of simple yes-no questions that are implied by Q. 3. If Q is risky but dQ is nite, then Q is reducible to a nite set of conditional yes-no questions [i.e., questions of the form ?fA ^ B; A ^ :B g] that are implied by Q. 4. If Q is risky but there is a d-wff B such that (i) B is entailed by every A in dQ and (ii) B mc-entails dQ, then Q is reducible to some set of conditional yes-no questions that are implied by Q. Wisniewski proves these and other theorems using straightforward modeltheoretic arguments, and it might seem that such arguments are the only

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means available for establishing reducibility results. Without making any claims, we conjecture that there might also be exotic methods available, perhaps dierent methods for dierent types of question. We mention two examples: The rst is the set of techniques suggested in the paper of Todt and Schmidt-Radefeldt [1979]. Among the primitive signs of the language adopted are > (for the true) and ? (for the false). This allows interrogatives of the form ?b (b $ A) which literally say `Which Boolean truth-value is equivalent to the statement A?' or colloquially Ìs it the case that A?'. It remains to be seen how much can be done with this sort of apparatus, and in general what its advantages are. The second is the methodology suggested by Leszko [1980]. According to that work certain types of question can be represented via graphs. (Leszko concentrates on the Kubinski questions noted in Section 4.12 above, questions like `For which n x's and m y's is it the case that . . . ?'.) Once a question type has been represented via graphs, we can study the questions by studying the matrices associated with the graphs. See also Leszko [forthcoming].

7.5 Sequencing and Programming The general problems here are to evaluate sequences of questions with respect to their answer-yield, their safety, or other properties, and to compare sequences with respect to various concepts of containment, implication, and equivalence. One special class of problems concerns question trees, including the case of trees in which the nodes represent questions and the branches represent answers. Such a question tree can be used to represent a strategy or plan for asking questions one at a time, where at any time the choice of question depends on what previous questions have been asked and what answers have been given. Some problems for study are: Under what conditions does a question tree represent a safe plan? Under what conditions are two trees equivalent? Under what conditions is a tree equivalent to a single question?


49

Not much has been done on these matters in general, but a few useful concepts have been developed. In Belnap and Steel [1976], p. 138, a sequence of interrogatives (I1 ; : : : ; In ) is said to be a direct partition of an interrogative I if the conjunctive interrogative (I1 ^ : : : ^ In ) is erotetically equivalent to I , and a subdirect partition if (I1 ^ : : : ^ In ) is erotetically equivalent to (I ^ : : : ^ I ), i.e. I conjoined n times. One special kind of sequence that has received some attention is the `corrections-accumulating' sequence. The basic idea was presented in Stahl [1962]. Consider the three questions: 1. Which are the two primes between 13 and 17? 2. Which are at least two primes between 13 and 17? 3. Is there any prime between 13 and 17, or not? Stahl pointed out that in natural language we occasionally put the sequence 1-2-3 by saying `1, or 2, or 3,' where the or is understood to be noncommutative. Stahl generalized and formalized as follows [recall Section 2.2 above]: An inferential question series is a series in which the rst question is relative to some S and the n + 1st question is relative to S [ fAn g, where An is a suÆcient answer to the nth question, An is not a consequence of S , and all the suÆcient answers to the nth question which neither imply An nor are consequences of S are not compatible with An . A suÆcient answer of the nth degree is a wff which either is a consequence of S or implies a conjunction (A1 ^ A2 ^ : : : ^ An 1 ^ B ) which is consistent with S , where B is a suÆcient answer to the nth question which does not imply An . The intention is to yield the theorem: A suÆcient answer of nth degree is incompatible with suÆcient answers of lower degree, unless these are consequences of S . Aqvist develops this idea within his framework as follows. Let Q be a QIE-question, and let fp1; : : : ; pn g be a nite set of ordinary statements. De ne Q[p1 ; : : : ; pn ] as (p1 ^ : : : ^ pn ^ Q), de ne its Core as (p1 ^ : : : ^ pn ^ CoreQ), and de ne its Pres as (p1 ^ : : : ^ pn ^ PresQ). Recall that CorrecX = :PresX . Now let S = fQ1 ; : : : ; Qn g be any nite set of QIE-questions. Form out of S the n! distinct n-termed sequences such that each member of S occurs exactly once in the sequence. Next, name these sequences and arrange them in some xed order: S1 = hQ1 ; : : : ; Qn i; S2 = hQ1 ; : : : ; Qn i; : : : ; Sn! = hQ1 ; : : : ; Qn i: Then, for each Sj , de ne the simplest corrections-accumulating sequence associated with Sj (or sca(Sj ) for short) as hQ1 ; Q2 [CorrecQ1 ]; : : : ; 1

1

2

2

n!

j

j

n!

j

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Qn [CorrecQ1 ; CorrecQ2 ; : : : ; CorrecQ(n 1) ]i: Finally, for each Sj , let its QIE-translation = the QIE-translation of sca(Sj ) = j

j

j

j

!(CoreQ1 _ Core(Q2 [CorrecQ1 ]) _ : : : _ Core(Qn [CorrecQ1 ; : : : ; CorrecQ(n 1) ])): j

j

j

j

j

j

Let S = hQ1 ; : : : ; Qn i be a question sequence. We say that S is corrections-accumulable iff, for all 1 i < j n, PresQj does not entail PresQi . It is successively presupposition-containing iff, for all 1 i < j n, PresQi entails PresQj . It is quite reasonable iff all the disjuncts inside the ! in the

QIE-translation of sca(S ) are consistent. It turns out that, for each nite question-set S = fQ1 ; : : : ; Qn g, there is at most one sequence Sj (1 j n!) formable out of S that is both corrections-accumulable and successively presupposition-containing. In the example above, the sequence would be h1; 2; 3i. Also, for a sequence to be quite reasonable it is necessary that it be corrections-accumulable. One other fact: A corrections-accumulable sequence can have a safe question Q (i.e. with valid PresQ) only in its nal position. (For all of the above, see Aqvist [1969].) Similar concepts and constructions can be developed in Belnap's framework by using his conditional and given-that questions. We form the appropriate sequence of interrogatives and then construct their union. See Belnap [1969]. Picard [1980] studies question sequences in the light of practical considerations like probability, cost, and utility. The general problem is how to replace a single complex and costly question Q by a questionnaire Q0 (which is a sequence of simple which and whether questions), such that Q0 will yield the true answer to Q but asking Q0 will be more eÆcient and economical than asking Q. Questionnaires are represented as weighted nite circuitless graphs meeting certain conditions. (Think of a questionnaire as a bush | starting from one node | whose non-terminal nodes are questions and whose branches are the direct answers.) Each answer is assigned a probability. Answers and questions can be assigned utilities and costs. For more in this area see, e.g., Kampe de Feriet and Picard (eds.) [1974].

7.6 Comparison of Approaches Is there a correct approach to the theory and logic of questions? It is not yet clear that we have a correct approach to clarifying and answering that question; especially, it is not yet clear that we have a correct criterion for recognizing a correct answer. What is clear is that much more work remains to be done in comparing dierent approaches and evaluating their respective advantages. A little


51

work of this sort has been done, but most of it has been done on small aspects and points of detail. What we oer below are some rough surmises. We don't claim that these are accurate or correct; we do hope that they are clear enough to stimulate further study. 1. The systems that can provide the most questions are those that assume questions as metaphysical or intensional entities | as in the theories of Tichy [1978] or Higginbotham [1993]. 2. Is one approach better than others at providing interrogatives? There is some appeal in the idea noted in 6.8 above, that we can adopt a very rich language and thus have interrogatives that specify a detailed description of what is wanted and how to search for it. On the other hand, interrogatives are instruments for communication, and our choice of interrogative system is in uenced by the purposes at hand. Thus it is meaningful or useful to compare interrogative systems relative to speci c motivations (e.g. the motivation to model the question-and-answer system of the Danish people, or the motivation to construct an information-retrieval system for the Yale Medical School Library), but not useful to make comparisons otherwise. 3. The systems that would be most useful in machine-assisted interactions, and especially in formal systems of information retrieval, will have the eectiveness properties emphasized in Belnap and Steel [1976]. 4. For empirical models of the question-and-answer process in natural language several dierent kinds of system will be needed, including not only those of the MMK approach (designed to t the `standard' situation, as noted in 5.1 above) but also others (designed to t other types of situation). 5. Theorizing about questions requires theorizing about interrogatives. We can be con dent that an intensional theory of questions is complete and correct only if we are con dent that we have a complete and correct theory of human concepts and intentions, and we can be con dent of the latter only if we are con dent that we have a complete and correct theory of human language, including a complete and correct theory of interrogatives.

7.7 General Erotetic Logic: Motivations There are several motivations for generalizing from erotetic logic in the narrow sense, concerned with question and answer, to erotetic logic in a broader sense, concerned with all the kinds of expression that call for reply. First consider mixed sentences | e.g.:

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1. `The old machine is broken, or does it need fuel?' 2. `The new one is missing, but do we need it?' 3. `What is wrong with the old one? or nd the new one.' To provide for such sentences we want a logic that will specify for declaratives, imperatives, interrogatives, . . . (?) what compounds are permissible, what expressions call for replies, and what replies are called for. Second, consider vectored sentences such as Às Provost, I ask you, Dean Smith, will the plan be approved?' This may be construed as a sentence that, at some level of analysis, consists of two parts: 1. the body (`Will the plan be approved?'), and 2. the vector (indicating that the message comes from Ì' qua Provost and is for Smith qua Dean). For discussions of vectored sentences see Harrah [1994]. To provide for such sentences we want a logic that will specify what expressions count as vectored sentences, and what expressions (vectored or unvectored) count as replies. Third, consider vectored messages such as the formal memos used in large organizations and the formal letters used in commercial and legal correspondence. These have a vector that speci es a to, a from, a when, and possibly other parameters, and a body that may contain any number of sentences of various kinds. For such messages we want a logic that speci es what expressions count as messages and, for each message, what counts as a suÆcient reply. In 7.8 below we outline a way of developing a logic that provides for mixed sentences, vectored sentences, and vectored messages. This logic may be viewed as a system of general erotetic logic, and our sketch of it should serve to indicate what a general erotetic logic is. From another perspective it may be viewed as a logic of message and reply, or a communicational logic. Perhaps the concepts of general erotetic logic and communicational logic coincide, for in both cases the essential concern is with (a) a set of expressions and (b) for each expression, its set of suÆcient replies.

7.8 General Erotetic Logic: Systems In this section we outline a way of developing a particular system of general erotetic logic. It will be clear that we are in eect describing a class of systems, and indeed a fairly wide approach. We don't claim that this is the only correct or fruitful approach. The motivation for this approach is


53

both empirical and engineering; the aim is to construct systems that will be useful in connection with human communication. (For more on motivation, empirical grounding, and details of development, see Harrah [1985, 1987, 1994].) In the paragraphs below we rst describe the part of the system that handles unvectored sentences, and then the part for vectored sentences and vectored messages. We begin with a standard rst-order system having identity, descriptions, and some nonlogical axioms for set theory and syntax. We write U and E for its quanti ers and F , G, H , . . . for its wffs, which we call d-wffs. We add an in nite stock of speech act operators O, O0 , O00 , . . . . A basic speech act wff (or bsa-wff ) is an expression OV Y such that O is a speech act operator, V is a (possibly empty) string of distinct variables, and Y is a list (Y1 ; : : : ; Yn ) in which each Yi is a term or d -wff. The basic wffs (or b-wffs ) are the d -wffs and bsa -wffs. To every b -wff F we assign a d -wff CA, a d -wff CP , a set IR, and a set W R, such that: 1. IR consists of d -wffs (the indicated replies to F ). 2. CA (the core assertion in F ) is implied by every d -wff in IR. 3. W R (the wanted replies to F ) is a subset of IR. 4. CP (the core projection in F ) is implied by every d -wff in W R. 5. CA is implied by CP . Given a bsa -wff F , with its CA and CP , we say that: The negative reply to F is :CP . The corrective reply to F is :CA. The direct replies to F are: 1. the wanted replies, and :CP , if W R is nonempty; 2. the indicated replies, and :CA, if W R is empty but IR is not; 3. CP , (CA ^ :CP ), and :CA, if IR is empty. The full replies to F are the d -wffs that imply direct replies. The partial replies to F are the d -wffs that are implied by direct replies. The relevant replies to F are the full replies plus the partial replies. To give examples, we use the signs !u ; !c; !d ; :d ; :as ; :an ; ?w ; ?1

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to refer to distinct speech act operators (expressing respectively ultimatum, command, directive, declaration, assertion, announcement, whetherquestion, one-example question). To the eight kinds of bsa -wffs at the left below, content may be assigned as follows: CA CP IR WR !u (G) (G _ G0 ) G fGg fGg !c (G) (G _ G0 ) G fGg !d (G) (G _ G0 ) G :d (G) G G fGg fGg :as (G) G G fGg :an (G) G G ?w (G; G0 ) (G _ G0 ) (G _ G0 ) fG; G0 g fG; G0 g ?1 x(Gx) ExGx ExGx fGa; : : :g fGa; : : :g For smoothness, in the case of each d -wff F , we say that CP (F ) = CA(F ) = F , and the W R(F ) = IR(F ) = . (Note: occasionally, as here, we use `CA', . . . , `W R' as functors.) An erotetic wff (or e -wff) is a bsa -wff F such that W R(F ) 6= . The speech act wffs (or sa -wffs) are de ned recursively: 1. Every b -wff is an sa -wff. 2. If F and G are sa -wffs and x is a variable, then (F ^ G), (F _ G), UxF , ExF are sa -wffs, and (F ! G) is an sa -wff if F is a d -wff. A proper sa -wff is an sa -wff that is not a d -wff, and a non-basic sa -wff is an sa -wff that is not a b -wff. If F is an sa -wff, then G is the core assertion in F (or CA(F ) for short) iff G is like F except that, wherever F contains a bsa -wff H , G contains CA(H ). Similarly for CP (F ). To any non-basic sa -wff F : 1. The negative reply is :CP (F ). 2. The corrective reply is :CA(F ). 3. The direct replies are CP (F ), (CA(F ) ^ :CP (F )), and :CA(F ). 4. The full replies are the d -wffs that imply direct replies.

5. The partial replies are the d -wffs implied by direct replies. 6. The relevant replies are the full replies plus the partial replies. (Note that the non-basic sa -wffs do not have indicated or wanted replies.) We assume that for the d -wffs we have a rst-order predicate logic of the usual kind, and that in addition there might be axioms for set theory, syntax, or the like. To provide for analysis of sa -wffs we add the following rules of ca-derivation :


55

(1) (F ^ G) ` F (2) (F ^ G) ` (G ^ F ) (3) ((F ^ G) ^ H ) ` (F ^ (G ^ H )) (4) F , G ` (F ^ G) (5) (F _ F ) ` F (6) (F _ G) ` (G _ F ) (7) ((F _ G) _ H ) ` (F _ (G _ H )) (8) (F _ G) ` (CA(F ) ! F ) (9) :G, (G _ F ) ` F (10) G, (G ! F ) ` F (11) UxF x ` F t (12) ExF x ` (CA(F t) ! F t) (13) F ` G, where G is any one-step alphabetic variant of F (14) F ` CA(F ) Z is a ca-derivation from S iff Z is a nite nonempty sequence of sa -wffs such that, for every member F of Z , either F is an axiom, F is a member of S , or F comes from preceding members of Z by a rule of ca -derivation. If F is the last member of Z , we say that Z is a ca -derivation of F from S and that F is ca -derivable from S , and we write S `ca F . The following theorem shows that ca -derivation is conservative with respect to the derivation of d -wffs. Let F be any d -wff, let S be any set of sa -wffs, and let CA(S ) be the set of d -wffs that are the core assertions in the members of S . Let us say that F is standardly derivable from a set S 0 just in case there is a nite nonempty sequence Z of d -wffs G (including F ) such that every G either is in S 0 or is an axiom or comes from preceding members of Z by some rule of rst-order predicate logic. Then: THEOREM 4. F is ca-derivable from S iff F is standardly derivable from CA(S ). Various types of content are now de nable. E.g. the assertive commitment of S is the set of all d -sentences F such that S `ca F ; the projective commitment of S is the set of all d -sentences F such that S 0 `ca F (where S 0 is the union of S and the set of core projections in members of S ); and the erotetic commitment of S is the set of all e -sentences F such that S `ca F . Let Z be a ca -derivation from S , and let S 0 be a nite set of closed terms. Then Z is ca-complete for S relative to S 0 iff all the members of S have been put into Z and all the rules that can be applied have been applied. More precisely: 1. For any proper sa -wff (F ^ G), if it is in Z , so is F . [and analogously for ca -rules 2, 3, 5, 6, 7, 9, 10, and 14] 2. For any proper sa -wff (F _ G), if it is in Z , and some X is such that X is a proper sa -wff, X is either F or G, and S `ca CA(X ), then, for at least one such X , (CA(X ) ! X ) is in Z .

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3. For any proper sa -wff (G _ F ), if it is in Z , and S `ca :G, then :G is in Z . 4. For any proper sa -wff (G ! F ), if it is in Z , and S `ca G, then G is in Z . 5. For any proper sa -wff UxF x, if it is in Z , and t is a closed term in S 0 , then F t is in Z . 6. For any proper sa -wff ExF x, if it is in Z , and some closed term t in S 0 is such that S `ca CA(F t), then, for at least one such term t, (CA(F t) ! F t) is in Z . Where S is a nite set of sa -wffs, a suÆcient reply to S is constructed in the following way: First nd a ca -derivation Z from S that is ca -complete for S relative to the set of closed terms that occur in members of S . Choose b -sentences F1 ; : : : ; Fn that occur in Z , provided that all the e -sentences in Z are included among F1 ; : : : ; Fn . Then choose G1 ; : : : ; Gn such that each Gi is a direct reply to Fi . Then (G1 ^ : : : ^ Gn ) is a suÆcient reply to S . Unfortunately ca -complete derivations are not eectively recognizable as such, so suÆcient replies are not eectively recognizable as such. On the other hand, by making certain additions and changing some details, we can make the reply process more eective in certain respects. The key is to extend the language by adding a stock of reply indicators ri and r -wffs of the form (rj F1 : G1 ) ^ : : : ^ (rj Fn : Gn ) 1

n

Roughly, each (ri F : G) says that G is a reply to F of the kind i. In particular, a suÆcient reply to S would have the form displayed above, where each ri would be an indicator for direct reply. For details, see Harrah [1985]. Concerning vectored sentences and vectored messages: Each such expression X consists of a body B and a vector V ; the content of X is a function of the content of B and the content of V . To simplify here we assume that the body of a vectored message is a nite nonempty string of sa -sentences; thus each vectored sentence counts as a vectored message, but a vectored sentence cannot occur inside a vectored message. Vectors are expressions of various kinds, and each kind of vector brings certain presumptions. Example: `To: Jane Smith, Dean of the College' brings the presumption `Jane Smith is Dean of the College'


57

(For discussion of vectors and presumptions, see Harrah [1994].) The content of the vector is determined by these presumptions. We assume that each vectored X has nitely many presumptions, that the presumptions are d -sentences, and that each presumption of X is eectively recognizable from `X . For message analysis: Let M be a vectored message, and let S (M ) be the set consisting of (1) the sa -sentences in the body of M and (2) the presumptions of M . Then Z is an ma-derivation from M iff Z is a ca derivation from S (M ), and Z is ma-complete for M iff Z is ca -complete for S (M ) relative to the set of terms that occur in M . We construct a suÆcient reply to M in either of three ways: Option I: First nd an ma -derivation Z from M that is ma -complete for M . Choose b -sentences F1 ; : : : ; Fn that occur in Z and include all the e -sentences in Z . Then choose direct replies Gi to these Fi and form (G1 ^ : : : ^ Gn ) as a suÆcient reply to M . Option II: Find a d -sentence F that is ca -derivable from the set of presumptions of M (an F that you believe is false). Then the negation :F is a vector-challenge to M and may be given as a suÆcient reply to M . Option III: Find a d -sentence F such that (1) F is ma -derivable from M , and (2) every ma -derivation of F from M contains at least one presumption of M . Then the negation :F is a vector-challenge and is a suÆcient reply to M .

University of California, Riverside, USA. BIBLIOGRAPHY [ Aqvist, 1965] L. Aqvist. A New Approach to the Logical Theory of Interrogatives. Almqvist & Wiksell, Uppsala, 1965. [ Aqvist, 1969] L. Aqvist. Scattered topics in interrogative logic. In J. Davis, D. J. Hockney, and W. K. Wilson, editors, Philosophical Logic, pages 114 { 121. D. Reidel, Dordrecht, 1969. [ Aqvist, 1971] L. Aqvist. Revised foundations for imperative epistemic and interrogative logic. Theoria, 37:33 { 73, 1971. [ Aqvist, 1983] L. Aqvist. On the \Tell Me Truly" approach to the analysis of interrogatives. In F. Kiefer, editor, Questions and Answers, pages 9 { 14. D. Reidel, Dordrecht, 1983. [Belnap and Steel, 1976] N. Belnap and T. Steel. The Logic of Questions and Answers. Yale, New Haven, 1976. [Belnap, 1963] N. Belnap. An analysis of questions: preliminary report. Technical Report 7 1287 1000/00, System Development Corporation, Santa Monica, CA, 1963. [Belnap, 1969] N. Belnap. Aqvist's corrections-accumulating question-sequences. In J. Davis, D. J. Hockney, and W. K. Wilson, editors, Philosophical Logic, pages 122 { 134. D. Reidel, Dordrecht, 1969. [Belnap, 1981] N. Belnap. Questions and answers in Montague grammar. In S. Peters and E. Saarinen, editors, Processes, Beliefs, and Questions. Essays on Formal Semantics of Natural Language and Natural Language Processing, pages 165 { 198. D. Reidel, Dordrecht, 1981.

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[Belnap, 1983] N. Belnap. Approaches to the semantics of questions in natural language. In R. Bauerle et al., editors, Meaning, Use, and Interpretation of Language, pages 21 { 29. Walter de Gruyter, Berlin, 1983. [Berkov, 1979] V. Berkov. Nauchnaya problema (logiko-metodologicheskiy aspekt). Izdatel'stvo BGY im. V.I. Lenina, Minsk, 1979. [Bogdan, 1987] R. Bogdan, editor. Jaakko Hintikka. D. Reidel, Dordrecht, 1987. [Bolinger, 1978] D. Bolinger. Yes-no questions are not alternative questions. In H. Hi_z, editor, Questions, pages 87 { 105. D. Reidel, Dordrecht, 1978. [Bromberger, 1992] S. Bromberger. On What We Know We Don't Know: Explanation, Theory, Linguistics, and How Questions Shape Them. University of Chicago Press, Chicago, 1992. [Carlson, 1983] L. Carlson. Dialogue Games. D. Reidel, Dordrecht, 1983. [Cohen, 1929] F. Cohen. What is a question? The Monist, 39:350 { 364, 1929. [Cresswell, 1965] M. Cresswell. On the logic of incomplete answers. The Journal of Symbolic Logic, 30:65 { 68, 1965. [Dacey, 1981] R. Dacey. An interrogative account of the dialectical inquiring system based upon the economic theory of information. Synthese, 47:43 { 55, 1981. [Davis et al., 1969] J. Davis, D. J. Hockney, and W. K. Wilson, editors. Philosophical Logic. D. Reidel, Dordrecht, 1969. [Ficht, 1978] H. Ficht. Supplement to a bibliography on the theory of questions and answers. Linguistiche Berichte, 55:92 { 114, 1978. [Finn, 1974] V. Finn. K logiko-semioticheskoy teorii informatsionnogo poiska. In Informatsionnye voprosy semiotiki, lingvistiki i avtomaticheskogo perevoda, volume 5. Vsesoyuzyy Institut Nauchnoy i Tekhnicheskoy Informatsii, ANSSSR, Moscow, 1974. [Gornstein, 1967] I. Gornstein. The logical analysis of questions: a historical survey. 1967. [Grewendorf, 1983] G. Grewendorf. What answers can be given? In F. Kiefer, editor, Questions and Answers, pages 45 { 84. D. Reidel, Dordrecht, 1983. [Groenendijk and Stokhof, 1984] J. Groenendijk and M. Stokhof. Studies on the Semantics of Questions and the Pragmatics of Answers. Academisch Proefschrift, Amsterdam, 1984. [Hamblin, 1958] C. Hamblin. Questions. The Australasian Journal of Philosophy, 36:159 { 168, 1958. [Hamblin, 1967] C. Hamblin. Questions. In P. Edwards, editor, The Encyclopedia of Philosophy. Macmillan, New York, 1967. [Hand, editor, 1994] M. Hand, editor. Game theoretical semantics. Synthese, 99:311 { 456, 1994. [Hand, 1988] M. Hand. Game-theoretical semantics, Montague semantics, and questions. Synthese, 74:207 { 222, 1988. [Harrah, 1961] D. Harrah. A logic of questions and answers. Philosophy of Science, 28:40 { 46, 1961. [Harrah, 1963a] D. Harrah. Communication: A Logical Model. MIT, Cambridge, MA, 1963a. [Harrah, 1963b] D. Harrah. Review of Stahl [1962]. The Journal of Symbolic Logic, 28:259, 1963b. [Harrah, 1969] D. Harrah. On completeness in the logic of questions. American Philosophical Quarterly, 6:158 { 164, 1969. [Harrah, 1975] D. Harrah. A system for erotetic sentences. In A. Anderson et al., editors, The Logical Enterprise, pages 235 { 245. Yale, New Haven, 1975. [Harrah, 1979] D. Harrah. Critical study of Hintikka [1976]. Noûs, 13:95 { 99, 1979. [Harrah, 1980] D. Harrah. On speech acts and their logic. Paci c Philosophical Quarterly, 61:204 { 211, 1980. [Harrah, 1981] D. Harrah. The semantics of question sets. In D. Krallmann and G. Stickel, editors, Zur Theorie der Frage. Gunter Narr Verlag, Tubingen, 1981. [Harrah, 1981a] D. Harrah. On the complexity of texts and text theory. Text, 1:83 { 95, 1981a.


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[Harrah, 1982] D. Harrah. Guarding what we say. In T. Pauli, editor, Philosophical Essays dedicated to Lennart Aqvist, pages 119 { 131. University of Uppsala, Uppsala, 1982. [Harrah, 1985] D. Harrah. A logic of message and reply. Synthese, 63:275 { 294, 1985. [Harrah, 1987] D. Harrah. Hintikka's theory of questions. In R. Bogdan, editor, Jaakko Hintikka, pages 199 { 213. D. Reidel, Dordrecht, 1987. [Harrah, 1994] D. Harrah. On the vectoring of speech acts. In S. Tsohatzidis, editor, Foundations of Speech Act Theory: Philosophical and Linguistic Perspectives, pages 374 { 390. Routledge, London, 1994. [Harris, 1994] S. Harris. GTS and interrogative tableaux. Synthese, 99:329 { 343, 1994. [Higginbotham, 1993] J. Higginbotham. Interrogatives. In K. Hale and S. Keyser, editors, The View from Building 20: Essays in Linguistics in Honor of Sylvain Bromberger, pages 195 { 227. The MIT Press, Cambridge, MA, 1993. [Hintikka and Halonen, 1995] J. Hintikka and I. Halonen. Semantics and pragmatics for why-questions. The Journal of Philosophy, 92:636 { 657, 1995. [Hintikka, editor, 1988] J. Hintikka, editor. Knowledge-seeking by questioning. Synthese, 74:1 { 262, 1988. [Hintikka, 1976] J. Hintikka. The Semantics of Questions and the Questions of Semantics: Case Studies in the Interrelations of Logic, Semantics and Syntax. NorthHolland, Amsterdam, 1976. [Hintikka, 1983] J. Hintikka. New foundations for a theory of questions and answers. In F. Kiefer, editor, Questions and Answers, pages 159 { 190. D. Reidel, Dordrecht, 1983. [Hintikka, 1992] J. Hintikka. The interrogative model of inquiry as a general theory of argumentation. Communication and Cognition, 25:221 { 242, 1992. [Hi_z, 1962] H. Hi_z. Questions and answers. The Journal of Philosophy, 59:253 { 265, 1962. [Hi_z, 1978] H. Hi_z, editor. Questions. D. Reidel, Dordrecht, 1978. [Hoepelman, 1983] J. Hoepelman. On questions. In F. Kiefer, editor, Questions and Answers, pages 191 { 227. D. Reidel, Dordrecht, 1983. [Kampe de Feriet and Picard, 1974] J. Kampe de Feriet and C. Picard, editors. Theories de l'Information (Lecture notes in Mathematics, Volume 398). Springer-Verlag, Berlin, 1974. [Karttunen, 1978] L. Karttunen. Syntax and semantics of questions. In H. Hi_z, editor, Questions, pages 165 { 210. D. Reidel, Dordrecht, 1978. [Keenan and Hull, 1973] E. Keenan and R. Hull. The logical presuppositions of questions and answers. In J. Peto and D. Franck, editors, Prasuppositionen in Philosophie und Linguistik, pages 441 { 466. Athenaum, Frankfurt/M., 1973. [Kiefer, 1983] F. Kiefer, editor. Questions and Answers. D. Reidel, Dordrecht, 1983. [Koj and Wisniewski, 1989] L. Koj and A. Wisniewski. Inquiries into the generating and proper use of questions. Wydawnictwo Naukowe UMCS, Realizm Racjonalnosc Relatywizm, 12, 1989. [Koura, 1988] A. Koura. An approach to why-questions. Synthese, 74:191 { 206, 1988. [Krallmann and Stickel, 1981] D. Krallmann and G. Stickel, editors. Zur Theorie der Frage. Gunter Narr Verlag, Tubingen, 1981. [Kripke, 1975] S. Kripke. Outline of a theory of truth. The Journal of Philosophy, 72:690 { 716, 1975. [Kubinski, 1960] T. Kubinski. An essay in the logic of questions. Atti del XII Congr. Intern. di Filoso a (Firenza), 5:315 { 322, 1960. [Kubinski, 1967] T. Kubinski. Some observations about a notion of incomplete answer. Studia Logica, 21:39 { 42, 1967. [Kubinski, 1980] T. Kubinski. An Outline of the Logical Theory of Questions. AkademieVerlag, Berlin, 1980. [Lehnert, 1978] W. Lehnert. The Process of Question Answering. Wiley, New York, 1978. [Leszko, 1980] R. Leszko. Wyznaczanie Pewnych Klas Pytan Przez Grafy i Ich Macierze. Wy_zsza Szkola Pedagogiczna, Zielona Gora, 1980.

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[Leszko, forthcoming] R. Leszko. Graphs and matrices of compound numerical questions. Acta Universitatis Wratislaviensis, forthcoming. [Lewis and Lewis, 1975] D. Lewis and S. Lewis. Review of Olson and Paul, contemporary philosophy in scandinavia. Theoria, 41:39 { 60, 1975. [Materna, 1981] P. Materna. Question-like and non-question-like imperative sentences. Linguistics and Philosophy, 4:393 { 404, 1981. [Mayo, 1956] B. Mayo. Deliberative questions: a criticism. Analysis, 16:58 { 63, 1956. [Meyer, 1988] M. Meyer, editor. Questions and Questioning. Walter de Gruyter, Berlin, 1988. [Picard, 1980] C. Picard. Graphs and Questionnaires. North-Holland, Amsterdam, 1980. [Prior and Prior, 1955] M. Prior and A. Prior. Erotetic logic. Philosophical Review, 64:43 { 59, 1955. [Shoesmith and Smiley, 1978] D. Shoesmith and T. Smiley. Multiple-conclusion Logic. Cambridge University Press, Cambridge, 1978. [Stahl, 1956] G. Stahl. La logica de las preguntas. Anales de la Universidad de Chile, 102:71 { 75, 1956. [Stahl, 1962] G. Stahl. Fragenfolgen. In M. Kasbauer and F. Kutschera, editors, Logik und Logikkalkul, pages 149 { 157. Alber, Freiburg/Munich, 1962. [Szaniawski, 1973] K. Szaniawski. Questions and their pragmatic value. In R. Bogdan and I. Niiniluoto, editors, Logic, Language, and Probability, pages 121 { 123. D. Reidel, Dordrecht, 1973. [Tichy, 1978] P. Tichy. Questions, answers, and logic. American Philosophical Quarterly, 15:275 { 284, 1978. [Todt and Schmidt-Radefeldt, 1979] G. Todt and J. Schmidt-Radefeldt. Wissensfragen und direkte Antworten in der Fragelogik LA? . Linguistiche Berichte, 62:1 { 24, 1979. [Todt and Schmidt-Radefeldt, 1981] G. Todt and J. Schmidt-Radefeldt. Review of Belnap and Steel [1976]. The Journal of Pragmatics, 5:95 { 101, 1981. [Wachowicz, 1978] K. Wachowicz. Q-morpheme hypothesis. In H. Hi_z, editor, Questions. D. Reidel, Dordrecht, 1978. [Wheatley, 1955] J. Wheatley. Deliberative questions. Analysis, 15:49 { 60, 1955. [Wisniewski, 1994a] A. Wisniewski. Erotetic implications. Journal of Philosophical Logic, 23:173 { 195, 1994a. [Wisniewski, 1994b] A. Wisniewski. On the reducibility of questions. Erkenntnis, 40:265 { 284, 1994b. [Wisniewski, 1995] A. Wisniewski. The Posing of Questions: Logical Foundations of Erotetic Inferences. Kluwer, Dordrecht, 1995. [Wisniewski, editor, 1997] A. Wisniewski and J. Zygmunt, editors. Erotetic Logic, Deontic Logic, and Other Logical Matters: Essays in Memory of Tadeusz Kubinski. Wydawnictwo Uniwersytetu Wroclawskiego, Wroclaw, 1997.

HEINRICH WANSING SEQUENT SYSTEMS FOR MODAL LOGICS

INTRODUCTION [T]he framework of ordinary sequents is not capable of handling all interesting logics. There are logics with nice, simple semantics and obvious interest for which no decent, cut-free formulation seems to exist : : :. Larger, but still satisfactory frameworks should, therefore, be sought. A. Avron [1996, p. 3] This chapter surveys the application of various kinds of sequent systems to modal and temporal logic, also called tense logic. The starting point are ordinary Gentzen sequents and their limitations both technically and philosophically. The rest of the chapter is devoted to generalizations of the ordinary notion of sequent. These considerations are restricted to formalisms that do not make explicit use of semantic parameters like possible worlds or truth values, thereby excluding, for instance, Gabbay's labelled deductive systems, indexed tableau calculi, and Kanger-style proof systems from being dealt with. Readers interested in these types of proof systems : are referred to [Gabbay, 1996], [Gore, 1999] and [Pliuskeviene, 1998]. Also Orlowska's [1988; 1996] Rasiowa-Sikorski-style relational proof systems for normal modal logics will not be considered in the present chapter. In relational proof systems the logical object language is associated with a language of relational terms. These terms may contain subterms representing the accessibility relation in possible-worlds models, so that semantic information is available at the same level as syntactic information. The derivation rules in relational proof systems manipulate nite sequences of relational formulas constructed from relational terms and relational operations. An overview of ordinary sequent systems for non-classical logics is given in [Ono, 1998], and for a general background on proof theory the reader may consult [Troelstra and Schwichtenberg, 2000]. In this chapter we shall pay special attention to display logic, a general proof-theoretic approach developed by Belnap [1982]. Two applications of the modal display calculus are included as case studies: the formulas-as-types notion of construction for temporal logic and a display calculus for propositional bi-intuitionistic logic (also called Heyting-Brouwer logic). This logic comprises both constructive implication and coimplication (see, for example, [Gore, 2000], [Rauszer, 1980], [Wolter, 1998]), and its sequent-calculus presentation to be given is based on a modal translation into the temporal propositional logic S4t.1 1 The chapter consists of revised and expanded material from [Wansing, 1998] and includes the contents of the unpublished report [Wansing, 2000] on formulas-as-types for temporal logics. Moreover, the sequent calculus for bi-intuitionistic logic and subsystems of bi-intuitionistic logics in Section 3.8 and the translation of multiple-sequent systems into higher-arity sequent systems in Section 4.1 are new. D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, Volume 8, 61{145.


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A note on notation. In the present chapter, both classical and constructive logics will be considered. Therefore it makes sense to re ect this distinction in the notation for the logical operations. In particular, the following symbols will be used: B (constructive, intuitionistic implication), J (coimplication), (Boolean implication), a (intuitionistic negation), ` (conegation), : (Boolean negation). 1 ORDINARY SEQUENT SYSTEMS The presentation of normal modal logics as ordinary (standard) sequent systems has turned out to be problematic for both technical and philosophical reasons. The technical problems chie y result from a lack of exibility of the ordinary notion of sequent for dealing with the multitude of interesting and important modal logics in a uniform and perspicuous way. In this section a number of standard Gentzen systems for normal modal propositional logics is reviewed in order to give an impression of what has been and what can be done to present normal modal logics as ordinary Gentzen calculi. An ordinary Gentzen system is a collection of rule schemata for manipulating Gentzen sequents; these are derivability statements of the form ! ; where and are nite, possibly empty sets of formulas. The set terms `' and ` ' are called the antecedent and the succedent of ! , respectively. Often, a sequent fA1 ; : : : ; Am g ! fB1 ; : : : ; Bn g is written as A1 ; : : : ; Am ! B1 ; : : : ; Bn . This notation supports viewing the `,' (the comma) as a structure connective in the language of sequents. Indeed, the sequent arrow in Gentzen's [1934] denotes a derivability relation between nite sequences of formulas separated by the comma. Gentzen, however, postulated structural rules that justify thinking of antecedents and succedents as denoting sets: (permutation) ; A; B; ! ! ; A; B; ; B; A; ! ! ; B; A; (contraction) ; A; A; ! ! ; A; A; ; A; ! ! ; A; Gentzen also postulated (monotonicity) ; ! ! ; ; A; ! ! ; A; These three rules are structural in the sense of exhibiting no operation from an underlying logical object language. If the polymorphic comma is interpreted as a binary structure connective that may or may not be associative, the antecedent and the succedent of a sequent are Gentzen terms,

SEQUENT SYSTEMS FOR MODAL LOGICS

63

and in generalized sequent calculi, the sequents display Gentzen terms or other, much more complex data structures. We shall use ``' to denote the derivability relation in a given axiomatic system or a consequence relation between nite sets of sequents and single sequents satisfying identity, cut, and monotonicity. In other words, if and are nite sets of sequents and s, s0 are sequents, then we assume that fsg ` s, `s `s [ fsg ` s0 : and [ fs0g ` s [ ` s0

1.1 Ordinary Gentzen systems for normal modal logics The syntax of modal propositional logic (in Backus-Naur form, see for example [Goldblatt, 1992, p. 3]) is given by: A ::= p j t j f j :A j A ^ B j A _ B j A B j A B j 3A j 2A: The smallest normal modal propositional logic K admits a simple presentation as an ordinary Gentzen system (see, for instance, [Leivant, 1981], [Mints, 1990], [Sambin and Valentini, 1982]). In the language with 2 (\necessarily") as the only primitive modal operator and 3A (\possibly A") being de ned as :2:A, one may just add the rule (! 2)1 ! A ` 2 ! 2A to the standard sequent system LCPL for classical propositional logic CPL, where 2 = f2A j A 2 g. A sequent calculus LK4 for K4 can be obtained by supplementing LCPL with the rule (! 2)2 ; 2 ! A ` 2 ! 2A (see [Sambin and Valentini, 1982]). In [Goble, 1974] it is shown that the pair of modal sequent rules (! 2)1 and (2 !)1 ; A ! ; ` 2; 2A ! ;

yields a sequent system for KD (where `;' denotes the empty set) and that a sequent calculus for KD4 is obtained, if (! 2)1 is replaced by the rule (! 2)3 0 ! A ` 2 ! 2A; where 0 results from by pre xing zero or more formulas in by 2. Shvarts [1989] gives a sequent calculus formulation of KD45 by adjoining to LCPL the following rule for 2: [2] 21 ; 2 ! 2 1 ; 2 ` 21 ; 22 ! 2 1 ; 2 2 ; where 2 contains at most one formula. If in addition 1 and to be non-empty, this results in a sequent system for K45.

2 are required

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Among the most important modal logics are the almost ubiquitous systems S4 and S5. Standard sequent systems for the axiomatic calculi S4 (= KT4) and S5 (= KT5 = KT4B) were studied by Ohnishi and Matsumoto [1957]. They considered the following schematic sequent rules for 2 and 3: (! 2)0 2 ! 2 ; A ` 2 ! 2 ; 2A; (2 !)0 ; A ! ` ; 2A ! ; (! 3)0 ! ; A ` ! ; 3A; (3 !)0 3 ; A ! 3 ` 3 ; 3A ! 3; where 3 = f3A j A 2 g. If either the rules (! 2)0 and (2 !)0 or the rules (! 3)0 and (3 !)0 are adjoined to LCPL, then the result is a sequent calculus LS5 for S5. If is empty in (! 2)0 or (3 !)0 , this yields a sequent calculus LS4 for S4. Several other modal logics can be obtained by imposing suitable constraints on the structures exhibited in (! 2)0 and (3 !)0 , respectively. Ohnishi and Matsumoto show that if (! 2)0 and (3 !)0 are replaced by (! 2)1 and (3 !)1 A ! ` 3A ! 3 ; one obtains a Gentzen-system LKT for KT (= T). Kripke [1963] noted that the equivalences between 2A and :3:A and between 3A and :2:A cannot be proved by means of Ohnishi's and Matsumoto's rules. In the case of S4, Kripke suggested remedying this by using sequent rules which exhibit both 2 and 3, namely in addition to (2 !)0 and (! 3)0 the rules (! 2)0 2 ! A; 3 ` 2 ! 2A; 3 and (3 !)0 A; 2 ! 3 ` 3A; 2 ! 3: Such rules fail to give a separate account of the inferential behaviour of 2 and 3, since only the combined use of these operations is speci ed. Another problem with Ohnishi's and Matsumoto's sequent rules for S5 is that the cut-rule ! ; A; ; A ! ` ; ! ; cannot be eliminated: the system without cut allows proving less formulas than the full system containing cut. Ohnishi and Matsumoto [1959] give the following counter-example to cut-elimination: 2p ! 2p ; ! :2p; 2p p ! p ; ! 2:2p; 2p 2p ! p ; ! 2:2p; p A solution to the problem of de ning a cut-free ordinary Gentzen system for S5 has been given in [Brauner, 2000].2 The logic S5 can be faithfully 2 Another, perhaps less convincing solution has been presented by Ohnishi [1982]. De ne the degree deg(A) of a modal formula in the language with 2 primitive as follows:


65

embedded into monadic predicate logic, the rst-order logic of unary predicates, under a translation t employing a single individual variable x, see for instance [Mints, 1992]. The translation t assigns to every propositional variable p an atomic formula P (x), and for compound formulas it is de ned as follows: t(t) = t, t(:A) = :t(A), t(A]B ) = t(A) ] t(B ), for ] 2 f; ^; _g, t(2A) = 8xt(A), t(3A) = 9xt(A). THEOREM 1. A modal formula A is provable in S 5 if and only if t(A) is

provable in monadic predicate logic.

The familiar cut-free sequent calculus for monadic predicate logic can serve as a starting point for de ning a cut-free ordinary sequent system for S5 with side-conditions on the introduction rules for 2 on the right and 3 on the left of the sequent arrow. The side conditions are simple, though their precise formulation requires some terminology that will be useful also in other contexts. An inference inf is a pair (; s), where is a set of sequents (the premises of inf ) and s is a single sequent (the conclusion of inf ). A rule of inference R is a set of inferences. If inf 2 R, then inf is said to be an instantiation of R. The rule R is an axiomatic rule, if = ; for every (; s) 2 R. We assume that inference rules are stated by using variables for structures (in the present case nite sets of formulas) and formulas. Every structure occurrence in an inference inf (a sequent s) is called a constituent of inf (s). The parameters of inf 2 R are those constituents which occur as substructures of structures assigned to structure variables in the statement of R. Constituents of inf are de ned as congruent in inf if and only if (i) they are occupying similar positions in occurrences of structures assigned to the same structure variable, in the present case i they belong to a set assigned to the same set variable. DEFINITION 2. Two formula occurrences are immediately connected in a proof i contains an inference inf such that one of the following 1. deg(p) = 0, for every propositional variable p; 2. deg(:A) = deg(A); 3. deg(A ^ B ) = max(deg(A), deg(B )); 4. deg(2A) = deg(A) + 1. Ohnishi adds to (2 !)0 and (! 2)0 two further rules that deviate considerably from familiar introduction schemata:

; A ; ! ` ; A; ! and

! ; A ; ` ! ; A; ;

where the formula A is de ned in such a way that (i) A and A are equivalent in S5 and (ii) deg(A ) 1:

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conditions holds: 1. both occurrences are non-parametric, one in the conclusion and the other in a premise of inf ; 2. inf belongs to an axiomatic sequent rule and both occurrences are non-parametric in inf ; 3. inf 2 cut and both occurrences are non-parametric in inf ; 4. the occurrences are parametric and congruent in inf . A list of formula occurrences A1 ; : : : ; An in a proof is called a connection between A1 and An in i for every i 2 f1; : : : ; n{1g, the occurrences Ai and Ai+1 are immediately connected in . A formula is said to be modally closed if every propositional variable in the formula occurs in the scope of an occurrence 3 or 2. DEFINITION 3. Two formula occurrences in a proof are said to be dependent on each other in i there exists a connection between these occurrences that does not contain any modally closed formula. The sequent system LS5 extends LCPL by (2 !)0 , (! 3)0 and the rules: (! 2)00 ! ; A ` ! ; 2A and (3 !)00 ; A ! ` ; 3A ! ; where applications of (! 2)00 and (3 !)00 in a proof must be such that in none of the formula occurrences in and depends on the displayed occurrence of A. A cut-free proof of the notorious sequent ; ! 2:2p; p is then easily available (as it is also in Ohnishi's [1982] calculus):

p!p 2p ! p ; ! :2p; p ; ! 2:2p; p THEOREM 4. ([Brauner, 2000]) A sequent ! is provable in LS 5 i V W is provable in S 5. Avron [1984] (see also [Shimura, 1991]) presents a sequent calculus LS4Grz for S4Grz (= KGrz). He replaces the rule (! 2)0 in Ohnishi and Matsumoto's sequent calculus for S4 by the rule (! 2)4 2(A 2A); 2 ! A ` 2 ! 2A exhibiting both 2 and . In [Takano, 1992], Takano de nes sequent calculi LKB, LKTB, LKDB, and LK4B for KB, KTB (= B), KDB, and K4B.


67

The systems LKB and LK4B are obtained from LCPL by including the rules (! 2)B ! 2; A ` 2 ! ; 2A and (! 2)4BE ; 2 ! 2; 2; A ` 2 ! 2; ; 2A respectively. LKTB and LKDB result from LKB by adjoining (2 !)0 and (2 !)D ! 2 ` 2 ! respectively. Standard sequent systems for several other modal logics can be found in [Gore, 1992] and [Zeman, 1973]. The sequent calculus for S4.3 (= S4 + 2(2A B ) _ 2(2B A)) in [Zeman, 1973] results from LS4 by the addition of the axiomatic sequent 2(A _ 2B ); 2(2A _ B ) ! 2A; 2B: Shimura [1991] obtains a cut-free sequent system LS4.3 by adding to LCPL the rules (2 !)0 and (! 2)5 2 ! (2) r f2A1 g : : : 2 ! (2) r f2An g ` 2 ! 2; where = fA1 ; : : : ; An g and r is set-theoretic dierence.

1.2 Ordinary Gentzen systems for normal temporal logics The syntax of temporal propositional logic is given by: A ::= p j t j f j :A j A ^ B j A _ B j A B j A B j hP iA j [P ]A j hF iA j [F ]A: Also a number of normal temporal propositional logics have been presented as ordinary sequent calculi. Nishimura [1980], for example, de nes sequent systems LKt and LK4t for the minimal normal temporal logic Kt and the tense-logical counterpart K4t of K4. The sequent calculus LKt comprises the following introduction rules for forward-looking necessity [F ] (\always in the future") and backward-looking necessity [P ] (\always in the past"):3 (! [F ]) ! A; [P ] ` [F ] ! [F ]A; ; (! [P ]) ! A; [F ] ` [P ] ! [P ]A; ; where [F ] = f[F ]A j A 2 g and [P ] = f[P ]A j A 2 g. In K4t, these rules are replaced by the following pair of rules: (! [F ])4 [F ] ; ! A; [P ]; [P ] ` [F ] ! [F ]A; ; [P ]; (! [P ])4 [P ] ; ! A; [F ]; [F ] ` [P ] ! [P ]A; ; [F ]:

3 Nishimura allows in nite sets in antecedent and succedent position. It is proved, however, that if a sequent ! is provable, then there are nite sets 0 and 0 such that the sequent 0 ! 0 is provable.

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In both systems, hP i (\sometimes in the past") and hF i (\sometimes in the future") are treaded not as primitive but as de ned by hP iA := :[P ]:A and hF iA := :[F ]:A: Note also that this approach gives completely parallel rules for [F ] and [P ] and that these rules do not exploit the interrelation between the backward and the forward-looking modalities, that shows up, for instance, in the provability of A [F ]hP iA and A [P ]hF iA: In summary, it may be said that many normal modal and temporal logics are presentable as ordinary Gentzen calculi, and that in some cases suitable constraints on the structures exhibited in the statement of the sequent rules for the modal operators allow for a number of variations. However, no uniform way of presenting only the most important normal modal and temporal propositional logics as ordinary Gentzen calculi is known. Further, the standard approach fails to be modular : in general it is not the case that a single axiom schema is captured by a single sequent rule (or a nite set of such rules). In the following section a more philosophical critique of ordinary Gentzen systems is advanced.

1.3 Introduction schemata and the meaning of the logical operations The philosophical (and methodological) problems with applying the notion of a Gentzen sequent to modal logics have to do with the idea of de ning the logical operations by means of introduction schemata (together with structural assumptions about derivability formulated in terms of structural rules). This ànti-realistic' conception of the meaning of the logical operations is often traced back to a certain passage on natural deduction from Gentzen's Investigations into Logical Deduction [Gentzen, 1934, p. 80]: [I]ntroductions represent, as it were, the `de nitions' of the symbols concerned, and the eliminations are no more, in the nal analysis, than the consequences of these de nitions. To qualify as a de nition of a logical operation, an introduction schema must satisfy certain adequacy criteria. Such conditions are discussed, for instance, by Hacking [1994]. Following Hacking, if introduction rules are to be regarded as de ning logical operations, these rules must be such that the structural rules monotonicity (also called weakening, thinning, or dilution), re exivity, and cut can be eliminated. Hacking claims that [i]t is not provability of cut-elimination that excludes modal logic, but dilution-elimination : : :. The serious modal logics such as T, S4 and S5 have cut-free sequent-calculus formalizations, but the rules place restrictions on side formulas. Gentzen's rules for sentential connections are all `local' in that they concern


69

only the components from which the principal formula is built up, and place no restrictions on the side formulas. Gentzen's own rst-order rules, though not strictly local, are equivalent to local ones. That is why dilution-elimination goes through for rst-order logic but not for modal logics ([Hacking, 1994, p. 24]). By dilution-elimination Hacking means that the monotonicity rules !

` ; A ! ;

!

` ! ;A

may be replaced by atomic thinning rules !

` ; p ! ;

!

` ! ; p:

without changing the set of provable sequents. Similarly, re exivity-elimination amounts to replaceability of ` A ! A by ` p ! p. The term \cutelimination" is reserved for something stronger than replaceability of cut by the atomic cut-rule ! ; p;

;p !

` ; ! ; :

A cut-elimination proof shows the admissibility of cut: the rule has no eect on the set of provable sequents. The introduction rules for 2 in LS4 prevent dilution-elimination. Obviously, the sequent 2B; 2A ! 2A, for example, cannot be proved using only these rules and atomic thinning. A problem with the requirement of dilution-elimination is the weak status monotonicity has acquired as a de ning characteristic of logical deduction. In view of the substantial work on relevance logic, many other substructural logics, and a plethora of nonmonotonic reasoning formalisms extending a monotonic base system, monotonicity of inference is not generally viewed as a touchstone of logicality anymore. Moreover, also re exivity and cut have been questioned. Unrestricted transitivity of deduction as expressed by the cut-rule does not hold, for instance, in Tennant's intuitionistic relevant logic [1994], and both re exivity and cut fail to be validated by Update-to-Test semantic consequence as de ned in Dynamic Logic, see [van Benthem, 1996]. Re exivity-elimination and cut-elimination are, however, important. According to Belnap [1982, p. 383], the provability of A ! A constitutes half of what is required to show that the \meaning" of formulas : : : is not context-sensitive, but that instead formulas \mean the same" in both antecedent and consequent position. (The [Cut] Elimination Theorem : : : is the other half of what is required for this purpose). A similar remark can be found in [Girard, 1989, p. 31]. Cut-elimination is indispensable, because it amounts to the familiar non-creativity requirement

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for de nitions (see, for instance, [Hacking, 1994], [von Kutschera, 1968]). If one adds introduction rules for a ( nitary) operation f to a sequent calculus, this addition ought to be conservative, so that in the extended formalism, every proof of an f -free formula A is convertible into a proof of A without any application of an introduction rule for f . There are other reasons why the eliminability of cut is a desirable property. Usually, cut-elimination implies the subformula property: every cutfree proof of a sequent s contains only subformulas of formulas in s. In sequent calculi for decidable logics, the subformula property can often be used to give a syntactic proof of decidability. According to Sambin and Valentini [1982, p. 316], it is usually not diÆcult to choose suitable [sequent] rules for each modal logic if one is content with completeness of rules. The real problem however is to nd a set of rules also satisfying the subformula-property. The sequent calculi for S5 in [Mints, 1970], [Sato, 1977], and [Sato, 1980], although admitting cut-elimination, do not have the subformula property. In a sequent calculus with an enriched structural language, the subformula property need not be accompanied by a substructure property. In such systems the subformula property for the logical vocabulary need neither imply nor be of direct use for syntactic decidability proofs. Avron [1996, p. 2] requires of a decent sequent calculus simplicity of the structures employed and a `real' subformula property. But even without the substructure property, the subformula property may be useful, for instance in proving conservative extension results, see also Section 3.8. It is well-known that cut-elimination itself does not guarantee eÆcient proof search (see [D'Agostino and Mondadori, 1994], [Boolos, 1984]), so that it may be attractive to work with an analytic, subformula property preserving cut-rule, if possible. An application of cut ! ; A

;A !

` ; ! ;

is analytic (see [Smullyan, 1968]), if the cut-formula A is a subformula of some formula in the conclusion sequent ; ! ; . Let Sub() denote the set of all subformulas of formulas in . Applications of the sequent rules (! 2)B ! 2; A ` 2 ! ; 2A (! 2)4BE ; 2 ! 2; 2; A ` 2 ! 2; ; 2A and (2 !)D ! 2 ` 2 ! may be said to be analytic if 2 Sub( [fAg), 2 Sub(2 [2[fAg), and 2 Sub( ), respectively. Takano [1992] shows that the cut-rule in

SEQUENT SYSTEMS FOR MODAL LOGICS LS5 , LKB, LKTB, LKDB, LK4B, LKt

71

and LK4t can be replaced by the analytic cut-rule: every proof in these sequent calculi can be transformed into a proof of the same sequent such that every application of cut (and, moreover, every application of the rules (! 2)B , (! 2)4BE , and (! 2)D ) in this proof is analytic. Although admissibility of analytic cut is a welcome property, in general, unrestricted cut-elimination is to be preferred over elimination of analytic cut. Admissibility of cut has great conceptual signi cance. The cut-rule justi es certain substitutions of data; in particular it justi es the use of previously proved formulas. Moreover, if the cut-rule is assumed, the noncreativity requirement for de nitions implies that cut must be eliminable. There are other nice properties of introduction schemata as de nitions in addition to enabling cut-elimination and re exivity-elimination. The assignment of meaning to the logical operations should, for instance, be non-holistic, and hence sequent rules like the above (! 2)0 and (3 !)0 are unsuitable. If (the statement of) an introduction rule for a logical operation f exhibits no connective other than f , the rule is called separated, see [Zucker and Tragesser, 1978]. An even stronger condition is segregation, requiring that the antecedent (succedent) of the conclusion sequent in a left (right) introduction rule must not exhibit any structure operation. Segregation has been suggested (although not under this name) by Belnap [1996] who explains that [t]he nub is this. If a rule for only shows how A B behaves in context, then that rule is not merely explaining the meaning of . It is also and inextricably explaining the meaning of the context. Suppose we give suÆcient conditions for

A B; ! in part by the rule !A B! A B; ! Then we are not explaining A B alone. We are simultaneously involving the comma not just in our explicans (that would surely be all right), but in our explicandum. We are explaining two things at once. There is no way around this. You do not have to take it as a defect, but it is a fact. : : : If you are a `holist', probably you will not care; but then there is not much about which holists much care. [Belnap, 1996, p. 81 f.] (notation adjusted)

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Moreover, the rules for f may be required to be weakly symmetrical in the sense that every rule should either belong to a set of rules (f !) which introduce f on the left side of ! in the conclusion sequent or to a set of rules (! f ) which introduce f on the right side of ! in the conclusion sequent. The introduction rules for f are called symmetrical, if they are weakly symmetrical and both (! f ) and (f !) are non-empty. The sequent rules for f are called weakly explicit, if the rules (! f ) and (f !) exhibit f in their conclusion sequents only, and they are called explicit, if in addition to being weakly explicit, the rules in (! f ) and (f !) exhibit only one occurrence of f on the right, respectively the left side of !. Separation, symmetry, and explicitness of the rules imply that in a sequent calculus for a given logic , every connective that is explicitly de nable in also has separate, symmetrical, and explicit introduction rules. These rules can be found by decomposition of the de ned connective, if it is assumed that the deductive role of f (A1 ; : : : ; An ) only depends on the deductive relationships between A1 ; : : : ; An . It is therefore desirable to have introduction rules for 2, 3, hP i, [P ], hF i and [F ] as primitive operations, so that the familiar mutual de nitions are derivable. A further desirable property, reminiscent of implicit de nability in predicate logic, is the unique characterization of f by its introduction rules. Suppose that is a logical system with a syntactic presentation S in which f occurs. Let S be the result of rewriting f everywhere in S as f , and let be the system presented by the union SS of S and S in the combined language with both f and f . Let Af denote a formula (in this language) that contains a certain occurrence of f , and let Af denote the result of replacing this occurrence of f in A by f . The connectives f and f are said to be uniquely characterized in i for every formula Af in the language of , Af is provable in SS i Af is provable in SS . Dosen [1985] has proved that unique characterization is a non-trivial property and that the connectives in his higher-level systems S4p/D and S5p/D for S4 and S5, respectively, are uniquely characterized. As we have seen, the standard sequent-style proof-theory for normal modal and temporal logic fails to be modular. The idea that modularity can be achieved by systematically varying structural features of the derivability relation while keeping the introduction rules for the logical operations untouched can be traced back to Gentzen [1934] and has been referred to as Dosen's Principle in [Wansing, 1994]. In [Dosen, 1988, p. 352], Dosen suggests that \the rules for the logical operations are never changed: all changes are made in the structural rules." This methodology is adopted, for example in Dosen's [1985] higher-level sequent systems for S4 and S5, Blamey and Humberstone's [1991] higher-arity sequent calculi for certain extensions of K, Nishimura's [1980] higher-arity sequent systems for Kt and K4t, and the presentation of normal modal and temporal logics as cut-free display sequent calculi.


73

Another methodological aspect is generality. Is there a type of sequent system that allows not only a uniform treatment of the most important modal and temporal logics but also a treatment of substructural logics, other non-classical logics and systems combining operations from dierent families of logics and that, moreover, is rich enough to suggest important, hitherto unexplored logics? The framework of display logic to be presented in the next section has been devised explicitly as an instrument for combining logics (see [Belnap, 1982]), and has been suggested, for example, as a tool for de ning subsystems of classical predicate logic (see [Wansing, 1999]). In addition to generality, a `real' subformula property, and Dosen's principle, Avron [1996] requires of a good sequent calculus framework also semantics independence. The framework should not be so closely tied to a particular semantics that one can more or less read o the semantic structures in question. Moreover, the proof systems instantiating the framework should lead to a better understanding of the respective logics and the dierences between them. Note that each of the ordinary sequent systems presented in the present section fails to satisfy some of the more philosophical requirements mentioned. The same holds true for the ordinary sequent systems for various non-normal, classical modal logics investigated in [Lavendhomme and Lucas, 2000]. There are thus not only technical but also methodological and philosophical reasons for investigating generalizations of the notion of a Gentzen sequent. 2 GENERALIZED SEQUENT SYSTEMS In this section the application of a number of generalizations of the ordinary notion of sequent to normal modal propositional and temporal logics is surveyed.

2.1 Higher-level sequent systems Dosen [1985] has developed certain non-standard sequent systems for S4 and S5. In these Gentzen-style systems one is dealing with sequents of arbitrary nite level. Sequents of level 1 are like ordinary sequents, whereas sequents of level n + 1 (0 < n < !) have nite sets of sequents of level n on both sides of the sequent arrow. The main sequent arrow in a sequent of level n carries the superscript n , and ; is regarded as a set of any nite level. The rules for logical operations are presented as double-line rules. A double-line rule s1 ; : : : ; sn s0

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HEINRICH WANSING

involving sequents s0 ; : : : ; sn , denotes the rules s1 ; : : : ; sn s0 s ; ;:::; 0: s0 s1 sn Dosen gives the following double-line sequent rules for 2 and 3:

X + f; !1 fAgg !2 X2 + fX3 !1 X4 g X1 !2 X2 + fX3 + f2Ag !1 X4 g

X1 + ffAg !1 ;g !2 X2 + fX3 !1 X4 g ; X1 !2 X2 + fX3 !1 X4 + f3Agg where + refers to the union of disjoint sets. If these rules are added to Dosen's higher-level sequent calculus Cp/D for CPL, this results in the sequent system S5p/D for S5. The sequent calculus S4p/D for S4 is then obtained by imposing a structural restriction on the monotonicity rule of level 2:

X !2 Y

` X [ Z1 !2 Y [ Z2 : The restriction is this: if Y = ;, then Z2 must be a singleton or empty; if Y = 6 ;, then Z2 must be empty. If the same restriction is applied to

monotonicity of level 1 in Cp/D, then this gives a higher-level sequent system for intuitionistic propositional logic IPL. Note that 3 and 2 are interde nable in S4p/D and S5p/D. The doubleline rules for 2 and 3, however, do not satisfy weak symmetry and weak explicitness, but the upward directions of these rules can be replaced by:

; !1 fAg ` ; !1 f2Ag

and

fAg !1 ; ` f3Ag !1 ;:

Whereas cut can be eliminated at levels 1 and 2, cut of all levels fails to be eliminable [Dosen, 1985, Lemma 1]. Moreover, in Dosen's higher-level framework it is not clear how restrictions similar to the one used to obtain S4p/D from S5p/D would allow to capture further axiomatic systems of normal modal propositional logic.

2.2 Higher-dimensional sequent systems A `higher-dimensional' proof theory for modal logics has been developed by Masini [1992; 1996]. This approach is based on the notion of a 2-sequent. In order to de ne this notion, various preparatory de nitions are useful. Any nite sequence of modal formulas is called a 1-sequence. The empty 1-sequence is denoted by . A 2-sequence is an in nite `vertical' succession of 1 sequences, = fi g0 i : k = g. A 2-sequent is an expression ! , where and


75

are 2-sequences. The depth of ! ($ ! )) is de ned as max(\ ; $. If ! is a 2-sequent and A an occurrence of a modal formula in ! , then A is said to be maximal in ! , if A is at level k in or in and k = \( ! ). A is the maximum in ! , if A is the unique maximal formula in ! . The sequent rules for 2 are based on the idea of \internalizing the level structure of 2-sequents" [Masini, 1992, p. 231]:

(2 !)

(3 !)

; A 0

!

; 2A 0

A

(! 2)

!

!

; 3A

!

! !

! (! 3)

!

A ; 2A ; A 0 ; 3A 0

where , , , and denote arbitrary 1-sequences, and A must be the maximum of the premise 2-sequent in (! 2) and (3 !). According to Masini, these introduction rules give rise to a \general basic proof theory of modalities" [Masini, 1992, p. 232]. If added to a 2-sequent calculus for CPL, the above rules result, however, in a sequent calculus for KD instead of the basic system K. This sequent system for KD admits cut-elimination, 2 and 3 are interde nable, and the introduction rules are separate, symmetrical, and explicit, but no indication is given of how to present axiomatic extensions of KD as higher-dimensional sequent systems. Moreover, it is not clear how Masini's framework may be modi ed in order to obtain a 2-sequent calculus for K.

2.3 Higher-arity sequent systems In search of generalizations of the standard Gentzen-style sequent format, it is a natural move to consider consequence relations with an arity greater than 2. It seems that the rst higher-arity sequent calculus was formulated by Schroter [1955], see also [Gottwald, 1989]. This formalism is a natural generalization of Gentzen's sequent calculus for CPL to truth-functional

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n-valued logic. The intended truth-functional reading of a Gentzen sequent s = ! is given by a translation of s into a formula: ( ! ) =

^

_

;

The sequent s thus is true under a given interpretation if either some formula in is false, or some formula in is true, and the two places of the sequent arrow correspond to the two truth-values of classical logic. In general, in n-valued logic (with 2 n) one obtains n-place sequents s = 1 ; 2 ; : : : ; n , with the understanding that s is true under an interpretation if for every i n, some formula in i has truth-value i; for a comprehensive treatment of sequent calculi for truth-functional many-valued logics see [Zach, 1993]. We shall here briefly review some relevant parts of the work of Blamey and Humberstone [1991], who investigate an application of three-place and, ultimately, four-place sequent arrows to normal modal logic. This approach is congenial to display logic with respect to a realization of the Dosen-Principle insofar as Blamey and Humberstone emphasize that distinctions between various well-known normal modal logics can \be re ected at the purely structural level, if an appropriate notion of sequent " is adopted [Blamey and Humberstone, 1991, p. 763]. Let , , , and range over nite sets of formulas in the modal propositional language with 2 as primitive. The four-place sequent

! has the following heuristic reading: ^

(

^

^ 2) (

_

_

_

2):

This kind of sequent had independently been used by Sato [1977], where a cut-free sequent calculus for S5 is presented containing a left introduction rule for 2 that fails to be weakly explicit. Blamey's and Humberstone's introduction rules for 2 are: (2 #)0 ` ; !;A 2A (2 ")0 ` 2A !A; ;: In order to obtain a sequent calculus for K the following structural rules are assumed: (R) ` A !;; A (vertical R) ` ; !AA ; (M ) ! ` ; 0 !;;00 ; 0 (undercut) !;; A !0 ;A ` !;0 (T ) ; A ! ! A; ` ! (vertical T ) !;A ! ;A ` ! :


77

Against the background of these rules, the introduction rules (2 #)0 and (2 ")0 are interreplaceable with the following rules, respectively: (2 #) ; 2A ! ` !;A (2 ") !;A ` ; 2A ! : The introduction rules for the Boolean operations are adaptations of the familiar rules to the higher-arity case. Here is a simple example of a derivation in this formalism (using some obvious notational simpli cations): 2A; 2B ! 2A ^ 2B (2#) 2A !B 2A ^ 2B (2#) A^B ! A A ^ B ! A; B ; !A;B 2A ^ 2B (undercut) twice ; !A^B 2A ^ 2B (2") 2(A ^ B ) ! 2A ^ 2B The axiom schemata D, T , 4, and B are captured by purely structural rules not exhibiting any logical operations: D !;; ; ` ; !; ; T ` ; !;A A 4 ! A !0 ;A ` !;0 B ! !;A ` ! : ; A Since Blamey and Humberstone are primarily interested in semantical aspects of their sequent systems, they do not consider cut-elimination. Although their calculi satisfy Dosen's Principle, it remains unclear whether their approach is fully modular for the most important systems of normal modal propositional logic. They do not present a structural equivalent of the 5-axiom schema, but rather treat S5 as KTB4. In [Nishimura, 1980], Nishimura uses six-place sequents 1 ; ; 2 ! 1 ; ; 2 : These higher-arity sequents can intuitively be read as follows: ^ ^ ^ _ _ _ ( [P ]1 ^ ^ [F ]2 ) ( [P ]1 _ _ [F ]2 ): Nishimura de nes introduction rules for the tense logical operations [F ] and [P ], which are explicit in the sense of Section 1.3: (! [F ])0 1; ; 2 ! 1 ; ; A; 2 1; ; 2 ! 1 ; ; [F ]A; 2 ([F ] !)0 1; ; A; 2 ! 1 ; ; 2 1; ; [F ]A; 2 ! 1 ; ; 2 0 (! [P ]) 1; ; 2 ! 1 ; A; ; 2 1; ; 2 ! 1 ; [P ]A; 2 ([P ] !)0 1; A; ; 2 ! 1 ; ; 2 1; [P ]A; ; 2 ! 1 ; ; 2

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In accordance with the Dosen Principle, these rules are held constant in sequent systems for Kt and K4t. The dierence between these logics is accounted for by dierent structural rules, namely (r-trans)

;; ; ; ! ; A; ; ;; ;; ! ;; ; A

(l-trans)

;; ; ; ! ;; A; ; ;; ; ! A; ; ;

in the case of Kt and (r-trans)4

;; ; ! ; ; A; ; ;; ;; ! ; ; A

(l-trans)4

; ; ; ! ;; A; ; ; ;; ; ! A; ;

in the case of K4t. Nishimura observes that although in the introduction rules for hF i and hP i subformulas are preserved from premise sequent to conclusion sequent, cut-elimination fails to hold in the six-place sequent systems for Kt and K4t. There is, for instance, no cut-free proof of ; p; ! ; [F ]:[P ]:p;.4

2.4 Multiple-sequent systems Indrzejczak, in [1997; 1998], suggested non-standard sequent systems for certain extensions of the minimal regular modal logic C using three sequent arrows !, 2! , and 3! . These sequent arrows denote binary relations between nite sets of S -formulas, where the set of S -formulas is de ned as the union of the set of modal formulas and f A j A is a modal formulag. As before, we shall use A, B , C , : : : to denote modal formulas. The symbol ` ' is a unary structure connective that may not be nested, and the sequent arrows 2! and 3! are auxiliary in the sense that they fail to represent consequence relations, because (in general) neither ` A2! A nor ` A3! A. The logics presented by such multiple-sequent systems are given by the set of provable sequents ! . The intended meaning of a sequent is captured by a translation from sequents into ordinary sequents using a translation Æ from S -formulas to modal formulas. For every modal formula A, Æ( A) := :A and Æ(A) := A. The translation is de ned as follows:

( ! ) = Æ( ) ! Æ() V W ( 2! ) = Æ( ) ! 2 Æ() V W ( 3! ) = 3 Æ( ) ! Æ() V

W

Here Æ( ) := fA j A 2 g [ f:A j A 2 g. For every modal formula A, A is de ned as A and A as A. If is a set of S -formulas, := 4 Note that Nishimura allows in nite sets in antecedent and succedent position. It is, however, shown that if a sequent 1 ; ; 2 ! 1 ; ; 2 is provable, then there are nite sets 0i i , 0i i , (i = 1; 2), 0 , and 0 such that the sequent 01 ; 0 ; 02 ! 01 ; 0 ; 02 is provable.


79

Let (!) be any of !, 2! , 3! . The following re exivity and monotonicity rules are assumed:

fA j A 2 g [ f A j A 2 g. ` A ! A;

(!)

` (!) ; A;

(!)

` ; A (!) :

Next, there are further structural rules called shifting rules: [! ] A; ! ` ! ; A [ !] ! ; A ` ; A ! [TR] 2! ` 3! 3! ` 2!

The introduction rules for ^, _, and : are formulated for arbitrary sequent arrows. Whereas the rules for ^ and _ are versions of the familiar introduction rules, the rules for : and can be formulated such that they make use of the structure connective : ; A (!) ` ; :A (!) (!) ; A ` (!) ; :A ; A (!) ; B (!) ` ; ; A B (!) ; (!) ; A; B ` (!) ; A B The introduction rules for the modal operators are not formulated for arbitrary sequent arrows: [22! ] A ! ` 2A2! [! 2] 2! A ` ! 2A [33! ] A3! ` 3A ! [3! 3] ! A ` 3! 3A [3! 3] A; 3! ` 3! ; 3A [22! ] 2! ; A ` 2A2! The above collection of sequent rules forms a multiple-sequent calculus MC for the system C. An axiomatization of C can be obtained by replacing the necessitation rule in the familiar axiomatization of K by the weaker rule (RR) if (A ^ B ) C is provable, then so is (2A ^ 2B ) 2C; see [Chellas, 1980]. The necessitation rule and the modal axiom schemata D, T , and 4 can be captured in a modular fashion by pairs of sequent rules: [nec] ! ; ` 3! ; ; ! ` ;2! [D] 2! ; ` ! ; ;3! ` ; ! [T ] 2! ` ! 3! ` ! [4] ! ` 2! ! ` 3! ; where in rule [4], every S -formula in has the shape 2A or 3A and every S -formula in has the shape 3A or 2A. All sequent systems obtained in this way satisfy a generalized subformula property: for every modal formula A, it holds that if A or A is used in a proof of ! , then A is a subformula of [ (where the notion of a subformula of an S -formula is de ned in the obvious way). Indrzejczak does not investigate the admissibility of

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cut for ! or the admissibility of cut for 2! and 3! in extensions of CT (where ` A2! A and ` A3! A). Note that the introduction rules for the modal operators fail to be symmetrical, since there are no introduction rule for 2 on the left and 3 on the right of !. Moreover, the side conditions on [4] are such that the status of this rule as a purely structural rule is doubtful. The multiple-sequent systems for extensions of KB make use of denumerably many sequent arrows n! (n 0), where logics are de ned by the provable sequents 0! . The introduction rules

A n! ` 2A n+1! A n+1! ` 3A n!

n+1! A ` n! 2A n! A ` n+1! 3A

fail to introduce 2 on the left and 3 on the right of 0!, so that also these rules are not symmetrical. In Section 4.1, we shall point to a simple relation between Indrzejczak's multiple-sequent systems and higher-arity sequent systems for modal logics.

2.5 Hypersequents Hypersequents were introduced into the literature by Pottinger [1983], and have later systematically been studied by Avron [1991; 1991a; 1996]. A hypersequent is a sequence 1

! 1 j

2

! 2 j : : : j

n

! n

of ordinary sequents (or, more generally, sequents in which i and i are sequences of formula occurrences) as their components. The symbol `j' in the statement of a hypersequent enriches the language of sequents and is intuitively to be read as disjunction. This expressive enhancement \makes it possible to introduce new types of structural rules, and : : : to allow greater versatility in developing interesting logical systems" [Avron, 1996, p. 6]. In particular, a distinction may be drawn between internal and external versions of structural rules. The internal rules deal with formulas within a certain component, whereas the external rules deal with components within a hypersequent. Let G, H , H1 , H2 etc. be schematic letters for possibly empty hypersequents. External monotonicity, for instance, can be contrasted with internal monotonicity:

H1 j H2 ` H1 j G j H2 vs. H1 j

! j H2 ` H1 j A; ! j H2 :

Cut only has an internal version: G1 j 1 ! 1 ; A j H1 G2 j A; 2 ! 2 j H2 G1 j G2 j 1 ; 2 ! 1 ; 2 j H1 j H2 The use of hypersequents allows a cut-free presentation GS5 of S5 satisfying the subformula property. The system GS5 consists of hypersequential


81

versions of the rules of LS4, in particular, external and internal versions of contraction and monotonicity, the above cut-rule, and a structural rule of a new kind, namely the modalized splitting rule: (MS )

G j 2 1;

2

! 21 ; 2 j H ` G j 2 1 ! 21 j

2

! 2 j H:

In the next section we shall de ne display sequents, and in Section 4.2 we shall de ne a translation of hypersequents into display sequents. 3 DISPLAY LOGIC We shall develop display logic only to the extent needed to cover a variety of normal modal and temporal logics based on classical or intuitionistic logic. A more comprehensive presentation of display logic and its application to modal and non-classical logics can be found in [Belnap, 1982], [Belnap, 1990], [Belnap, 1996], [Gore, 1998], [Kracht, 1996], [Restall, 1998], [Wansing, 1998]. Note that except for the substructure property, all requirements examined in the previous sections are satis ed by the display sequent systems to be presented.

3.1 Introduction rules through residuation Whereas the ordinary sequent systems for temporal logics presented in Section 1.2 fail to exploit the interaction between the backward and the forward looking modalities, the modal display calculus is based on observing that the operators hP i and [F ] form a residuated pair. The following de nition is taken from Dunn [1990, p. 32]: DEFINITION 5. Consider two partially ordered sets A = (A; ) and B = (B; 0 ) with functions f: A ! B and g: B ! A: The pair (f; g) is called residuated i (fa 0 b i a gb); a Galois connection i (b 0 fa i a gb); a dual Galois connection i (fa 0 b i gb a); a dual residuated pair i (b 0 fa i gb a): Obviously, (hP i; [F ]) forms a residuated pair with respect to the provability relation in normal extensions of Kt, and (:hF i; :hP i) is a Galois connection.5 These ideas of residuation and Galois connection can be generalized. In [Dunn, 1990], [Dunn, 1993], Dunn has de ned an abstract law of

5 The fact that hP i and [F ] form a residuated pair is also used in Kashima's [1994] sequent calculi for various normal temporal logics. The approach of Kashima is similar to the modal display calculus and the modal signs approach developed by Cerrato [1993; 1996] insofar as the structural language of sequents is extended by unary structure operations. Whereas nesting of these operations is not allowed in Cerrato's sequent systems for normal modal propositional logics, Kashima allows iteration. Kashima inductively de nes a notion of sequent as follows:

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residuation for n-place connectives f and g. The formulation of this principle refers to traces of operations and assumes the presence (or de nability) of a truth constant t and a falsity constant f . We shall use A a` B to express that A and B are interderivable in a given axiom system. DEFINITION 6. An n-place connective f (n 0) has a trace (1 ; : : : ; n ) 7! + (in symbols T (f ) = (1 ; : : : ; n ) 7! +) i

f (A1 ; : : : ; t; : : : ; An ) a` t; if i = + (the indicated t is in position i); f (A1 ; : : : ; f ; : : : ; An ) a` t; if i = (the indicated f is in position i); if A ` B and i = +; then f (A1 ; : : : ; A; : : : ; An ) ` f (A1 ; : : : ; B; : : : ; An ); if A ` B and i = ; then f (A1 ; : : : ; B; : : : ; An ) ` f (A1 ; : : : ; A; : : : ; An ): The operation f has a trace (1 ; : : : ; n ) 7!

(T (f ) = (1 ; : : : ; n ) 7! ) i

f (A1 ; : : : ; f ; : : : ; An ) a` f ; if i = + (the indicated f is in position i); f (A1 ; : : : ; t; : : : ; An ) a` f ; if i = (the indicated t is in position i); if A ` B and i = +; then f (A1 ; : : : ; B; : : : ; An ) ` f (A1 ; : : : ; A; : : : ; An ); if A ` B and i = ; then f (A1 ; : : : ; A; : : : ; An ) ` f (A1 ; : : : ; B; : : : ; An ): In Kt, : has traces 7! + and + 7! , whereas [F ] has trace + 7! + and hP i has trace 7! . DEFINITION 7. Two n-place operations f and g are contrapositives in place j i T (f ) = (1 ; : : : ; j ; : : : ; n ) 7! implies T (g) = (1 ; : : : ; ; : : : ; n ) 7! j , where + = and = +. DEFINITION 8. Let B ` f (A1 ; : : : ; An ) if T (f ) = (: : :) 7! + S (f; A1 ; : : : ; An ; B ) i f (A1 ; : : : ; An ) ` B if T (f ) = (: : :) 7! 1. every temporal formula is a sequent; 2. if is a sequent, then so is P f g and F f g; 3. if n 0 and every i (1 i n) is a sequent, then so is 1 ; : : : ; i . The intuitive meaning of a sequent is given by the following inductively de ned translation () from sequents into formulas: 1. ( ) = A, if is the formula A; 2. (P f g) = [P ]P ( ) ; (F f g) = [F ]F ( ) ; W 3. if n > 0, then ( 1 ; : : : ; n) = f( 1 ) ; : : : ; ( n ) g; 4. ( ) = (p ^ :p), for some atom p. Residuation then shows up in Kashima's \turn rules":

;F fg ` P ; ;

;P fg ` F ; :

Most of Kashima's sequent rules used to capture various structural properties of accessibility either fail to be explicit or separated in the sense of Section 1.3. Cut-elimination for these systems is shown semantically, i.e., in a non-constructive way.


83

A pair of n-place connectives f and g satis es the abstract law of residuation just in case for some j (1 j n), f and g are contrapositives in place j , and

S (f; A1 ; : : : ; Aj ; : : : ; An ; B ) i S (f; A1 ; : : : ; B; : : : ; An ; Aj ): OBSERVATION 9. The abstract law of residuation holds for the pairs (t, f ), (:; :), (hP i; [F ]), (^; B), (J; _), (^; : _ :::), and ( ^ ::::; _), where B is intuitionistic implication and J is coimplication. Coimplication J is characterized by

A ` B _ C; i A J B ` C; : In classical logic, the residual of disjunction is de nable, since

A ` B _ C; i A ^ :B ` C; i

:(A B ) ` C;

but in bi-intuitionistic logic it is not, see Section 3.8. For each of the pairs (t, f ), (:; :), (hP i; [F ]), (^; B), (J; _), the structural language of display sequents contains one structure connective. Since in classical logic ^ and _ are interde nable using :, the pairs (^; : _ :::) and ( ^ ::::; _) require only a single structure connective in addition to the unary structure operation associated with (:; :). We shall use X , Y , Z (possibly with subscripts) as variables for structures. A display sequent is an expression X ! Y ; X is called the antecedent and Y is called the succedent of X ! Y . The structures are de ned by:

X ::= A j I j X j X j X Æ Y j X o Y j X n Y: The association of structure connectives with pairs of operations satisfying the abstract law of residuation is accomplished by the following translations 1 of antecedents and 2 of succedents into formulas:

1 (A) 1 (I) 1 (X ) 1 (X ) 1 (X o Y ) 1 (X n Y ) 1 (X Æ Y )

= = = = = = =

A

t

:2 (X ) hP i1 (X ) 1 (X ) ^ 1 (Y ) 1 (X ) J 1 (Y ) 1 (X ) ^ 1 (Y )

2 (A) 2 (I) 2 (X ) 2 (X ) 2 (X o Y ) 2 (X n Y ) 2 (X Æ Y )

= = = = = = =

A

f

:1 (X )

[F ]2 (X ) 2 (X ) B 2 (Y ) 2 (X ) _ 2 (Y ) 2 (X ) _ 2 (Y )

Under these translations, the following basic structural rules are valid ((1){ (4) in normal temporal logic; (5) and (6) in bi-intuitionistic logic) if ! is

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HEINRICH WANSING

understood as provability: Basic structural rules (1) X Æ Y ! Z a` X ! Z Æ Y a` Y ! X Æ Z (2) X ! Y Æ Z a` X Æ Z ! Y a` Y Æ X ! Z (3) X ! Y a` Y ! X a` X ! Y (4) X ! Y a` X ! Y (5) X o Y ! Z a` Y ! X o Z a` X ! Y o Z (6) X ! Y n Z a` X n Y ! Z a` X n Z ! Y; where X1 ! Y1 a` X2 ! Y2 abbreviates X1 ! Y1 ` X2 ! Y2 and X2 ! Y2 ` X1 ! Y1 . If two sequents are interderivable by means of (1){ (6), then these sequents are said to be structurally or display equivalent. The following pairs of sequents, for example, are display equivalent on the strength of (1){(3): X Æ Y ! Z Z ! Y Æ X ; X ! Y Æ Z Z Æ Y ! X ; X!Y Y ! X ; X ! Y Y ! X ; X!Y X ! Y: The name `display logic' derives from the fact that any substructure of a given display sequent s may be displayed as the entire antecedent or succedent of a structurally equivalent sequent s0 . In order to state this fact precisely, we de ne the notion of a polarity vector and antecedent and succedent part of a sequent (cf. [Gore, 1998]). DEFINITION 10. To each n-place structure connective c we assign two polarity vectors ap(c; 1 ; : : : ; n ) and sp(c; 1; : : : ; n), where i 2 f+; g and 1 i n: ap(; ) ap(; +) ap(Æ; +; +) ap(o; +; +) ap(n; +; ) sp(; ) sp(; +) sp(Æ; +; +) sp(o; ; +) sp(n; +; +) We write ap(c; j; ) and sp(c; j; ) to express that c has antecedent, respectively succedent polarity at place j . DEFINITION 11. Let s = X ! Y . The exhibited occurrence of X is an antecedent part of s, and the exhibited occurrence of Y is a succedent part of s. If c(X1 ; : : : ; Xn ) is an antecedent [succedent] part of s, then the substructure occurrence Xj (1 j n) is 1. an antecedent [succedent] part of s if ap(c; j; +) [sp(c; j; +)]; 2. a succedent [antecedent] part of s if ap(c; j; ) [sp(c; j; )]. THEOREM 12. (Display Theorem, Belnap) For each display sequent s and each antecedent [succedent] part X of s there exists a display sequent s0 structurally equivalent to s such that X is the entire antecedent [succedent] of s0 .


85

The theorem was rst proved in [Belnap, 1982]; we shall follow the proof in [Restall, 1998]. A context results from a structure by replacing one occurrence of a substructure by the `Void' (in symbols ` '). If f is a context and X is a structure, then f (X ) is the result of substituting X for the Void in f . A context f is called antecedent positive (negative) if the indicated X is an antecedent part (a succedent part) of f (X ) ! Y ; f is said to be succedent positive (negative) if the indicated X is a succedent part (an antecedent part) of Y ! f (X ). A contextual sequent has the shape f ! Z or Z ! f , and a pair of contextual sequents is said to be structurally equivalent if the sequents are interderivable by means of rules (1){(6). The Display Theorem then follows from the following lemma. LEMMA 13. (i) Suppose f is a context in antecedent position. If f is antecedent positive, then f (X ) ! Y is structurally equivalent to X ! f a(Y ), where f a is a context obtained by unraveling the Void in f . If f is antecedent negative, then f (X ) ! Y is structurally equivalent to f a (Y ) ! X . (ii) Suppose f is a context in succedent position. If f is succedent positive, then Y ! f (X ) is structurally equivalent to f c (Y ) ! X , where f c is a context obtained by unraveling the Void in f . If f is succedent negative, then Y ! f (X ) is structurally equivalent to X ! f c(Y ). The proof is by induction on the complexity of contexts. Case 1: f = . Then f is antecedent and succedent positive, and f a (Y ) = f c(Y ) = Y . Case 2: f = g. Then f (X ) ! Y is structurally equivalent to g(X ) ! Y , and Y ! f (X ) is equivalent to Y ! g(X ). By the induction hypothesis, these sequents are equivalent to X ! f a (Y ), f a (Y ) ! X , f c (Y ) ! X , or X ! f c(Y ). Hence f a = ga ( ) and f c = gc( ). Case 3: f = g. Then f (X ) ! Y is equivalent to Y ! g(X ). Depending on whether g is succedent positive or negative, f (X ) ! Y is structurally equivalent to gc(Y ) ! X or to X ! gc(Y ). Therefore, by the induction hypothesis, f a = gc ( ). Similarly, f c = ga ( ). Case 4: f = Z Æ g. Then f (X ) ! Y is equivalent to g(X ) ! Z Æ Y . By the induction hypothesis, this sequent is equivalent to X ! ga (Z Æ Y ) or ga(Z Æ Y ) ! X , and hence f a = ga (Z Æ ). Similarly, f c = ga ( Æ Z ). Case 5: f = g Æ Z . Similar to Case 4. Case 6: f = g o Z . Then Y ! f (X ) is equivalent to g(X ) ! Y o Z , and by the induction hypothesis, the latter is equivalent to X ! ga(Y o Z ) or to ga (Y o Z ) ! X . Thus f c = ga ( o Z ). Similarly, f a = gc(Z o ). Case 7: f = Z o g. Analogous to the previous case. Cases 8 and 9: f = g n Z and f = Z n g. Analogous to Cases 6 and 7. Proof.

If (for suitable notions of structural equivalence, antecedent part, and succedent part) a sequent calculus satis es the Display Theorem, it is said to enjoy the display property. Note that the set of rules (1){(6) is not the only

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(! f ) (f !) (! t) (t !) (! :) (: !) (! ^) (^ !) (! _) (_ !) (!) (!) (!) (!) (! [F ]) ([F ] !) (! hF i) (hF i !) (! [P ]) ([P ] !) (! hP i) (hP i !) (! ^)0 (^ !)0 (!B) (B!) (! _)0 (_ !)0 (!J) (J!)

truth and falsity rules X!I`X !f `f !I Ì!t I!X`t!X Boolean introduction rules X ! A ` X ! :A A ! X ` :A ! X X !A Y !B `X ÆY !A^B AÆB !X À^B ! X X !AÆB `X !A_B A!X B !Y À_B !X ÆY X ÆA !B `X !A B X ! A B ! Y ` A B ! X Æ Y X ÆA !B X ÆB !A`X !AB X ! A B ! Y X ! B A ! Y ` A B ! X Æ Y tense logical introduction rules X ! A ` X ! [F ]A A ! X ` [F ]A ! X X ! A ` X ! hF iA A ! Y ` hF iA ! Y X ! A ` X ! [P ]A A ! X ` [P ]A ! X X ! A ` X ! hP iA A ! X ` hP iA ! X nonclassical introduction rules X !A Y !B `X oY !A^B AoB !X À^B !X X !AoB `X !A BB X !A B !Y À BB !X oY X !AnB `X !A_B A!X B !Y À_B !X nY X !A B !Y `X nY !AJB AnB !X ÀJB !X Table 1. Introduction rules.


87

(IÆ+ )

X ! Z ` IÆX ! Z X !Z `X ÆI!Z X !Z `X !Z ÆI X !Z `X ! IÆZ (IÆ ) I Æ X ! Z ` X ! Z X ÆI!Z `X !Z X !Z ÆI `X !Z X !IÆZ `X !Z (I) I!X `Z !X X !I`X!Z (I ) I ! X a` I ! X X ! I a` X ! I (PÆ) X1 Æ X2 ! Z ` X2 Æ X1 ! Z Z ! X1 Æ X2 ` Z ! X2 Æ X1 (CÆ) X Æ X ! Z ` X ! Z Z !X ÆX `Z !X (EÆ) X ! Z ` X Æ X ! Z Z !X `Z !X ÆX (MÆ) X1 ! Z ` X1 Æ X2 ! Z X1 ! Z ` X2 Æ X1 ! Z Z ! X1 ` Z ! X1 Æ X2 Z ! X1 ` Z ! X2 Æ X1 (AÆ) X1 Æ (X2 Æ X3 ) ! Z a` (X1 Æ X2 ) Æ X3 ! Z Z ! X1 Æ (X2 Æ X3 ) a` (X1 Æ X2 ) Æ X3 ! Z (MN) I ! X ` I ! X X ! I ` X ! I I ! X ` I ! X X ! I ` X ! I Table 2. Additional structural rules. possible choice of display rules warranting the display property, see [Belnap, 1996] and [Gore, 1998].6 The display property allows an \ èssentials-only' proof of cut elimination relying on easily established and maximally general properties of structural and connective rules" [Belnap, 1996, p. 80]. Further, the display property enables a statement of the introduction rules that satis es the segregation requirement. Belnap emphasizes that the display property may be used to keep certain proof-theoretic components as separate as possible. In a sequent calculus enjoying the display property, the behaviour of the structural elements can be described by the structural rules, and the right (left) introductions rules for an n-place logical operation f can be formulated with f (A1 ; : : : ; An ) standing alone as the entire succedent (antecedent) of the conclusion sequent. Since f (A1 ; : : : ; An ) plays no inferential roles beyond being derived and allowing to derive, these left and right rules provide a complete explanation of the inferential meaning of f . The constant I induces introduction rules for t and f . The operations and Æ give rise to introduction rules for the Boolean connectives. The structure operation permits formulating introduction rules for the modal6 Gore [Gore, 1998] introduces binary structure connectives < and > to be interpreted as directional versions of implication in succedent position and coimplication in antecedent position. The display property is guaranteed by the following structural rules (notation adjusted): X ! Z < Y a` X Æ Y ! Z a` Y ! X > Z Z < Y ! X a` Z ! X Æ Y a` X > Z ! Y:

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ities, whereas o and n give rise to introduction schemata for conjunction, disjunction, implication, and coimplication in bi-intuitionistic logic. These introduction rules are assembled in Table 1. The further structural rules in Table 2 contain many redundancies when they are assumed as a set. Such a rich inventory of structural inference rules is, however, an advantage in a treatment of substructural subsystems of normal modal and temporal logics, see [Gore, 1998]. In addition to a set of structural rules and a set of introduction rules, every display sequent system contains two logical rules exhibiting neither structural nor logical operations, namely re exivity for atoms (alias identity) and cut: (id)

`p!p

and (cut) X ! A A ! Y

` X ! Y:

The identity rule (id) can be generalized to arbitrary formulas from temporal or bi-intuitionistic logic. OBSERVATION 14. For every formula A, ` A ! A. Proof.

The proof is by induction on the complexity of A. For example,

A!A [P ]A ! A [P ]A ! [P ]A

A!A A ! hP iA A ! hP iA hP iA ! hP iA

A!A B!B AnB !AJB A J B ! A J B:

DEFINITION 15. The display sequent system DCP L is given by (id), (cut), the Boolean rules, and the structural rules exhibiting I, , and Æ. The system DKt consists of DCP L plus the tense logical rules and the structural rules exhibiting . The system DK results from DKt by removing the introduction rules for [P ] and hP i. A sequent rule is invertible if every premise sequent can be derived from the conclusion sequent. OBSERVATION 16. The following holds in every purely structural extension of DKt and DK . (i) The logical operations are uniquely characterized. (ii) The introduction rules for :, ^, and _, the left introduction rules for t, hP i, and hF i, and the right introduction rules for f , , , [P ], and [F ] are invertible. (iii) The modalities [F ] and hF i ([P ] and hP i) are interde nable using :. Note that there exist various duality and symmetry transformations on proofs in display logic, see [Gore, 1998], [Kracht, 1996].


89

3.2 Completeness We shall rst consider weak completeness of DKt and DK, that is, the coincidence of Kt (K) and DKt (DK) with respect to provable formulas. We shall then strengthen this result and in Section 3.4 turn to axiomatic extensions of K and Kt. THEOREM 17. (i) If ` A in Kt, then ` I ! A in DKt. (ii) If ` X ! Y in DKt, then 1 (X ) ` 2 (Y ) in Kt. (i) We may take any axiomatization of Kt and show that the axiom schemata are provable in DKt, and the proof rules preserve provability in DKt. The following is a cut-free proof of the K axiom schema for [F ]; the proof for [P ] is analogous: Proof.

A!A [F ]A ! A [F ](A B ) Æ [F ]A ! A ([F ](A B ) Æ [F ]A) ! A B ! B A B ! ([F ](A B ) Æ [F ]A) Æ B [F ](A B ) ! ( ([F ](A B ) Æ [F ]A) Æ B ) [F ](A B ) Æ [F ]A ! ( ([F ](A B ) Æ [F ]A) Æ B ) ([F ](A B ) Æ [F ]A) ! ([F ](A B ) Æ [F ]A) Æ B ([F ](A B ) Æ [F ]A) Æ ([F ](A B ) Æ [F ]A) ! B ([F ](A B ) Æ [F ]A) ! B [F ](A B ) Æ [F ]A ! [F ]B [F ](A B ) ! [F ]A [F ]B I Æ [F ](A B ) ! [F ]A [F ]B I ! [F ](A B ) [F ]A [F ]B Necessitation for [F ] and [P ] is taken care of by the (MN) rules. It remains to derive the tense logical interaction schemata A [F ]hP iA and A [P ]hF iA:

A!A A ! hP iA A ! [F ]hP iA

A!A A ! hF iA hF iA ! A hF iA ! A A ! hF iA A ! [P ]hF iA

(i) In Kt, ` A i ` I ! A in DKt. (ii) In K , ` A i

(ii) By induction on the complexity of proofs in DKt. COROLLARY 18. ` I ! A in DK .

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(i) By the previous theorem. (ii) This follows from the fact that every frame complete normal propositional tense logic is a conservative extension of its modal fragment. Proof.

LEMMA 19. In every extension of DKt by structural inference rules, it holds that ` X ! 1 (X ) and ` 2 (X ) ! X . Proof.

By induction on the complexity of X .

This lemma allows one to prove strong completeness. THEOREM 20. In DKt, ` X ! Y i 1 (X ) ` 2 (Y ) in Kt. ()): This is Theorem 17, (ii). ((): Suppose that in Kt, 1 (X ) ` 2 (Y ). Hence `Kt 1 (X ) 2 (Y ). By Corollary 18, `DKt I ! 1 (X ) 2 (Y ) and thus `DKt 1 (X ) ! 2 (Y ): Since by Lemma 19, ` X ! 1 (X ) and ` 2 (Y ) ! Y in DKt, an application of cut gives ` X ! Y . Proof.

COROLLARY 21. DK is strongly sound and complete with respect to K . COROLLARY 22. DCP L is strongly sound and complete with respect to

CP L.

3.3 Strong cut-elimination A remarkable quality of display logic is that a strong cut-elimination theorem holds for every properly displayable and every displayable logic. Proper displayability and displayability are easily checkable properties. A proper display calculus is a calculus of sequents whose rules of inference satisfy the following eight conditions (recall the terminology from Section 1.1): C1 Preservation of formulas. Each formula which is a constituent of some premise of inf is a subformula of some formula in the conclusion of inf. C2 Shape-alikeness of parameters. Congruent parameters are occurrences of the same structure. C3 Non-proliferation of parameters. Each parameter of inf is congruent to at most one constituent in the conclusion of inf. C4 Position-alikeness of parameters. Congruent parameters are either all antecedent or all succedent parts of their respective sequents. C5 Display of principal constituents. A principal formula of inf is either the entire antecedent or the entire succedent of the conclusion of inf. C6 Closure under substitution for consequent parts. Each rule is closed under simultaneous substitution of arbitrary structures for congruent formulas which are consequent parts.


91

C7 Closure under substitution for antecedent parts. Each rule is closed under simultaneous substitution of arbitrary structures for congruent formulas which are antecedent parts. C8 Eliminability of matching principal formulas. If there are inferences inf1 and inf2 with respective conclusions (1) X ! A and (2) A ! Y with A principal in both inferences, and if cut is applied to obtain (3) X ! Y , then either (3) is identical to one of (1) or (2), or there is a proof of (3) from the premises of inf1 and inf2 in which every cut-formula of any application of cut is a proper subformula of A. Obviously, every display calculus satisfying C1 enjoys the subformula property, that is, every cut-free proof of any sequent s contains no formulas which are not subformulas of constituents of s. If a logical system can be presented as a proper display calculus, it is said to be properly displayable. Belnap [1982] showed that in every properly displayable logic, a proof of a sequent s can be converted into a proof of s not containing any application of cut (1) X ! A (2) A ! Y : (3) X ! Y The proof of strong cut-elimination reveals that every suÆciently long sequence of steps in a certain process of cut-elimination terminates with a cut-free proof. The elimination process consists of various kinds of actions, principal moves, parametric moves, and a combination of parametric and principal moves. If the cut-formula A is principal in the nal inference in the proofs of both (1) and (2), a principal move is performed. Otherwise, if there is no previous application of cut, a parametric move or a combination of parametric and principal moves is executed. According to this distinction we de ne primitive reductions of proofs ending in an application of cut. Recently, Jeremy Dawson and Rajeev Gore discovered a gap in the proof of strong normalization presented in [Wansing, 1998]. To avoid the problem, the primitive reduction steps have to be rede ned. Let i be the proof of (i) we are dealing with, (i = 1; 2). Principal moves. By C8, there are two subcases: Case 1. (3) is the same as (i): 1(3)2 ; i Case 2. There is a proof of (3) from the premises s1 ; : : : ; sn of (1) and s01 ; : : : ; s0m of (2) in which every cut-formula of any application of cut is a proper subformula of A: 1 2 1 2 0 s1 ; : : : ; sn s1 ; : : : ; s0m ; (1) (2) (3) (3)

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Parametric moves. The parametric moves modify proofs on a larger scale

than the principal moves. The parametric moves show that applications of structural rules need never immediately precede applications of cut. Suppose that A is parametric in the inference ending in (1). The case for (2) is completely symmetrical. In order to de ne the parametric moves, we inductively de ne a set Q of occurrences of A, called the set of `parametric ancestors' of A (in 1 ), cf. [Belnap, 1982, p. 394]. We start with putting the displayed occurrence of A in (1) into Q. Then, by working up 1 , we add for every inference inf in 1 each constituent of a premise of inf which is congruent (with respect to inf ) to a constituent of the conclusion of inf already in Q. What we obtain is a nite tree of parametric ancestors of A rooted in the displayed occurrence of A in (1). This tree and the tree of parametric ancestors of the displayed occurrence of A in (2) either contain an application of cut or not. If so, we do not perform a reduction, but instead consider one of these applications of cut above (1) or (2) for reduction. If not, that is, if there is no application of cut in the trees of parametric ancestors, then for each path of parametric ancestors of A in 1 , we distinguish two subcases. Let Au be the uppermost element of the path and let inf be the inference ending in the sequent s which contains Au . Case 1. Au is not parametric in inf. By C4 and C5, it is the entire consequent of s. We cut with 2 and replace every parametric ancestor of A below Au in the path by Y . Case 2. Au is parametric in inf. Then, with respect to inf, Au is congruent only to itself, and we just replace every parametric ancestor of A below Au in the path by Y . Moreover, we delete 2 , which is now super uous. Call the result of simultaneously carrying out these operations for every path of parametric ancestors of A in 1 and removing the initial occurrence of (3) (since now (2) = (3)) l . If the tree of parametric ancestors of the displayed occurrence of A in (1) contains at most one element Au that is not parametric in inf, reduces to l : ; l . Typically we have the situation of Figure 1. By C3 and the bottom-up de nition of Q, for every inference inf in 1 , Q must contain the whole congruence class of inf, if Q is inhabited at all. By C4, Q only consists of consequent parts. Hence, by C2 and C6, the result of such a reduction is in fact a proof of (3), since on the path from (1) to Z ! A we have the same sequence of inference rules being applied as on the path from (3) to Z ! Y . If the cut-formula A is parametric in the inference ending in (2), we rely on C7 instead of C6 and obtain a proof r . If the tree of parametric ancestors of the displayed occurrence of A in (1) contains more than one element Au that is not parametric in inf, parametric and principal moves have to be combined. If A is non-parametric in the nal inference of 2 , we apply to l a principal move on every cut with 2 . Call


HH

HH

Z!A

HH (1) (3)

Z!A Z!Y

;

93

2

HH

2

(3)

Figure 1. the resulting proof l : ; l . If A is parametric in the nal inference of 2 , consider lr . We apply to lr a principal move on every cut with any subproof of 2 ending in a sequent containing a parametric ancestor Au . Call the resulting proof lr : ; lr . Thus, if the tree of parametric ancestors of the displayed occurrence of A in (1) contains more than one element Au that is not parametric in inf, the primitive reduction of gives a proof that is calculated via some intermediate steps. Moreover, instead of a cut with cut-formula A, we obtain several cuts with subformulas of A as the cut-formula. Here is a worked out example:

=

1 (A Æ B ) Æ X ! (A Æ B ) (A Æ B ) Æ X ! (A _ B ) (A _ B ) Æ X ! (A Æ B ) 2 2 (A _ B ) Æ X ! (A _ B ) A ! Y B!Z X ! (A _ B ) Æ (A _ B ) (A _ B ) ! (Y Æ Z ) X ! (A _ B ) (A _ B ) ! (Y Æ Z ) Æ W X ! (Y Æ Z ) Æ W 1

1

l =

(A Æ B ) Æ X ! (A Æ B ) (A Æ B ) Æ X ! (A _ B ) 2 (A Æ B ) Æ X ! (Y Æ Z ) Æ W ((Y Æ Z ) Æ W ) Æ X ! (A Æ B ) ((Y Æ Z ) Æ W ) Æ X ! (A _ B ) 2 ((Y Æ Z ) Æ W ) Æ X ! (Y Æ Z ) Æ W X ! ((Y Æ Z ) Æ W ) Æ ((Y Æ Z ) Æ W ) X ! ((Y Æ Z ) Æ W )

2

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1 2 2 (A Æ B ) Æ X ! (A Æ B ) A!Y B!Z (A Æ B ) Æ X ! (A _ B ) (A _ B ) ! (Y Æ Z ) (A Æ B ) Æ X ! (Y Æ Z ) (A Æ B ) Æ X ! (Y Æ Z ) Æ W 2 2 A!Y B!Z lr = ((Y Æ Z ) Æ W ) Æ X ! (A Æ B ) ((Y Æ Z ) Æ W ) Æ X ! (A _ B ) (A _ B ) ! (Y Æ Z ) ((Y Æ Z ) Æ W ) Æ X ! (Y Æ Z ) ((Y Æ Z ) Æ W ) Æ X ! (Y Æ Z ) Æ W X ! ((Y Æ Z ) Æ W ) Æ ((Y Æ Z ) Æ W ) X ! ((Y Æ Z ) Æ W ) 1

2

1

2

1 (A Æ B ) Æ X ! (A Æ B ) 2 ((A Æ B ) Æ X ) Æ B ! A A ! Y ((A Æ B ) Æ X ) Æ B ! Y 2 Y Æ ((A Æ B ) Æ X ) ! B B!Z Y Æ ((A Æ B ) Æ X ) ! Z (A Æ B ) Æ X ! (Y Æ Z ) (A Æ B ) Æ X ! (Y Æ Z ) Æ W 2 ; lr = ((Y Æ Z ) Æ W ) Æ X ! A Æ B (((Y Æ Z ) Æ W ) Æ X ) Æ B ! A A ! Y (((Y Æ Z ) Æ W ) Æ X ) Æ B ! Y 2 Y Æ (((Y Æ Z ) Æ W ) Æ X ) ! B B!Z Y Æ (((Y Æ Z ) Æ W ) Æ X ) ! Z ((Y Æ Z ) Æ W ) Æ X ! (Y Æ Z ) ((Y Æ Z ) Æ W ) Æ X ! ((Y Æ Z ) Æ W ) X ! ((Y Æ Z ) Æ W ) Æ ((Y Æ Z ) Æ W ) X ! ((Y Æ Z ) Æ W ) 1

2

1

2

THEOREM 23. Every proper display calculus enjoys strong cut-elimination. Proof.

See Appendix A.

COROLLARY 24. Cut is an admissible rule of every proper display

calculus.

Theorem 23 can straightforwardly be applied to DK and DKt. It can easily be checked that in these systems conditions C1 { C7 are satis ed.


95

Veri cation of C8 is also a simple exercise. We have for instance:

X ! A A ! Y X ! [F ]A [F ]A ! Y X ! Y

X ! A A ! Y X ! Y X ! Y A ! Y Y ! A Y ! A X!A A ! Y X ! A A ! Y X ! hAi hAi ! Y ; X ! Y X ! Y Y ! X Y ! X X ! Y: THEOREM 25. Strong cut-elimination holds for DK and DKt. COROLLARY 26. DKt is a conservative extension of DK . ;

We shall now brie y consider generalizations of Theorem 23. By conditions C6 and C7, the inference rules of a proper display calculus are closed under simultaneous substitution of arbitrary structures for congruent formulas. The proof of strong normalization can be generalized to logics which for formulas of a certain shape satisfy closure under substitution either only for congruent formulas (of this shape) which are consequent parts or only for congruent formulas (of this shape) which are antecedent parts. In order to extend the proof of strong cut-elimination to such systems, C6 and C7 have to be replaced by the more general condition of regularity, see [Belnap, 1990]. A formula A is de ned as cons-regular if the following holds: (i) if A occurs as a consequent parameter of an inference inf in a certain rule R, then R contains also the inference resulting by replacing every member of the congruence class of A in inf with an arbitrary structure X , and (ii) if A occurs as an antecedent parameter of an inference inf in a certain rule R, then R contains also the inference resulting by replacing every member of the congruence class of A in inf with any structure X such that X ! A is the conclusion of an inference in which A is not parametric. The notion of ant-regularity is de ned in exactly the dual way. The new condition on rules then is C6/C7 Regularity. Every formula is regular. A display calculus simpliciter is a calculus of sequents satisfying C1 - C5, C6/7, and C8. If a logic can be presented as a display calculus, then it is said to be displayable. Obviously, every properly displayable logic is displayable. Also the parametric moves must be rede ned. Suppose in what follows that the cut-formula A is parametric in both the nal inference of 1 and the nal inference of 2 . Moreover, suppose that the trees of parametric ancestors

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HEINRICH WANSING

of A in 1 and in 2 do not contain any application of cut. If Au is the tip of a path of parametric ancestors of A in i , let inf be the inference ending in the sequent which contains Au . Let us call Au signi cant, if it is not parametric in inf. Then, in a proper display calculus we may choose whether we cut every signi cant tip Au in the tree of parametric ancestors of A in 1 with 2 or whether we cut every signi cant tip Au in the tree of parametric ancestors of A in 2 with 1 to obtain l or r . Both operations form an essential part in the de nition of certain primitive reductions. In a display calculus simpliciter this indeterministic choice has to be abandoned. If the cut-formula is cons-regular, we cut with 2 , and if the cut-formula is ant-regular, we cut with 1 . This further restriction on parametric moves does not aect the proof of strong cut-elimination. THEOREM 27. Every displayable logic enjoys strong cut-elimination. A further strengthening of the strong cut-elimination theorem has recently been proved in [Demri and Gore, 1999], where it is shown that condition C8 may be relaxed. A proof ending in a principal application of cut may also be replaced by a proof 0 of the same sequent if the degree of any application of cut in 0 is the same as the degree of the cut-formula in , and in 0 , every inference except possibly one falls under a structural rule with a single premise. Moreover, in [Demri and Gore, 1999] a display sequent calculus for the minimal nominal tense logic is de ned, and it is shown that every extension of this calculus by structural rules satisfying conditions C1 { C7 enjoys strong cut-elimination.

3.4 Kracht's algorithm The class of all properly displayable normal propositional tense logics has been characterized by Kracht [1996]. The idea is to obtain a canonical way of capturing axiomatic extensions of Kt by purely structural inference rules over DKt. DEFINITION 28. Let Kt + be an extension of Kt by a tense logical axiom schema , and let DKt + 0 be an extension of DKt by a set 0 of purely structural inference rules. Kt + is said to be properly displayed by DKt + 0 if (i) DKt + 0 is a proper display calculus and (ii) every derived rule of Kt + is the -translation of a sequent rule derivable in DKt + 0 . Now, every axiom schema is equivalent to a schema of the form A B , where A and B are implication-free. The schema A B has the same deductive strength as the rule

B ! X ` A ! X: Moreover, if A and B are only built up from propositional variables, t, ^,


97

_, hF i, and hP i, then by classical logic and distribution of hF i and hP i over disjunction, we have

A

_

C and B im i

_

D; j n j

where every Ci and Dj is only built up from t, ^, hF i, and hP i. Therefore A B may as well be replaced by the rule schemata D1 ! Y : : : Dn ! Y Ci ! Y: These rule schemata can now be translated into purely structural display sequent rules, using the following translation from formulas of the fragment under consideration into structures: (p) = p (t) = I (hF iA) = (A) (hP iA) = (A) (A ^ B ) = (A) ^ (B ) The resulting structural rules

(D1 ) ! Y : : : (Dn ) ! Y (Ci ) ! Y may still violate condition C3. In order to avoid this obstruction of proper display, it must be required that in the inducing schema A B , the schematic formula A contains each formula variable only once. A tense logical formula schema is then said to be primitive if it has the form A B , A contains each formula variable only once, and A, B are built up from t, ^, _, hF i, and hP i. LEMMA 29. Every extension of Kt by primitive axiom schemata can be

properly displayed.

Next, if DKt + 0 properly displays Kt + , by condition (ii) of De nition 28, the structural rules in 0 may all have the form X1 ! Y : : : Xn ! Y Z ! Y: This rule has the same deductive strength as the axiom schema

1 (Z )

_

(X ); i 1 i

which is a primitive formula schema. THEOREM 30. (Kracht) An axiomatic extension of Kt can be properly

displayed in precisely the case that it is axiomatizable by a set of primitive axiom schemata.

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HEINRICH WANSING

The question whether an axiomatically presented normal temporal logic is properly displayable thus reduces to the question whether can be axiomatized by primitive axioms over Kt. The implicit use of tense logic in the structural language of sequents may help to nd simple structural sequent rules expressing less simple modal axiom schemata. The following example is taken from [Kracht, 1996]. The :3 axiom schema 2(2A 2B ) _ 2(2B 2A) has the primitive modal equivalent (3A ^ 3B ) ((3(A ^ 3B ) _ 3(B ^ 3A)) _ 3(A ^ B ));

which in tense logic is equivalent to the simpler primitive schema

hP ihF iA ((hF iA _ A) _ hP iA): Application of Kracht's algorithm results in the following structural rule:

X !Y

X ! Y X ! Y ` X ! Y:

Kracht also proves a semantic characterization of the properly displayable tense logics. Let F be a class of Kripke frames hW; R; R 1 i for temporal logics, where R 1 is the inverse of R (i.e., R = f(x; y) j (y; x) 2 Rg). A rst-order sentence (open formula) over two binary relation symbols R and R 1 is said to be primitive if it has the form (8)(9)A, where every quanti er is restricted with respect to R or R 1, and A is built up from ^, _, and atomic formulas x = y, xRy, xR 1 y, where at least one of x, y is not in the scope of an existential quanti er. THEOREM 31. (Kracht) A class F of Kripke frames for temporal logics is

describable by a set of primitive rst-order sentences i the tense logic of F can be properly displayed.

The characteristic axiom schemata of quite a few fundamental systems of modal and tense logic are equivalent to primitive schemata, and therefore these systems can be presented as proper display calculi, cf. Table 3.7 A set of structural sequent rules 0 is said to correspond to a property of an accessibility relation R (with a modal or tense logical axiom schema ) i under the -translation the rules in 0 are admissible just in the event that

7 Gore recently observed that Theorem 20 in [Kracht, 1996] is incorrect. This theorem states that an axiomatic extension of K can be properly displayed i it is axiomatizable by a set of primitive modal axiom schemata. There are, however, rst-order frame properties that correspond to a primitive tense logical schema but fail to correspond to a primitive modal axiom schema. An example of such a frame property is weak directedness:

8s8t8u(sRt ^ sRu 9v(tRv ^ uRv)): Weak directedness corresponds to the .2 schema 32A 23A (alias hF i[F ]A [F ]hF iA). Although .2 has no primitive modal equivalent, it has a primitive tense logical equivalent, namely hP ihF iA hF ihP iA: The latter schema induces a structural rule that may be added to display calculi for (extensions of) K. Therefore, K.2 is properly displayable, although .2 is not primitive.


99

R enjoys the property (the rules in 0

have the same deductive strength as ). Every axiom schema in Table 3 corresponds to a purely structural sequent rule 0 which can directly be determined from , see Table 4. schema primitive equivalent hF iA hF i T [F ]A A A hF iA 4 [F ]A [F ][F ]A hF ihF iA hF iA 5 hF iA [F ]hF iA hP ihF iA hF iA B A [F ]hF iA (A ^ hF iB ) hF i(B ^ hF iA) Alt1 hF iA [F ]A (hF iA ^ hF iB ) hF i(A ^ B ) c T A [F ]A hF iA A c 4 [F ][F ]A [F ]A hF iA hF ihF iA :2 hF i[F ]A [F ]hF iA hP ihF iA hF ihP iA :3 [F ]([F ]A [F ]B ) _ [F ]([F ]B [F ]A) hP ihF iA ((hF iA _ A) _ hP iA) linf hF iA [F ]((hF iA _ A) _ hP iA) hP ihF iA ((hF iA _ A) _ hP iA) linp hP iA [P ]((hP iA _ A) _ hF iA) hF ihP iA ((hP iA _ A) _ hF iA) V [F ]A hP i A Dp [P ]A hP iA hP i Tp [P ]A A A hP iA 4p [P ]A [P ][P ]A hP ihP iA hP iA 5p hP iA [P ]hP iA hF ihP iA hP iA Bp A [P ]hP iA (A ^ hP iB ) hP i(B ^ hP iA) Alt1p hP iA [P ]A (hP iA ^ hP iB ) hP i(A ^ B ) c Tp A [P ]A hP iA A c 4p [P ][P ]A [P ]A hP iA hP ihP iA Vp [P ]A hF i A D

[F ]A

t

t

t

t

t

t

Table 3. Axioms and primitive axioms. Let () be the set of all (all purely modal) axiom schemata from Table 3, , , 0 = f0 j 2 g, and 0 = f0 j 2 g. THEOREM 32. In DKt[ 0 , ` X ! Y i ` 1 (X ) 2 (Y ) in Kt[ . In DK [0 , ` X ! Y i ` 1 (X ) 2 (Y ) in K [.

This follows from axiomatizability by primitive schemata. THEOREM 33. Strong cut-elimination holds for DKt [ 0 and DK [0 . Proof. The rules in 0 and 0 satisfy conditions C2 { C7. 0 0 COROLLARY 34. DKt [ is a conservative extension of DK [ . Kracht's algorithm can be dualized. Every schema A B is interreplaceable with the rule X ! A ` X ! B: Proof.

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HEINRICH WANSING

D0 T0 40 50 B0

Alt10 T c0 4c0 :20 :30

linf 0 linp 0 V0 Dp 0 Tp 0 4p 0 5p 0 Bp 0

Alt1p 0 Tpc0 4cp 0 Vp 0

I ! Y ` I ! Y X ! Y ` X ! Y X ! Y ` X ! Y X ! Y ` X ! Y (X Æ Y ) ! Z ` Y Æ X ! Z (X Æ Y ) ! Z ` X Æ Y ! Z X ! Y ` X ! Y X ! Y ` X ! Y X !Y `X !Y X ! Y X ! Y X ! Y `X ! Y = :30 X ! Y X ! Y X ! Y X ! Y ` I ! Y I ! Y ` I ! Y X ! Y ` X ! Y X ! Y ` X ! Y X ! Y ` X ! Y (X Æ Y ) ! Z ` Y Æ X ! Z (X Æ Y ) ! Z ` X Æ Y ! Z X ! Y ` X ! Y X ! Y ` X ! Y X ! Y ` I ! Y

`X !Y

Table 4. Structural rules corresponding to axiom schemata. If A and B are only built up from propositional variables, f , ^, _, [F ], and [P ], then by classical logic and distribution of [F ] and [P ] over conjunction, we have ^ ^ A im Ci and B jn Dj ;

where every Ci and Dj is only built up from f , _, [F ], and [P ]. Therefore A B may be replaced by the rule schemata X ! C1 : : : X ! Cm X ! Dj : These schemata are translatable into purely structural sequent rules using the following translation 0 from formulas of the fragment under consideration into structures: 0 (p) = p 0 (f ) = I 0 0 0 ([F ]A) = (A) ([P ]A) = 0 (A) 0 (A _ B ) = 0 (A) _ 0 (B )


101

The resulting structural rules X ! 0 (C1 ) : : : X ! 0 (Cm ) X ! 0 (Dj ) again may still violate condition C3. In order to avoid the obstruction of proper display, it must be required that in the inducing schema A B , the schematic formula B contains each formula variable only once. A tense logical formula schema is then said to be dually primitive if it has the form A B , B contains each formula variable only once, and A, B are built up from f , ^, _, [F ], and [P ]. THEOREM 35. An axiomatic extension of Kt can be properly displayed i

it is axiomatizable by a set of dually primitive axiom schemata. For instance, rule T 0 is equivalent to X ! Y ` X ! Y and 40 with X ! Y ` X ! Y . Moreover, D0 is equivalent to X Æ Y ! I ` X ! Y , Alt10 with X ! Y ` X ! Y , and V 0 with ` I ! X , see

[Wansing, 1994]. The properly displayable modal and tense logics satisfy Dosen's Principle. They are all based on the same set of left and right introduction rules, so that the logical operations indeed have the same proof-theoretic, operational meaning in each of these systems. Kracht's characterization results show that many interesting and important intensional logics admit a cut-free display sequent calculus presentation. In Sections 3.8 and 4 other applications of the display calculus are pointed out. Display sequent systems for various non-normal modal logics may be found in [Belnap, 1982].

3.5 Formulas-as-types for temporal logics It is well-known that every derivation in Gentzen's natural deduction calculus for intuitionistic implicational logic can be encoded by a typed -term, and vice versa [Howard, 1980]. In particular, every natural deduction proof can be encoded by a closed term, and every closed term encodes a proof. It is also well-known that every pair of non-convertible typed -terms de nes dierent functionals of nite type [Friedman, 1975]. Every type A is associated with an in nite set DA , every term variable xA of type A denotes an element from DA , and every term M (A B) of type A B B denotes an element from the set (DB )D of all functions from DA to DB . Together with the encoding, this interpretation results in a set-theoretic semantics of proofs in intuitionistic implicational logic. In this section, we shall develop a set-theoretic interpretation of sequent proofs in the ft; [F ]; hP i; B; ^g{ fragment of the smallest normal temporal intuitionistic (or, for that purpose, minimal) logic IntKt. The interpretation is based on the observation that the modalities hP i and [F ] form a residuated pair with respect to derivability. The encoding of proofs by typed terms should be such that B

A

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HEINRICH WANSING

proof-simpli cation (or normalization) corresponds with a suitable reduction relation on terms, and therefore the set-theoretic semantics of terms has to validate the equalities underlying the reduction rules. The principal cut-elimination steps for hP i and [F ] reveal that two pairs of term forming operations o1 and o2 are needed such that o1 (o2 (M )) = M . We shall use the following identities: S

Pa = a

and

T

S a = a;

where P is the familiar powerset operation and S a =def fb j a bg). Since in general S a is a proper class, we shall restrict the denotations of terms to the universe V! . This is enough to accommodate the sets used as domains of the intended models in Section 3.7. We shall rst de ne a display sequent system DIntKt for the fragment of IntKt under consideration, and then present an extension t of the typed -calculus. The set of types in t is the set of all formulas in the language L = ft; [F ]; hP i;B;^g based on a denumerable set Atom of propositional variables. In Section 3.6 it is proved that term reduction is a homomorphic image of proof-simpli cation. Next, an encoding of terms by proofs is presented. A set-theoretic semantics of proofs in DIntKt is obtained in Section 3.7 by showing that every pair of non-convertible t -terms de nes dierent sets in the set-theoretic universe under consideration. In particular, every term M [F ]A denotes an element from fP a j a 2 DA g, and every term M hP iA denotes an element from fS a j a 2 DA g: Also the formulas-astypes notion of construction for various extensions of DIntKt is dealt with and remarks on some related work about formulas-as-types for modal logics are made. First, we shall de ne the sequent system DIntKt. We assume the following language of structures: X ::= A j I j X j X o Y: A sequent now is an expression X ! Y , provided Y 6= I. The declarative meaning of the structure connectives can be made explicit by a translation from the set of sequents into the set of L-formulas: (X ! Y ) := 1 (X ) B 2 (Y ); where i (i = 1; 2) is de ned as follows: i (A) = A 1 (I) = t 1 (X o Y ) = 1 (X ) ^ 1 (Y ) 2 (X o Y ) = 1 (X ) B 2 (Y ) 1 (X ) = hP i1 (X ) 2 (X ) = [F ]2 (X ) Given this understanding of the structure connectives, the basic structural rules (4) and (5) from Section 3.1 are assumed. Clearly, the Display Theorem holds for this structural language and calculus. 1


103

DEFINITION 36. The display sequent calculus DIntKt is given by the logical rules (id) and (cut), the basic structural rules (4) and (5), the introduction rules for t, B, hP i, [F ], and the rules (! ^)0 and (^ !)0 , together with the following structural rules: (empty structure) X ! Y ` I o X ! Y; X ! Y ` X o I ! Y I o X ! Y ` X ! Y; X o I ! Y ` X ! Y (associativity) (X1 o X2 ) o X3 ! Y a` X1 o (X2 o X3 ) ! Y (permutation) X o Y ! Z ` Y o X ! Z (contraction) X o X ! Y ` X ! Y (expansion) X ! Y ` X o X ! Y (monotonicity) X ! Z ` X o Y ! Z; X ! Z ` Y o X ! Z (necessitation) I ! X ` I ! X: To show that DIntKt is a display calculus for IntKt, we de ne an axiomatic calculus HIntKt. DEFINITION 37. The system HIntKt consists of the axiom schemata and rules of the ft; ^; Bg{fragment of positive intuitionistic logic, together with 1. ([F ]A ^ [F ]B ) B [F ](A ^ B ) 2. [F ]t

3. A B [F ]hP iA 4. 5.

ÀBB ` [F ]A B [F ]B ÀBB ` hP iA B hP iB

The relational semantics to be presented is a straightforward adaptation of the semantics developed by Bosic and Dosen [1984]. A comprehensive survey of intuitionistic modal logics and their algebraic and relational semantics is [Wolter and Zakharyaschev, 1999]. A temporal frame is de ned as a structure hW; RI ; RT i, where W is a non-empty set (of states), RI and RT are binary relations on W , RI is both re exive and transitive, and, moreover, (i) RI RT RT RI (i.e. the composition of RT and RI is a subset of the composition of RI and RT ) and (ii) RI 1 RT 1 RT 1RI 1 . If F = hW; RI ; RT i is a temporal frame, the temporal model based on F is the structure hF ; vi, where v is a function from Atom W into f0; 1g satisfying: (Heredity) (v(p; u) = 1 and uRI t) implies v(p; t) = 1:

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HEINRICH WANSING

Let M = hW; RI ; RT ; vi be a temporal model. Veri cation of a formula A at a state u 2 W (M; u j= A) is inductively de ned as follows:

M; u j= p M; u j= t M; u j= A ^ B M; u j= A B B M; u j= [F ]A M; u j= hP iA

i v(p; u) = 1 i M; u j= A and M; u j= B i (8t 2 W ) uRI t implies [M; t 6j= A or M; t j= B ] i (8t 2 W ) uRT t implies M; t j= A i (9t 2 W ) tRT u and M; t j= A

For every formula A, if A is veri ed at state u and uRI t, then A is also veri ed at t. Condition (i) ensures this general heredity property for formulas [F ]A, and condition (ii) ensures it for formulas hP iA. A formula A is true in a model hW; RT ; RI ; vi if A is veri ed at every u 2 W , and A is said to be true on a frame F , if A is valid in every model based on F . If K is a class of models (frames), A is said to be valid in K i A is valid in every model (valid on every frame) in K. THEOREM 38. HIntKt is sound and complete with respect to the class of all temporal frames, i.e. for every L{formula A, A is provable in HIntKt i A is valid in the class of all temporal frames. Soundness is shown by induction on proofs in pleteness see Appendix B. Proof.

HIntKt;

for com-

LEMMA 39. (1) If ` A in HIntKt, then ` I ! A in DIntKt, and (2) If ` X ! Y in DIntKt, then ` (X ! Y ) in HIntKt. (1) By induction on proofs in HIntKt. We shall consider only two example cases: Proof.

A!A A ! hP iA A ! [F ]hP iA A o I ! [F ]hP iA I ! A o [F ]hP iA I ! A B [F ]hP iA

B!B A!A [F ]A ! A [F ]B ! B [F ]A o [F ]B ! A [F ]A o [F ]B ! B ([F ]A o [F ]B ) ! A ([F ]A o [F ]B ) ! B ([F ]A o [F ]B ) o ([F ]A o [F ]B ) ! A ^ B ([F ]A o [F ]B ) ! A ^ B ([F ]A o [F ]B ) ! [F ](A ^ B ) ([F ]A ^ [F ]B ) ! [F ](A ^ B ) ([F ]A ^ [F ]B ) o I ! [F ](A ^ B ) I ! ([F ]A ^ [F ]B ) o [F ](A ^ B ) I ! ([F ]A ^ [F ]B ) B [F ](A ^ B )

(2) By induction on proofs in DIntKt.


105

COROLLARY 40. In HIntKt, ` A i ` I ! A in DIntKt. By induction on the complexity of X , one can prove the following LEMMA 41. In every extension of DIntKt by structural rules, it holds that ` X ! 1 (X ) and ` 2 (X ) ! X . THEOREM 42. In DIntKt, ` X ! Y i ` (X ! Y ) in HIntKt. Proof.

Analogous to the proof of Theorem 20.

Since DIntKt is a proper display calculus, we have the following THEOREM 43. DIntKt enjoys strong cut-elimination. Take any terminating cut-elimination algorithm elim c for DIntKt. We may also de ne a binary relation s on the set of proofs in DIntKt by the following stipulations:

A!A B!B AoB !A^B A^B !A^B

s

A^B !A^B

A!A B!B ABB !AoB s ABB!ABB ABB!ABB If s 0 , we say that in 0 a redundant part of has been removed. Let elim r denote the terminating algorithm that removes redundant parts of a proof in top-down left to right order, so that a redundant part is removed only if it has no redundant part above it. Obviously, in any extension of DIntKt, every proof of a sequent s can be converted into a proof of s containing no redundant part. Let elim denote elim r elim c , i.e. the composition of elim r and elim c . The algorithm elim is the process of proof simpli cation to be considered. We assume that elim() = if contains no application of (cut) and no redundant part.

3.6 The typed -calculus t

The set T of type symbols (or just types) is the set of all L-formulas. The set V of term variables is de ned as fviA j 0 < i 2 !; A 2 T g. DEFINITION 44. The set Term of typed terms is de ned as the smallest set such that 1. V

;

2. if M A , N B 2 , then hM A ; N B i(A^B) 2 ; 3. M (A^B) 2 , then (M (A^B) )A0 , (M (A^B))B1 2 ;

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HEINRICH WANSING

4. if xA 2 V and M B 2 , then (xA M B )(A B) 2 ; B

5. if M (A B), N A 2 , then (M (A B) ; N A )B 2 ; 6. if M A 2 , then (P M )[F ]A, (S M )hP iA 2 ; B

B

7. if M [F ]A 2 , then ([M [F ]A )A 2 ; 8. if M hP iA 2 , then (\M hP iA )A 2 .

A term M A is said to be a term of type A; obviously, every term has a unique type. If confusion is unlikely to arise, we shall often write M instead of M A and omit parentheses not needed for disambiguation. The set fv (M ) of free variables of M , the set of subterms of M , and M [xA := N A ], the result of substituting term N of type A for every occurrence of xA in M are inductively de ned in the obvious way. If a variable x in M is not an element of fv (M ), x is said to be a bound variable of M . The set of bound variables of M is denoted as bv (M ). We shall also write M (xA1 ; : : : ; xAn ) to express that x1 ; : : : ; xn 2 fv (M ). If M (xA1 ; : : : ; xAn ) and N1 ; : : : ; Nn are terms of types A1 ; : : : ; An , then M (N1 ; : : : ; Nn ) is the result of substituting in M the variables xi by Ni . We shall use `' to denote syntactic identity between term. 1

1

n

n

DEFINITION 45. The typed -calculus t consists of the following rules and axiom schemata: 1. xA M = (yA M [x := y]), if y 62 (fv (M ) [ b v (M )); 2. x(M; x) = M , if x 62 fv (M ); 3. (xM )N = M [x := N ], if b v (M ) [ fv (N ) = ;; 4. (hM0 ; M1i)i = Mi ; 5. h(M )0 ; (M )1 i = M ; 6. 7.

[P M = M ; \S M = M ;

8. M A = M A;

` N = M ; M = N; N = G ` M = G; M = N ` (G; M ) = (G; N ); M = N ` (M; G) = (N; G); M = N ` xM = xN ; M = N ` P M = P N ; M = N ` [M = [N .

9. M = N 10. 11. 12.


107

DEFINITION 46. The binary relations on Term, !r (one-step reduction), r (reduction), and =r (equality) are de ned as follows:

1.

2.

x(Mx) !r M , if x 62 fv (M ); (xM )N !r M [x := N ], if b v (M ) [ fv (N ) = ;; (hM; N i)0 !r M ; (hM; N i)1 !r N ; h(M )0 ; (M )1 i !r M ; [P M !r M ; \S M !r M ; if M A B !r N A B , then (M; GA ) !r (N; G); if M A^B !r N A^B , then (M )i !r (N )i ; if M A !r N A , then xM !r xN , (GA B M ) !r (GN ), hM; Gi !r hN; Gi, hG; M i !r hG; N i, P M !r P N , S M !r S N , \M !r \N , [M !r [N . B

B

B

r is the re exive transitive closure of !r ;

3. =r is the equivalence relation generated by r . DEFINITION 47. t -terms x(Mx) (where x 62 fv (M )), (xM )N (where b v(M ) [ fv (N ) = ;), (hM; N i)0 ; (hM; N i)1 ; h(M )0 ; (M )1 i; [P M; and \S M are called redexes. A term M is a normal form (nf ) if it has no redex as a subterm, and M has a nf if there is a nf N such that M =r N . M is said to be strongly normalizable with respect to r (sn(M )) if every sequence of reduction steps starting at M is nite. THEOREM 48. Every M 2 Term is strongly normalizable with respect to

r .

Proof.

See Appendix C.

Let norm(M ) refer to the iterated contraction of the leftmost redex in M . Since by the previous theorem, every reduction starting at M is nite, norm is a terminating normalization algorithm with respect to r . We shall now encode proofs by giving recipes for building up constructions of sequents. Every formula occurring in an antecedent part of a sequent s is said to be an antecedent formula component of s. DEFINITION 49. A construction of a sequent s is a term M A such that an occurrence of A is the succedent part of s, and every type of a free variable of M is an antecedent formula component of s. This notion of construction is a straightforward adaptation of the notion of construction for ordinary natural deduction and sequent calculi. The set of types of the free variables occurring in the term encoding a derivation is a subset of the set of assumptions on which depends. Therefore applications

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HEINRICH WANSING

of structural inference rules are not re ected by term modi cations, and variations of structural rules are captured by imposing conditions on variable binding and occurrences of free variables in the encoding terms (see, for instance, [van Benthem, 1986, Chapter 7], [van Benthem, 1991], [Helman, 1977], [Wansing, 1992]). OBSERVATION 50. Given a proof in DIntKt of a sequent s, one can nd a construction M of s. We de ne a function f from the set DIntKt of proofs in DIntKt to T erm such that f () is a construction of the conclusion sequent of . The pairs of sequent rules and terms or term construction rules in Table 5 amount to an inductive de nition of f . The variables newly introduced into the conclusion of a term construction rule are the numerically rst variables of the types indicated not occurring in the premise term. Proof.

Clearly, norm is a function on Term . Let + DIntKt denote the set of all proofs in DIntKt containing an application of (cut) or a redundant part, and let DIntKt denote the set of all cut-free proofs in DIntKt containing no redundant part. Let +Term denote the set of all terms that are not normal forms, and let Term denote the set of all terms that are normal forms. THEOREM 51. Let A = hDIntKt; e limi and B = hTerm ; norm i. The function f de ned in the proof of Observation 50 is a homomorphism from

A to B. Proof.

See Appendix D.

Under the encoding of proofs by terms, surjective pairing (h(M )0 ; (M )1 i !r M ) and {reduction (x(Mx) !r M , if x 62 fv (M )) correspond with replacing proofs

A!A B!B AoB !A^B A^B !A^B

and

A!A B!B A BB !AoB ABB!ABB

by the axiomatic sequents A^B ! A^B and A B B ! A B B , respectively. Note that there are no analogues of surjective pairing and {reduction that correspond with a replacement of proofs of [F ]A ! [F ]A and hP iA ! hP iA from A ! A by the axiomatic sequents [F ]A ! [F ]A and hP iA ! hP iA. Moreover, since in the encoding applications of structural rules are not re ected by term formation steps, it is in general not the case that if M = f (), can be uniquely reconstructed from M .


A!A X !A A!Y X !Y s s0

!t I!X t!X I

Logical rules

Structural rules

v1A M A N (xA ) N [x := M ]

M M

Intuitionistic connective rules

X !A Y !B XoY !A^B AoB !X A^B !X X !AoB X!ABB X !A B!Y ABB !X oY

v1t M M

MA NB hM; N i M (xA ; yB ) M ((z A^B )0 ; (z A^B )1 ) M (xA ) xA M MA N (xB ) N [x := (y(A B); M )] B

Modal connective rules X ! A M X ! [F ]A PM M (xA ) A!X [F ]A ! X M ([y[F ]A) X !A M SM X ! hP iA A ! X M (xA ) hP iA ! X M (\yhP iA )

Table 5. Sequent rules and term construction rules.

109

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HEINRICH WANSING

3.7 A denotational semantics of proofs We shall now de ne models for t . The completeness proof to be given straightforwardly extends H. Friedman's [1975] completeness proof for typed {calculus. The plan of the proof is as follows: rst it is shown that t is sound and complete with respect to the class of all models. This is achieved by de ning a canonical model that itself characterizes t . Then a notion of intended model is de ned. In such models the typed terms have their intended set-theoretic interpretation. In order to characterize provable equality of terms in t by validity in all intended models, it is shown that for every intended model M, there exists a `partial homomorphism' from M onto the canonical model. Since such partial homomorphisms turn out to preserve validity, t is sound and complete with respect to the class of all intended models. DEFINITION 52. A structure F = hfDA g; fAPA;B g; fPRO0A;B g, fPRO1A;B g; fPAIRA;B g, fPAg, fSA g, fP#Ag, fS#Ag i is called a type structure frame (or just a frame) i for all types A, B : 1. DA (the domain of type A) is a non-empty set; 2. APA;B : D(A B) DA ! DB , PRO0A;B : D(A^B) ! DA , PRO1A;B : D(A^B) ! DB , PAIRA;B : DA DB ! D(A^B) , PA : DA ! D[F ]A ; SA : DA ! DhP iA ; P#A : D[F ]A ! DA ; S#A : DhP iA ! DA ; B

3. (extensionality) if a; b 2 D(A B) and (8c 2 DA ) we have (APA;B (a; c) = APA;B (b; c)), then a = b; B

4. (pro) for all a 2 DA , b 2 DB : PRO0A;B (PAIRA;B (a; b)) = a, PRO1A;B (PAIRA;B (a; b)) = b; 5. (pair) for all a 2 DA^B : PAIRA;B (PRO0A;B (a); Pro1A;B (a)) = a; 6. (future) for all a 2 DA : P# (Pa) = a; 7. (past) for all a 2 DA : S# (Sa) = a.

An assignment in a frame hfDA g, fAPA;B g, fPRO0A;B g, fPRO1A;B g; fPAIRA;B g, fPAg, fSA g, fP#Ag, fS#A g i is a function f de ned on the set V of term variables such that f (xA ) 2 DA . The set of all assignments in a given frame is denoted by Asg . If y 2 V , then fay is de ned by fay (x) = f (x), if x 6= y, fay (y) = a.


111

DEFINITION 53. Suppose that F = hfDA g, fAPA;B g, fPRO0A;B g, 1 fPROA;B g; fPAIRA;B g, fPAg, fSA g, fP #Ag, fS #A g i is a frame. Then hF ; val i is said to be a type structure model (or justSa model) based on F i val is the valuation function from Term Asg to A2T DA such that: 1. val (x; f ) = f (x); 2. APA;B (val ((xM ); f ); a) = val (M; fax ), 8a 2 DA ; 3. val ((M (A B) ; N B ); f ) = APA;B (val (M; f ); val (N; f )); B

4. val (hM A ; N B i; f ) = PAIRA;B (val (M; f ); val (N; f )); 5. val ((M (A^B))i ; f ) = PROiA;B (val (M; f )), i = 0; 1; 6. val ((P M A)[F ]A ; f ) = PA (val (M; f )); 7. val ((S M A )hP iA ; f ) = SA (val (M; f )); 8. val (([M [F ]A )A ; f ) = P#A (val (M; f )); 9. val ((\M hP iA )A ; f ) = S#A (val (M; f )). Let M = hF ; val i be a model. x LEMMA 54. (1) val (M [x := N ]; f ) = val (M; fval (N;f ) ), if bv (M ) \ fv (N ) = y x ;. (2) val (M [x := y]; fa ) = val (M; fa ), if y 62 bv (M ) [ fv (M ). Proof.

(1) By induction on M , for xed N ; (2) by (1).

The equality M = N is said to hold in M under assignment f (M; f j= M = N ) i val (M; f ) = val (N; f ). M = N is called valid in M (M j= M = N ) i M; f j= M = N , for all f 2 Asg . M = N is said to be valid in a class K of models, if M j= M = N , for each M 2 K. OBSERVATION 55. (Soundness) If M = N is provable in t , then M = N is valid in the class of all models. Proof. By induction on proofs in t . We must show that every axiom is valid in every model, and that the rules of inference preserve validity. We shall consider two cases not already dealt with in [Friedman, 1975].

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HEINRICH WANSING

h(M )0 ; (M )1 i = M : val ((hM; N i)0 ; (hM; N i)1 i; f ) = PAIR(val ((hM; N i)0 ; f ); val ((hM; N i)1 ; f )) = PAIR(PRO0 (PAIR(val (M; f ); val (N; f ))), PRO1 (PAIR(val (M; f ); val (N; f )))) = PAIR(val (M; f ); val (N; f )) = val (hM; N i; f ). \S M = M : val (\S M; f ) = S# (val (S M; f ) = S# (S(val (M; f )) = val (M; f ) Next, we de ne the frame F0 on which the canonical model is based. Let j M j = fN j` M = N g; j M j is the equivalence class of M with respect t

to provable equality in t . DEFINITION 56. F0 = hfDA g, fAPA;B g, fPRO0A;B g, fPAIRA;B g, fPAg, fSA g, fP#A g, fS#A g i is de ned as follows:

fPRO1A;B g;

DA = fj M

j j M is of type Ag; APA;B (j M A B j; j N A j) = j (M; N ) j; PRO0A;B (j M A^B j) = j (M )0 j; PRO1A;B (j M A^B j) = j (M )1 j; PAIRA;B (j M A j; j N B j) = j hM; N i j; PA (j M A j) = j P M j; SA (j M A j) = j S M j; P#A (j M A j) = j [M j; S#A (j M A j) = j \M j.

LEMMA 57.

B

F0 is a frame.

Clearly, DA is a non-empty set, and APA;B , PRO0A;B , PRO1A;B ; PAIRA;B , PA , SA , P#A , and S#A are functions with appropriate domain and range, for all types A and B . For (extensionality) see [Friedman, 1975]. For (pro), (pair), (future), and (past), use the obvious equalities. Proof.

A function g : V ! Term is called a substitution, if g(x) and x are of the same type. A substitution is called regular, if for pairwise distinct variables x; y, fv (g(x)) \ fv (g(y)) = ;. Let M (g) denote the result of simultaneously


113

replacing in M every free occurrence of each variable x by g(x). It can easily be shown that if M 2 Term and is a nite set of variables, then there is an N such that ` M = N , fv (M ) = fv (N ), and bv (N ) \ = ;. DEFINITION 58. Suppose f is an assignment in F0 and g is a regular substitution such that f (x) = j g(x) j, for every x 2 V . For a given term M , choose a term N such that ` M = N and for every x 2 fv (N ), bv (N ) \ fv (g(x)) = ;. Then val (M; f ) is de ned by val (M; f ) = j N (g) j. S It can be shown that val : Term Asg ! A DA , and ` M = N implies val (M; f ) = val (N; f ), cf. [Friedman, 1975]. LEMMA 59. M0 = hF0 ; val i is a type structure model. Proof. We consider those conditions not already assumed in Friedman's paper. Let g be a regular substitution and f (x) = j g(x) j, for f 2 Asg . Choose M1 , N1 such that ` M = M1 , ` N = N1 , and bv (M1 ) \ fv (g(x)) = bv (N1 ) \ fv (g(x)) = ;, for every x 2 fv (M1 ) [ fv (N1 ). t

t

t

t

t

4 : val (hM; N i; f ) = j hM1 ; N1 i(g) j = PAIR(j M1(g) j; j N1 (g) j) = PAIR(val (M; f ); val (N; f )).

5 : val ((M )i ; f ) = j (M1 )i (g) j = PROi (j M1 (g) j) = PROi (val (M; f )).

6 : val ((P M A )[F ]A ; f ) = j P M1 (g) j = PA (j M1(g) j) = PA (val (M; f )).

8 : val (([M [F ]A )A ; f ) = j [M1 (g) j = P#A (j M1 (g) j) = P#A (val (M; f )).

THEOREM 60. (Completeness) If M = N is valid in the class of all models, then ` M = N . Proof. Suppose 6` M = N . Choose M1 , N1 such that ` M = N1 , ` N = N1 , and bv (M1 ) \ fv (M1 ) = bv (N1 ) \ fv (N1 ) = ;. Then val (M; f ) = j M1 j = 6 j N1 j = val (N; f ), for f (x) = j id(x) j, for all x 2 V , where id is the identity function on V . Thus, M0 6j= M = N . 7 and 9 : analogous to the previous two cases. t

t

t

c

We now de ne the intended models. Following the terminology of Friedman, we shall call the frames underlying an intended model `full temporal type structures over in nite sets'. DEFINITION 61. A type structure frame F = hfDA g, fAPA;B g, fPRO0A;B g, fPRO1A;B g; fPAIRA;B g, fPA g, fSAg, fP#A g, fS#Agi is said to be a full temporal type structure over in nite sets, if

Dt is in nite, and for every p 2 Atom ; Dp is in nite; DA^B = DA DB ; DA B = (DB )D ; B

A

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HEINRICH WANSING

D[F ]A = fP a j a 2 DA g; DhP iA = fS a j a 2 DA g; APA;B (a; b) = a(b); PRO0A;B (ha; bi) = a; PRO1A;B (ha; bi) = b; PAIRA;B (a; b) = ha; bi; PA (a) = P a; SA (a) = S a; P#A (a) = [a; S#A (a) = \a: DEFINITION 62. Let F = h fDA g, fAPA;B g, fPRO0A;B g, fPRO1A;B g; fPAIRA;B g, fPA g, fSA g, fP #Ag, fS #A g i, F = hfDA g, fAPA;B g, 0 g, fPRO1 g; fPAIR g, fP g, fS g, fP# g, fS# g i be frames, fPROA;B A;B A;B A A A A and let M = hF ; val i and M = hF ; val i be models. A family of functions ffAg is called a partial homomorphism from M onto M i 1. for each type A, fA is a partial function from DA onto DA ; 2. if fA B (a) exists, then fB (APA;B (a; b)) = APA;B (fA B (a); fA (b)), for all b in the domain of fA, 3. if fA(a), fB (b) exist, then fA^B (PAIRA;B (a; b)) = PAIRA;B (fA (a); fB (b)); 4. if fA^B (a) exists, then fA(PRO0 (a)) = PRO0 (fA^B (a)); B

B

A;B fB (PRO1A;B (a))

A;B 1 (f PROA;B A^B (a));

5. if fA^B (a) exists, then = 6. if fA(a) exists, then f[F ]A (PA (a)) = PA (fA (a)); fhP iA (SA (a)) = SA (fA (a)); 7. if f[F ]A(a); fhP iA (b) exist, then fA (P#A (a)) = P#A (f[F ]A (a)); fA (S#A (b)) = S#A (fhP iA (b)). LEMMA 63. Let M, M be as in the previous de nition, and let ffAg be a partial homomorphism from M onto M. If g, g are assignments in F and F respectively, and fA(g(xA )) = g(x), then fA(val (M A ; g)) = val (M; g ). Proof. By induction on M . We consider the cases not already dealt with in [Friedman, 1975]. Note that we may assume fA(g(xA )) = g (x), since fA is onto.

M hN A ; GB i: fA^B (val (hN; Gi; g)) = fA^B (PAIR(val (N; g); val (G; g))) = PAIRA;B (fA (val (N; g)); fB (val (G; g))) = PAIRA;B (val (N; g ); val (G; g )) by the induction hypothesis = val (hN; Gi; g ).


115

M (NiA^B )i : f (val ((N )i ; g)) = f (PROi (val (N; g))) = PRO (fA^B (val (N; g))) = PROi (val (N; g )) by the induction hypothesis = val ((N )i ; g). M P N A : f[F ]A(v al(P N; g)) = f[F ]A(PA (v al(N; g))) = PA (fA (v al(N; g))) = PA (v al (N; g )) = v al (P N; g ). M [N [F ]A : fA (v al([N; g)) = fA (P #A (v al(N; g))) = P#A (f[F ]A(v al(N; g))) = P#A (v al (N; g )) = v al ([N; g ). M S N; \N : analogous to the previous two cases. COROLLARY 64. Let M = hF ; v ali, M = hF ; v al i be models. If there is a partial homomorphism from M onto M , then M j= M = N implies M j= M = N . Proof. Suppose M j= M B = N B , ffA g is a partial homomorphism from M onto M , and g is an assignment in M . We choose an assignment g in M such that for every A 2 T , g (x) = fA (g(xA )). By the previous lemma, v al (M; g ) = fB (v al(M; g) = fB (v al(N; g) = v al (N; g ) THEOREM 65. Let M be a model based on a full temporal type structure over in nite sets. Then ` M = N i M j= M = N . Proof. It suÆces to show that M j= M = N implies M0 j= M = N . To prove this, we de ne by induction on A a partial homomorphism ffAg from M onto M0 as follows: A = p, A = t, p 2 Atom : fA is any function from DA onto M0 's domain DA . t

(Such a function exists, since DA is in nite and DA is denumerable.)

A = (B ^ C ):

If fB (b), fC (c) exist, then fB^C (hb; ci) = fB^C (PAIR(b; c)) is de ned as PAIRB;C (fB (b); fC (c)):

A = (B B C ):

fB C (a) is de ned as the unique member of D(B C ) (if it exists) such that fC (a(b)) = APB;C (fB C (a); fB (b)), for all b in the domain of fB . B

B

B

A = [F ]A:

f[F ]A(a) = f[F ]A(PA (b)) for some b 2 DA is de ned as PA (fA (b)) if fA (b) exists.

A = hP iA:

fhP iA (a) = fhP iA (SA (b)) for some b 2 DA is de ned as SA (fA (b)) if fA (b) exists.

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HEINRICH WANSING

That ffA g is a partial homomorphism follows from the de nition of ffAg and the following equations:

PRO

0A;B (ha; bi)) A( fA (a) PRO0A;B (PAIRA;B (fA (a); fB (b))) PRO0A;B (fA^B ( A;B (a; b))) PRO0A;B (fA^B (ha; bi)) f

= = = =

=

=

PAIR

= =

P

A ( #A (P a)) A (a) P #A (f[F ]A(P a)) f

=

=

PAIR

=

S

A ( #A (S a)) A (a) S #A (fhP iA (S a)) f

=

f

PRO

1A;B (ha; bi)) B( fB (b) PRO1A;B (PAIRA;B (fA (a); fB (b))) PRO1A;B (fA^B ( A;B (a; b))) PRO1A;B (fA^B (ha; bi)) f

f

It remains to be shown that fA is onto, for every type A. For A = t and A = p 2 Atom, this follows from the de nitions of ft , fp and F0 . For the remaining cases we consider two examples. A = [F ]B . Assume d = j P M j 2 D[F ]B . Choose a 2 D[F ]B such that a = P b for b 2 DB and b = fB 1 (j M B j). Since fB is onto, such an element a from D[F ]B exists. Then f[F ]B (a) = f[F ]B (PB (b)) = PB (fB (b)) = j P M j = d. Consider now A = (B B C ), and assume d 2 D(B C ) . Choose a 2 D(B C ) such that for every b in the domain of fB , a(b) 2 fC 1 (Ap(d; fB (b))). Then f(B C )(a) = d. Since fC and fB may be assumed to be onto, the set of such a 2 D(B C ) is non-empty. B

B

B

B

Whereas the encoding of substructural subsystems of DIntKt obtained by giving up all or part of DIntKt's structural rules will require modi cations of the notion of construction, in order to encode structural extensions of DIntKt, the notion of construction need not be altered. Various extensions of HIntKt can be presented as structural extensions of DIntKt. The following axiom schemata are those schematic axioms from Table 3, which are in L. Each axiom schema Ax in this table corresponds with the associated structural rule Ax0 in the sense that an L{formula A is provable in HIntKt + Ax i I ! A is provable in DIntKt + Ax0 . In the literature, several proposals have been made to extend the formulasas-types notion of construction from positive logic to modal logics based on it. We shall here brie y point to ve such approaches. Gabbay and de Queiroz [1992] interpret the necessity modality 2 \as a sort of second-order universal quanti cation (quanti cation over structured collections of formulas)" [Gabbay and de Quieroz, 1992, p. 1359]. Using the framework of Labelled Natural Deduction [de Queiroz and Gabbay, 1999], proofs in various modal logics are encoded by imposing conditions on abstraction over possible-world variables [de Queiroz and Gabbay, 1997]. However, Gabbay and de Queiroz do not consider a Friedman-style completeness proof for the {calculi under consideration. 1.

SEQUENT SYSTEMS FOR MODAL LOGICS name axiom schema T

4

[F ]A

V

[F ]A

T

A

c c 4

B B[

[F ]A

A F ][F ]A

F ]A

B[ ] B h it Bh i h ih i B h i ( ^h i )Bh [F ][F ]A

T

t

name structural rule 0 X ! Y ` X ! Y T 0 4 X ! Y ` X ! Y 0 X !Y ` !Y V c 0 X ! Y ` X ! Y T c 0 X ! Y ` X ! Y 4 0 !Y ` !Y Dp 0 Tp X ! Y ` X ! Y 0 X ! Y ` X ! Y 4p 0 (X o Y ) ! Z ` Y o X ! Z Bp Alt 1p 0 (X o Y ) ! Z ` X o Y ! Z c 0 X ! Y ` X ! Y Tp c 0 X ! Y ` X ! Y 4p

I

B[

F A

p P A p A 4p P P A P A Bp A P B P i(B ^ hP iA) Alt 1p (hP iA ^ hP iB ) B hP i(A ^ B ) c hP iA B A Tp c hP iA B hP ihP iA 4p D

117

P

I

I

Table 6. Axioms in L. Borghuis [1993; 1994; 1998] investigates the formulas-as-types-notion of construction for several normal modal propositional logics based on CPL. Fitch-style natural deduction proofs in these modal logics are interpreted in a second-order {calculus. In this approach, unary type-forming operators are introduced to encode applications of import and export rules for 2 in Fitch-style natural deduction. The operations k^ and k encoding the export and import rules for 2 in the smallest normal modal logic K, for example, ) !r M . Borghuis proves strong satisfy the following reduction rule: k^(kM normalization results for the modal typed {calculi under consideration. However, the term-forming operations used to encode applications of import and export rules for 2 are not provided with a set-theoretic interpretation. 2.

Martini and Masini [1996] consider formulas-as-types for 2-sequent calculi, cf. Section 2.2. They introduce two unary term-forming operations gen and ungen to encode applications of 2{introduction and 2{elimination rules. A strong normalization theorem is proved for the typed {calculus encoding proofs in the 2-sequent calculus for the modal logic S4. However, the typed terms do not receive a set-theoretic interpretation. 3.

Recently, Sasaki [1999] suggested understanding a {term of type 2A as either denoting an element from the domain associated with A, or being unde ned. A term M A 2B would then denote a partial function from DA to DB . Sasaki de nes an extended typed {calculus with various formation rules for obtaining terms of type 2A. Moreover, natural deduction proofs in the extension of the intuitionistic modal logic IntK by the axiom schemata 4.

B

Tc A B 2A and 4c 22A B 2A

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HEINRICH WANSING

are encoded by terms in the extended typed {calculus. Unfortunately, no denotational semantics for this {calculus is developed. The approach that comes closest to the one presented here is Restall's [1999, Chapter 7], who also applies Belnap's display calculus. Introductions of [F ] on the right (left) of the sequent arrow are encoded using a unary operator up (down), lifting (lowering) terms of type A ([F ]A) to terms of type [F ]A (A), just like the operation P ([). Backward-looking possibility is treated quite dierently. Introductions of hP i (in Restall's notation { ) on the right are encoded using a unary type-lifting operation (not to be confused with the structure connective ). Introductions on the left are encoded by a unary term-forming operation turning terms N B , M hP iA into the term let M be x in N of type B . Whereas the term down upN reduces in one step to N , let G be x in N reduces in one step to N [x := G]. Restall proves normalization for the extended typed {calculus under consideration, however, no set-theoretic interpretation of up, down, , and let M be x in is suggested. 5.

In the literature on functional programming there are various proposals for providing an operational semantics of proofs in modal logics, notably in intuitionistic S4. Natural deduction in the framework of Martin-Lof's type theory is considered in [Davis and Pfenning, 2000] and [Pfenning, 2000]. Also, further references can be found in these papers.

3.8 Bi-intuitionistic logic Suppose a connective f1 is introduced in a nite-set-to-formula sequent calculus, whereas another connective f2 is introduced in a formula-to- niteset sequent system. Then the right introduction rules for f1 and the left introduction rules for f2 satisfy the segregation condition. However, if we just combine the sets of rules of both sequent calculi, neither Af1 B nor Af2 B is introduced in the most general context, namely in an arbitrary nite set of formulas, because there are no structure operations like in display logic that allow keeping track of succedent (antecedent) formulas on the left (right) of !. This leads to a problem encountered in formulating an ordinary sequent calculus for bi-intuitionistic logic BiInt, the combination of intuitionistic logic and dual-intuitionistic logic. It can be shown that in the ordinary nite-set-to-formula sequent calculus no binary operation ] is de nable such that ] satis es (in the nite-set-to-formula setting) the dual Deduction Theorem characteristic of coimplication: A ! B i A]B ! ;, see [Gore, 2000]. Bi-intuitionistic logic extends the language of intuitionistic logic by coimplication, the residual of disjunction, and conegation. The syntax of BiInt is given by:

A ::= p j aA j À j A ^ B j A _ B j A B B j A J B:


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In the presence of a falsity constant f , intuitionistic negation a can be de ned by aA := (A B f ), and in the presence of a truth constant t, conegation ` can be de ned by À := (t J A). Bi-intuitionistic logic has a natural algebraic and possible-worlds semantics, see [Rauszer, 1980]. The possible-worlds semantics adds to Kripke models for intuitionistic logic evaluation clauses for conegation and coimplication. A frame is a pair hI; vi, where I a is non-empty set (of states), and v is a re exive and transitive binary relation on I . A structure hI; v; vi is a bi-intuitionistic model if v is a function assigning to every propositional variable p a subset v(p) of I and, moreover, for every t; u 2 I , if t v u and t 2 v(p), then u 2 v(p). Veri cation of a formula A in the model M = hI; v; vi at state t (in symbols M; t j= A) is inductively de ned as follows:

M; t j= p i t 2 v(p); for every propositional variable p; M; t j= aA i for all u 2 I; t v u implies M; u 6j= A M; t j= À i there exists u 2 I; u v t; and M; u 6j= A M; t j= A ^ B i M; t j= A and M; t j= B ; M; t j= A _ B i M; t j= A or M; t j= B ; M; t j= A B B i for all u 2 I; if t v u then M; u 6j= A orM; u j= B ; M; t j= A J B i there is a u 2 I; u v t M; u j= A and M; u 6j= B ; where M; t 6j= A is the (classical) negation of M; t j= A. A formula A is valid in M = hI; v; vi if for every t 2 I , M; t j= A; and A is valid on a frame F = hI; vi if A is valid in every model hF ; vi based on F . A formula A is said to be valid in a class K of models (frames) if A is valid in every model (frame) from K. The axiomatic system HBiInt consists of axiom schemata for intuitionistic logic Int, modus ponens, the rule from A infer aÀ and the following axiom schemata: 1. A B (B _ (A J B ))

2. (A J B ) B `(A B B )

3. ((A J B ) J C ) B (A J (B _ C )) 4. 5. 6. 7. 8.

a(A J B ) B (A B B ) (A B (B J B )) B aA aA B (A B (B J B )) ((B B B ) J A) B À À B ((B B B ) J A)

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THEOREM 66. A formula A in the language of BiInt is valid in the class of all models i A is provable in HBiInt. In the present section, we shall apply the modal display calculus and use a modal translation of BiInt into S4t to give a display sequent calculus for BiInt based on the structure connectives I, , Æ, and , cf. [Gor e, 1995], [Wansing, 1998, Chapter 10]. A direct display sequent system for BiInt not relying on a modal translation has been presented in [Gore, 2000]. Sometimes making a detour via a modal translation may be useful. In [Wansing, 1999], a modal translation into S4 has been used to give a cut-free display sequent calculus for a certain constructive modal logic of consistency, for which no other proof system is known. In view of the possible-worlds semantics for BiInt and the familiar modal translation of Int into S4 (see [Godel, 1933]), a faithful modal translation m of BiInt into S4t can be straightforwardly de ned as follows: 1. m(p) = [F ]p, for every propositional variable p; 2. m(t) = t; 3. m(f ) = f ; 4. m(A]B ) = m(A)]m(B ), ] 2 f^; _g; 5. m(A B B ) = [F](m(A) m(B )); 6. m(A J B ) = hP i:(m(A) m(B )). THEOREM 67. ([Lukowski, 1996]) A formula A in the language of BiInt is provable in HBiInt i m(A) is provable in S 4t. DEFINITION 68. The display sequent system DBiInt consists of (id), (cut), the basic structural rules (1) { (4) of Section 1.3, rules (! t), (t !), (! f ), (f !), (! ^), (^ !), (! _), (_ !), the structural rules from Table 2 and: (! a) (a !) (! `) (` !) (!B)m (B!)m (!J)m (J!)m (p ersistence) (r eflexivity) (t ransitivity)

X ! A ` X ! aA A ! X ` aA ! X X ! A ` X ! À A ! X ` À ! X X Æ A ! B ` X ! A B B X ! A B ! Y ` A B B ! (X Æ Y ) X ! A B ! X ` X ! A J B X ÆA!B ÀJB !X p ! X ` p ! X X ! Y ` X ! Y X ! Y ` X ! Y


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It can be shown that the persistence rule for arbitrary formulas is an admissible rule of DBiInt. This can be used to prove weak completeness of DBiInt with respect to HBiInt. LEMMA 69. In DBiInt, A ! X ` A ! X: Proof.

By induction on A; for example:

A!A A ! A aA ! A aA ! A aA ! A aA ! A aA ! aA aA ! X

aA ! X (cut) A!A A ! A B!B A ! A Æ B B ! A Æ B (A Æ B ) ! A B ! (A Æ B ) (A Æ B ) ! A J B (A Æ B ) ! (A J B ) (A Æ B ) ! (A J B ) (A J B ) ! A Æ B A Æ (A J B ) ! B (A J B ) Æ A ! B A J B ! (A J B ) (A J B ) ! A J B AJB!X (A J B ) ! X

THEOREM 70. In DBiInt ` I ! A i in HBiInt ` A:

(: By induction on proofs in HBiInt. As an example, we here consider only the proof of one axiom schema of HBiInt: Proof.

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B!B

IÆB ! B A!A B JB ! I A B (B J B ) ! (A Æ I) A B (B J B ) ! A Æ I (reflexivity) A B (B J B ) ! I Æ A I Æ (A B (B J B )) ! A I Æ (A B (B J B )) ! A (persistence) (I Æ (A B (B J B ))) ! A I Æ (A B (B J B )) ! aA I ! (A B (B J B )) B aA ): We de ne the translations 1 and 2 from structures into tense logical formulas as in Section 1.3, except that now 1 (A) = 2 (A) = m(A). By induction on proofs in DBiInt, it can be shown that ` X ! Y in DBiInt implies ` 1 (X ) 2 (Y ) in S4t. Therefore, ` I ! A in DBiInt implies ` m(A) in S4t. By the previous theorem we have ` A in HBiInt. THEOREM 71. Strong cut-elimination holds for DBiInt. is a proper display calculus. As to the ful llment of condition C8, the derivation on the left, for example, reduces to the derivation on the right, using contraction: Proof. DBiInt

X ! A B ! X Y Æ A ! B X ! A J B AJB!Y X ! Y

Y ÆA !B X ! A A ! Y Æ B X ! Y Æ B Y Æ X ! B B ! X Y Æ X ! X X ! Y Æ X X Æ X ! Y X ! Y X ! Y

COROLLARY 72. DBiInt

DBiInt and DS 4t.

[ DS 4t is a conservative extension of both

As in Section 3.1, let for modal formulas A the translations i (i = 1, 2) be de ned by i (A) = A. LEMMA 73. In DBiInt [ DS 4t, (i) ` X ! 1 (X ) and (ii) ` 2 (X ) ! X.


123

Both (i) and (ii) are proved simultaneously by induction on X . In particular we have to verify that for every formula of the language of BiInt, ` A ! m(A) and ` m(A) ! A. But this is the case, see for example: Proof.

A ! m(A) m(B ) ! B m(A) m(B ) ! A Æ B (A Æ B ) ! (m(A) m(B )) (A Æ B ) ! :(m(A) m(B )) (A Æ B ) ! hP i:(m(A) m(B )) (A Æ B ) ! hP i:(m(A) m(B )) hP i:(m(A) m(B )) ! A Æ B hP i:(m(A) m(B )) ! B Æ A hP i:(m(A) m(B )) Æ A ! B A J B ! hP i:(m(A) m(B ))

m(A) ! A B ! m(B ) m(A) ! A Æ m(B ) B Æ m(A) ! m(B ) A Æ m(A) ! m(B ) B ! m(A) m(B ) A ! m(A) m(B ) (m(A) m(B )) ! B (m(A) m(B )) ! A :(m(A) m(B )) ! B :(m(A) m(B )) ! A B ! :(m(A) m(B )) :(M(A) m(B )) ! A J B :(M(A) m(B )) ! (A J B ) hP i:(M(A) m(B )) ! A J B

THEOREM 74. In DBiInt ` X ! Y i 1 (X ) 2 (Y ) is valid on every frame (understood as a frame for S 4t). Proof. ()): This follows by induction on proofs in DBiInt. ((): Suppose that 1 (X ) 2 (Y ) is valid on every frame. Hence 1 (X ) 2 (Y ) is a theorem of S4t and hence ` 1 (X ) ! 2 (Y ) in DBiInt [ DS 4t. By the previous lemma, ` X ! Y in DBiInt [ DS 4t and by Corollary 72, ` X ! Y in DBiInt. One advantage of the translation-based sequent system DBiInt is that by abandoning combinations of the structural rules (p ersistence); (r eflexivity), and (t ransitivity), one obtains cut-free sequent calculus presentations of the subsystems of BiInt that arise from giving up the corresponding semantic requirements: persistence of atomic information, re exivity, and transitivity of the relation v. Also seriality of v, a weakening of re exivity, is expressible by a purely structural sequent rule, see condition D0 in Table 4.

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4 INTERRELATIONS AND EXTENSIONS While the existence of a rich inventory of types of proof systems for modal and other logics may be welcomed, for instance, from the point of view of designing and combining logics, there also exists the need of comparing dierent approaches and investigating their interrelations and their relative advantages and disadvantages. Mints [1997], for example, presents cutfree systems of indexed sequents for certain extensions of K and de nes a translation of these sequent systems into equivalent display calculi. In this nal section a translation of multiple-sequent systems into higher-arity sequent systems and a translation of hypersequents into display sequents are de ned, showing that multiple-sequent systems can be simulated within higher-arity proof systems and that the method of hypersequents can be simulated within display logic. Moreover, one interesting aspect of extending the sequent-style proof systems for modal and temporal propositional logics to sequent calculi for modal and temporal predicate logics is considered, namely avoiding the provability of the Barcan formula and its converse. We also brie y refer to recent work on display calculi for extended modal languages. Finally, the relation between display logic and Dunn's Gaggle Theory is pointed out.

4.1 Translation of multiple-sequent systems The translation in Section 2.4 reveals a straightforward relation between Indrzejczak's multiple-sequent systems and higher-arity sequent systems for modal logics. The intended meaning of the multiple-sequents can be expressed by four-place sequents using a translation : ( ! ) = Æ( ) !;; Æ() W ( 2! ) = Æ( ) !; Æ() ; V ( 3! ) = ; !:; Æ( ) Æ(): If S is a multiple-sequent system, then let (S) be the result of the translation of the rules of S. Let denote the translation of four-place sequents into modal formulas stated in Section 2.3. If s1 ; : : : ; sn =s is a rule of MC, then ((s1 )); : : : ; ((sn ))= ((s)) is validity preserving in C. For the rule [T R], for instance, we have (([T R])) =

Æ() 2 VÆ( ) 3: Æ( ) : Æ() V

W

W

=

Æ() 2W Æ( ) 3 Æ( ) Æ( ) V

W

V

Moreover, (RR) is derivable and CPL is contained in (MC). Hence, OBSERVATION 75. The system (MC ) is sound and complete with respect to C : ` ! in (MC ) i (( ! )) is valid in C .


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The translation is also faithful for the extension of MC by the rules [nec], [D], [T ], and [4] and extensions of C by the necessitation rule and the axiom schemata D, T and 4.

4.2 Translation of hypersequents In order to characterize various non-classical logics by means of hypersequential calculi, Avron [1996] uses dierent semantical readings of hypersequents. Basically a distinction can be drawn between interpreting the sequent arrow of a component in a hypersequent as material implication or as a constructive implication not de nable in terms of Boolean negation and disjunction. This dierence in interpretation requires dierent translations of hypersequents into display sequents. If the sequent arrow is interpreted constructively, a suitable translation may, for example, exploit a faithful embedding of the logic under consideration into a normal modal or temporal logic. In such a case, the sequent arrow is interpreted as strict material implication. In [Wansing, 1998, Chapter 11] translations of hypersequents into display sequents are de ned that simulate hypersequents in Avron's hypersequential calculi GL3, GS5, and GLC for Lukasiewicz 3valued logic L3, S5, and Dummett's superintuitionistic logic LC, also called Godel-Dummett logic. We shall here consider only the translations suitable for S5 and LC. The treatment of GL3 is slightly more involved, because L3 comprises connectives from dierent `families' of logical operations. To deal with this composite character of L3 in display logic, the structure connective Æ is replaced by two binary structure operations Æc and Æi , see [Wansing, 1998]. If = fA1 ; : : : ; An g, let = fA1 ; : : : ; An g. Since Æ is assumed to be associative and commutative, we may put (Æ) = A1 Æ : : : Æ An . If = ;, let = (Æ) = I. Recall the notion of hypersequent from Section 2.5. DEFINITION 76. The translation 0 of ordinary sequents into display structures is de ned by

0 ( ! ) = ((Æ ) Æ (Æ )); and the translation of non-empty hypersequents into display sequents is de ned by

(s1 j : : : j sn ) = I ! 0 (s1 ) Æ : : : Æ 0 (sn ): THEOREM 77. For every hypersequent H ,

GS 5.

` (H ) in DS 5 i ` H in

In the hypersequential system GLC the components of a hypersequent are restricted to be ordinary Gentzen sequents with at most a single conclusion. Dummett's LC is the logic of linearly ordered intuitionistic Kripke

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models. An axiomatization of LC is obtained from an axiomatization HInt of Int by adding the axiom schema (A B B ) _ (B B A). It is well-known that the modal translation m de ned in Section 3.8 (restricted to the language of intuitionistic logic, i.e. the language of LC) is a faithful embedding of LC into S4.3, the logic of linearly ordered modal Kripke models. THEOREM 78. For every formula A in the language of LC , ` A in LC i ` m(A) in S 4:3. DEFINITION 79. The translation 0 of a single-conclusion ordinary sequent s = A1 ; : : : ; An ! B is de ned by

0 (s) = (A1 Æ (A2 Æ : : : (An Æ B ) : : :)): If s = A1 ; : : : ; An ! ;, then (s) = (A1 Æ (A2 Æ : : : (An Æ I) : : :)): If s = ; ! B , then (s) = (I Æ B ), and if s = ; ! ;, (s) = (I Æ I). The translation of hypersequents with at most single-conclusion components into display sequents is de ned by (s1 j : : : j sn ) = I ! 0 (s1 ) Æ : : : Æ 0 (sn ): THEOREM 80. For every hypersequent H with at most single-conclusion components, ` (H ) in DLC i ` H in GLC .

4.3 Predicate logics and other logics Modal predicate logic is still a largely unexplored area. As to sequent systems for modal predicate logics, one notorious problem is providing introduction rules for the modal operators and the quanti ers such that neither the Barcan formula (BF) 8x2A 28xA nor its converse (BFc) 28xA 8x2A are provable on the strength of only these rules. It is wellknown that (BF) corresponds to the assumption of constant domains and (BFc) to the persistence of individuals along the accessibility relation; cf. for example [Fitting, 1993]. One way of avoiding the provability of the Barcan formula and its converse is described in [Wansing, 1998, Chapter 12]. The idea is to exploit the well-known similarity between 2 [3] and 8x [9x] to develop display introduction rules for 8x [9x]; i.e., instead of thinking of the modal operators as quanti ers, one thinks of the quanti ers as modal operators, see also [Andreka et al., 1998]. The addition of quanti ers to display logic is brie y discussed in [Belnap, 1982]: Quanti ers may be added with the obvious rules:

`X (UQ) (xAa )Ax ` X

X ` Aa X ` (x)Ax

provided, for the right rule, that a does not occur free in the conclusion. : : : The rule for the existential quanti er would be


127

dual. : : : [A]s yet this addition provides no extra illumination. I think that is because these rules for quanti ers are \structure free" (no structure connectives are involved; : : :). One upshot is that adding these quanti ers to modal logic brings along Barcan and its converse : : : willy-nilly, which is an indication of an unre ned account; alternatives therefore need investigating. [Belnap, 1982, p. 408 f.] Using the structure-independent rules (UQ), we would have the following proofs of (BF) and (BFc):

A!A 2A ! A (UQ) 8x2A ! A 8x2A ! A (UQ) 8x2A ! 8xA 8x2A ! 28xA I Æ 8x2A ! 28xA I ! 8x2A 28xA

A ! A (UQ) 8xA ! A 28xA ! A 28xA ! A 28xA ! 2A (UQ) 28xA ! 8x2A I Æ 28xA ! 8x2A I ! 28xA 8x2A

Structure-dependent introduction rules for 8x and 9x are, however, available. For every binary relation Rx on a non-empty set S of states, we may de ne the following functions on the powerset of S :

8xA := fa j 8b (aRxb ) b 2 A)g; 9xA := fa j 9b (bRx a & b 2 A)g; 8xA := fa j 8b (bRxa ) b 2 A)g; 9xA := fa j 9b (aRx b & b 2 A)g: We then have

9xA B i A 8xB;

9xA B i A 8x B;

and for every individual variable x, we may introduce a structure connective x, which in succedent position is to be understood as 8x and in antecedent position as a backward-looking existential quanti er 9x. Semantically, what is required to account for these quanti ers is a generalization of the Tarskian semantics for rst-order logic, see [Andreka et al., 1998]. Let M be any rstorder model and let , , : : : range over variable assignments in M. Tarski's truth de nition for the existential quanti er is:

M j= 9xA[]

i for some assignment on jMj: =x and M j= A[ ];

where =x means that and dier at most with respect to the object assigned to x. In the more general semantics the concrete relations =x between variable assignments are replaced by abstract binary relations Rx of `variable update' between `states' , , , : : : from a set of states S .

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Assuming an interpretation of atoms containing free variables, the truth de nition for the existential quanti er becomes:

M; j= 9xA

i for some 2 S : Rx and M; j= A

Thus, to every individual variable x there is associated a transition relation Rx on states. The resulting minimal predicate logic, KFOL, is nothing but the !-modal version of the minimal normal modal logic K. In order to obtain an axiomatization of KFOL, one may just take any axiomatic presentation of K and replace every occurrence of 3 and 2 by one of 9x and 8x, respectively. The basic structural rules for the structure connective x are:

X ! x Y

a` x X ! Y: In analogy to the case for 2 and 3, we obtain the following structuredependent introduction rules for 8x and 9x: (! 8x) xX ! A ` X ! 8xA (! 9x) X ! A ` x X ! 9xA (8x !) A ! X ` 8xA ! x X (9x !) x A ! X ` 9xA ! X

In addition to these introduction rules we need further structural assumption in order to take care of the necessitation rules in axiomatic presentations of normal modal and tense logics: (MN x)

I

! X ` I ! xX

X ! I ` X ! xI

The structural account of the quanti ers as modal operators blocks the above proofs of (BF) and (BFc). In the presence of additional structural sequent rules, however, these schemata become derivable: OBSERVATION 81. BF and BFc correspond to the structural rules rBF X ! x Y ` X ! x Y ; rBFc X ! x ` X ! x Y: The apparatus of display logic has also been applied to other extensions of normal modal propositional logic. A result of Kracht concerns the undecidability of decidability of display calculi. Consider the fusion or ìndependent sum' of Kf and Kf, i.e. the bimodal logic Kf Kf of two functional accessibility relations R1 , R2 . In this system there are two pairs of modal operators, say, [1], h1i and [2], h2i each satisfying the D and the Alt1 axiom schemata. The structural language of sequents for this logic comes with two unary operations 1 and 2 satisfying the display equivalence

i X ! Y a` X ! i Y; i = 1; 2: Clearly, Kf Kf has many properly displayable extensions. Using an encoding of Thue-processes into frames of Kf Kf, Grefe and Kracht [1996] have proved a theorem about the undecidability of decidability.


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THEOREM 82. (Grefe and Kracht) It is undecidable whether or not a dis-

play calculus is decidable.

According to Kracht, Theorem 82 indicates a serious weakness of display logic. In any case, the theorem provides insight into the expressive power of display logic; it shows that the subformula property and the strong cut-elimination theorem for displayable logics fail to guarantee decidability. Undecidability of the decidability of properly displayable extensions of Kf Kf is a remarkable property of this particular family of bimodal logics, but is not a defect of the modal display calculus, at least insofar as the proof of the theorem also shows that it is undecidable whether or not a nite axiomatic calculus is decidable. Would it be desirable to have a proof-theoretic framework in which only decidable logics can be presented? A weakness of display logic is that it does not lend itself easily to obtain decidability proofs. Restall [1998] uses a display presentation to prove, among other things, decidability of certain relevance logics which are not known to have the nite model property. In [Wansing, 1998, Chapter 6] display logic is used to prove decidability of Kf and deterministic dynamic propositional logic without Kleene star. Display calculi for logics with relative accessibility relations can be found in [Demri and Gore, 2000] and for nominal tense logics in [Demri and Gore, 1999]. In both cases the calculi are obtained using modal translations.

4.4 Gaggle Theory The generality of display logic has been highlighted by Restall [1995], who observes a close relation between display logic and J. Michael Dunn's Gaggle Theory [1990; 1993; 1995]. The relation between gaggle theory and display logic has also been investigated and worked out by Gore [1998]. A gaggle is an algebra G = hG; ; OP i, where is a distributive lattice ordering on G, and OP is a founded family of operations. The latter means that there is an f 2 OP such that for every g 2 OP, f and g satisfy the abstract law of residuation, see Section 3. If one only requires that is a partial order, and every f 2 OP has a trace, then G is said to be a tonoid. Restall de nes the notion of mimicing structure. An n-place logical operation f mimics antecedent structure if there is a possibly complex n-place structure connective ] such that the following rules are admissible:

s = ](A1 ; : : : ; An ) ! X ` f (A1 ; : : : ; An ) ! X C (X1 ; A1 ) : : : C (Xn ; An ) ` ](A1 ; : : : ; An ) ! f (A1 ; : : : ; An ) where ](A1 ; : : : ; An ) is an antecedent part of s, C (Xi ; Ai ) = Xi ! Ai , if Ai is an antecedent part of ](A1 ; : : : ; An ), and C (Xi ; Ai ) = Ai ! Xi , if Ai is a succedent part of ](A1 ; : : : ; An ). Dually, f mimics succedent structure

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if there is a possibly complex n-place structure connective ] such that the following rules are admissible:

s = X ! ](A1 ; : : : ; An ) ` X ! f (A1 ; : : : ; An ) C (X1 ; A1 ) : : : C (Xn ; An ) ` f (A1 ; : : : ; An ) ! ](A1 ; : : : ; An ) where ](A1 ; : : : ; An ) is a succedent part of s, C (Xi ; Ai ) = Xi ! Ai , if Ai is an antecedent part of ](A1 ; : : : ; An ), and C (Xi ; Ai ) = Ai ! Xi , if Ai is a succedent part of ](A1 ; : : : ; An ). THEOREM 83. (Restall [1995]) If a logical operation f in a display calculus presentation D of a logic mimics structure, then f is a tonoid operator on the Lindenbaum algebra of . If every logical operation of D mimics structure, mutual provability is a congruence relation and has an algebraic semantics. Dunn's representation theorem for tonoids supplies also a Kripke-style relational semantics. 5 APPENDICES

5.1 Appendix A The proof of Theorem 23 takes its pattern from the proof of strong normalization for typed -calculus (see for instance [Hindley and Seldin, 1986, Appendix 2]) and follows the argument given in [Roorda, 1991, Chapter 2, reprinted in [Troelstra, 1992]]. This proof has been extracted from the proof of strong cut-elimination for classical predicate logic in [Dragalin, 1988, Appendix B]. Suppose that is a proof containing an application of cut. A (one-step) reduction of is the proof resulting by applying a primitive reduction to a subproof of . If reduces to , this is denoted by > (or < ). is said to be reducible i there is a such that > . LEMMA 84. If a proof cannot be reduced, then it is cut-free. Since the case distinction in the de nition of primitive reductions is exhaustive, every proof that contains an application of cut is reducible. Proof.

DEFINITION 85. We inductively de ne the set of inductive proofs. a b

c

Every instantiation of an axiomatic rule is an inductive proof. If ends in an inference inf dierent from cut, and every premise si of inf has an inductive proof i in , then is inductive. = 1(3)2

cut

is inductive, if every such that > is inductive.


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LEMMA 86. If is inductive, and > , then is inductive. By induction on the construction of . If is inductive by a, then no reduction can be performed. If is inductive by b, then every reduction on takes place in the i 's, which are inductive. Hence, by the induction hypothesis, is inductive due to b. If is inductive by c, then is inductive by de nition. Proof.

DEFINITION 87. Let be an inductive proof. The size ind() of is inductively de ned as follows (the clauses correspond to those in the previous de nition): a

ind() = 1;

b

ind() =

P

ind() =

P

c

i ind(i ) + 1;

< ind() + 1.

A proof is said to be strongly normalizable i every sequence of reductions starting at terminates. LEMMA 88. Every inductive proof is strongly normalizable. By induction on ind(). If ind() = 1, no reduction is feasible. If is inductive by b, then every reduction is in the premises i , and we can apply the induction hypothesis. If is inductive by c, then every proof to which reduces is inductive and therefore every such proof is strongly normalizable, by the induction hypothesis. But then is also strongly normalizable. Proof.

LEMMA 89. Let be an inductive proof and let inf be the nal inference of . If > 0 by reducing a proof j of a premise sequent of inf, then ind() > ind(0 ). By induction on ind(). If ind() = 1, then cannot be reduced. Whence is inductive by b or c. If is inductive by c, then by de nition, ind() > ind(0 ). If is inductive by b, then j is inductive by de nition. If j is inductive by a, it cannot be reduced. If j is inductive by b, then the reduction of j to 0j takes place in the proof of some premise sequent of the nal inference of j . By the induction hypothesis, ind(j ) > ind(0j ). Hence ind() > ind(0 ). If j is inductive by c, then by de nition, ind(j ) > ind(0j ) and thus ind() > ind(0 ). Proof.

LEMMA 90. Suppose ends in an application inf of cut, and 1 and 2 are the proofs of the premises of inf. If 1 and 2 are inductive, then so is .

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HEINRICH WANSING

We must show that every < is inductive. For this purpose, we de ne two complexity measures for : r(), the rank of , and h(), the height of . r() is the number of symbols in the cut-formula. h() is de ned by: h() = ind(1 ) + ind(2 ): Proof.

We use induction on r() and, for xed rank, induction on h(). Case 1. is obtained by reduction in 1 or 2 , say 1 > 01 . It follows from Lemma 89 that ind(01 ) < ind(1 ). Then h() < h(). Since 1 and 2 are inductive, by Lemma 86, has inductive premises, and by the induction hypothesis for h(), is inductive. Case 2. is obtained by reducing inf. Then this reduction was either a principal or a parametric move. Principal move.

Case 1. Since proves one of (1) or (2), is inductive by assumption. Case 2. Since for every new proof 0 ending in an application of cut, r() > r(0 ), is inductive by the induction hypothesis for r(). Suppose A is parametric in the inference ending in (1) (the case for (2) is analogous). If the tree of parametric ancestors of the displayed occurrence of A in (1) contains at most one element Au that is not parametric in inf, we have Figure 1, and we may assume that there is no application of cut on the path from (1) to Z ! A. Parametric move.

1 1 : 0 Let = Z ! A 2 and 00 = Z ! A Z!Y Consider and 0 . Clearly, r() = r(0 ), hence we use induction on the height. Since both 1 and 00 are inductive by b, ind(00 ) < ind(1 ): Hence h(0 ) < h(). By the induction hypothesis for h(), 0 is inductive, and thus is inductive by de nition. If the primitive reduction of to requires cutting with 2 more than once, analogously every new 0 and hence can be shown to be inductive. If the tree of parametric ancestors of the displayed occurrence of A in (1) contains more than one element Au that is not parametric in inf, = l or = lr . Since for every new proof 0 ending in an application of cut, r() > r(0 ), is inductive by the induction hypothesis for r(). COROLLARY 91. Every proof is inductive. Now Theorem 23 follows by Lemma 88 and Corollary 91, and cut is an admissible rule by Lemma 84.


133

5.2 Appendix B To prove completeness of HIntKt with respect to the class of all temporal models we shall adopt completely standard methods as applied, for example, in [Schutte 1969, pp. 48{51]. Suppose and are nite sets of formulas, where is empty or a singleton, and let p be a new propositional variable not already in Atom. The formula B is de ned as follows: B =

B B if 6= ;; = fB g if = ;; = fB g B p if 6= ;; = ; tBp if = = ;

8 V > > < V > > :

tBB

The pair (; ) is said to be consistent if B is unprovable in HIntKt based on L+ = L [ fpg. In what follows, let A 2 L. Let sub (A) denote the nite set of all subformulas of A. If C = (A1 B : : : (An 1 B An ) : : :), S then sub (fC g) = ( 1in sub (Ai )) n fpg; sub (;) = ;. The pair (; ) is called A-complete, if [ sub ( ) = sub (A). A pair ( ; ) is called an expansion of (; ), if is a nite superset of , and either = or has the shape (A1 B : : : (An 1 B An ) : : :) and n > 1. LEMMA 92. If (; ) is consistent, then so is ( [ fAg; ) or (; fA B B g), where B = p if = ;, and = fB g otherwise. Suppose neither ( [ fVAg; ) nor (; fA B B g) are consistent. V Then both ( ^ A) B B and B (A B B ) are derivable in HIntKt V based on L+ . But then also B B is derivable, and hence (; ) is not consistent; a contradiction. Proof.

COROLLARY 93. Every consistent pair (; ) such that ; sub ( ) sub (A) can be expanded to an A{complete consistent pair. Let sub (A): Then is said to be A{designated, if some A{complete pair (; ), where sub ( ) = sub (A) n is consistent. By soundness of HIntKt based on L+ , the formula t B p fails to be provable. Therefore (;; ;) is consistent. By the previous corollary, for every formula A, (;; ;) can be expanded to an A{complete consistent pair. Hence, for every A, the set D(A) of all A{designated subsets of sub (A) is non-empty. LEMMA 94. If C 2 sub (A), then C belongs to an A{designated set i B fC g is provable in HIntKt. If C 2 , then clearly B fC g is provable in HIntKt. If C 62 , then C 2 sub (A) n , and since is A{designated, (; fC g) is consistent. In other words, B fC g is not provable in HIntKt. Proof.

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HEINRICH WANSING

DEFINITION 95. For every formula A, the structure RTA ; vA i is called the canonical model for A if

WA RIA uRTAt A v (p; u) = 1

MA

= hW A ; RIA;

= D(A) = i [F ]B 2 u implies B 2 t i p 2 u:

As we have seen, the set W A is non-empty, and it can easily be shown that MA is indeed a temporal model. LEMMA 96. Let u; t 2 D(A). For every formula B , ([F ]B 2 u implies B 2 t) i for every formula C , (C 2 u implies hP iC 2 t). First, suppose (i) for all B , [F ]B 2 u implies B 2 t but (ii) there is a formula C 2 u such that hP iC 62 t. By (i), [F ]hP iC 62 u. By the previous lemma, u B [F ]hP iC is not provable in HIntKt. Since C B [F ]hP iC is provable, also u B C fails to be provable. But then, by the previous lemma, C 62 u, which contradicts (ii). Suppose now (iii) for all C , C 2 u implies hP iC 2 t but (iv) there is a formula [F ]B 2 u such that B 62 t. By (iii), hP i[F ]B 2 t, and by the previous lemma, t B hP i[F ]B is provable in HIntKt. Since hP i[F ]B B B is provable, also t B B is provable. Hence B 2 t, a contradiction with (iv). Proof.

LEMMA 97. (Veri cation Lemma) Consider MA = hW A ; RIA ; RTA; vA i. For every C 2 sub (A) and every u 2 D(A), MA ; u j= C i C 2 u. V

By induction on C . We shall consider only V two cases. Let u denote t, if u = ;; and note that for all BV2 u, ` hP i u B hP iB . Hence for every u; t 2 W A we have: (*) if hP i u 2 t, then for every B 2 u, hP iB 2 t. 1. C = [F ]B . ): Suppose [F ]B 62 u. This is the case i Proof.

u B [F ]B cannot be proved i hP i VVu B B cannot be proved i (hP i u; fB g) is consistent V i (9t 2 D(A)) u t; hP i u 2 t; B 62 t by Corollary 93 only if (9t 2 D(A)) uRTA t; B 62 t by Lemma 96 and (*) i M; u 6j= [F ]B by the ind. hyp. V

(: Suppose [F ]B 2 u. Then for all t 2 W A , uRTAt implies B 2 t. induction hypothesis, Mc ; u j= [F ]B:

By the


135

2. C = hP iB . ): Suppose MA ; u j= hP iB: This is the case i (9t 2 W A ) tRTA u and MA ; t j= B only if (9t 2 W A ) (B 2 t implies hP iB 2 u); B 2 t by Lem. 96; ind. hyp. only if hP iB 2 u: (: Suppose hP iB 2 u. Put t0 := fC j hP iC 2 ug. Clearly, the pair (t0 ; ;) is consistent. Hence V (9t 2 W A ) t0 t; t0 2 t by Corollary 85 only if (9t 2 W A ) tRTA u and MA ; t j= B by Lemma 88 and the ind. hyp. i MA ; u j= hP iB COROLLARY 98.

in HIntKt.

If A is valid in every temporal model, then A is provable

Suppose A is not provable in HIntKt. Then the pair (;; fAg) is consistent, and, by the previous corollary, there exists a u 2 D(A) such that A 62 u. By the Veri cation Lemma, MA; u 6j= A: Proof.

COROLLARY 99. Proof.

HIntKt

is decidable.

This follows easily by the fact that sub (A) is nite.

5.3 Appendix C In order to prove strong normalization for t , we shall follow R. de Vrijer's [1987] proof of strong normalization for typed {calculus with pairing and projections satisfying surjective pairing. Let h(M ) (the height of the reduction tree of M ) be the length of a reduction sequence of M that has maximal length. DEFINITION 100. M A 2 Term is said to be computable i 1. sn(M ); 2. if A = B B C , M r N1 , and N2B is computable, then (N1 ; N2)C is computable; 3. if A = B ^ C and M r hN1 ; N2 i, then N1B ; N2C are computable; 4. if A = [F ]B and M r P N , then N B is computable; 5. if A = hP iB and M r S N , then N B is computable.

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HEINRICH WANSING

The set of all computable terms is denoted by C. By this de nition, every computable term is strongly normalizable. The aim is to show that every term is computable. LEMMA 101.

(a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k)

If M 2 C and M r N , then N 2 C. C is closed under repeated formation of application terms (M; N ). If x 2 V , then x 2 C. If for every N A 2 C, (M (A B) ; N ) 2 C; then M 2 C: If (M A^B )0 2 C and (M A^B )1 2 C, then M 2 C. If N1 , N2 2 C, and G 2 C, for every G such that hN1 ; N2 i !r G, then hN1 ; N2 i 2 C. If N 2 C and G 2 C, for all G such that (N )i !r G, then (N )i 2 C. If N 2 C, and G 2 C, for all G such that P N !r G, then P N 2 C. If N 2 C, and G 2 C, for all G such that S N !r G, then S N 2 C. If N 2 C, and G 2 C, for all G such that [N !r G, then [N 2 C. If N 2 C, and G 2 C, for all G such that \N !r G, then \N 2 C. B

(a): By induction on h(M ). (b) By re exivity of r and Clause 2 in the de nition of C. (c): By induction on A 2 T . If A = B B C , the claim follows by (b). (d): If for every N A 2 C, (M; N ) 2 C, then sn (M ), since by (c) and the assumption (M; xB ) 2C. Now suppose M r N1, N2 2 C, and for every N , (M; N ) 2 C. Then (M; N2 ) (N1 ; N2 ) and, by (a), (N1 ; N2 ) 2 C. Thus M 2 C. (e): Since sn((M )i ), also sn(M ). Suppose M r hN0 ; N1 i. Then (M )i r (hN0 ; N1 i)i !r Ni . Since (M )i 2 C and C is closed under r , also Ni 2 C. (f): Obviously, for every M , s n(M ) i s n(N ), for each N such that M !r N . Moreover, suppose that hN1 ; N2 i r hG1 ; G2 i. This is the case i hN1 ; N2 i hG1 ; G2 i or there is a term M such that hN1 ; N2 i !r M , and M r hG1 ; G2 i. In both cases G1 ; G2 2 C. (g): By induction on the type A of (N )i . If A is atomic, Clauses 2{5 in the de nition of C hold trivially. A = hP iB : Suppose (N )i r S M . If N hM1 ; M2 i, then (N )i !r Mi , and Mi 2 C. If S M 6 Mi , then Mi r S M , and S M 2 C, by closure of C under r . If N 6 hM1 ; M2 i, then there is a term M 2 C such that (N )i !r M and M r S M . In each subcase, M 2 C. The cases A = [F ]B and A = B ^ C are analogous. If A = B B C , we may use closure of C under application. (h): Suppose P N r P G. This holds i N G or there is a term M such that P N !r M and M r P G. In both cases G 2 C . (i): Analogous to (h). (j): By induction Proof.


137

on the type A of [N . The only interesting case is A = [F ]B . Suppose [N r P M . If N P N1 , then [N !r N1 and N1 2 C. If P M 6 P N1, then N1 r P M , and P M 2 C. In each case M 2 C. (k): Analogous to (j).

THEOREM 102. If M

2 Term is {free, then M 2 C.

By induction on M . (1): M is a variable: Lemma 101 (c). (2) M (N1 ; N2 ): Lemma 101 (b) and the induction hypothesis. (3) M = hN1A ; N2B i: In view of Lemma 101 (f), it is enough to show that G 2 C, for every G such that hN1 ; N2 i !r G. There are tow subcases. (i): N1 (G)0 and N2 (G)1 . Then the claim follows by (e). (ii): G hN1 ; N i and N2 !r N or G hM ; N2 i and N1 !r M . We may use induction on h(N1 ) + h(N2 ). (4) M (N )i . In view of Lemma 101 (g), it is enough to show that G 2 C, for every G such that M !r G. There are tow cases. (i) N hN0 ; N1 i and G Ni . Then we may use the induction hypothesis. (ii) G (N )i , N !r N , and we may use induction on h(N ). (5) M P N : In view of Lemma 101 (h), it is enough to show that G 2 C, for every G such that M !r G. If M !r G, then G P N , N !r N , and we may use induction on h(N ). (6) M S N : Analogous to (5), using Lemma 101 (i). (7) M \N : Given Lemma 101 (k), it suÆces to show that G 2 C, for every G such that M !r G. There are two cases. (i) N S G1 and G G1 . Then we may use the induction hypothesis. (ii) G \N , N !r N , and we may use induction on h(N ). (8) M [N : Analogous to (7), using Lemma 101 (j). Proof.

Strong normalizability of all terms is derived from computability of all terms under substitution. DEFINITION 103. M A 2 Term is said to be computable under substitution i any substitution of free variables in M by computable terms of suitable type results in a computable term. Let Cs denote the set of all terms computable under substitution. THEOREM 104. Every t {term M is computable under substitution. By induction on M . For term variables the claim is obvious. Moreover, since C is closed under application, Cs is also closed under application. If M hN1 ; N2 i, M (N )i , M P N , or M S N , the claim follows by the induction hypothesis. If M xA N , it must be show that xN 2 Cs if N 2 Cs . Suppose that xN is the result of substituting a computable term for a free variable in xN , and suppose that GA is a computable term such that (M; G) does not have a type B B C . Then, by Lemma 101 (f) { (k), ((xN )G) 2 C, if for every term H , ((xN )G) !r H implies H 2 C. Since by assumption N 2 Cs , we have N 2 C. Therefore we may use induction on h(N ) + h(G) to show that ((xN )G) 2 C. There are three Proof.

138

HEINRICH WANSING

Term

DIntKt + DIntKt

DIntKt

u elim

?u

f f

-u norm

- ?u

+T erm

T erm

Figure 2. Normalization as a homomorphic image of proof-simpli cation. subcases. (i) H N [x := G] and x 2 fv (N ). Then N 2 C implies H 2 C. (ii) H N [x := G] and x 62 fv (N ). Then H N 2 C. (iii) H is obtained from ((xN )G) by executing one reduction step either in N or G. In this case we may use the induction hypothesis. COROLLARY 105. If M is a t {term, then M is strongly normalizable.

5.4 Appendix D It has to be shown that f is a homomorphism from A to B, i.e., for every 2 + DIntKt, we have f (elim ()) = norm (f ()), see Figure 2. The proof is by induction on . If the rule applied to obtain the conclusion sequent sc of is an axiomatic sequent A ! A, then f (elim ()) = f (), and f () is a nf. If the rule applied to obtain sc is such that the term construction step associated with it cannot generate a redex, we may apply the induction hypothesis. We shall consider the remaining cases. 0 Case 1. = A o B ! X A^B !X A redex could be generated if the free variables xA , yB in the construction of A o B ! X occur in the context hx; yi. But then X = A ^ B , A o B ! X has been derived from fA ! A; B ! B g, and elim() = A ^ B ! A ^ B . The claim holds, since h(v1A^B )0 ; (v1A^B )1 i !r v1A^B . 0 Case 2. = X ! A o B X!ABB A redex could be generated if the free variable xA in the construction of X ! A o B occurs in the context (N A B ; xA ). But then X = A B B , B


139

X ! A o B has been derived from fA ! A; B ! B g, and elim() = A B B ! A B B . The claim holds, since v1A (v1A B ; v1A ) !r v1A B . B

B

1 2 Case 3. = X ! A A ! Y X!Y Suppose the exhibited application of cut in is not principal. If this application is reduced in one step, either the f {images of the resulting proof and are the same, or some principal cuts have been performed on subformulas of A. Thus, there are ve remaining cases to be considered. Case 3.1 (t):

!X I!t t!X I!X I

is converted into

#f

I

!X #f

M

v1t M

!r

M

M

Case 3.2 (^): 3 AoB !Z X !A A!B oZ 1 2 3 X !BoZ X !A Y !B A o B ! Z is conv. into 2 X oB ! Z X oY ! A^B A^B !Z Y !B B !X oZ X oY !Z Y !X oZ X oY !Z 1

#f

#f N (xA ; yB ) N 1 N (M1 ) N (M1 ) M2B N (M1 ) N (M1 ; M2 ) N (M1 ; M2 )

MA M1A M2B N (xA ; yB ) A hM1 ; M2 i N ((z ^B )0 ; (zA^B )1 ) N ((hM1 ; M2 i)0 ; (hM1 ; M2 i)1 )

r

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HEINRICH WANSING

Case 3.3 (B): 1 2 3 X !AoB Y ! A B !Z X !A BB A BB !Y oZ X ! Y oZ

1 X ! AoB 3 XoA!B B !Z X oA !Z 2 is conv. into Y! A A!X oZ Y !X oZ X oY !Z X !Y oZ

#f

#f M B (xA ) MB

M B (xA ) N1A N2 (yB ) xA M N2 (z ABB ); N1 ) N2 (xA M; N1 )

!r

N2 (yB ) N2 (M ) N2 (M ) N1A N2 (M (xA )) N2 (M (N1 )) N2 (M (N1 )) N2 (M (N1 ))

Case 3.4 ([F ]): 1 2 1 2 X ! A A!Y X ! A A !Y X ! Y X ! [F ]A [F ]A ! Y is converted into X ! Y X ! Y

#f

#f

M A N (xA ) MA N (xA ) N (M ) P M N ([y[F ]A) N ([P M ) !r N (M ) Case 3.5 (hP i): analogous to the previous case. ACKNOWLEDGEMENT I would like to thank Dov Gabbay for inviting me to contribute this chapter to the Second Edition of the Handbook of Philosophical Logic.

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[Demri and Gore, 2000] S. Demri and R. Gore, Display calculi for logics with relative accessibility relations, Journal of Logic, Language and Information 9 (2000), 213{236. [Dosen, 1985] K. Dosen, Sequent-systems for modal logic, Journal of Symbolic Logic 50 (1985), 149{159. [Dosen, 1988] K. Dosen, Sequent systems and groupoid models I, Studia Logica 47 (1988), 353{389. [Dragalin, 1988] A. Dragalin, Mathematical Intuitionism. Introduction to Proof Theory, American Mathematical Society, Providence,1988. [Dunn, 1990] J. M. Dunn, Gaggle Theory: An Abstraction of Galois Connections and Residuation with Applications to Negation and Various Logical Operations, in: J. van Eijk (ed.), Logics in AI, Proc. European Workshop JELIA 1990, Lecture Notes in Computer Science 478, Springer-Verlag, Berlin, 1990, 31{51. [Dunn, 1993] J.M. Dunn, Partial-Gaggles Applied to Logics with Restricted Structural Rules, in: P. Schroeder-Heister and K. Dosen (eds.), Substructural Logics, Clarendon Press, Oxford, 1993, 63{108. [Dunn, 1995] J.M. Dunn, Gaggle Theory Applied to Modal, Intuitionistic, and Relevance Logics, in: I. Max and W. Stelzner (eds.), Logik und Mathematik: FregeKolloquium 1993, de Gruyter, Berlin, 1995, 335{368. [Fitting, 1993] M. Fitting, Basic Modal Logic, in: D. Gabbay et al. (eds), Handbook of Logic in Arti cial Intelligence and Logic Programming, Vol. 1, Logical Foundations, Oxford UP, Oxford, 1993, 365{448. [Friedman, 1975] H. Friedman, Equality between functionals, in: R. Parikh (ed.), Logic Colloquium Boston 1972{73, Springer Lecture Notes in Mathematics Vol. 453, Springer-Verlag, Berlin, 1975, 22{37. [Gabbay, 1996] D. Gabbay, Labelled Deductive Systems: Volume 1. Foundations, Oxford University Press, Oxford, 1996. [Gabbay and de Quieroz, 1992] D. Gabbay and R. de Quieroz, Extending the CurryHoward interpretation to linear, relevant and other resource logics, Journal of Symbolic Logic 57 (1992), 1319{1365. [Gentzen, 1934] G. Gentzen. Investigations into Logical Deduction, in: M. E. Szabo (ed.), The Collected Papers of Gerhard Gentzen, North Holland, Amsterdam, 1969, 68-131. English translation of: Untersuchungen uber das logische Schlieen, Mathematische Zeitschrift 39 (1934), I 176{210, II 405{431. [Girard, 1989] J.-Y. Girard, Y. Lafont, and P. Taylor, Proofs and Types, Cambridge University Press, Cambridge, 1989. [Goble, 1974] L. Goble, Gentzen systems for modal logics, Notre Dame Journal of Formal Logic 15 (1974), 455{461. [Godel, 1933] K. Godel, Eine Interpretation des intuitionistischen Aussagenkalkuls, Ergebnisse eines mathematischen Kolloquiums 4 (1933), 39{40. Reprinted and translated in: S. Feferman et al. (eds.), Kurt Godel. Collected Works. Vol. 1, Oxford University Press, Oxford, 1986, 300{303. [Goldblatt, 1992] R. Goldblatt, Logics of Time and Computation, CSLI Lecture Notes 7, Stanford, CSLI Publications, 2nd revised and expanded edition, 1992. [Gore, 1992] R. Gore., Cut-Free Sequent and Tableau Systems for Propositional Normal Modal Logics, PhD thesis, University of Cambridge Computer Laboratory, Technical Report No. 257, 1992. [Gore, 1995] R. Gore, Intuitionistic Logic Redisplayed, Technical Report TR-ARP-11995, Australian National University, 1995. [Gore, 1998] R. Gore, Substructural Logics on Display, Logic Journal of the IGPL 6 (1998), 451{504. [Gore, 1998] R. Gore, Gaggles, Gentzen and Galois: How to disply your favourite susbstructurla logic, Logic Journal of the IGPL 6 (1998), 669{694. [Gore, 1999] R. Gore, Tableau Methods for Modal and Temporal Logics, in: M. D'Agostino, D. Gabbay, R. Hahnle, and J. Posegga (eds), Handbook of Tableau Methods, Kluwer Academic Publishers, Dordrecht, 1999, 297{396. [Gore, 2000] R. Gore, Dual Intuitionistic Logic Revisited, in: R. Dyckho (ed.), Proceedings Tableaux 2000, LNAI 1847, Springer-Verlag Berlin, 2000, 252{267. [Gottwald, 1989] S. Gottwald, Mehrwertige Logik, Akademie-Verlag, Berlin, 1989.


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LENNART AQVIST

DEONTIC LOGIC

I. INTRODUCTION 1 THE PROTAGORAS PARADOX: AN EXERCISE IN ELEMENTARY LOGIC FOR LAWYERS AND MORALISTS An ancient paradox is about the famous Greek law teacher Protagoras and goes like this: Protagoras and Euathlus agree that the former is to instruct the latter in rhetoric and is to receive a certain fee which is to be paid if and only if Euathlus wins his rst court-case (in some versions: as soon as he has won his rst case). Well, Euathlus completed his course but did not take any law cases. Some time elapsed and Protagoras sued his student for the sum. The following arguments were presented to the judge in court. Protagoras: If I win this case, then Euathlus has to pay me by virtue of your verdict. On the other hand, if he wins the case, then he will won his rst case, hence he has to pay me, this time by virtue of our agreement. In either case, he has to pay me. Therefore, he is obliged to pay me my fee. Euathlus: If I win this case, then, by your verdict, I don't have to pay. If, however, Protagoras wins the case, then I will not yet have won my rst case, so, by our agreement, I don't have to pay. Hence I am not obliged to pay the fee. Let us now raise two questions: Who was right? Could deontic logic, in the sense of the logical theory of norms and normative systems, be helpful in providing a solution to this problem, or kind of problem? In this chapter we shall not attempt to answer the rst question, but just refer the reader to the attempts made by Lenzen [1977], Smullyan [1978] and Aqvist [1981]. But we shall indeed argue for an aÆrmative answer to the second question, agreeing with the following statement made by Bertrand Russell in Òn Denoting': A logical theory may be tested by its capacity for dealing with puzzles, and it is a whole-some plan, in thinking about logic, to stock the mind with as many puzzles as possible, since these serve much the same purpose as is served by experiments in physical science. As we shall see in Section 9 below, however, our aÆrmative answer will have to be carefully quali ed as a result of the examination we undertake D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, Volume 8, 147{264.


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in Part II of a number of paradoxes and dilemmas that have beset recent developments in deontic logic in the last thirty years. First of all, though, let us have a somewhat closer look at the subject as it stands nowadays. 2 THE IMPORTANCE OF VON WRIGHT'S AND ANDERSON'S WORK What is deontic logic? It is tempting to answer with Rescher [1966], a propos the closely related area of the logic of imperatives and commands, that it is a eld with the property that there is virtually no single issue in it upon which a settled consensus has been reached. Resisting that temptation, though, we say that deontic logic, broadly conceived, is the logical study of the normative use of language and that its subject matter is a variety of normative concepts, notably those of obligation (prescription), prohibition (forbiddance), permission and commitment. The rst one among these concepts is often expressed by such words as `shall', òught' and `must', the second by `shall not', òught not' and `must not', and the third one by `may'; the fourth notion amounts to an idea of conditional obligation, expressible by ìf..., then it shall (ought, must) be the case that '. A powerful trend of research in the area was initiated by the famous contribution of Von Wright [1951], where the formal properties of monadic (ùnconditional', àbsolute') normative concepts were systematically explored. Certain paradoxical results were seen to arise in Von Wright's monadic deontic logic, however, which led him to propose systems for and permissiondyadic (`conditional', `relative') normative concepts, where the notions of obligation, permission etc. are made relative to, or conditional on, certain circumstances. Thus, the dyadic deontic logic of Von Wright [1956] was proposed as a reaction to the Prior [1954] paradoxes of commitment (`derived obligation'), and that of Von Wright [1964; 1965] as a reaction to the Chisholm [1963] contrary-to-duy imperative paradox. One major problem-area, with which we shall deal in this chapter, concerns the mathematical structure and interpretation of the Von Wright-type deontic logics just mentioned, whether they be of the monadic kind or the dyadic one. In Anderson [1956] the author interestingly argued that the study of normative concepts undertaken by deontic logic could pro t a good deal from our considering their behavior in the context of normative systems, like systems of ethics (moral theories) and systems of positive law. He then, naturally, noticed and emphasized the role played by sanctions or penalties in actual normative systems, and went on to de ne the deontic or normative notions of obligation, forbiddance etc. along the following lines: letting S be a constant proposition, describing a situation which will count as a penalty or sanction relatively to the normative system under investigation, we say that a state-of-aairs p is obligatory if and only if (i) the absence

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of p entails the sanction S ; that p is forbidden i p (itself) entails the sanction S ; and that p is permitted i it is possible that p obtains without the sanction S being realized. (For graphic pictures of essentially these de nitions, in the style of traditional `squares of opposition', see Anderson [1968] and Aqvist [1987, Section 5.3].) Anderson [1956; 1958] then sets out to add these de nitions to various systems of alethic modal logic (i.e. the logic of òrdinary' necessity and possibility), whereby he achieves a kind of reduction of monadic deontic logic to alethic modal logic, provided only that the alethic system is supplemented with the constant S and (possibly) with some axiom governing S . Anderson [1956; 1959] also suggests a de nition in terms of S and alethic modal notions of the dyadic concept of commitment. A second major problem-area which will occupy us in the present work, concerns the mathematical structure and interpretation of Anderson-style systems of alethic modal logic with a propositional constant added to their basic machinery. Also, we shall be highly interested in the relation of such Anderson-style systems to Von Wright-type deontic logics of the two sorts mentioned above. Let me now brie y say something about the current state of the subject of deontic logic. I think it is only fair to claim with Von Wright [1977] (his introduction to the proceedings of the international and interdisciplinary Bielefeld Colloquium in March 1975 | presumably the rst one of its kind) that the widespread and intense interest aroused by deontic logic indicates that we have to deal with a new logical discipline, which has come to stay and is not just èine vorubergehende Erscheinung'. On the other hand, he points out, it is still a relatively poorly developed branch of exact research for the following reasons: (i) The number of open problems is very big. (ii) There is a good deal of controversy and disagreement about fundamental matters in the area, e.g. about the interpretation and the validity of its basic principles. (iii) The high expectations as to the applicability of deontic logic to actual and potential normative systems, notably in the areas of ethics and legal theory, can hardly be said to have been satis ed to more than a very slight and modest degree. The energy and dedication with which Von Wright himself has, since the ' fties, labored to improve and re ne on his originally proposed systems, bear out conclusions (i) and (ii) very clearly in my opinion. And there is no doubt that the optimism in regard to applications | so characteristic of Anderson | has turned into pessimism. My third main concern in this chapter will then be to do something to remove that pessimism. Suppose we want to achieve a logical analysis [Oppenheim, 1944] or a rational reconstruction [Wedberg, 1951] of some system

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of positive law, say, some relevant part of any existing commercial or criminal code. It is then clear that the languages of the current systems of deontic logic, which we will encounter below, are almost totally inadequate for the formulation of even very simple rules of the system. A main reason for this being so is that these languages are just propositional and thus lack quanti cational resources of expression. What is even worse, they lack explicit temporal resources, which fact makes them especially useless from the legal point of view. In Section 9 below, we follow Van Eck [1981] in arguing that, for any serious purposes of application, the expressive resources of deontic languages must be enriched so as to include temporal and quanti cational ones. If this is done, the hope of deontic logicians and others concerned to be able to contribute substantively to ethical and legal theory might be regained. Maybe also to a linguistically important branch such as speech act theory. Let me close this introductory section with a cursory historical note. Suggestions about a logic of normative concepts and sentences (including one for imperatives and commands) may be found in Aristotle, in the Stoics (see Rescher [1966]), in medieval philosophers (see Knuuttila [1981]), in Leibniz, as well as in Bentham and his followers in legal philosophy (see Lindahl [1977]). The rst systematic attempt to build a formal theory of normative concepts is due to Mally [1926] (for a nice exposition of Mally's Deontik, see Fllesdal and Hilpinen [1971], who also cover later twentiethcentury developments in an exemplary way). 3 PLAN OF THIS CHAPTER The present work can be divided into two. The rst deals with certain muchdiscussed diÆculties in connection with the application of formal systems of deontic logic to a natural language such as English (Part II: Paradoxes and Dilemmas), and the other half is devoted to the purely mathematical presentation and elaboration of a number of formal systems of deontic logic (Parts III{VI). In Part III we deal with ten systems of monadic deontic logic, which go back to the work of Timothy J. Smiley in [1963], to that of William H. Hanson in [1965], and, in the three cases of OM+ , OS4+ and OS5+, to the mixed alethic-deontic systems OM, OM' and OM00 of Anderson [1956]; among other things, Smiley [1963] proved the three former systems to be identical with the deontic fragments of the three latter ones. The ten Smiley{Hanson systems, as I pertinently call them, are studied both from a proof-theoretical or axiomatic point of view (Section 10.2) and from that of model-theoretical `possible worlds' semantics in a sense deriving from Hintikka [1957], Kanger [1957], Montague [1960] and, above all, Kripke [1963] (Section 10.3). In Section 11 we prove the semantic soundness and

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completeness of the ten Smiley{Hanson systems, replacing the method of semantic tableaux used by Kripke [1963] and Hanson [1965] with the Henkin technique of maximal consistent (`saturated') sets, as transferred to modal logic by Makinson [1966] (see also Lemmon and Scott [1966]). In Part IV we introduce ten Anderson-style systems of alethic modal logic with a propositional constant Q, interpreted as the negation of the Andersonian sanction S and due to Kanger [1957]. Each of these systems is supplemented by the famous Anderson-style de nitions of monadic deontic operators, expressing obligation, permission and prohibition, respectively: OA = df (Q ! A) P A = df (Q ^ A) F A = df (Q ! :A): Several of these `mixed' alethic-deontic systems were considered by Smiley [1963], to whom, essentially, we owe what I take to be one of the main mathematical results on propositional monadic deontic logic: the Translation Theorem stated in Theorem 45 and proved (in very broad outline) in the proof thereafter. The gist of that result is that the ten Smiley{Hanson systems are, in the well de ned sense of De nition 44, the deontic fragments of the corresponding alethic systems, as supplemented with Q and with the above de nitions of O; P and F: In Parts V and VI we try to pursue the very same line of thought in the area of propositional dyadic deontic logic, for which a good deal of motivation was provided in Part II, notably by Prior's paradoxes of commitment and Chisholm's contrary-to-duty imperative paradox (Sections 7 and 8). Thus, we are concerned about logics of conditional obligation and permission, expressed by such dyadic forms as OB A and PB A (Section 15), and also about the possibility of representing such dyadic logics in systems of alethic modal logic to which a monadic, or one-place, Q-connective is added as well as the de nitions given in Section 15.1: OB A = df (QB ! A) PB A = df (QB ^ A) FB A = df (QB ! :A): We are then able, in Part V, to extend the Smileyan Translation Theorem to the following pairs of logical systems: Ody S4 and S4Qmo (1) O S5 and S5 (Section 16) dy Qmo (2)

Ody S5N

and

S5N Qmo

(Section 17)

where the rst member in each pair is a system of (propositional) dyadic deontic logic and where the second one is a (propositional) alethic modal

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logic supplemented with the monadic Q-operator and with the above Section 15.1 de nitions of dyadic deontic operators. In Part VI we consider certain axiomatic extensions of the systems Ody S5N (Section 18) and S5N Qmo (Section 19). In Section 20 we then obtain a weakened version of the projected Smileyan Translation theorem, but have to leave open the problem of establishing it fully. In Sections 21{22 we make an attempt to reconstruct and identify three dyadic deontic logics due to Bengt Hansson [1969] on the basis of certain further extensions of the calculi Ody S5N and S5nQmo . A highly interesting feature of those extensions is that, in their semantics, we work with an explicitly speci ed preference relation with which we connect the remaining items in the models considered. Again, in Section 23, we deal with the completeness problem for the most important one among the extensions of Ody S5N , called the `strongly normal' core system G, and oer a positive solution to that problem. Finally, in three concluding sections (Sections 24{26) we present some further quite recent results on that `core' system G, which were obtained in Aqvist [1996; 1997]. The idea of basing Dyadic Deontic Logic, or the Logic of Conditional Obligation/Permission, on some kind of preference theory was proposed by several writers in the late 1960s and early 1970s. Pioneering contributions are due to Sven Danielsson [1968] and to Bengt Hansson [1969]; of these two, [Hansson, 1969] is more easily accessible from a mathematical standpoint, whence our aforementioned attempt to reconstruct his systems in our own framework. Danielsson's work is considered in the Appendix of [ Aqvist, 1987], where it is compared with that of Bas C. van Fraassen, Franz von Kutschera and David Lewis. So much for the logical technicalities of various systems of formal deontic logic. Going back to Part II, then, we start out with certain preliminary considerations of the relation between formal languages (like that of the Smiley{Hanson systems of monadic deontic logic) and natural languages such as English. Thus, in Section 5, we explain in some detail how a system of formal deontic logic can be supplemented with de nitions of locutions in ordinary English in much the same way as systems of alethic modal logic were extended with de nitions of deontic modalities by Anderson; e.g. we stipulate the following: (D6) It is obligatory that A = df OA (D7) It is permitted that A = df P A (D8) You post the letter = df p where p is the rst proposition letter in the formal language, which, on the basis of de nition-theoretical considerations, we regard as a propositional constant and hence as a logical symbol. We then end up with a `mixed' formal-English system which contains, on the basis of the de nitions added,

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certain expressions that count as reasonably good English sentences. In Section 5.1 and 5.2 we then show how our approach leads to (i) a fragment of normative English generated by those de nitions, and (ii) a translation (formalization, symbolization) of that fragment into the original formal language | a translation which is in an obvious sense induced by those de nitions. After having generalized these ideas in a straightforward way, we go on to introduce (in Section 5.3) the conception of a natural deontic logic over the English fragment just generated; this conception is based on an admittedly vague and imprecise notion of logical validity (truth), which is supposedly applicable to some major part of the English language or, at the very least, to the sentences of the fragment. In Section 5.5 we consider one well known attempt to make such an intuitive notion of validity more precise, viz. the so called Bolzano criterion. The whole argument of the crucially important Section 5 provides a basis for comparing and contrasting our natural deontic logic with systems of formal deontic logic, like the Smiley{Hanson ones. As appears from Section 5.3, such a comparison will, in general, lead to one of two results: either (i) the natural deontic logic is `perfectly matched' by the formal one, or (ii) it is not so `perfectly matched', because there are `clashes', or `discrepancies' between the two. In turn, such a clash may be of two dierent kinds, at least (as is explained under (I) and (II) in Section 5.3). All this leads up to the three notions of adequacy and faithful representation de ned in Section 5.4. That section and its de nitions are the main outcome of the discussion of formalization and translation in Section 5. We regard that preparatory discussion as indispensable to any orderly presentation of the deontic paradoxes: it gives us a framework, viz. de nitional extensions of formal theories, which enables us to present those paradoxes with a suÆcient degree of mathematical precision and, at the same time, is of some independent methodological interest to the study of the connections between natural and formal languages in general. In Section 6{8 we proceed to an exposition of the familiar puzzles known respectively as Alf Ross' paradox, Arthur N. Prior's paradoxes of commitment, and Roderick M. Chisholm's contrary-to-duty imperative paradox. A vital reason for dealing with the latter two is to show how the idea of dyadic deontic logic, originally proposed by Von Wright [1956], naturally suggests itself in any attempt to overcome them. In Section 9, however, we follow Van Eck [1981] in giving a survey of various problems which, as it seems, cannot be satisfactorily handled on the dyadic approach, nor, for that matter, on the monadic one. In Section 9.2 we diagnose the current state of the subject as a whole by agreeing with Van Eck [1981] that there is an urgent need in the area for explicitly temporal and quanti cational resources in the basic languages of workable and useful deontic logics; as far as the latter sort of resources are concerned, the point has been made by quite a few writers, of course, e.g. Anderson [1956, p. 200]. In Section

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9.2.1, we comment on a powerful research trend in present-day deontic logic, which combines our subject with the logic of action. As for problems not dealt with in this essay, but which are nevertheless of a very fundamental nature, let us just mention the following two: (i) The problem of truth-or-falsity of deontic (normative) sentences: are there any true-or-false deontic sentences, expressing obligations, permissions or prohibitions? If not, why? If yes, what are the conditions under which such sentences are true/false, respectively? The discussion in the early twentieth century of this basic issue led some moral and legal philosophers as well as precursors of modern Von Wrightstyle deontic logic to emphasize a highly important distinction, which may be found in Bentham and which is nowadays usually credited to Ingemar Hedenûs [1941]. So our second problem is: (ii) The problem of explicating formally the Hedenius [1941] distinction between `genuine' and `spurious' deontic sentences. According to Hedenius [1941], a sentence like `You shall not kill!' normally directly expresses a prohibition against killing and is then a genuine deontic sentence. But the very same sentence, when uttered, e.g. by a Swedish lawyer, may well be interpreted as an elliptical formulation for Àccording to Swedish law in 1982 you shall not kill' and function as a spurious deontic sentence, which just asserts the existence of a norm prohibiting killing within a speci ed legal system (without `directly expressing' that prohibition, as it were). So the present problem concerns the formal explication of the Hedenius genuine vs. spurious distinction. We note here that Wedberg [1951] draws a similar distinction in dividing deontic sentences relatively to a given legal system into such as are internal and such as are external to the system. Again, Stenius [1963] stresses a modal vs. factual distinction applicable to interpretations of normative sentences, Hansson [1969] an analogous imperative vs. descriptive one. Finally, Von Wright [1963] distinguishes norms from norm-propositions. As a quick reaction to this second problem, I recommend that deontic logicians consider more seriously the scarce attempts in the literature to construct logics of commanding as opposed to logics of commands, e.g. Fisher [1961a], Hanson [1966] and Bailhache [1981], where authorities and addressees are explicitly brought to the fore. These attempts look very promising indeed for future developments of our subject.

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4 ELEMENTARY PROPERTIES OF SOME VON WRIGHT-TYPE DEONTIC LOGICS In this section we shall rst introduce two notions of a normal Von Wrighttype deontic logic, secondly, comment on our suggested de nition of those notions and, thirdly, list some obvious properties of these logics. Consider the formal language of the ten Smiley{Hanson systems of propositional monadic deontic logic studied in Parts III and IV below. Its set of we formed sentences is de ned as the smallest set which (i) has every proposition letter as an element, and (ii) is closed under the usual truthfunctional connectives including the constants verum and falsum as well as under the two primitive monadic deontic operators O (for obligation) and P (for permission). We then propose the following: DEFINITION 1. Let L be any subset of . Then: (I)

L is a normal propositional monadic Von Wright-type deontic logic i (a) every thesis, i.e. provable sentence, of the system OK (Section 10.2) is a member of L, and (b) L is closed under uniform substitution for proposition letters, detachment for material implication and the rule of O-necessitation (Section 10.2).

(II)

L is a strongly normal (propositional monadic) Von Wright-type deontic logic i

(a) every thesis of the system OK+ (Section 10.2) is a member of L, and (b) L is closed under substitution, detachment and O-necessitation. REMARK 2. (i) The system OK is determined as follows:

Axiom schemata: (A0)

All tautologies over

(A1)

P A , :O:A

(A2)

O(A ! B ) ! (OA ! OB ):

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Rules of proof: A; A ! B (modus ponens, detachment). B A (O-necessitation): (R2) OA (R1)

In the spirit of Section 10.2 we then de ne the set of OK-provable sentences (of OK-theses) as the smallest set which contains every instance of the schemata (A0){(A2) as its member and which is closed under the rules (R1) and (R2). Since in this work we usually identify a logic(al system) with the set of its theses, we may even say that the system OK is identical to that set. Note also that our use of axiom schemata instead of single axioms guarantees that OK is closed under (uniform) substitution (for proposition letters), so no primitive rule of proof to that eect is needed. The system OK+ results from OK by adding to the latter every instance of the schema (A3) OA ! P A (whatever is obligatory is also permitted) as a new axiom; for sure, OK+ is to remain closed under (R1) and (R2). (ii) Our de nition of `normality' is meant to harmonize with the de nition of a normal (alethic) modal logic given, e.g. by Makinson [1966], Segerberg [1971a] and Hansson and Gardenfors [1973]. Many writers, including Von Wright [1951], Prior [1955] and Anderson [1956], are likely to regard this notion of normality (i.e. the one de ned in (I)) as too weak, since not every instance of the schema (A3) is provable in OK; they are then likely to prefer, other things being equal, OK+ to OK and our concept of strong normality as the better notion. But quite a few authors, say, Erik Stenius [1963, (interesting argument on p. 254)] and Manfred Moritz [1963] (to mention just one specimen from a large production), have reacted against accepting (A3) (or its equivalent OK+1 stated in Section 4.1 below) as a valid principle of deontic logic which is satis ed by every existing system of norms (at best, according to Stenius, (A3) is satis ed by every system of norms which is possible to obey); such authors are likely to prefer our weaker notion of normality. Our de nition of normality might still seem to be objectionable on the ground that every normal logic is required to be closed under the rule (R2) of O-necessitation. Roughly speaking, accepting this rule commits us to the position that every logically true proposition

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(e.g. every tautology) is obligatory and, dually, every logical falsehood (contradictory state-of-aairs) is forbidden. Prior [1955] nds `no evident reasonableness' in this position, and Von Wright [1951] explicitly rejects it when he proposes a principle of deontic contingency to the eect that such schemata as

O(A _ :A) and

:P (A ^ :A)

should not be accepted as valid. It seems to me, however, that the rule of O-necessitation (or something very much like it) has been successfully defended by Stenius [1963, p. 253], and by Anderson [1956, pp. 181{183]. Also, in Anderson [1956, Section IX], he outlines an interesting way of doing justice to the intuitions underlying Von Wright's principle of deontic contingency. Essentially, his method amounts to this: we may accept a system admitting rule R2 as our basic (monadic) deontic logic; then, if such a system has resources for expressing alethic contingency in the sense of absence of necessity and of impossibility, we could de ne in it new concepts of obligation, permission and prohibition, which will apply only to contingent propositions (state-of-aairs). Clearly, the logic of these new concepts will not be normal in the sense of our de nition above; but it may be developed as a de nitional extension of a normal deontic logic in our sense. And this, I take it, is highly advantageous from a methodological standpoint. (iii) Somewhat cautiously, we speak in the de nition above of normal (strongly normal) propositional monadic Von Wright-type deontic logics for the following reasons. Although our notions are ultimately inspired by the pioneering contribution of Von Wright [1951], the main dierence between his original system and those discussed in this essay is that for Von Wright O and P are deontic predicates, which form sentences when applied to names of acts (in the sense of àct-types'), whereas for us, and many others, O and P are deontic modalities (modal operators or connectives), which form sentences when applied to sentences (which may possibly assert that such and such an act is performed, though). An advantage of viewing O and P as modalities is that questions concerning the status and acceptability of `mixed formulae', like OA ! A, and of formulae involving iterated modalities, like O(OA ! A), can be meaningfully raised and discussed; on Von Wright's original approach, such questions were ruled out at the outset, since those kinds of formulae did not even count as well formed sentences.

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(iv) Our proposed concepts of normality deviate from the Andersonian notion of a normal deontic logic, de ned in Anderson [1956, p. 168], in the following respects: (a) Whereas we require closure under the rule of O-necessitation, Anderson only requires closure under a rule allowing for the intersubstitutability of provably equivalent expressions from the classical two-valued propositional calculus. (b) As was pointed out under (ii) above, he wants the schema (A3) to be provable in any normal deontic logic; we make this a condition of strong normality. (c) In addition to requiring certain schemata to be provable in any normal deontic logic Anderson also requires that certain schemata should not be provable in such logics, e.g..

P A ! A (whatever is permitted is the case) A ! P A (whatever is the case is permitted). Now, I think our considerations under (ii) above explain suÆciently well why we prefer to deviate from the Andersonian concept of normality in the respects (a) and (b). As far as (c) goes, I take his suggestion that certain schemata be unprovable in normal deontic logics to be perfectly sound; negative requirements of the sort may well be used to de ne new and stronger notions of normality, if properly defended, that is to say. Anderson's list of ùnprovables' could even be extended with a huge number of diÆcult items; let me just pick a few from the vast literature: (1) OA , A

(whatever is obligatory is the case and conversely). (2) O(A _ B ) ! (OA _ OB ) (if A-or-B is obligatory, then A is obligatory or B is obligatory). (3) (P A ^ P B ) ! P (A ^ B ) (if A is permitted and B is permitted, then A-and-B is permitted). (4) (OA ^ (A ! OB )) ! OB (if A is obligatory and if A then it is obligatory that B , then B is obligatory). The strange result (1) is provable in the Mally [1926] system, see also Fllesdal and Hilpinen [1971, p. 4]. For interesting comments on the Mally system from the fresh standpoint of deontic temporal

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logic, see Van Eck [1981, p. 81 f]. The unacceptable results 2 and 3 would seem to be provable in natural extensions of the Kalinowski [1953] systems K1 and K2, which are based on a suggestive (2 + 3) valued matrix; see Prior [1956]. Again, (4), due to Prior [1955], is criticized by Hintikka [1957] and [1971]; but it receives an interesting vindication in terms of the Hintikka notion of deontic (as opposed to logical) consequence. Note that this notion of Hintikka's is very de nitely based on a conception of O and P as modalities as opposed to predicates (see remark (iii) above). (v) We readily verify that the ten Smiley{Hanson systems (Section 10) are normal propositional monadic Von Wright-type deontic logics in the sense of clause (I) of our de nition; and that the ve +-systems (starting with OK+ ) are strongly normal ones. The question now arises: can we extend the notions of normality to the systems of dyadic deontic logic studied in Parts V and VI? My answer will be a bit tentative, because dyadic deontic logic does not yet appear to be a suÆciently well established discipline; as will be clear from the Appendix of Aqvist [1987], there is still too much controversy and disagreement about fundamentals to justify a rm answer. Nevertheless, I believe that our argument in Parts V and VI shows that the basic language of dyadic deontic logic must contain the operators N and M of universal necessity and universal possibility (for this terminology, see Scott [1970, p. 157]. The set of sentences of such a language will then be 2O;N (Section 17) rather than 2O (Section 15), and the weakest dyadic logic over 2O;N is the system Ody S5N (see again Section 17). So we propose to de ne a normal propositional dyadic Von Wright-type deontic logic as any subset of 2O;N which contains every thesis of Ody S5N and which is closed under its rules of inference (detachment and N necessitation; as usual, closure under substitution is guaranteed by the use of axiom schemata). Again, a strongly normal dyadic logic of this sort will, we propose, have to contain the system G (Sections 22 and 23) and to be closed under the above rules. Note that Ody S5N and G are much richer theories than, e.g. the Smiley{Hanson systems, in point of expressive and deductive power.

4.1 Theorems and rules of OK and OK+ We now list some theorem-schemata, or thesis-schemata, of OK, i.e. schemata of which each instance (in ) is provable in OK. By our de nition of normality, they will then be provable in every normal monadic deontic logic as well.

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160 OK1. OK2. OK3. OK4. OK5. OK6. OK7. OK8. OK9. OK10. OK11. OK12. OK13. OK14. OK15.

OA , :P :A: :OA , P :A: O:A , :P A: O>: O(A ^ B ) , (OA ^ OB ): P (A _ B ) , (P A _ P B ): OA ^ P B ! P (A ^ B ): OA _ OB ! O(A _ B ): P (A ^ B ) ! (P A ^ P B ): (OA ^ O(A ! B )) ! OB: (P A ^ O(A ! B )) ! P B: (O:B ^ O(A ! B )) ! O:A (O(A ! (B _ C )) ^ (O:B ^ O:C )) ! O:A: OB ! O(A ! B ): O:A ! O(A ! B ):

Suggested readings of many of these items can be found in Anderson [1956, pp. 180 ]. Note that the compound operator O: may be read as ìt is forbidden that'. Furthermore, OK (and hence any normal monadic logic) is closed under the following rules of proof: OKa. OKb. OKc. OKd.

A!B : OA ! OB A,B : OA , OB A!B : PA ! PB A,B : PA , PB

Again, the following schemata are provable in OK+ and, hence, in any strongly normal monadic calculus:

DEONTIC LOGIC OK+ 1. OK+ 2. OK+ 3. OK+ 4. OK+ 5.

161

OA ! :O:A: :(OA ^ O:A): P A _ P :A: P >: :(O(A _ B ) ^ (O:A ^ O:B )):

Schema OK+ 5 may be taken to assert that it is impossible to be obliged to choose between forbidden alternatives (see, e.g. Von Wright [1951]). Note that OK+5 is not a theorem-schema of OK; hence, it is not forthcoming in every normal monadic calculus. Finally, OK+ is closed under the following rule of proof: A OK+ a. : PA whereas OK fails to be closed under that rule. II. PARADOXES AND DILEMMA`S 5 PRELIMINARIES ON FORMALIZATION AND TRANSLATION Consider the formal, or symbolic, language common to the ten Smiley{ Hanson systems of monadic deontic logic to be studied in Part III. That formal language can be conceived of as a structure where:

L = hBas, LogCon, Aux, Senti

(i) Bas (= the set of basic sentences of L) is a denumberable set Prop of proposition letters p; q; r; p1 ; p2 ; : : : . (ii) LogCon (= the set of primitive logical connectives or constants of L) is the set f>; ?; :; ^; _; !; ,; O; P g. (iii) Aux (= the set of auxiliary symbols of L) is the set consisting of the left parenthesis and the right parenthesis; thus, Aux = f(; )g: (iv) Sent (= the set of all well formed sentences of L) is identical to the set as de ned Section 10.1 i.e. the smallest set S such that (a) every proposition letter in Prop is in S , (b) > and ? are in S , (c) if A is in S , then so are :A; OA and P A,

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(d) if A; B are in S , then so are (A ^ B ); (A _ B ); (A (A , B ).

! B)

and

This `recursive' or ìnductive' de nition of may be summarized by saying that the set of sentences of L is the smallest set which (i) contains every basic sentence in Bas as an element, this according to clause (a), and (ii) is closed under every logical connective in LogCon, this being the joint eect of clauses (b){(d). Note that these connectives are of dierent degrees, which are revealed, so to speak, by the way they are used to form new sentences : by clause (c), :; O and P do so when applied to one single sentence as their argument and, hence, are said to be of degree 1; by clause (d),^; _; ! and , do so when applied to any two sentences as their arguments, and are, consequently, said to be of degree 2; nally, by clause (b), > and ? require no argument at all when used to form sentences, hence, they are said to be of degree 0 and to be propositional constants (as opposed to so called propositional variables. Now, assume that we are interested in theories, or logics, formulated in the language L as just described; for instance, any of the Smiley{Hanson systems to be studied below. Generally speaking, such a logic is a subset of determined by a nite number of classes of axioms having a common and peculiar form or Gestalt (these classes are usually known as àxiom schemata') as well as by certain rules of inference (perhaps more appropriately: rules of proof). Now, why have deontic logicians and moral philosophers alike paid so much attention to systems of the Smiley{Hanson kind rather than to other subsets of that might just as well have been selected for attention? An obvious reason appears to be this: certain de nite ordinary language readings are associated with the logical connectives in L and, on these readings, the principles of these logics have more or less good claims to being true, valid, correct, acceptable, or whatnot, in a pre-systematic, informal or intuitive sense. Which English readings are we then to associate with the connectives in LogCon? Here is a familiar list:

:: ^: _: !: ,:

not (more fully: it is not the case that) , and...) and (more fully: both or (more fully: either , or...) if , then... if and only if (alternatively: if and only if , then...) O; it is obligatory that (alternatively: it ought to be that) P : it is permitted that (alternatively: it may be that).

Thus, :; ^; _; ! and , are to be symbols for the ve best known socalled truth-functions of classical propositional logic, viz. negation, con-

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junction, disjunction, material implication and material equivalence, respectively. Again, O and P are to symbolize the normative (or deontic) notions of obligation and permission. Finally, > (verum in Latin) symbolizes some arbitrary, but xed logical truth or tautology, and ? (falsum in Latin) some arbitrary, but xed logical falsehood, contradiction or absurdity; their precise reading in English is less important. We may think of the list above as presenting items in a little logical dictionary or lexicon: they tell us both (i) how to translate connectives in LogCon into plain English, and (ii) how to translate certain English locutions, viz. those appearing to the right in the list, `back' into the formal language L. In this way, we suggest, we get an idea about the intended interpretation of the language L or, at least, of its logical connectives. But it still remains unclear how to understand the list of lexical items (the `logical dictionary') itself; what is its status in a formal theory, like any one of the Smiley{ Hanson systems of monadic deontic logic? What does it mean to assign an English reading to a connective in L on the basis of a lexical item in the list? In answer to these questions, we propose to equate that list with the following series of de nitions, applicable to any L-sentences A and B : (D1) (D2) (D3) (D4) (D5) (D6) (D7)

It is not the case that A =df:A. Both A and B = df (A ^ B ). Either A or B =df (A _ B ). If A then B =df (A ! B ). If and only if A then B =df (A , B ). It is obligatory that A =df OA. It is permitted that A =df P A.

Let us now distinguish some eects of adding this series D1{D7 to our language L or to any theory formulated in L. (I)

We obtain a new language, call it L(D1{D7), which is like L in having (i) the same set Bas of basic sentences, viz. the set Prop, and (ii) the same set Aux consisting of the two brackets. But L(D1{D7) diers from L in having (i) a larger set LogCon than L and (ii) a larger set of sentences than L; because the seven de ned English connectives ìt and...' etc. will be among the logical is not the case that', `both connectives of L(D1{D7), though not among those of L; and because

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the set of sentences of L(D1{D7), call it (D1{D7), will be closed, not only under the old, `symbolic' connectives of L, but under the new, de ned English ones as well. For example, the following strings: it is not the case that p (3) it is obligatory that q (where p; q are in Prop) are sentences of L(D1{D7), though not of L. We shall call the new language L(D1{D7) a de nitional enrichment of L (reserving the more familiar label `de nitional extension' for theories, or logics, formulated in L). REMARK 3. A description of the rst sentence above, which is agreeable to fans of use and mention, is this: the result of writing the pre x ìt is not the case that' immediately in front of the proposition letter in Prop that is denoted by `p'. In this work we mostly stick to the familiar convention of using logical connectives autonomously, i.e. as names of themselves. (II) Usually, one conceives of a de nition as something added to a theory rather than to a `bare' language (see e.g. Suppes [1957, p. 152 f.]). Suppose, then, that we add the series D1{D7 to any of the Smiley{ Hanson systems below, e.g. to the logic OK as described in Section 10.2. How are we to understand D1{D7 within OK, then? In particular, what does the symbol `=df' mean? We suggest here that `=df' can throughout be replaced by the sign , for material equivalence or `biconditionality', and that every resulting sentence is to be regarded as a new axiom which is added to those of the logic OK. (We assume then that the outermost pair of parentheses has been dropped from D1{D7, complying with the customary convention.) From the standpoint of the logic OK, D1{D7 dier from its `proper' axiom schemata A0{A2 chie y in respect of introducing into it new symbols that are not already in its language L, viz. the seven English connectives de ned by D1{D7 (they also dier from A0{A2 in other respects not commented upon here), Thus, the logic OK supplemented with D1{ D7 as here understood, call it OK(D1{D7), will be a certain subset of the larger sentence-set (D1{D7), not of , and is determined not only by A0{A2 and the rules of inference R1 and R2, but by the `definitional axiom schemata' D1{D7 as well. Finally, just as L(D1{D7) was said above to be a de nitional enrichment of the language L, we now say that the logic OK(D1{D7) is a de nitional extension of the logic OK; the same thing goes for any theory over, i.e. formulated in, L, to which D1{D7 are added. (III) We observed a moment ago that such strings as is not the case that p (3) it it is obligatory that q (with p; q in Prop)

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count as sentences in the enrichment L(D1{D7) of L. Now, these strings are hardly acceptable as sentences of English, although they surely involve nice English components, viz. the de nienda of D1 and D6, respectively. At best, they could count as sentences of pseudoEnglish or quasi-English. The reason for this being so is that, as such, p and q fail to make any sense in English; as one often says, they are just empty place-holders for genuine English sentences. But couldn't we begin to assign English readings to these letters, just as we did to the logical connectives in L by means of D1{D7? If this were possible, we might be able to generate some genuinely English sentences within some de nitional enrichment of L. Let us try. Consider the English sentence guring in the Alf Ross paradox, viz. (0) You post the letter. Again, let p be the rst proposition letter in an assumed enumeration, or ordering, of the set Prop. We now lay down the de nition: (D8) You post the letter =df p which, in accordance with the decisions taken under (II) above, is to be understood as a single axiom: You post the letter , p (or, perhaps, as a degenerate axiom schema having that single axiom as its only instance). We must now raise a vitally important question: is D8, thus understood, acceptable from a standpoint of the theory of de nitions? Well, its answer depends on which of the following alternatives applies: (A) In D8, p is, a propositional variable, and any theory T formulated in the language L enriched by D8, call it L(D8), is closed under a rule of substitution for propositional variables. (B) In D8, p is, and indeed has to be, a propositional constant, which is syntactically and grammatically on a par with the logical 0place connectives > and ? of L. Otherwise, D8 is altogether unacceptable from the point of view of de nition theory. Let us explore these two alternatives in turn. Alternative A. What does it mean to say that any theory over L(D8) is closed under substitution for propositional variables? At least the following, I suggest: (1) For each sentence A of L(D8): You post the letter , A is provable (as a `thesis' or `theorem') in any such theory.

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More leisurely expressed: D8 is an axiom of any theory T over L(D8). Hence, D8 is provable in T . And so is any result of substituting a sentence A of L(D8) for the variable p in D8. Now, an obvious consequence of adding D8 to L is this: (2) The strings: You post the letter : You post the letter are sentences of L(D8). Therefore, by (1) and (2), we conclude that the string ($) You post the letter , : You post the letter is not only a sentence of L(D8) but also provable in any theory over L(D8), since ($) is the result of substituting the L(D8)sentence : You post the letter for p in D8. But ($) is obviously a contradiction in any such theory (of any interest), hence, any such theory is inconsistent. Hence, alternative A must be rejected. Another way of explaining its failure is as follows. On alternative A, D8 violates a basic de nition-theoretic principle according to which no proper de nition of a sentence-forming connective is allowed to use any free (i.e. not bound to any quanti er) variable in the de niens which does not occur already in the de niendum. Well, in D8, on this alternative, p is a free propositional variable which obviously does not occur in the English sentence (or sentence-forming connective of degree 0) `You post the letter' (although, for sure, the 16th letter `p' of the English alphabet does). For a statement of the de nition-theoretic principle in a somewhat dierent context, see Suppes [1957, p. 156 f.]; for its motivation in the present context, recall that we just proved every theory over L(D8) to be inconsistent. Note also that the principle is a consequence of more general criteria in de nition theory known as those of non-creativity and relative consistency (Suppes [1957, pp. 154{5]). Alternative B. Having rejected A, this is the alternative we must accept in order to protect D8. Let us pay attention to two consequences of regarding p in D8 as a constant, and indeed a logical one. (Note that in the context we disregard any subdivision of constants into logical vs. non-logical (descriptive) ones; this is a deliberate, but remediable oversimpli cation.) (i) The disastrous sentence ($) will not be provable in any consistent theory over L(D8), or over L(D1{D7, D8), because D8 is now

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a proper de nition which conforms to the rules and criteria of de nition theory. In particular, there is nothing like a `rule of substitution' for logical constants. (ii) Let us continue to think of the set Prop of `proposition letters' as a set of free propositional variables, which form the stock of nonlogical symbols of L and its enrichments. But then, obviously, the rst letter p, used in D8 as now understood, is wrongly classi ed, being a constant symbol. So in any enrichment of L containing D8, p has to be re-classi ed as a logical constant, i.e. moved from Prop into the set LogCon of the enrichment. Similarly, the English sentence `You post the letter', de ned by D8, should be classi ed as a constant symbol and placed in the set LogCon of such enrichments. EXERCISE 4. Consider the language L(D1{D7, D8) and think of it as a structure L(D1{D7, D8) = hBas, LogCon, Aux, Senti. Specify the components of that structure, observing the instructions just given under (ii) above! REMARK 5. One might object to our argument concerning the status of p in D8 that we have overlooked a third alternative: p is a propositional parameter. Here is my answer: either a parameter is a free variable and the present alternative amounts to Alternative A and is òut'; or a parameter is a constant, so the present alternative amounts to Alternative B and we gladly accept it. Is the objection thereby met?

5.1 On the fragment of normative English constructible in L(D1{D7, D8). Consider the language L(D1{D7, D8). The following are sentences of L(D1{D7, D8): It is obligatory that you post the letter. Either it is not the case that you post the letter or it is permitted that you post the letter. If it is obligatory that you post the letter then it is permitted that you post the letter. which are also, indeed, reasonably good sentences of English. (We don't deny that they can be improved from the standpoint of stylistic elegance, using devices like better punctuation, inversion of word order, pronominalization, attaching some negation- or obligation-expressing phrase to the main verb in the in nitive, and so on.) Now, is it possible somehow to characterize rigorously the set of those English sentences which we can generate

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in the language L(D1{D7, D8)? Yes, easily, as follows: it is the smallest set containing (i) the English sentence (0) (= `You post the letter'), de ned in D8, and which (ii) is closed under the seven English connectives de ned in the series D1{D7. We now propose the following, somewhat more comprehensive notion: DEFINITION 6 (FNE(D1{D7, D8)). Let L(D1{D7, D8) be the de nitional enrichment of L that arises from adding D1{D7, D8 to L. Think of it as a structure whose components are speci ed as in the Exercise above! Then, by the fragment of normative English constructible in L(D1{D7, D8) we shall mean the structure FNE(D1{D7, D8) = hBas, LogCon, Aux, Senti where: (i) Bas = (ii) LogCon =

(iii) (iv)

REMARK 7.

fYou post the letterg. fit is not the case that, both

and..., either or..., if then..., if and only if then..., it is obligatory that, it is permitted thatg. Aux = ; (i.e. the empty set). Sent = (as usual) the smallest set which contains every basic sentence in Bas as an element and is closed under every logical connective in LogCon.

(a) The basic sentence as well as the logical connectives are used autonymously in this description, i.e. as names of themselves. Hence, no quotation marks are needed. (b) Note that, although the English sentence `You post the letter' was classi ed as a logical connective (of degree 0) in L(D1{D7, D8), it is not so classi ed in the fragment FNE(D1{D7, D8) but as a basic sentence. The reason for this decision is that we want Sent to be de nable as the result of closing a non-empty set Bas under certain connectives. (c) There is no need for parentheses in the fragment. This is due to the `smart' reading in English of the binary connectives that is codi ed by the de nitions D2{D5; the point being that every English sentence in the fragment, which has any of the de nienda of D2{D5 as its principal sign or main connective, begins with that very connective.

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Thus, no ambiguities of grouping are forthcoming in the fragment, and its notation is Polish or Lukasiewiczean.

5.2 The translation induced by D1{D7, D8; An extension of that translation Let us quickly rehearse what has been done so far. We started out by considering the purely symbolic, or formal language L common to the Smiley{ Hanson systems of monadic deontic logic and its set of purely symbolic sentences. Then, we added to L eight de nitions D1{D7, D8, thereby obtaining a richer language L(D1{D7, D8), whose set LogCon contained eight fresh English connectives, viz. the de nienda of D1{D8; also, the rst proposition letter in an assumed ordering of Prop, i.e. p, was moved into that set LogCon. The set of sentences of that richer language, call it SentL(D1 D7;D8) , was then seen not only to be larger than that of L, but indeed to contain as a subset a certain class of `reasonably good' sentences of English, viz. the set of sentences of the fragment FNE(D1{D7, D8) of Normative English constructible in L(D1{D7, D8), as de ned above. Call that class either by the fancy name of SentF NE(D1

D7;D8)

(=Sent, as de ned by clause (iv))

or call it simply . Now, can we say anything informative about the relation of the original sentence-set to that of the fragment FNE, i.e. , apart from the claim that the fragment and its sentences were somehow generated by the addition of D1{D7, D8 to the original language L? (Assume, for safety, that the letter p is reclassi ed and moved into LogCon already in L, so that we are free to add D8 to L.) Well, we can obviously assert the following: There exists a translation (formalization, symbolization) of the fragment into the original formal language L in the sense of a function t which maps one-to-one into , where t is de ned by the following recursive conditions: (i) t(`You post the letter') = p; cf. D8. Again, assume that t has been de ned for any English sentences A; B in . Then: (ii) t(it is not the case that A) = :t(A); cf. D1. (iii) t(both A and B )=(t(A) ^ t(B )); cf. D2. (iv) t(either A or B )=(t(A) _ t(B )); cf. D3.

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(v) t(if A then B )=(t(A) ! t(B )); cf. D4.

(vi) t(if and only if A then B )=(t(A) , t(B )); cf. D5. (vii) t(it is obligatory that A)=(Ot(A); cf. D6. (viii) t(it is permitted that A)=(P t(A); cf. D7. It is obvious that, as de ned, t is a function from into . It is almost just as obvious that t is one-one in the sense that for any two distinct sentences A; B in we have that t(A) and t(B ) are distinct sentences in ; the inductive proof of this fact is rather tedious, although basically easy to grasp. We see that each clause in the de nition of t corresponds to exactly one member in the series D1{D8. Hence, it is appropriate to speak of the translation t as being induced by that series. Again, in the present case of and , we speak alternatively of t as a formalization or a symbolization, because is the set of sentences of a fragment of a natural language, viz.English, and is the set of sentences of a purely symbolic language, viz. L, and t translates the former into the latter. Not every translation function has this special character, of course. Suppose now that we add further de nitions in the style of D8 to the language L(D1{D7, D8), of the general form: (D9) (D10) . . . (D8+k.)

=df p1 : =df p2 : =df pk :

where: (i) k is a positive integer 1. (ii) p1 ; : : : ; pk are distinct proposition letters that have been reclassi ed already in the original language L and moved into its set LogCon, where we also nd p which is distinct from all of them. (iii) The blanks are lled by distinct English sentences considered as unanalyzed wholes (ùngetrenntes Ganzes' in the terminology of Hilbert and Ackermann [1928]), and all distinct from `You post the letter'. Then, the fragment of normative English constructible in the enrichment L(D1{D7, D8{D8+k) is easy to identify: its set Bas of basic sentences contains exactly k + 1 English members and, as usual, its total set Sent of sentences is the result of closing Bas under the by now familiar seven

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English connectives. Now, let be that total set, this time, and let still be the set of purely symbolic sentences of L (with p; p1; : : : ; pk reclassi ed as indicated above). How are we to de ne the extended translation, let us still call it t, which is induced by the series D1{D7, D8{D8+k? Very easy: just add to our earlier de nition of t the following k clauses in the recursion (or induction) basis: (i1 ) t(de niendum of D9) = p1 ; cf. D9. (i2 ) t(de niendum of D10) = p2 ; cf. D10. . . . (ik ) t(de niendum of D8+k) = pk ; cf. D8+k. EXERCISE 8. Verify that, as just de ned, the extended translation t remains a one-one mapping of , as presently understood, into !

5.3 On the natural deontic logic over FNE(D1{D7, D8{D8+k) Let FNE(D1{D7, D8{D8+k), or FNEk for short, be the fragment of normative English constructible in the enrichment L(D1{D7, D8{D8+k) of L, as described above. Let still be the set of sentences of FNEk ; thus, all members of are English sentences (some of them, though, less then fully elegant from the stylistic point of view). Then, by the Natural Deontic Logic over FNEk we shall mean a certain subset NDL of , which is vaguely characterized as follows: NDL = the set of sentences of FNEk which are logically valid or logically correct (or, if you have no qualms about applying that notion to normative sentences: logically true). Suppose, at least for the time being and for the sake of argument, that this admittedly vague characterization of NDL makes `some reasonable' sense to you and to me. Let A be any sentence in English. We then have the following immediate result:

A 2 NDL i A belongs to and A is logically valid. Writing ` A' for À is logically valid', we may alternatively express the result as a set-theoretical identity: NDL = fA 2 : Ag Some further notions can now be de ned:

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DEFINITION 9 (Logical consequence, inconsistency and consistency in NDL). Let be a set of sentences in , and let A be a sentence in . Then we say: (i) A is a logical consequence in NDL of (in symbols: kNDL A) i there are sentences B1 ; B2 ; : : : ; Bn in , with n 0, such that

If both B1 and both B2 and ... and both Bn

1

and Bn , then A

(i.e. that sentence is to be logically valid). (ii)

is inconsistent in NDL i there is a sentence B in such that the sentence Both B and it is not the case that B is a logical consequence in NDL of .

(iii)

is consistent in NDL i

is not inconsistent in NDL.

The three notions just introduced should be compared with the prooftheoretical concepts of derivability, inconsistency and consistency in certain formal deontic logics L (see Section 10.2.1 below). Also, we observe that

the usefulness of the de nition above rests on our having recourse to a viable notion of logical validity, which is applicable to English sentences in and, hopefully, even to larger sets of English sentences. We now face the diÆculty that such an intuitive, natural-language-oriented conception is vague, imprecise and the source of endless disputes and disagreement; in this respect it diers unfavorably from the well-de ned notions of provability and validity in a speci ed formal system of deontic logic. The diÆculty is interestingly illustrated by the so-called `paradoxes of deontic logic' in the following way. Let L be some formal system of deontic logic over our language L, to which de nitions D1{D7, D8{D8+k are added, so that the fragment FNEk and its sentence-set are available as well as the natural deontic logic NDL over that fragment. Consider the set of all sentences in whose t-translations are provable in L; thus = fA 2 : t(A) is provable in Lg: Now, suppose we start to compare NDL and . If we nd that they are identical, then the notion of provability in L `matches perfectly' the intuitive conception of logical validity determining NDL and everything is ne; there is no clash between the well-de ned formal notion and the intuitive conception. This possibility is not very likely to arise. It is much more likely that we face one of the following possibilities:

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(I) We nd a sentence in which is such that (a) it is not in NDL because we feel that it is not logically valid, but (b) its t-translation is indeed provable in L (which is usually easy to verify); so the sentence is in . Then we have a clash between the formal notion of provability in L and the intuitive concept of validity of a kind I call below failure of rightto-left adequacy. Roughly speaking, the clash is due to the fact that L sanctions more English sentences as valid than our `logical intuition' is willing to accept as members of NDL. Most deontic paradoxes will be seen to be of this kind in subsequent sections. (II) We nd a sentence in which is such that (a) it is logically valid from an intuitive viewpoint and hence is in NDL, but (b) its t-translation fails to be provable in L (as is usually easy to verify); so the sentence is not in . This situation illustrates an opposite sort of clash which I call below failure of left-to-right adequacy. The import of such a clash is then that there are intuitive validities in NDL, which fail to be representable in the formal system L. In Section 9.2.2 below I argue that the Good Samaritan paradox illustrates this failure with respect to the Smiley{Hanson systems of monadic deontic logic; admittedly, it is usually cited as an instance of the former kind of failure, just as the majority of paradoxes are. These considerations were designed to show how the vagueness and impreciseness of an intuitive, natural-language-oriented notion of logical validity leads to `clashes' when confronted with formal systems of deontic logic. In order to eliminate to some extent the vagueness from which our characterization of NDL thus suers we propose below (Section 5.5) a criterion of validity known as the Bolzano Criterion, which will be seen àt work' in Section 7. Other criteria are possible, however, as will be seen in Section 6. Since the concepts of (failure of) adequacy met with in (I) and (II) above are, we suggest, highly important to any orderly discussion and even presentation of the deontic paradoxes, we shall now introduce them in a rigorous way.

5.4 Some notions of adequate translation and faithful representation DEFINITION 10 (Three concepts of adequacy). Let L, to begin with, be any of the ten Smiley{Hanson systems of monadic deontic logic, characterized in Section 10.2 below in proof-theoretical terms. We write `j L A' to indicate that the formal sentence A (in ) is provable in L. Consider the Natural Deontic Logic NDL ( ) over the English fragment FNEk as well as the extended translation t from into . We then say the following:

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(i) t is left-to-right adequate with respect to NDL and L, i, for each A in , if A 2 NDL then j L t(A):

(ii) t is right-to-left adequate with respect to NDL and L, i, for each A in , if A 62 NDL then 6 j L t(A); where `6 j L ' means `not provable in L'.

(iii) t is fully adequate with respect to NDL and L, i, for each A in ; A 2 NDL if and only if j L t(A): REMARK 11. (a) Clearly, t is fully adequate w.r.t. NDL and L, just in case t is both left-to-right and right-to-left adequate w.r.t. NDL and L. Again, in order to show that t is not fully adequate w.r.t. NDL and L, it is enough to show that either t fails to be left-to-right adequate or t fails to be right-to-left adequate w.r.t. NDL and L. (b) We may use clauses(i){(iii) in the present de nition as a basis for introducing the following notions of faithful representation, applicable to L: (i0 ) L is a left-to-right faithful representation of NDL under t, i, t is leftto-right adequate w.r.t. NDL and L. (ii0 ) L is a right-to-left faithful representation of NDL under t, i, t is right-to-left adequate w.r.t. NDL and L. (iii0 ) L is a faithful representation of NDL under t, i, t is fully adequate w.r.t. NDL and L.

5.5 On the Bolzano criterion of logical validity for sentences in natural languages Consider the fragment FNEk and its sentence-set . As applied to members of , what I shall call the Bolzano criterion of logical validity asserts the following: BCLV. Let A 2 : Then, A is logically valid ( A) i (i) A is true, and (ii) every result of uniformly substituting a sentence in for any basic sentence in A is true as well. For examples of this criterion ìn use', see Section 7 below. The present version of the Bolzano criterion is, of course, a bit restrictive, since it only applies to members of . Inspired by Quine [1963], Fllesdal and Hilpinen [1971, p. 1] suggest a less restrictive version:

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A deontic sentence is a truth of deontic logic if it is true and remains true for all variations of its non-logical and non-deontic words (that is, expressions which are not logical or deontic words). Again, Kanger [1957, p. 50] gives the following version of the Bolzano criterion: By a logically true statement we understand a statement A such that the result of generalizing all extralogical constants in A is true. If we assume all deontic words in the sense of Fllesdal and Hilpinen to be logical constants in the sense of Kanger, we could perhaps prove their respective versions of the Bolzano criterion to be equivalent. For our purposes in the following Sections, however, the version BCLV given above should hopefully turn out to be suÆcient. 6 ALF ROSS'S PARADOX Consider the English sentences: (0) You post the letter. (1) You burn the letter. Consider also the enrichment L(D1{D7,D8,D9) of L, where D8 and D9 are as follows: (D8) You post the letter = df p. (D9) You burn the letter = df p1 . Again, let FNE1 be the fragment of normative English constructible in that enrichment, where, speci cally, Bas = f(0); (1)g and = the smallest superset of Bas closed under our seven English connectives de ned in D1{ D7. The translation t induced by the series D1{D9 is then de ned as in Section 5.2 above, where, in particular, we have the following clauses in the recursion basis: (i) t(`You post the letter') = t((0)) = p: (i1 ) t(`You burn the letter') = t((1)) = p1 . The Alf Ross paradox, rst presented in Ross [1944], is to be taken, we suggest, as an argument against t being fully adequate with respect to the natural deontic logic NDL over FNEk and any Smiley{Hanson system L of monadic deontic logic. In order to state the argument let us rst consider the sentence

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(1.1) It is obligatory that either you post the letter or you burn the letter which is in ; a nicer stylistic variant of (1.1) in the sense of Kalish and Montague [1964, pp. 10f.] is perhaps the following: (1.2) You ought to post the letter or burn it Furthermore, we consider the sentences: (0.1) It is obligatory that you post the letter, (0.2) If it is obligatory that you post the letter then it is obligatory that either you post the letter or you burn the letter, which are both in . Note that (0.2) is the conditional having (0.1) as its antecedent and (1.1) as its consequent. Now, the `paradox' starts out with the following claim, which we shall treat as an hypothesis in the proof-theoretical sense by so indicating its status to the right: 1. The sentence (1.1) is not a logical `claim' or hypothesis consequence in NDL of the unit set of (0.1). In symbols: f(0:1)g k6 NDL (1.1) By our de nition of logical consequence in NDL and the fact that (0.1) is in and is the only sentence in f(0:1)g, we then obtain from the hypothesis 1: 2. (0.2) is not logically valid. from 1 by the de nition of kNDL and the fact etc. In symbols: 6 (0.2). where we indicate to the right how this line 2 is obtained. Hence: 3. (0.2) 62 NDL from 2 by the de nition of NDL. Again, let L be any of the Smiley{Hanson systems of monadic deontic logic. The following is a demonstration that the L-sentence Op ! O(p _ p1 ) is provable in L; where we write `j L ' for `provable in L': 4. j L p ! (p _ p1 ) since all tautologies over L are provable in L by virtue of axiom schema A0. 5.

j L Op ! O(p _ p1 )

from 4 by the fact that the set of L-provable L-sentences is closed under the rule of inference A!B : OA ! OB

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Here, to say that the set of L-provable L-sentences is closed under that rule of inference means that for all L-sentences A; B : if j L A ! B , then j L OA ! OB . We continue the argument by making the following straightforward observation: 6. t((0:2)) = Op ! O(p _ p1 ) by the de nition of t, clauses (i),(i1 ),(iv),(vii) and (v). Therefore: 7. j L t((0:2)) from 5 and 6 by the logic of =. 8. There is a sentence A in , viz. from 3 and 7 by adjunction and (0.2), such that A 62 NDL but existential generalization. j L t(A) 9. t is not right-to-left adequate from 8 by the de nition of rightwith respect to NDL and L to-left adequacy. 10. t is not fully adequate w.r.t. from 9 by the de nition of full NDL and L adequacy. Thus, on the basis of the hypothesis 1, we have used the conditional sentence (0.2) to show that t fails to be right-to-left, and hence fully, adequate w.r.t. NDL and any Smiley{Hanson system L of monadic deontic logic. In other words, no such system is a faithful representation of NDL under t. Before embarking on a discussion of this argument, we stick in a little exercise: EXERCISE 12. (i) Prove that the set of L-provable L-sentences is closed under the rule A ! B=OA ! OB , for any of the ten Smiley{Hanson systems L, as described in Section 10 below! (ii) Give a rigorous proof in full detail of line 6 above! Let us now try to assess the above argument. Clearly, the proof of lines 9 and 10 rests, or is conditional, on the hypothesis 1, which is to the eect that (1.1) is not a logical consequence (in NDL) of (the unit set of) (0.1). What motivation or reason could then be given for this claim? The following line of thought is indicated and discussed by Wedberg [1969, p. 217], and by Hansson [1969, p. 383]. A way for the addressee of (1.1) to obey the command expressed by it is surely that he burns the letter. By so doing, however, he does not obey the

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command expressed by (0.1), which he even positively disobeys. So there is a way of obeying (1.1), which is at the same time a way of disobeying (0.1) and, hence, a way of not obeying it. Therefore, (1.1) cannot be a logical consequence of (0.1). (The following terminological replacements are all right with me: `norm' for `command', `satisfy' or `ful l' for òbey', `dissatisfy' or `violate' for `disobey'.) As is clearly enough indicated by the aforementioned authors, this argument is rather confused. I would like to add the following point to their valuable discussion. The criterion of logical consequence to which the argument tacitly appeals appears to be this: LC0. Let A; B be any command-expressing sentences in English. Then, B is a logical consequence of A i every way of obeying B is a way of obeying A. Putting (1.1) = B and (0.1) = A in LC0, we indeed arrive at the strange result under debate. But hasn't LC0 got things turned upside down? According to criteria in terms of obedience or satisfaction (suggested, e.g. by Von Wright [1955] and Rescher [1966]), we should rather have something like: LC1. B is a logical consequence of A i every way of obeying A (= the implicans) is a way of obeying B (= the implicatum). Using LC1 in the place of LC0, we cannot derive hypothesis 1 any longer. On the contrary, using LC1, we have that (1.1) is indeed a logical consequence of (0.1), since every way of posting the letter is a way of either-posting-orburning it. Are we then entitled to dismiss the Ross Paradox on the basis of having removed one basic confusion that seems to underlie it? I don't think so, nor does, e.g. Von Wright [1968, pp. 21 ], where an interesting argument in favor of hypothesis 1 is indicated, having the form of a reductio ad absurdum of its negation, which is to the eect that (1.1) is a logical consequence of (0.1). Von Wright's reductio is presented and discussed in some detail in Aqvist [1987, Section 5]. Here we just call the reader's attention to some main conclusions emerging from that discussion. (i) The argument of Von Wright [1968, pp. 21 ], is seen to depend crucially on a so-called principle of free choice permission, the status and acceptability of which has since been the object of a still lively and intensive debate. We mention the following contributions from the literature: Von Wright [1971], Fllesdal and Hilpinen [1971], Kamp [1973; 1979], Lewis [1978], Hilpinen [1979; 1981] as well as Nute [1981]. (ii) Although the principle of free choice permission fails to be valid for the weak notion of permission re ected by the `standard' P-operator

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in normal monadic Von-Wright-type deontic logics (Section 4 above), there remains the possibility of de ning notions of strong permission for which that principle is indeed valid | after all, permissive phrases are well known to be ambiguous in ordinary discourse. An interesting attempt to de ne such a notion of strong permission was made by Von Wright [1971]; the idea is based on the Andersonian reduction of deontic logic to alethic modal logic with a propositional constant. We also mention that Anderson [1968] uses the very same idea to develop what he calls eubouliatic logic, in the sense of a logic of prudence, safety, risk and related concepts of a decision-theoretic brand. (iii) Contrary to the view of its originator, the Alf Ross paradox does not seem to be a serious threat to the very possibility of constructing a viable deontic logic. But it usefully directs our attention to the ambiguity of normative phrases in natural language as a possible source of error and confusion | in viable deontic logics we should be able to express, to do justice to, and to pinpoint such ambiguities. For this reason I agree with Von Wright in claiming that the puzzle deserves serious consideration. 7 ARTHUR N. PRIOR'S PARADOXES OF DERIVED OBLIGATION (`COMMITMENT') Consider the English sentences: (2) John Doe impregnates Suzy Mae. (3) John Doe marries Suzy Mae. as well as the enrichment L(D1{D11) of L, where D10 and D11 as follows: (D10) John Doe impregnates Suzy Mae =df p2 . (D11) John Doe marries Suzy Mae =df p3 . We let FNE3 be the fragment of normative English constructible in L(D1{ D11), where Bas contains (2) and (3) as new members and where is de ned as usual. Fresh clauses in the basis of the recursive de nition of the translation t: (i2 ) t(`John Doe impregnates Suzy Mae') = p2 . (i3 ) t(`John Doe marries Suzy Mae') = p3 . The paradoxes of commitment, or derived obligation, go back at least to Prior [1954] and can be viewed as arguments against t being fully adequate with respect to NDL and any Smiley{Hanson system L of monadic deontic logic. Consider the sentences:

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(2.1) If it is not the case that John Doe impregnates Suzy Mae then if John Doe impregnates Suzy Mae then it is obligatory that John Doe marries Suzy Mae. (2.2) If it is obligatory that John Doe marries Suzy Mae then if John Doe impregnates Suzy Mae then it is obligatory that John Doe marries Suzy Mae. (2.3) If it is obligatory that it is not the case that John Doe impregnates Suzy Mae then it is obligatory that if John Doe impregnates Suzy Mae then John Doe marries Suzy Mae. (2.4) If it is obligatory that John Doe marries Suzy Mae then it is obligatory that if John Doe impregnates Suzy Mae then John Doe marries Suzy Mae. Although they are all in , the sentences (2.1){(2.4) give a somewhat queer impression and may be diÆcult to understand. So here are their t-translations into L:

t((2.1)) :p2 ! (p2 ! Op3 ). t((2.2)) Op3 ! (p2 ! Op3 ). t((2.3)) O:p2 ! O(p2 ! p3 ). t((2.4)) Op3 ! O(p2 ! p3 ). We now state four paradoxes of commitment in òne fell swoop': 1. None of the sentences (2.i), for `claim' or hypothesis i = 1; 2; 3; 4 are logically valid. In symbols: 6 (2.i). Now, let L be any of the Smiley{Hanson systems of Monadic Deontic Logic. Then:

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j L t((2.i)), for all i = 1; : : : ; 4

181

exercise

3. There are sentences Ai (with i = from 1 and 2 by the de nition of 1; : : : ; 4) in , viz. (2.i), such NDL, adjunction and existential that Ai 62 NDL but j L t(Ai ) generalization 4. t is not right-to-left adequate from 3 by the de nition of rightwith respect to NDL and L to-left adequacy 5. t is not fully adequate w.r.t. from 4 by the de nition of full NDL and L adequacy Before trying to assess this argument, we stick in a little exercise: EXERCISE 13. (i) Prove assertion 2 in the above argument! (ii) Find more idiomatic stylistic variants of the English (?) sentences (2.i), for i = 1; 2; 3; 4! What motivation could reasonably be given for the claim 1 in this argument? I suggest the following: the common consequent of (2.1) and (2.2) as well as that of (2.3) and (2.4) may both be said to involve the notion of commitment or conditional (`derived') obligation in such a way that (2.5) Impregnating Suzy Mae commits John to marrying her is a stylistic variant (in the sense of Kalish and Montague [1964]) of both these common consequents. Let us now consider four cases in turn, with a view to illustrate the Bolzano criterion of logical validity (see above Section 5.5).

Case I. Suppose that (2.1) is logically valid. Then, by the Bolzano criterion,

not only is (2.1) itself true, but so is every result of uniformly substituting a sentence of the fragment FNE3 for any basic sentence in (2.1). Therefore, the following sentence in must be true: (2.1.1) If it is not the case that John Doe impregnates Suzy Mae, then if John Doe impregnates Suzy Mae then it is obligatory that it is not the case that John Doe marries Suzy Mae the t-translation of which is

t((2:1:1)) = :p2 ! (p2 ! O:p3 ):

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But this result is unpalatable for the following reason: on the basis of it we may infer from the mere fact that John does not impregnate Suzy that impregnating her commits John both to the act of marrying her as well as to the act of not marrying her. There are certainly legal systems where such an inference is rejected as absurd. Hence, (2.1) cannot be logically valid.

Case II. Here we consider an extended fragment FNE4 , whose set Bas contains the new sentence

(4) John Doe kills Suzy Mae which is introduced by a de nition D12 in the obvious way so as to yield the fresh clause for t: (i4 ) t(`John Doe kills Suzy Mae') = p4 . Now, suppose that (2.2) is logically valid. Then, by the Bolzano criterion, not only is (2.2) true itself, but so is the result of substituting (4) for (2) in (2.2). Call this result (2.2.1) and observe that

t((2:2:1)) = Op3 ! (p4 ! Op3 ) Again, (2.2.1) being true is a strange result, because it seemingly entitles us to say that if John has a duty to marry Suzy, then, by logic alone, he has such a duty (even) if he kills her. But, we are told by some plain man in the street, when she is dead, John cannot marry her and so is not obliged or committed to marry her. For, as the saying goes, ought implies can. Hence, to sum up, if (2.2) is logically valid, then (2.2.1) is true. But (2.2.1) is not true (for the reasons just given); therefore, by modus tollens, (2.2) is not logically valid.

Case III. Suppose that (2.3) is logically valid. Then, by the Bolzano crite-

rion, not only is (2.3) true itself, but so is the result of inserting the word ìt is not the case that' immediately in front of (3) in (2.3). Call that result (2.3.1) and note that

t((2:3:1)) = O:p2 ! O(p2 ! :p3 ): The argument against (2.3.1) being true is similar to the one used in Case I: on the basis of it we may infer from the mere fact that it is forbidden for John to impregnate Suzy that impregnating her commits John both to the act of marrying her and to that of not marrying her. But this is absurd for the same reason as in Case I.

Case IV. The point of departure here is the assumption that (2.4) is logically

valid. The refutation of that assumption proceeds (rightly or wrongly) along

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the lines followed in Case II. The result analogous to (2.2.1) is called (2.4.1) where

t((2:4:1)) = Op3 ! O(p4 ! p3 ): We have now got an idea about what kind of counterexamples people are apt to give against the alleged logical validity of the sentences (2.i). We may then try to pin down certain patterns of ìntuitive reaction' to these paradoxes of commitment, which are fairly well discernible or recognizable in the vast literature on the subject. (Incidentally, they will also be seen to apply to the Alf Ross paradox.) Let me distinguish the following two tendencies: (i) the deprivation-of-counterintuitive-force tendency, and (ii) the improved-formalization tendency, and brie y comment on them in turn.

7.1 The deprivation-of-counterintuitive-force tendency One argues as follows. Admittedly, formulations like (2.i) are ambiguous in ordinary English, so there are dierent ways of understanding or interpreting (2.i). Now, each of the Smiley{Hanson systems L is based on a clearcut model-theoretic semantics, which provides a mathematically precise interpretation of its logical constants in terms of truth conditions stated relatively to certain set-theoretical structures called models (see Section 10.3 below). Via our de nitions D1{D7 etc. generating fragments of normative English, this precise interpretation is automatically transferred to every sentence of such an English fragment. So, if we just stick to that interpretation of (2.i) and, most importantly, do not `read into' them anything `beyond' it, their counterintuitive appearance will simply vanish. '-connective in any of (2.i) To illustrate a bit: since every ìf ... then is intended to mean the same as the arrow ! of material implication, as ordinarily understood in the classical propositional calculus (on which all the Smiley{Hanson systems are based), we have the following results on the t-translations of (2.i):

j L t((2:1)) j L t((2:2)) j L t((2:3)) j L t((2:4))

, , , ,

(:p2 ! (:p2 _ Op3 )) (Op3 ! (:p2 _ Op3 )) (O:p2 ! O(:p2 _ p3 )) (Op3 ! O(:p2 _ p3 ))

According to the strategy of interpretation just outlined, we are to understand the sentences (2.i) to mean exactly what is meant by the corresponding

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right members in these equivalences according to the semantics for L; most importantly, we are not to read anything more into them. We then observe the following: (i) Sentences (2.1) and (2.2) as well as their t-translations are instances of familiar tautologies, which cannot fail to be true by virtue of the usual truth-table test. Hence, their counterintuitive force disappears as soon as this fact is grasped and strictly adhered to. (ii) Sentences (2.3) and (2.4) as well as their t-translations are not instances of any truth-table tautologies. Instead, their validity in L depends on and is due to the model-theoretic truth condition for the O-operator (in terms of `possible worlds', the relation of `deontic accessibility' etc. ; see Section 10.3 below). Once this fact is grasped and strictly adhered to, their counter-intuitive force will vanish. Note here that, on the basis of the last two of the equivalences above, the paradoxical conditionals (2.3) and (2.4) both reduce to Ross-paradoxical sentences, with which we have already dealt. Let us also note, nally, that the present Deprivation-of-CounterintuitiveForce Tendency is nicely illustrated, e.g. by Von Wright [1956, p. 508] and by Anderson [1956, p.185], as well as by Prior [1955, p. 224]. This tendency, however, is not the only one emerging from their valuable discussion, as we shall now see.

7.2 The improved-formalization tendency Even1 if the argument just reported is successful in èxplaining away' the counterintuitiveness of the sentences (2.i), it is diÆcult to remain satis ed with it às giving the full story of the matter'. As we said above, formulations like (2.i) are admittedly ambiguous in ordinary language, so the possibility of understanding their consequents as expressing some notion of commitment (as illustrated by (2.5)) remains an interesting fact of linguistic usage. But then the problem arises how to formalize the notion adequately, which is surely a problem in view of the diÆculties pinpointed in Cases I{IV above. Let us now quickly nd out what attitudes Von Wright and Anderson took on this issue. In the second part of Von Wright [1956] we are warned against interpreting the form

O(A ! B ) to mean `doing A commits us (morally) to do B ' (p. 508). Von Wright says that if we do so

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: : : then a `paradox' instantly arises. For then we should have to say that a forbidden act commits us to any other act (whether obligatory, permitted, or forbidden). And this, obviously, con icts with our ìntuitions' in the matter. Perhaps we can say that Von Wright's reason for his caveat amounts essentially to Case III. Again, Anderson [1956, p. 185], expresses a similar warning: \... we should be wary in interpreting OCpq as meaning `p commits us to q'." Being thus dissatis ed with O(A ! B ) (and with A ! OB , we may add in the case of Anderson) as adequately re ecting the everyday notion of commitment, our authors are naturally led to look for new ways of formalizing that notion. They tried dierent approaches, though, which we are now going to describe. Von Wright [1956] introduces a new primitive symbol P (p=c), to be read as: p is permitted under conditions c. He then de nes O(p=c) as :P (:p=c), for which he suggests the reading: p is obligatory under conditions c, or: c commits us to (do) p. Furthermore, he gives two axioms for the new primitive operator, on the basis of which, to the best of my knowledge, the rst known system of dyadic deontic logic was developed. Further additions to and re nements of the system were suggested in Von Wright [1964] and [1965]. This idea of using binary (i.e. two-place) primitives, expressing conditional or `relative' permission, prohibition and obligation, initiated a very fruitful line of research in the history of modern deontic logic. In eect, dyadic deontic logic as originating with Von Wright [1956], can be said to have dominated recent work in the eld up to this date. For the moment, we just remind the reader of the following contributions: Rescher [1958; 1962], Powers [1967], Danielsson [1968], Hansson [1969], Segerberg [1971], Fllesdal and Hilpinen [1971], Van Fraassen [1972], Von Kutschera [1973; 1974], Lewis [1974], Chellas [1974] and Spohn [1975]. Let us now turn to Anderson. He was able to make a dierent suggestion, because he had at his disposal a more powerful logical apparatus on which he wished to base the theory of deontic notions, including commitment: that of alethic modal logic with a propositional constant S symbolizing some penalty or sanction. So already in Anderson [1956] we nd him suggesting (p. 185) that an alternative (to OCpq) candidate for the formal analogue of commitment is C 0 pOq: `p entails that q is obligatory'. Here `C 0 denotes strict implication; using the symbolism adopted in the present essay, we write in the place of C 0 pOq:

(A ! OB ): In Anderson [1959], written as a reaction to the Rescher [1958] attempt to elaborate further the Von Wright [1956] proposal about dyadic deontic

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logic, Anderson repeats his suggestion and, highly importantly in my opinion, proceeds to lay down a number of adequacy criteria for a theory of commitment (or a viable analysis of that notion). Let us pause for a while to consider the import of the Anderson proposal. To start with, let us replace the Von Wright [1956] dyadic notation O(p=c) with the one adopted in the present essay, viz. OB A to be read: ìf B then it is obligatory that A' or: `B commits us to A'. For this notation, see Section 15 below. The Anderson proposal is then to the following eect: consider the following DEFINITION 14. Def com: OB A = df

(B ! OA):

Then, add Defcom to any suitable system K of alethic modal logic with the constant S (or with Kanger's Q), which already has de nitions of `standard monadic' obligation and permission (see below Sections 12 and 13). Suppose that K is well determined: we are then able to investigate the logic of commitment (i.e. the laws governing locutions of the form OB A) within K supplemented with Defcom . Having now broadly outlined the respective approaches of Von Wright and Anderson to the problem of formalizing a `reasonable' concept of commitment or conditional obligation, we have to face the obvious questions: What expectations are to guide us in this enterprise? What properties do we expect that concept to have (and not to have)? I call this the problem of adequacy criteria and will devote a special section to it.

7.3 Adequacy criteria for a theory of commitment or conditional obligation Consider the language of Dyadic Deontic Logic as described in Section 17 below. Its set of sentences is called 20;N and is such that, for any A; B in 20;N ; OB A and PB A are in 20;N as well. We now adhere to the just adopted reading of OB A as `B commits us to A'. We are looking for a theory L over this language in the sense of a proper subset of 20;N (why proper?), which is to serve as a viable and plausible logic of commitment. Let provability and unprovability in L be denoted by j L and 6 j L , respectively. Adequacy criteria for L may now be divided into (i) those to the eect that certain sentence schemata are to be unprovable in L, and (ii) those to the eect that certain schemata are to be provable in L. Let us begin by giving some examples of the rst category. In the light of our discussion of the sentences (2.i) (i = 1; : : : ; 4) we consider certain generalized sentence schemata arising from the translations t((2.i)) as follows. First, generalize p2 to an arbitrary formal sentence A

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and p3 to an arbitrary formal sentence B . We then obtain, e.g. from t((2.1)) the schema

:A ! (A ! OB ) Secondly, replace the consequent in this schema by the appropriate formal analogue of commitment available in our language of dyadic deontic logic, viz. OA B . We thus obtain in our present example:

:A ! OA B Performing the same operations with all the t(2.i)), we obtain three schemata which, in the light of the diÆcult Cases I{IV, we expect all to be unprovable in L. In other words, we expect L to satisfy the following three adequacy criteria belonging to the unprovability category: (C1) (C2) (C3)

6 j L :A ! OA B: 6 j L OB ! OA B: 6 j L O:A ! OA B:

where the monadic, i.e. without subscript, O-operator is de ned by

OB = df O> B where the constant >, known as verum, denotes some arbitrary tautologous condition. This proposal for handling monadic or àbsolute' obligations in dyadic deontic logic was made already by Von Wright [1956, p. 509]. I shall now make two observations. (i) All systems of Dyadic Deontic Logic dealt with in Sections 17{23 satisfy the criteria C1{C3. In particular, this is true of the strongly normal \core" system G. (see Section 23 below). (ii) Von Wright [1956] claims that his new deontic logic satis es C2 and C3. And Anderson [1959] explicitly adopts C3 (writing F p for O:p, where F means ìt is forbidden that'). EXERCISE 15. (1) Prove assertion (i) just made above! (2) Suppose Von Wright were to prove the claim reported in (ii) above; how should he go about doing so?

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As a further adequacy criterion for L, we consider (C4) 6 j L OB A ! OB^C A. A classical intuitive counterexample to the validity of the schema in C4 is provided by the following instance of it:

Op p3 ! Op ^p p3 which is read: if impregnating Suzy Mae commits John Doe to marrying her, then both-impregnating-and-killing-her commits John to marrying her. This is absurd, though, since the antecedent may be accepted as true while the consequent is rejected as false. This counterexample, or argument for C4, apparently originated with Powers [1967] and is elaborated by various subsequent writers, notably Danielsson [1968, p. 66 f], Hansson [1969, p. 392], Van Fraassen [1972, p. 418 f] and Van Eck [1981, p. 8]. The objectionable schema in C4 is pertinently called a principle of augmentation by Chellas [1974, p. 31]. EXERCISE 16. (after Danielsson [1968, p.67]). 2

2

4

(1) Suppose that our desired theory L satis es one of the following adequacy criteria belonging to the provability category:

j L OB A , O(B ! A). j L OB A , (B ! OA). Show that in each case L will violate C4! Show also that in each case L violates at least one of C1{C3! Let K be a system of alethic modal logic with the constant S (or Q) to which Defcom is added so that K satis es the following criterion: (C7) j KOB A , (B ! OA): Then, show that K or, strictly speaking, its deontic fragment, violates (C5) (C6)

(2)

C4!

(3) What assumptions concerning deducibility in L(K) have minimally to be made in order for your proofs to work? (4) Suppose that C1{C4 are accepted as reasonable adequacy criteria for L. What conclusion as to the status of C5{C7 are we to draw in the light of the above results? EXERCISE 17. Show that all systems of dyadic deontic logic considered in Sections 17{23 satis es the criterion C4! Hint: it is enough to prove that our strongly normal system G (Section 23 below) has this property.

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In Anderson [1959] it is suggested that our desired theory L of commitment is to meet the following six adequacy criteria, which are all to the eect that certain schemata should be provable in L (and thus belong to our second category): (C8) j L (A ^ OA B ) ! OB: (C9) j L (OA ^ OA B ) ! OB: (C10) j L (P A ^ OA B ) ! P B: (C11) j L (OA B ^ OB C ) ! OA C: (C12) j L OA B ! O(A ! B ): (C13) j L O:A A ! OA: EXERCISE 18. (1) Show that the system G, as described in Section 23 below, satis es the criteria C9,C10,C12 (which is a weakened version of C5) and C13! (2) Consider any system of dyadic deontic logic which is an axiomatic extension of Ody S5N (see Section 18 below) and is dealt with in Part VI. Determine, for each of the four criteria just mentioned, which is the weakest system satisfying that criterion! (3) Consider again our strongly normal system G (Section 23). Show that it neither satis es C8 nor C11! In Aqvist [1963] the following criticism was levelled against the Andersonian set C8{C13 of criteria. (On page 25, note 2 of that paper, the gist of the argument was credited to T. Dahlquist.) Suppose that L satis es both C8 and C9. Then, provided only that L possesses a certain minimal deductive power, L will satisfy the following condition C14: (C14) j L (OA B ^ O:A :B ^ OA ^ :A) ! (OB ^ O:B ): Suppose further that L meets this condition: (C15) j L :(OB ^ O:B ): Then, as we easily show using modus tollens, L will also meet this condition: (C16) j L :(OA B ^ O:A :B ^ OA ^ :A): The refutability in L of the schema inside the negation-sign in C16 now amounts to this: whatever be meant by A and B here, the following conjunction (or set) of assumptions is logically impossible or provably false

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in L:

A commits us to B (not: A) commits us to (not B )

and it is obligatory that A it is not the case that A. But, the argument goes on, this result is counterintuitive, because we can nd A and B , as well as English readings of them, for which that conjunction (or set) appears to be perfectly possible or consistent logically. A famous case in point is the so-called Chisholm contrary-to-duty imperative paradox, rst stated in Chisholm [1963] and later discussed by a number of authors, e.g. Von Wright [1964; 1965], Sellars [1967], Aqvist [1966; 1967], Powers [1967], Hansson [1969], Fllesdal and Hilpinen [1971], Mott [1973], al-Hibri [1978], Tomberlin [1981], Van Eck [1981], and presumably several others. We now address ourselves to that puzzle. 8 RODERICK M. CHISHOLM'S CONTRARY-TO-DUTY IMPERATIVE PARADOX Several versions of the puzzle are known in the literature. Following Van Eck [1981] I shall consider a Suzy Mae version of it, which explicitly involves the notion of commitment: I. II. III. IV.

It ought to be that John does not impregnate Suzy Mae. Not-impregnating Suzy Mae commits John to not marrying her. Impregnating Suzy Mae commits John to marrying her. John impregnates Suzy Mae.

Let C = fI,II,III,IVg. We note that the set C is, from an intuitive standpoint, both consistent in the sense that no contradiction follows from it and non-redundant in the sense that none of its members follows from the remainder of the set. We then expect any adequate formalization of I{IV to preserve both these properties. Let us call this adequacy criterion our requirement of consistency and non-redundancy. We now consider three attempts to formalize the sentences I{IV, using only the resources of Monadic Deontic Logic.

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First attempt: Ia. O:p2 : IIa. O(:p2 ! :p3 ): IIIa. O(p2 ! p3 ): p2 : Objection: Let L be any of the ten Smiley{Hanson systems. We have that IIIa is L-derivable (L-deducible) from Ia, although III does not follow logically from I. Hence, the non-redundancy part of our requirement is violated by this proposal. IVa.

Second attempt:

Ib(=Ia). O:p2 : IIb. :p2 ! O:p3 : IIIb. p2 ! Op3 : IVb(=IVa). p2 : Objection: IIb is L-derivable from IVb, although II is not a logical consequence of IV. Therefore, non-redundancy is not preserved by this formalization either, contrary to our requirement.

Third attempt: Ic(=Ia). IIc(=IIa). IIIc(=IIIb). IVc(=IVa).

O:p2 : O(:p2 ! :p3 ): p2 ! Op3 : p2 :

Objection: Let L be any of the Smiley{Hanson +-systems, having the characteristic axiom schema A3: OA ! P A, which, by A1, is equivalent to OA ! :O:A (see Section 10:2 below): We then observe that fIc,IIcg j L O:p3 and that fIIIc,IVcg j L Op3 . Hence fIc,IIc, IIIc,IVcg j L ?, so the present formalization fails to preserve the consistency of C , contrary to our requirement. EXERCISE 19.

(i) The objections to the three attempts just considered rest on claims about derivability in all, or certain, Smiley{Hanson systems L. Give careful proofs (in full detail) of these claims!

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(ii) What happens to the objection to the third attempt, if the restriction to the Smiley{Hanson +-systems is dropped? (iii) Consider the formal sentences Ia{IVa, IIb, IIIb, which are all in . Which English sentences in , i.e. the set of sentences of the fragment FNE3 (see Section 7 above), are such that these formal sentences are the t-translations of the latter English sentences, respectively? What relation do those English sentences bear to I{IV above? Let us now pass to consideration of certain attempts to formalize the sentences I{IV, using the stronger resources of dyadic deontic logic, i.e. the language described in Section 17 below and its set of sentences 20;N .

Fourth attempt: Id(=Ia). IId(=IIa). IIId. IVd(=IVa).

Fifth attempt: Ie(=Ia). IIe. IIIe(=IIId). IVe(=IVa).

O:p2 : O(:p2 ! :p3 ): Op p3 : p2 : 2

O:p2 : O:p :p3 : Op p3 : p2 : 2

2

The fourth and fth attempts to deal with Chisholm's puzzle give rise to the following result: THEOREM 20 (Contrary-to-duty imperative paradox). Let L be any of

the systems of dyadic deontic logic presented in Sections 15{23. Let

C d = fId; IId; IIId; IVdg C e = fIe; IIe; IIIe; IVeg: Then: (i) C d is L-consistent in the sense that ? (falsum) is not L-derivable from C d. In symbols: C d6 j L ?. (ii) C e is L-consistent in the same sense, i.e. C e6 j L ?.

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(iii) C d is L-non-redundant in the sense that none of its members is Lderivable from the remainder of C d. (iv) C e is L-non-redundant in the same sense. In short, the intuitive content of the theorem is to the eect that the fourth and fth attempts both satisfy our requirement of consistency and non-redundancy with respect to any dyadic system L of a certain kind. Proof.[Sketch]

It is enough to prove the points (i){(iv) for the case where To begin with, let us have a look at the diagram shown in Figure 1. The meaning of this is that it represents a set W of possible worlds (or situations), consisting of four distinct members x; y; z; u; which are ranked by a binary relation of strict preference or strict betterness. That relation is represented by , and the ranking order is from left to right, so that u is the best member of W; x the second best etc.Moreover, the proposition letters p2 ; p3 (read in accordance with D10 and D11, Section 7 above) are taken to be true/false at dierent worlds as shown by the diagram:

L = our strongly normal system G (why?).

p2 is true at x and y, but false at z and u p3 is true at x and z , but false at y and u.

u

Æ :p2 :p3 :p2 ! :p3

W

x Æ p2 p3

z

Æ :p2

y Æ

p3

p2

:p3

Figure 1. Molecular Boolean compounds of these àtoms' receive truth-values according to the familiar tables. But how do we handle sentences of the forms OA and OB A (expressing àbsolute' and `conditional' obligations, respectively)? The main suggestion embodied in the dyadic approach to deontic logic (and perhaps most clearly stated by Hansson [1969]) comes down to this. Letting B (A) be any sentence in 20;N , we mean by a B (A)-world any world in W at which B (A) is true; then, we propose the following truth condition for any sentences of the form OB A, relatively to any world w in W : TC.

OB A is true at w i all the best B -worlds are A-worlds.

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Consider now the point y in W , i.e. the `worst of all possible worlds in W ' according to the ranking . For each member of C d we want to gure out whether it is true at y or not: Id? Remembering the Von Wright [1956] type de nition of the monadic O-operator (Section 7.3 above)

OB = df O> B

we obtain by TC that O> :p2 is true at y i all the best >-worlds are :p2 worlds. But the set of >-worlds = W (why?) and the set of best >-worlds, according to , = fug, i.e. the unit set of u. Now, :p2 is true at u, so all the best >-worlds are :p2 -worlds. Hence, by TC,Id is true at y. IId? The same kind of argument is helpful in establishing the truth at y of IId. IIId? By TC we have that Op p3 is true at y i all the best p2 -worlds are p3 -worlds as well. Now the set of p2 -worlds = fx; yg and the set of best p2 -worlds = fxg. p3 is true at x, so all the best p2 -worlds are p3 -worlds. Hence, by TC,IIId is true at y. 2

IVd? By our diagram, p2 (=IVd) is true at y. Upshot so far: every sentence in the set C d is true at y. We can now proceed to establish point (i) of our theorem. First of all, we claim that the `model' pictured in Figure 1 can be used to de ne a strong deontic H3 -model U , in the strict sense introduced in Section 22 below, which is such that all members of C d are true at y in U . (The construction, or de nition of U is left as an exercise to the reader.) Suppose then, contrary to (i), that C d is not G-consistent. By the de nition of the latter notion (see already Section 10.2.1) we obtain:

j G (Id ^ IId ^ IIId ^ IVd) !? or simply:

j G :(Id ^ IId ^ IIId ^ IVd) which is to the eect that the negation of the conjunction of the members of C d is provable in G.

However, since that negation is provable in G, then, by the Soundness Theorem for G (Theorem 72 below), it is strongly deontically H3 -valid, which means in particular that it is true at y in our model U just constructed. But, as all members of C d are true at y in U , their conjunction must be true at y as well. Contradiction. This proves point (i) of the theorem.

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To deal with (ii) we only have to show that the sentence IIe in C e is true at y in the intuitive model picture in Figure 1. Well, O:p :p3 is, by TC, true at y i all the best :p2 -worlds are :p3 -worlds as well. Now, the set of :p2 -worlds =fu; z g and the set of best :p2 -worlds = fug. :p3 is true at u (by our diagram), so all the best :p2 -worlds are :p3 -worlds. Hence, by TC,IIe is true at y. The remainder of the proof of point (ii) parallels that of (i). Our strategy in dealing with the non-redundancy points (iii) and (iv) is the following. Suppose we want to show that in G IVd(=p2 ) is independent in C d in the sense that fId, IId, IIIdg6`GIVd. Suppose that we nd , i.e. are able to construct, a strong deontic H3 -model U and a world w in U such that Id,IId,IIId as well as the negation of IVd are all true at w in U . We then use the Soundness Theorem for G to conclude (exactly how?) that IVd is not G-derivable from fId,IId,IIIdg. Following this strategy, the proof of (iii) and (iv) is almost routine and can be left to the reader. Just a few hints: The case of the independence of IVd(=IVe) in C d and C e is particularly easy: use the same model as above, but consider the point z , at which p2 is false, instead of y. For the case of Id(=Ie): stick to y as the `point of evaluation' in the above model, but change in such a way that x is ranked above u. For the cases of IId and IIe: stick to y in the original model, but assume p3 to be true at u. And so on. This completes the outline of a proof of the Theorem on the contrary-toduty imperative paradox. 2

8.1 On the choice between the fourth and the fth attempt Suppose we grant that the formalizations of I{IV codi ed in the fourth and fth attempts are superior to their predecessors, because they preserve both the intuitive consistency and the intuitive non-redundancy of the set C . Which of C d and C e are we then to choose? I shall now oer an argument for taking a neutral position on this issue: it does not matter which one we choose, we may leave the choice open. Let us ask, to begin with: as C d and C e dier only with respect to their second member, what is the logical relation of IId to IIe? In answer to that question we state and prove the following result: LEMMA 21 (IId and IIe). Let L be any of the dyadic systems dealt with in

Part VI below. Then: (i) If L contains the system E (see Section 22 at the end), then

j L O:p :p3 ! O(:p2 ! :p3 ): 2

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(ii) If L contains the system G (Section 22 at the end), then

j L p> :p2 ! O(:p2 ! :p3 ) , O:p :p3 ): 2

Proof.

Ad (i): 1. O:p :p3

hypothesis

2

2. N (:p2 , (> ^ :p2 )) 3. O:p :p3 , O>^:p :p3 2

2

contains S5 for N from 2 by 0 (Section 18), which is an axiom schema in E, using E

modus ponens 4. O>^:p :p3

from 1 and 3 by propositional logic

5. O>^:p :p3 ! O> (:p2 ! :p3 )

instance of 2 (Section 18), which is an axiom schema in E

6. O> (:p2 ! :p3 )

from 4,5 by modus ponens

2

2

The sequence 1{6 is a deduction in E and hence in any L of the sort under consideration. Rewriting O> as O in 6, we obtain the desired result (i) by the rule of conditional proof (or, if you like, the Deduction Theorem), which is valid in (for) E, of course.

Ad (ii): 1. P> :p2 ! (O:p :p3 ! O> (:p2 ! :p3 ))

from (i) by propositional logic, since G contains E

2. P> :p2 ! (O> (:p2 ! :p3 ) ! O>^:p :p3 )

instance of 4 (Section 18), which is one of the characteristic axiom schemata in G

3. P> :p2 ! (O> (:p2 ! :p3 ) ! O:p :p3 )

from 2 and line 3 in the proof of (i) above, using propositional logic

4. P> :p2 ! (O> (:p2 ! :p3 ) , O:p :p3 )

from 1 and 3 by propositional logic

2

2

2

2

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The present sequence 1{4 is a rather fragmentary proof in G of its last line, 4. Rewriting O> as O in 4, we obtain the desired result (ii). COROLLARY 22. Suppose that L contains the system G. Then:

(iii) C e j L IId (iv) C d j L IIe Here, (iii) follows from (i) and the fact that G contains E. Again, (iv) is obtained from (ii) as follows: using 3 (Section 18), which is a somewhat controversial schema of G, together with the fact that j G M > (G contains S5 for M ), we see that the sentence Id entails P> :p2 , i.e. the antecedent of 4 above. So, using that result together with 4 and IId, we obtained the desired conclusion IIe. Proof.

Clearly, in spite of IId being in general weaker than IIe, (iii) and (iv) are jointly to the eect that it does not matter which of C d and C e we choose as the `correct' formalization of C ; provided, however, that G can be accepted as a satisfactory dyadic system. I have nothing against assuming this to be the case. Bearing in mind that G is a generalized version of Hansson's DSDL3, this attitude of mine should be shared, e.g. by Hansson [1969], Fllesdal and Hilpinen [1971] and Spohn [1975]. 9 PROBLEMS UNSOLVED BY THE DYADIC APPROACH; THE NEED FOR TEMPORAL AND QUANTIFICATIONAL RESOURCES IN THE BASIC LANGUAGE OF SATISFACTORY DEONTIC LOGICS; ON THE LOGIC OF ACTION; FAILURE OF LEFT-TO-RIGHT ADEQUACY Our discussion of Prior's paradoxes of commitment and Chisholm's contraryto duty imperative paradox was mainly designed to show how the idea of dyadic deontic logic naturally arises as an attempt to cope with these diÆculties. The Exercises on such adequacy criteria for a logic of commitments as C1{C13 (Section 7.3 above) as well as our Theorem on the Contraryto-Duty Imperative Paradox and Lemma on IId and IIe should have given the reader a fairly clear opinion of the virtues of the dyadic approach and a nice explanation why it has proved to be such a powerful trend of thought in the development of modern deontic logic. It is now time to turn to its vices, i.e. to certain problems or problem-areas which the dyadic approach appears unable to handle. Mainly following the admirable survey given in Van Eck [1981], we present a list of such problems or problem-areas: (I.)

The dilemma of commitment and detachment.

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(II.) Prima facie vs. actual obligation; the ceteris paribus proviso. (III.) The ought-implies-can problem. (IV.) The Good Samaritan paradox and the Jephta dilemma. (V.) The problem of the relationship of act-utilitarianism to deontic logic. I shall now brie y state these diÆculties. After having done so, I shall then, without going into details, indicate what I take to be the proper attitude to them and what conclusions are in my opinion `reasonably' to be drawn from them.

9.1 Survey of diÆculties (after Van Eck [1981]) I. The dilemma of commitment and detachment. Suppose we were to accept a dyadic system above):

L

satisfying the Andersonian criterion C8 (Exercise 18

(C8) j L (A ^ OA B ) ! OB so that L allows a principle of detachment to be valid for commitmentexpressing formulae of the type OA B . Then, provided only that L is suÆciently strong in other respects (which?), we quickly obtain results like

C d(e)j L Op3 C d(e)j L O:p3 ^ Op3 C d(e)j L ? in violation of our requirement that the consistency of C should be preserved. Hence, if L is of this kind, the fourth and fth attempts both break down as solutions to the Chisholm puzzle. Now, these attempts may be defended by claiming that any `correct' dyadic system must, like our G and others, not allow detachment for commitment; it must de nitely not satisfy the criterion C8. In short, in order for the dyadic solutions to work, detachment should not be possible. However, detachment is not so easily given up from an intuitive standpoint. Here are some voices from the literature: In nothing like schema (i) (sc. the one in C8) is valid, how can conditional obligation-sentences play the important role in normative argumentation which they seem to play? (Danielsson [1968, p. 66]).

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How can we take seriously a conditional obligation if it cannot, by way of detachment, lead to an unconditional obligation? (Van Eck [1981, p. 23]). So, on the other hand, we seem to feel that detachment should be possible after all. But we cannot have things both ways, can we? This is the dilemma on commitment and detachment.

II. Prima Facie vs. Actual Obligation; the Ceteris Paribus Proviso. Suppose

that at a given time t John promises Suzy to marry her at a certain later time t + 7. Assume that the promise gives rise to an obligation for John to marry Suzy at t + 7, and that this obligation comes into force at time t. Now, in the meantime between t and t + 7, various things might happen that make it impossible for John to ful ll his obligation. For instance, he learns that his mother in Australia is dying, whence there arises, say at time t + 3, an obligation for John to go and visit her in Australia immediately. John takes o, but is then unable to marry Suzy at t + 7, so he breaks his promise and violates his rst obligation. Following Hintikka [1971] we may characterize this situation as one where an earlier obligation, due to the promise, is overruled by a stronger obligation, which arises in the meantime between the moment at which the rst obligation comes into force (= t) and the moment of its ful llment (= t +7). Hintikka [1971] goes on to suggest that the famous prima facie vs. actual duty distinction (Sir David Ross [1930; 1939], Richard Price [1948]) should somehow be applicable to this situation. Following Van Eck [1981], then, I think we may say that at time t + 3 the earlier obligation, though still in force at that time, is a mere prima facie duty, whereas the later and stronger obligation has acquired the status of an actual duty of John's. Our present problem concerns the explication of this distinction and its formal representation in systems of deontic logic; apparently, the issue was rst raised by Hintikka [1971] and further discussed by Purtill [1973], Bergstrom [1974] and Van Eck [1981]. Furthermore, if we try to bring out the distinction by saying that John's rst prima facie duty carries an implicit or tacit ceteris paribus rider òther things being equal',whereas his second actual duty does not, we face the problem of analyzing the import of ceteris paribus provisos, in general as well as in the particular case at hand.

III. The Ought-Implies-Can Problem. My remarks on the celebrated Kan-

tian principle will by necessity be very brief and will fail to do justice to the impressive richness of the literature on it. We ask: does òught' imply `can'? then, if the answer is Yes, in what sense of (at least) (i) `can' and (ii) ìmply'? Again, there are at least two alternatives under each heading here: (i.i) `can' means logical possibility, and (i.ii) `can' means some stronger possibility of a more `practical' or `real' kind, which might be explicated as a temporally dependent possibility in a sense that seemingly originates with

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Montague [1968] and is further developed, e.g. by Chellas [1969] and Van Eck [1981]; moreover, ìmply' could mean (ii.i) ordinary logical consequence, or (ii.ii) some dierent form of consequence, say, the interesting notion of deontic consequence proposed in Hintikka [1957] and [1971]. Having thus surveyed some candidates in the area, I just think Van Eck [1981, II. Section 2.2], has given excellent reasons for regarding the combination (i.ii) with (ii.i) as providing the most viable and interesting interpretation of the Kantian principle: the alternative (i.i) seems to make it trivial, and the reasons against (ii.i) and for (ii.ii) are far from clear. But there are independent positive reasons as well for interpreting `can' as temporally dependent, or historical, possibility and ìmply' as ordinary logical consequence.

IV. The Good Samaritan Paradox ant the Jephta Dilemma. The rst puzzle

here goes back at least to Prior [1958] and has been discussed in a number of contributions, of which we only mention Danielsson [1968], Wedberg [1969], Van Fraassen [1972], Casta~neda [1968a; 1974], Tomberlin and McGuinness [1977] and Van Eck [1981]. Interestingly, though, Knuuttila [1981] points out that in the fourteenth century versions of the paradox were known to and dealt with by Roger Rosetus in his Commentary on the Sentences. Again, the Jephta Dilemma (see the Book of Judges) was taken by Von Wright [1965] to be an interesting problem case for deontic logic, which illustrates such notions as those of a predicament and a con ict of duty. It has later been discussed extensively by Van Eck [1981]. The following version of the Good Samaritan paradox is presented in Tomberlin and McGuinness [1977] and goes back to Casta~neda [1974]; consider the argument: (5) If Bob pays $500 to the man he will murder one week hence, then Bob will murder a man one week hence. (6) It ought to be that Bob pays $500 to the man he will murder one week hence (because Bob owes that amount of money to the latter).

Therefore: (7) It ought to be that Bob will murder a man one week hence. In this argument, the rst premiss, (5), may be taken to be, not only true, but even logically true. As for the second, (6), let us just assume it to be true. On the other hand, (7), is plainly false, in spite of the fact that the premisses (5) and (6) are both true. Hence, the argument must be invalid. But, if we translate it into the language of any of the Smiley{ Hanson systems L of Monadic Deontic Logic, which are all closed under the

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201

rule of inference:

A!B (see Exercise 12 above) OA ! OB we nd that the translation of (7) is L-derivable from the translations of the premisses (5) and (6). What has gone wrong here? The Jephta Dilemma is reminiscent of the famous Morning Star Paradox in quanti ed modal logic with identity (see, e.g. Kanger [1957a]) and can be stated as follows. Consider the argument: (8) Miriam (i.e. the daughter of Jephta) is identical to the rst being that will meet Jephta on his return home. (9) It ought to be that Jephta immolates the rst being that will meet him on his return home (because he has promised God to do so).

Therefore: (10) It ought to be that Jephta immolates Miriam (his own daughter). Again, here, as it seems, the premisses are true while the conclusion is false. So the argument must be invalid. But if we formalize it in a suitable system L of deontic logic, it may well turn out that the inference is countenanced as valid in L. How are we to account for this?

V. The problem of the relationship of Act-Utilitarianism to deontic logic.

In Casta~neda [1967; 1968], Casta~neda points out the following intriguing diÆculty for the familiar ethical theory known as Act-Utilitarianism. My statement of the problem will involve some amount of `precization'. Let X be any moral agent, let C be any situation or set of circumstances, and let A be any act open to X in C (in the sense that it is possible for X to do A in C ). Then, the following is a central thesis of Act-Utilitarianism: (U) X ought to do A in C i for each act A0 such that (i) A0 is open to X in C and (ii) A0 is an alternative to A in C and (iii) A0 is distinct from A we have that the consequences of X 's doing A in C are better than those of X 's doing A0 in C . It may be that condition (iii) in this formulation of (U) is redundant, because entailed by (ii). Next, consider this set of assumptions:

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(11) X ought to do P ^ Q in C (where P ^ Q is a certain conjunctive act open to X in C ) (12) The three acts P ^ Q; P; Q are all (i) open to X in C , (ii) alternatives to each other in C , and (iii) distinct from one another. (13) If X ought to do P ^ Q in C , then X ought to do P in C and X ought to do Q in C . THEOREM 23. (after Casta~neda [1968]): The set f(11),(12),(13),(U)g is

inconsistent.

Exercise. (If you are unable to do it, see Casta~neda [1968]!) What assumptions about the preference relation better than do we minimally have to make in order to establish the present result? Proof.

We should note here that (13) is reminiscent of the principle asserting that O is distributive over ^, which is valid in all the Smiley{Hanson systems of monadic deontic logic. Now, suppose we stick to the assumptions (11) and (12): then we face the tough choice, described by Wedberg [1969], between (i) maintaining (13) and rejecting the utilitarian thesis (U) already on grounds of deontic logic; and (ii) maintaining (U) and rejecting the deontic-logical principle (13). Perhaps this is a `false dilemma', though: why not abandon (12) and try to save both (13) and (U)? I shall not discuss here this proposal and others that might be or have been made. I just like to point out that in Aqvist [1969] an attempt was made to relate Casta~neda's problem to the interesting work done by Bergstrom [1966] on utilitarian and teleological ethics. The notion of the alternatives to an action is seen to play a crucial role in Bergstrom's analysis and was further scrutinized by Prawitz [1968; 1970], Bergstrom [1968] and by other contributors. The discussion quickly turned out to be surprisingly complex and it is diÆcult to give a fair assessment of its outcome.

9.2 Diagnosis I have now nished my survey of the problem areas I{V. In my opinion, the existence of these diÆculties, together with various unsuccessful attempts to overcome them, give considerable support to the diagnosis that the languages of the current systems of deontic logic are far too poor to function as a satisfactory medium for formulating cues for the moral agent (Van Eck [1981, p. 1]).

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I think this conclusion is born out in an especially clear way by the Van Eck treatment of problem II and his fascinating account of how prima facie obligations pass into actual duties as time elapses. The account is based on a system of temporal logic, more speci cally: a system of temporally relative modal and deontic predicate logic. In my view, the virtues of this system are shown by its capacity to handle, not only the paradoxes of commitment and the contrary-to-duty imperative, but, as Van Eck also shows, the four problem areas I{IV as well. Handle them in a more convincing way than has so far been done up to this date, that is to say. As Van Eck observes in the preface to his [1981], though, he is not alone in having conceived of the idea of constructing a semantics for a notion of temporally relative necessity and basing a semantics of temporal deontic notions upon it: similar ideas can be found in Aqvist and Hoepelman [1981], going back to Chellas [1969] and Montague [1968] (see problem III above). Also, Thomason [1981] (deriving from an original 1970 version) and [1981a] should be mentioned in the present context. Now, the Van Eck framework appears to be richer than these rival ones, because it uses temporal variables, constants and quanti ers in the object-language. And this, I think, makes it more useful for philosophical applications; an illustration of this claim is perhaps the little paper Aqvist [1981], where precisely a Van Eck-type framework is applied to the ancient so-called Protagoras paradox (see also Lenzen [1977] and Smullyan [1978]). Among recent contributions in the same vein, those of Bailhache [1991] (going back to work done in the early eighties) and [1993] strike me as particularly valuable. See also Aqvist [1997a]. There are strong reasons, then, for enriching the basic language of satisfactory deontic logics with explicit temporal resources. Moreover, problems IV and V nicely show, I take it, the need for quanti cational resources in that language as well; how could we otherwise even begin to state those problems in an intelligible way? The indispensability of quanti ers in deontic logic was, on general grounds, very well argued by Hintikka already in his [1957] (and later in his [1971]), when he comments on the fundamental work done by Von Wright [1951; 1951a] as well as by Prior [1955]. A system of deontic predicate logic (quanti cation theory) is also presented in Kanger [1957]; his system is one with identity and, interestingly, blocks deontic analogues of the Morning Star paradox such as the Jephta dilemma in the model-theoretic semantics given for it. Again, the need for quanti ers in deontic logic seems to be one of the main tenets of Casta~neda's in an impressively large number of contributions, of which we mention here only Casta~neda [1954; 1959; 1981]. We must brie y touch on the following question: given the indispensability of quanti ers in deontic logic, exactly over what sort of entities do we have to quantify? agents, patients, times, places, circumstances or what? Hintikka [1957] suggested individual acts; in Makinson [1981] this suggestion

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is shown to give rise to considerable interpretational diÆculties. See also Robison [1964] for further suggestions. In answer to this question I would like here to recommend a very broad, liberal and open-minded attitude: as deontic logicians, we should be prepared to quantify over whatever entities the ethical theory or normative (e.g. legal) system requires us to consider seriously, and to adjust our deontic predicate logic accordingly. We should not, I contend, worry too much about ontological commitments; in today's research situation the important thing is to get the right kind of structure going.

9.2.1 On the logic of action Having now stressed the importance of temporal and quanti cational machinery to viable deontic logics, there is a third research trend in our area, to which I like to draw the reader's attention. This trend claims that such logics ought to be combined with a logic of action; it is usually taken to have been initiated in Von Wright [1963] and followed up, e.g. in Von Wright [1967] (with comments by Chisholm [1967]) and Von Wright [1974]. However, if we consider the distinguishing mark of a logic of action to be the presence in its basic language of a special `causal' operator of agency (expressing that an agent brings it about, sees to it, makes it true that so-and-so is the case), we might just as well credit Kanger [1957], Anderson [1962] and Kanger and Kanger [1966] with the idea. Anyway, the latter authors apply it in attempts to reconstruct and to extend the Hohfeldian system of jural relationships as set forth by Hohfeld [1919]. In this endeavor they are followed by, notably, Porn [1970], Anderson [1971] and Lindahl [1977]. This movement is highly interesting and promising for the future, and any account of present-day deontic logic would be seriously incomplete if it did not mention it. Finally, I like to close this Part by making a remark on the Good Samaritan paradox, which is intended to illustrate the notion of left-to-right adequacy (Section 5.4 above) and its failure in connection with the Smiley{ Hanson systems of monadic deontic logic.

9.2.2 Remark on the Good Samaritan paradox Consider the de nitional enrichment L(D1{D14) of L, where D13 and D14 are as follows: (D13) Bob pays $500 to the man he will murder one week hence =df p5 (D14) Bob will murder a man one week hence =df p6 Extending the translation t in the obvious way, we obtain the following formalization of the sentence (5) in the paradox:

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t(5) = (p5 ! p6 ): Note, then, that (5) is in (= the sentence-set of the fragment FNE6 ) and that (5) is logically true (presumably). Hence, (5) is in NDL. On the other hand, t(5) is not provable in any of the Smiley{Hanson systems L (why?). So we have a counterexample to the left-to-right adequacy of t with respect to NDL and L. The import of this counterexample is that there are validities in natural deontic logic (NDL), or relations of `natural' logical consequence, which fail to be representable in certain formal deontic logics, such as L. The reason is, of course, that we need a quanti cational formal framework with de nite descriptions in order adequately to formalize such a sentence as (5); the propositional language L is simply not expressive enough. We should contrast this counterexample with those met with above, which purported to show that t was not right-to-left adequate with respect to NDL and L, in the sense that the latter sanctions more logical validities than the former. The present situation is precisely the opposite one. The Good Samaritan paradox is usually cited as an instance of failing right-toleft adequacy, just as those of commitment etc.I think it is of some interest to note that it could also be used to illustrate the opposite failure. III. TEN SMILEY{HANSON SYSTEMS OF MONADIC DEONTIC LOGIC 10 LANGUAGE, PROOF THEORY AND SEMANTICS

10.1 Language 10.1.1 Alphabet Our alphabet consists of (i) a denumerable set Prop of proposition letters p; q; r; p1; p2 ; : : : ; (ii) the primitive logical connectives > (verum), ? (falsum), : (negation), O (obligation), P (permission), ^ (conjunction), _ (disjunction), ! (material implication) and , (material equivalence); and (iii) the parentheses ( ).

10.1.2 Sentences (well formed formulas, ws): The set of all sentences of our language is de ned as the smallest set S such that

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(a) every proposition letter in Prop is in S , (b)

> and ? are in S ,

(c) if A is in S , then so are :A; OA and P A, (d) if A; B are in S , then so are (A ^ B ); (A ^ B ); (A ! B ) and (A , B ): The sentences under (a) and (b) are the atomic sentences of the language.

10.1.3 Degrees of logical connectives > and ? are of degree 0; :; O; P are of degree 1; and the remaining conectives are all of degree 2.

10.1.4 De nition F A = df :P A (alternatively: O:A): 10.1.5 Conventions for dropping brackets Brackets are omitted in accordance with these canons: (i) Connectives of degree 1 bind more strongly than connectives of degree 2. (ii) Among the latter, ^ and _ bind more strongly than ! and ,. (iii) Outer brackets are mostly dropped around sentences.

10.2 Proof theory The following two rules of inferences are common to all the ten Smiley{ Hanson systems of monadic deontic logic to be dealt with: (R1)

A; A ! B (modus ponens). B

A (O-necessitation). OA Consider next the following list A0{A7 of axiom schemata: (R2)

DEONTIC LOGIC

(A0) (A1) (A2) (A3) (A4) (A5) (A6) (A7)

207

All truth-functional tautologies (over our present language) P A , :O:A O(A ! B ) ! (OA ! OB ) OA ! P A OA ! OOA P OA ! OA O(OA ! A) O(P OA ! A)

The ten logics to be studied here are called OK,OM,OS4,OB,OS5, and OS5+ . They are de ned as follows (where R1 and R2 are assumed for all): OK+ , OM+ ,OS4+ ,OB+ ,

OK OM OS4 OB OS5

= = = = =

A0{A2 A0{A2,A6 A0{A2,A4,A6 A0{A2,A6,A7 A0{A2,A4,A5 (note that A6 and A7 are derivable in OS5)

Again, let L be any of these ve systems. Then:

L+ = L; A3 Of these ten deontic logics, OK is (apart from unessential dierences) identical to the system F of Hanson [1965], OK+ to his D,OB+ to his DB, whereas OB is discussed neither by Hanson [1965] nor by Smiley [1963]. The remaining six systems are named exactly as in Smiley [1963].

10.2.1 Provability and consistency Let L be any of the ten systems just de ned. Then, the set of L-provable sentences (or the set of L-theses) is the smallest set S such that (i) each instance of every axiom schema of L is in S , and (ii) S is closed under the rules R1 and R2. We write `j L A' to indicate that A is L-provable. Also, a set S of sentences is L-inconsistent i there are B1 ; : : : ; Bn in S (n 1) such that j L (B1 ^ ^ Bn ) !?; and S is L-consistent otherwise. Again, we say that a sentence A is L-derivable from a set S of sentences, in symbols: S j L A, just in case S [ f:Ag is L-inconsistent. Clearly, j L A i

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;j L A, i.e. the L-provable sentences are exactly those that are L-derivable from the empty set.

10.3 Semantics 10.3.1 Models By a model we mean a triple U = hW ; R; Vi where: (i) W is a non-empty set (heuristically, of `possible worlds' or `possible situations'). (ii) R W W (a binary relation on W , heuristically, of `deontic alternativeness' or `co-permissibility'). (iii) V is an assignment, which associates a truth-value 1 or 0 with each ordered pair hp; xi where p is a proposition letter and x is an element of W ; in technical jargon, V : Prop W ! f1; 0g.

10.3.2 Truth conditions Let U = hW ; R; Vi be any model, let x be any member of W , and let A be in . We want to de ne what it means for A to be true at x in U , in U symbols: = x A. As usual, the de nition is recursive on the length of A: =Ux p i V (p; x) = 1 (for any p in Prop).

=Ux >:

U not = x ?. U =Ux :A i not = x A.

=Ux OA i for every y in W such that xRy; =Uy A.

=Ux P A i for some y in W such that xRy; =Uy A.

U U B. =Ux (A ^ B ) i = x A and = x

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U (A _ B ) i =U A or = x x

209

U = x B (or both).

U U U A; then = = x (A ! B ) i if = x x B.

U (A , B ) i ( =U A i =U B ). = x x x

10.3.3 Conditions on R in a model Corresponding to the ve axiom schemata A3{A7 we now list ve conditions on the relation R in a model (where we assume the variables `x', `y', `z ' to range over W , and where we use the symbols &; ; 8 and 9 as a shorthand notation in the metalanguage in the obvious way): (R3) (R4) (R5) (R6) (R7)

R is serial in W : R is transitive in W : R is Euclidean in W : R is almost re exive in W : R is almost symmetric in W :

8x9y(xRy) 8x; y; z (xRy&yRz xRz ) 8x; y; z (xRy&xRz yRz ) 8x; y(xRy yRy) 8x; y; z (xRy (yRz zRy))

10.3.4 Classi cation of models We now use the restrictions on R just listed to obtain a subcategorization of the set of all models into various kinds. Thus, we stipulate that: = the class of all models (no condition on R being imposed). The class of OM-models = the class of all models with almost re exive R. The class of OS4-models = the class of all models with transitive and almost re exive R. The class of OB-models = the class of all models with almost symmetric and almost re exive R. The class of OS5-models = the class of all models with Euclidean and transitive R. The class of OK+ -models = the class of all models with serial R. The class of OM+-models = the class of all models with serial and almost re exive R. The class of OS4+-models = the class of all models with serial, transitive and almost re exive R. The class of OK-models

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The class of OB+ -models

= the class of all models with serial, almost symmetric and almost re exive R. The class of OS5+ -models = the class of all models with serial, Euclidean and transitive R. In this chain of de nitions, we always take the relevant restrictions on R to be relative to the world-set W in a model, so that `serial' means `serial in W ', and so on.

10.3.5 Validity and satis ability Let L be any of the ten systems OK,OM,OS4,OB,OS5, OK+ ,OM+, OS4+ , OB+ , OS5+ . We say that a sentence A is L-valid (in symbols: U = A) i = x A for all L-models U and for all x in W . Also, we say that a L set S of sentences is L-satis able i there is an L-model U and member x of W such that for all sentences A in S; =Ux A. Clearly, we have that = A i L the unit set f:Ag is not L-satis able. Again, we may introduce a semantic notion parallel to that of (prooftheoretic) derivability: we say that a sentence A is semantically L-entailed by a set S of sentences (in symbols: S =L A) i S [f:Ag is not L-satis able. We then have that = A i ; = A. L L 11 SEMANTIC SOUNDNESS AND COMPLETENESS OF THE SMILEY{HANSON SYSTEMS THEOREM 24 (Soundness Theorem). Let L be any of the systems OK, OM,OS4, : : : OS5+ . Then, for all A 2 , if j L A, then = A. In other L

words, all L-provable sentences are L-valid.

[Outlined] For each system L we must show that (i) every instance of every axiom schema of L is L-valid, and that (ii) the rules R1 and R2 preserve L-validity. Then, we can verify by inductions on the length of proof in L that if j L A, then = A. To do this is a bit tedious, for sure, but L entirely routine. Let us give just one example here in order to illustrate the methodology, or strategy, of argument. Proof.

EXAMPLE 25. Suppose we want to check that all instances of A5 are indeed OS5-valid. Assume otherwise, then, i.e. that there is a sentence A such that, for some OS5-model U = hW ; R; Vi and some x in W , we have:

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U (1) it is not the case that = x P OA ! OA

Applying relevant truth conditions suÆciently many times to (1), we reduce it to (2) =U P OA and =U P :A. x

x

Applying the truth condition for P , we obtain from (2): (3) =U OA, for some y in W with xRy y

as well as (4) =Uz :A, for some z in W with xRz: Since U is an OS5-model, R is Euclidean in W ; hence we obtain (5) yRz (because, by (3) and (4), xRy and xRz ). Then, applying the truth condition for O to (3), we get from (5): (6) =U A z

which result contradicts (4), as the latter gives us (7) not =U A z

by the truth condition for :. Contradiction. COROLLARY 26. Let L be as usual and let S be any set of sentences. Then, if S is L-satis able, then S is L-consistent. Assume otherwise, i.e. , that some S is L-satis able but not Lconsistent. Then, by the de nition of L-inconsistency, there are B1 ; : : : ; Bn in S such that j L (B1 ^ ^ Bn ) !?. Hence, by the Soundness Theorem, = (B ^ ^ B ) !?. But this means that for some x in the model U , whose n L 1 existence is guaranteed by S being L-satis able, we have =xU B1 ^ ^ Bn as well as =U (B ^ ^ B ) !?, hence =U ?. Contradiction. Proof.

x

1

n

x

THEOREM 27 (Completeness Theorem). Version I (strong completeness). Let L be as usual and let S . Then, if S is L-consistent, then S

is L-satis able. Version II (weak completeness). Let L be as usual and let

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A 2 . Then, if = A, then j L A. In other words, all L-valid sentences are L

L-provable.

Let us rst see how the weak version can be obtained as a corollary of the strong one. Assume, contrary to the weak version, that for some sentence A; = A but not j L A. Then, f:Ag must be L-consistent (otherwise, L we would have f:Agj L ?; j L :A !?, and j L A; but we assumed: not j L A). Therefore, by the strong version I, f:Ag is L-satis able, i.e. for some L U U = A. But this result con icts model U and for some x in W; = : A , so not x x Proof.

with = A. Contradiction. L

We are then justi ed in concentrating our eorts on establishing the strong version I of the Completeness Theorem. We begin by calling attention to the following de nitions and lemmata. DEFINITION 28 (L-saturated sets). Let L be as usual and let x be any set of sentences. We say that x is L-saturated i (i) x is L-consistent, and

(ii) for each sentence A, either A 2 x or :A 2 x. LEMMA 29 (L-saturated sets). Let x be any L-saturated set of sentences. Then, for all sentences A; B :

Every L-provable sentence is in x. x is closed under modus ponens (if A 2 x and A ! B 2 x, then B 2 x). (iii) T 2 x. (iv) ? 62x. (v) :A 2 x i A62x. (vi) A ^ B 2 x i A 2 x and B 2 x. (vii) A _ B 2 x i A 2 x or B 2 x. (viii) A ! B 2 x i if A 2 x then B 2 x. (ix) A , B 2 x i A 2 x if and only if B 2 x. (i) (ii)

Proof.

Familiar.

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213

LEMMA 30 (Lindenbaum's Lemma). Any L-consistent set x of sentences can be extended to an L-saturated set x+ with x x+ (L being as usual).

LEMMA 31 (Makinson's Lemma). Let L be as usual and let x be any L-saturated set of sentences. Let A be any sentence such that :OA 2 x. Let xA = fB 2 : OB 2 xg [ f:Ag. Then xA is L-consistent. Proof. (see Makinson [1966, p. 382]). Suppose xA is not L-consistent. Then there are sentences B1 ; : : : ; Bn (n 0) such that each OBi 2 x and such that j L (B1 ^ ^ Bn ^ :A) !?; by virtue of the fact that axiom schema A0 is in every L, then, such that j L (B1 ^ ^ Bn ) ! A: Consider rst the case where n = 0. This means that j L A. Then by the rule R2 of O-necessitation (common to all our L), we have j L OA. Hence, by the Lemma on L-saturated sets, OA 2 x. Thus, OA and :OA are both in x, so x is L-inconsistent. Contradiction. Consider next the case where n 1. Since j L (B1 ^ ^ Bn ) ! A, we have by a tautology under A0 and R1 that j L B1 ! (B2 ! : : : (Bn ! A) : : : ). Hence, by R2, j L O(B1 ! (B2 ! : : : (Bn ! A) : : : )). Hence, using axiom schema A2 (common to all our L) n times, together with R1 and appropriate tautologies under A0, we obtain that j L OB1 ! (OB2 ! : : : (OBn ! OA) : : : ). Hence, by the Lemma on L-saturated sets, that sentence is in x. But each OBi 2 x, so, by the same Lemma (clause (ii) applied n times, OA 2 x. Thus, OA and :OA are both in x, so x is L-inconsistent. Contradiction. DEFINITION 32 (Canonical L-models). Let L, as usual, be any of our ten monadic deontic logics, and let S be any L-consistent set of sentences, so that, by Lindenbaum's Lemma, S + is L-saturated and S S + . By the canonical L-model generated by S we mean the structure UL = hWL ; RL ; VL i Proof.

See, e.g. Makinson [1966, p. 381 f].

where: (i) WL = the smallest collection U of L-saturated sets such that: (a) S + is in U . (b) If x is in U , and A is a sentence with :OA 2 x, then (xA )+ is in U (where xA is de ned as in Makinson's Lemma). (ii) RL = the binary relation on WL such that for all x; y in WL : xRL y i for all sentences A, whenever OA 2 x then A 2 y.

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(iii) VL = the assignment de ned as follows: for each proposition letter p and each x in WL ; V (p; x) = 1 i p 2 x. LEMMA 33 (Veri cation Lemma). As just de ned, UL = hWL ; RL ; VL i is

an L-model.

LEMMA 34 (Coincidence Lemma). For each sentence A and for each x in U L WL (as de ned above), x A i A 2 x. We shall wait a little with the proofs of these two lemmata. Instead, let us see how they together yield Version I of the Completeness Theorem. [Completeness Theorem] (Version I): Letting L be as usual, assume S to be any L-consistent set of sentences. We are to show that S is Lsatis able. Well, by the Veri cation Lemma, UL (as just de ned) is an L-model. By Uthe Coincidence Lemma, we obtain in particular that for each sentence A; L A i A 2 S + (as S + , by de nition, belongs to WL ). Hence, S since S S + , we have UL A for every A in S . In other words, assuming Proof.

+

S

+

S to be any L-consistent set of sentences, we have constructed an L-model, viz. UL , such that for some x in WL , viz. S + , UxL A for each A in S ; i.e. we have shown S to be L-satis able. We still need one more lemma before being able to establish the Veri cation Lemma and the Coincidence Lemma (the proofs of which have not yet been given): LEMMA 35 (Saturation Lemma for canonical L-models). Let L be as usual, let S be any L-consistent set of sentences, and let UL be de ned as above. Then, WL is such that for all sentences A and all x in WL :

(i) OA 2 x i for all y in WL with xRL y; A 2 y. (ii) P A 2 x i there is a y in WL such that xRL y and A 2 y. (From now on we shall use &; ; 8; 9 etc. as metalinguistic shorthands with their familiar meanings and use `x',`y',`z ' as variables over WL .) Ad(i): The ònly if' part is easy | assume, for any x; y in WL : Proof.

1. OA 2 x 2. xRL y Then:

hypothesis hypothesis

DEONTIC LOGIC

3. 8B (OB 2 x B 2 y 4. A 2 y 5. OA 2 x (xRL y A 2 y) 6.

8x; y(OA 2 x (xRL y A 2 y))

215

from 2 by the de nition of RL 1,3, universal instantiation, modus ponens 1{4, rule of conditional proof, discharging 1 and 2 x; y any members of WL

Number 6 can easily be rewritten as the ònly if' half of (i) Again, to do the ìf' part, assume for any x in WL : 1. OA 62 x

hypothesis

Then: 2.

:OA 2 x

3.

:A 2 xA

4. 5.

:A 2 (xA )+ A 62 (xA )+

6.

(xA )+ 2 WL

7.

8B (OB 2 x B 2 (xA )+ )

8.

xRL (xA )+

9.

9y(xRL y&A 62 y)

10. OA 62 x 9y(xRL y&A 62 y)

from 1 by the Lemma on Lsaturated sets by the de nition of xA in Makinson's Lemma xA (xA )+ by Lindenbaum from 4 by the Lemma on Lsaturated sets by the de nition of WL and 2 by the de nition of xA in Makinson's Lemma from 7 by the de nition of RL from 5,6,8 by existential generalization, the y at issue being (xA )+ 1{9, conditional proof, discharging 1 from 10 by contraposition

11. 8y(xRL y A 2 y) OA 2 x where 11 is the desired ìf' half of (i). Ad (ii): The veri cation of (ii) can be left to the reader. Hint: appeal to the fact that every instance of A1, i.e. P A , :O:A, is in every L-saturated

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set. The proof of the Saturation Lemma for Canonical L-Models is complete.

Let us now deal with our unproven lemmas and start with the easiest one (or, at least, the one with the shortest proof): [The Coincidence Lemma] For each w A and each x in WL ; UxL A i A 2 x. The proof proceeds by induction on the length of A.

Proof.

Basis. A is either (a) >, or (b) ?, or (c) some proposition letter p. (a) UxL > and > 2 x (by the truth condition for > and by the Lemma on L-saturated sets), (b) not UL ? and ? 62x (correspondingly),

x

(c) UxL p i VL (p; x) = 1 i p 2 x (by the truth condition for proposition letters and by the de nition of VL ).

Induction Step. The inductive cases for :; ^; _; ! and , are trivial, using the Lemma on L-saturated sets. Consider then Case A = OB (for U L some w B ): We would like to argue that x OB i for all y in WL such that xRL y; UyL B , i, for all y in WL such that xRL y; B 2 y, i, OB 2 x. Well, the rst ì' holds by the de nition of truth, the second is guaranteed by the inductive hypothesis, and the third ì' is simply clause (i) of the Saturation Lemma for canonical L-models. Thus, we are done. Case A = P B : The reasoning is perfectly analogous, the third ì' being clause (ii) of that lemma. The proof of the Coincidence Lemma is complete.

Missing Proof of the Veri cation Lemma. As de ned in the De nition of Canonical L-Models, UL = hWL ; RL ; VL i is an L-model. We have to consider various cases in the proof, depending on how we identify the logic L 2 fOK,OM,OS4,OB,OS5,OK+,OM+,OS4+ ,OB+, OS5+ g. The detailed demonstration in each case will not be given here; instead the reader is referred to Aqvist [1987, Section 10.1.11] for desired details. This completes our account of the Completeness Theorem for the ten Smiley{Hanson systems of monadic deontic logic. Proof.

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217

IV. REPRESENTABILITY OF MONADIC DEONTIC LOGICS IN SYSTEMS OF ALETHIC MODAL LOGIC WITH A PROPOSITIONAL CONSTANT 12 TEN ALETHIC MODAL LOGICS WITH A PROPOSITIONAL CONSTANT In this section we de ne ten systems KQ ; MQ ; S4Q ; BQ ; S5Q ; K+Q ; M+Q ; + + S4+ Q ; BQ and S5Q of alethic modal logic with a prohairetic, i.e. preferencetheoretical, propositional constant Q (after Kanger [1957]). They are all based on a common formal language, which we are now going to describe. Its alphabet is like that of the language of the Smiley{Hanson systems except that: (i)

(necessity) and (possibility) replace O and P , respectively, among the primitive logical connectives of degree 1.

(ii) A propositional constant Q (for òptimality' or àdmissibility') is added to the primitive logical connectives of degree 0. The set of all sentences of our new language is then de ned as in the old language except that clause (b) reads: (b)

>; ? and Q are in S

and clause (c) reads: (c) if A is in S , then so are :A; A and A. We must point out here explicitly that the set Prop of our new alethic language is assume to be identical to the set Prop of our old, deontic language. DEFINITION 36.

OA = df (Q ! A) P A = df (Q ^ A) F A = df (Q ! :A) As for the proof theory of our ten alethic systems, the following two rules of inference are common to all of them: A; A ! B (modus ponens) (R1) B A (R20 ) (-necessitation) A

218

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Consider then the following list B0{B7 of axiom schemata: (B0) All truth functional tautologies (over our new language) (B1) A , ::A; (B2) (A ! B ) ! (A ! B ); (B3) Q; (B4) A ! A; (B5) A ! A; (B6) A ! A; (B7) A ! A; Assuming R1 and R2' for all ten alethic modal logics with Q, then, we de ne them as follows: KQ = B0{B2 MQ = B0{B2,B6 S4Q = B0{B2,B4,B6 BQ = B0{B2,B6,B7 S5Q = B0{B2,B5,B6 (note that B4 and B7 are derivable in S5Q ) Again let K be any of these ve systems. Then:

K+ = K; B3

We observe that, apart from the presence of Q in the alethic language, the rst ve systems are familiar from literature on basic modal logic (see e.g. Kripke [1963], Makinson [1966], Lemmon and Scott [1966], Hughes and Cresswell [1968]). The remaining ve logics, the + systems, are then formed by adding a consistency postulate for Q, viz. B3, just as in Smiley [1963]. Let K be any of the ten systems just de ned. We introduce the notions of K-provability, K-inconsistency, K-consistency and K-derivability in perfect analogy with the corresponding L-notions de ned for the Smiley{Hanson deontic logics. We write `j K A' and `S j K A' to indicate, respectively, that the sentence A is K-provable and that A is K-derivable from a set S of sentences. Turning then to the semantics for our ten alethic systems, we obviously need a fresh notion of model. So, by an alethic model (perhaps we should even say àlethic prohairetic model', but it is too long) we shall mean an ordered quadruple U = hW ; Rx ; opt; Vi

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219

where: (i) W is a non-empty set. (ii) Rx W W is a binary relation on W (of àlethic alternativeness' or àlethic accessibility'). (iii) opt W (heuristically, opt is to be the set of òptimal', `best' or `suÆciently good' elements of W according to some unspeci ed preference ordering on W ). (iv) V : Prop

W ! f1; 0g (as usual).

Now, let U be any alethic model, let x be any member of W , and let A be in . The following changes in the de nition of truth at x in U are then called for: replace the clauses for O and P by the following, respectively: U A i for every y in W with xRx y; U A.

x

U A x

y

i for some y in W with xRx y; Uy A.

Moreover, we add a clause governing the constant Q: U Q i x 2 opt. x

Conditions on Rx and opt in alethic models. Corresponding to the ve axiom schemata B3{B7 we now list ve conditions on Rx and opt in an alethic model (adhering to previously adopted notational conventions): r3. Rx is òpt-serial' in W : 8x9y(xRx y & y 2 opt) r4. Rx is transitive in W . r5. Rx is Euclidean in W . r6. Rx is re exive in W : 8x(xRx x) r7. Rx is symmetric in W : 8x; y(xRx y yRx x)

Classi cation of alethic models. We summarize our categorization of alethic models in the self-explanatory Table 1:

Validity and satis ability. Let K 2 fKQ ,MQ,S4Q ,BQ ,S5Q,K+Q ,M+Q ,S4+Q , + B+ Q , S5Q g. The notions of K-validity, K-satis ability, and semantic K-entailment are then de ned in perfect analogy with the corresponding L-notions, and the notations K A and S K A will be used with their obvious meaning.

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Table 1. Kind of alethic model Condition on Rx and opt No restriction on Rx or on opt Rx re exive (in W ) Rx transitive and re exive Rx symmetric and re exive Rx Euclidean and re exive Rx opt-serial (in W ) Rx opt-serial and re exive Rx opt-serial, transitive and re exive Rx opt-serial, symmetric and re exive Rx opt-serial, Euclidean and re exive (in W )

KQ MQ S4Q BQ S5Q K+Q M+Q S4+Q B+Q S5+Q

13 SEMANTIC SOUNDNESS AND COMPLETENESS OF THE TEN ALETHIC SYSTEMS THEOREM 37. Let K be any of the ten systems KQ ;MQ ; : : : ;S5+Q . Then,

all K-provable sentences are K-valid. Proof.[Outlined]

deontic logic.

Proceed just as in the case of the L-systems of monadic

EXAMPLE 38. Suppose we want to check that the axiom Q(=B3) is indeed K+Q -valid. Assume otherwise then, i.e. that for some K+Q -model U = hW ; Rx ; opt; Vi and some x in W , we have: (1) not U Q. x

By the truth conditions for and Q, (1) amounts to: (2) not 9y(xRx y & y 2 opt).

DEONTIC LOGIC

But, by the opt-seriality of Rx in have: (3) 9y(xRx y & y 2 opt).

K+ Q -models

221

(see the table above), we

Contradiction. Hence B3 is K+Q -valid. THEOREM 39 (Completeness Theorem). Version I (strong completeness). Let K be as usual and let S . Then, if S is K-consistent, then S is K-satis able. Version II (weak completeness). Let K be as usual. Then, all K-valid sen-

tences are K-provable.

(Outlined). Obtaining the weak version as a corollary of the strong one, we concentrate on the latter. The De nition of and the Lemma (Lss) on L-saturated sets are restated for the K-systems without any signi cant changes. Similarly for Lindenbaum's Lemma. In Makinson's Lemma we replace every reference to O by a reference to and use R20 and B2 in the place of R2 and A2; the Lemma then goes through nicely for the K-systems as well. We come next to canonical models: DEFINITION 40 (Canonical K-models). Let K be as usual and let S be any K-consistent set of sentences. By the canonical K-model generated by S we mean the structure UK = hWK ; RxK ; optK ; VKi Proof

where: (i) WK = the smallest collection U of K-saturated sets such that: (a) S + is in U . (b) If x is in U , and A is a sentence with :A 2 x, then (xA )+ is in U (where xA is de ned as in our reformulated Makinson's Lemma). (ii) RKx = the binary relation on WK such that for all x; y in WK : xRKx y i 8A(A 2 x A 2 y): (iii) optK = fx 2 WK : Q 2 xg: (iv) VK = the assignment de ned as follows: VK (p; x) = 1 i p 2 x (for all p 2 Prop and x 2 WK ):

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LEMMA 41 (Veri cation Lemma.). As just de ned, UK = hWK ; RxK ; optK ; VK i is a K-model. LEMMA 42 (Coincidence Lemma.). For each sentence A and each x in U K WK : x A i A 2 x.

Waiting a little with the proofs of these lemmata, we establish that they together yield the strong completeness of the systems K by an argument perfectly analogous to the one used in connection with the L-systems. Again, the crucial clauses of the Saturation Lemma for Canonical KModels are as follows: (i)

A 2 x i for all y in WK with xRKx y; A 2 y.

(ii) A 2 x i there is a y in WK such that xRKx y and A 2 y. The proof of this lemma parallels the one given in the L-case; just replace O by ; P by , and so on.

Proof of the Coincidence Lemma. There is a new case in the induction basis, viz. Case A = Q. We are to show that UxK Q i Q 2 x. Well, we have that U K Q i x 2 opt K i Q 2 x; where the rst ì' comes from the truth x condition for Q and the second from the de nition of optK in canonical K-models (clause (iii)). So we are done. The novel cases in the induction step are those where A = B and A = B ; they are handled in perfect analogy with the cases A = OB and A = P B in the corresponding proof for the L-systems. For the critical ì's, appeal to the Saturation Lemma for Canonical K-Models. Proof.

Proof of the Veri cation Lemma. The cases where K 2 fKQ ,MQ ,

Proof. S4Q , BQ , S5Q g

are familiar from the literature on basic modal logic (see, e.g. Makinson [1966] and Lemmon and Scott [1966]). The only new thing to be veri ed in these ve cases is that optK , as we have de ned it, is a subset of WK ; which is a completely trivial point in view of clause (iii) of our De nition of Canonical K-Models. As for the remaining ve alethic systems, consider: Case K = K+Q . We are required to show that the relation RKx is optserial in W in the sense that 8x9y(xRx y & y 2 opt ). Well, in regard +

Q

KQ

+

KQ

+

of any x in WK , we have by Lss: +

Q

KQ

+

DEONTIC LOGIC

1. 2.

Q 2 x 9y(xRKx y & Q 2 y)

3. 4.

9y(xRKx y & y 2 optK ) 8x9y(xRKx y & y 2 optK

+

Q

+

+

+

Q

B3 for K+Q from 1 by clause (ii) of the Saturation Lemma for Canonical KModels from 2 by the de nition of optK from 3 by universal generalization, x being any member of WK +

Q

Q

223

+

Q

)

Q

+

Q

where 4 = Q.E.D. The remaining cases present no novelties, so the proof of the Veri cation Lemma is complete. The Completeness Theorem for the ten alethic K-systems is thereby fully proved. 14 THE PROBLEM OF ISOLATING THE DEONTIC FRAGMENT OF THE K-SYSTEMS

14.1 Problem Let be the set of sentences of our alethic language (common to the Ksystems) and let 0 be the set of sentences of our deontic language (common to the Smiley{Hanson L-systems); we now need dierent labels for the two sets. Let K, as usual, be any of the ten alethic systems with the constant Q. Then, exactly which sentences in 0 are provable in K, using the Kde nitions of O and P ? where the latter are:

OA = df (Q ! A); P A = df (Q ^ A): In other words, the problem is to characterize, for each K, the set of deontic sentences which are provable in K on the basis of those two de nitions (meaning by `deontic sentence any member of 0 ) A third formulation of our task: for each K, isolate the deontic fragment of K! Now, the locution `sentence in 0 provable in K using the K-de nitions of O and P ', which crops up in these formulations, is not entirely clear, or, at least, could be made more precise. To that purpose, we suggest that the K-de nitions of O and P in eect amount to there being a certain function which maps our deontic language into the alethic one in the following way: DEFINITION 43 (The translation from 0 into ). For each sentence A in 0 , de ne (A) 2 by the following recursive conditions:

224

(i) (ii) (iii) (iv) (v) Similarly for (vi) (vii)

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(p) = p; for each proposition letter p, (>) = >; (?) =?; (:A) = :(A); (A ^ B ) = (A) ^ (B ): (A _ B ); (A ! B ) and (A , B ): (OA) = (Q ! (A)) (P A) = (Q ^ (A))

Clearly, (vi) and (vii) are the only interesting clauses in this de nition, because we easily verify by induction on the length of A that (A) = A, provided that A does not contain O or P . Note how (vi) and (vii) correspond to the K-de nitions of O and P . Note also the importance of our assumption that the alethic and the deontic language have the same set Prop of proposition letters (why is that assumption important in the present context?) In the sequel we shall often write A instead of (A). We need one more de nition in order to be able to give a precise formulation of our problem. DEFINITION 44 (The deontic fragment of K under ). Let K be as usual, and let be the translation from 0 into as just de ned. By the deontic fragment of K under (in symbols: DF(K; )) we mean the set of sentences A in 0 such that A is provable in K; more compactly expressed: DF (K; )= fA 2 0 : j K Ag: Since the translation is xed, we may drop the reference to it and speak simply of the deontic fragment of K,DF(K), in accordance with the convention DF(K) = DF(K; )

for K as usual. A precise version of the problem raised at the beginning of this section is then the following:

14.2 The problem restated

Let K be any of our ten alethic systems.Let L be any of the ten Smiley{ Hanson deontic logics and let us identify L with the set ot its theses so that L=fA 2 0 : j L Ag. Then, for which L, if any, do we have that L =DF(K)?

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Let us illustrate the import of the restated problem a little. Suppose that we claim that (the set of theses of) OM is in fact identical to the deontic fragment of MQ . What are we then claiming? According to our de nitions, the following: (1) OM= fA 2 0 : j OM Ag = fA 2 0 : j M Ag = DF (MQ ). Fortunately, a more intelligible rendering of (1) is available: (2) For each sentence A in 0 : j OM A i j M A i.e. in plain language, A is provable in OM i its translation A is provable in MQ (for any deontic sentence A). Smiley [1963] in eect proved this result (2), i.e. that OM = DF(MQ ). He also proved, among other things, that OS4 = DF(S4Q ), OS5 = DF(S5Q ), + + + + OM+ = DF(M+ Q ), OS4 = DF(S4Q ) and OS5 = DF(S5Q ), using algebraic techniques. We shall now restate these Smileyan results and extend them so as to obtain a full solution to the problem raised above. We do so by indicating how to prove a Translation Theorem for monadic deontic logic, applying the Henkin-style model-theoretic technique of saturated sets instead of the matrix method used by Smiley. We think that, by doing so, we not only facilitate the understanding of monadic deontic logics as such, but also will be able to see more clearly their connection with dyadic deontic logics (logics of conditional obligation and permission) and to understand better the transition from the former to the latter. First of all, let us correlate our alethic systems to the deontic ones by de ning a one-one function c from the former onto the latter. The de nition of c appears from the self-explanatory Table 2. We can now state a nice result on deontic logic: THEOREM 45 (Translation theorem for monadic deontic logic). (After Smiley [1963]). Let K be any of the ten alethic systems KQ , MQ , : : : ,S5+Q , and let c(K) be its correlate among the ten Smiley{Hanson systems according to the above table. We identify c(K) with the set of its theses. Then, c(K) = DF(K); i.e. for each sentence A in 0 : j c(K)A i j K A. The proof, which provides a solution to our present problem, is a bit lengthy, so we devote a special section to it. Q

Q

Proof. (The Translation Theorem) = KQ c( ) = OK.

(very broad outline)

Case K and K Ònly if' part: We are to show that j OK A only if j K A, for A 2 0 . We Q

so so by induction on the length of the supposed OK-proof of A. Basis. The length of the supposed OK-proof = 1, so A is an instance of one or other of the axiom schemata A0{A2. Suppose A is an axiom under A0 so that A is a tautology over the deontic language. Then A is a tautology over the alethic language (the detailed

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Table 2. Alethic system K Deontic system c(K) KQ MQ S4Q BQ S5Q K+Q M+Q S4+Q B+Q S5+Q

OK OM OS4 OB OS5 OK+ OM+ OS4+ OB+ OS5+

proof of this is left to the reader), hence A is an axiom under B0, hence j K A. Suppose A is an axiom under A1 so that A = P B , :O:B and A = (Q ^ B ) , :(Q ! (:B )), for some B 2 0 . The following is then a KQ -proof of A: Q

1. 2.

(Q ^ B ) , ::(Q ^ B ) ::(Q ^ B ) , :(Q ! :B )

3.

(Q ^ B ) , :(Q ! (:B ))

B1 from B0,B2,R1,R2' by various elementary steps 1,2,B0,R1, de nition of

where 3 = A. Hence, j K A, as desired. Again, suppose that A is an instance of A2 so that A = (Q ! (B ! C )) ! ((Q ! B ) ! (Q ! C )), for some B and C in 0 . The desired result to the eect that j K A is readily obtained form B0 and B2 by R2' and R1. Q

Q

Induction Step. There is an OK-proof of A of length > 1, and either (i)

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A is got by applying R1 to some OK-thesis B and B ! A, or (ii) A is of the form OB and is obtained by applying R2 to some OK-thesis B .

Case (i): By the induction hypothesis B and (B ! A) are both provable in KQ . But, by the de nition of , (B ! A) = B ! A, so that A follows by R1. Hence, j K A. Q

Case (ii): By the induction hypothesis we have j K B in this case. We then obtain j K (OB ) as follows: Q

Q

1. Q ! B 2. (Q ! B ) 3. (OB )

j K B; B 0; R1 Q

from 1 by R2' from 2 by the de nition of

where 3 = A. Hence j K A, as desired. Q

This completes the proof of the ònly if' part.

Ìf' part: We must show that if j K A, then j OK A, or, contrapositively, that if 6 j OK A, (A is not OK-provable), then 6 j K A (A is not KQ Q

Q

provable), for any sentence A in the deontic language. This part is harder, because proof-theoretical methods seem to be less natural here; however, in view of our soundness and completeness results for the L- and the Ksystems, the problem is not too diÆcult to cope with.

Strategy of argument. We would like to argue as follows: 1. 2.

6 j OK A 6 OK A

hypothesis from 1 by the completeness of

3.

6 Ux A, for some OK-model U = hW ; R; Vi and some x in W

from 2 by the de nition of OKvalidity.

OK

Consider that OK-model U . We claim that we can construct from it a corresponding KQ -model U = hW ; R; opt; Vi with the property that for all B 2 0 and all y in W : Uy B i Uy B . Then:

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4.

6 Ux A

5.

6 K A

6.

6 j K A

Q

Q

from 3 by the fact that U exists and has the property mentioned from 4 by the de nition of KQ validity, U being a KQ -model where A fails to be true at some x in W from 5 by the soundness of KQ

where 6 is our desired conclusion. The crux of this argument is obviously isolated at a single point, viz. the construction of the KQ -model U from the given OK-model U , and the proof that U has the desired property indicated above. On the basis of that construction and proof, the crucial step from 3 to 4 is fully justi ed and the ìf' part is seen to go through in the present case. What remains to be done, then, is to state a de nition and to prove a couple of nice lemmata. DEFINITION 46 (of U ). Let U = hW ; R; Vi be any OK-model. We de ne U to be the structure hW; R ; opt; V i where: (i) R = R. (ii) opt = fy 2 W : for some x in W; xRyg. Note that W and V are common to U and U . As for V , this is made possible by our assumption that the alethic and the deontic language have the same set Prop of proposition letters. We may also remark that opt is here de ned to be what is known as `the converse domain' of the relation R in OK-models. LEMMA 47 (Easy Lemma). As de ned, U is a KQ -model. Appealing to the de nition of a KQ -model, we see that it is enough to show (i) that R W W , and (ii) that opt W , there being no further restrictions on Rx and opt in such alethic models. These points are immediate in view of (i) and (ii) in the de nition of the structure U . LEMMA 48 (On relations). Let U and U be as in the De nition of U . Then, for all x; y; in W : Proof.

xRy i xR y and y 2 opt: Proof.

Left-to-right: Assume, for any x; y in W :

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229

hypothesis

Then: 2. 9x(xRy) 3. y 2 opt 4. xR y 5. xR y&y 2 opt

from 1 by existential generalization from 2 by the de nition of opt in U from 1 by de nition of R in U 3,4, adjunction

where 5 is the desired conclusion. Right-to-left. This direction is immediate by the de nition of R .

LEMMA 49 (Crucial Lemma). Let U and U be as in the De nition of U . Then, for all A 2 0 and for all x in W : Ux A i Ux A. By induction on the length of A. By the de nition of the translation , the three cases in the induction basis are seen to be trivial. For the same reason, the inductive cases involving truth-functional connectives go through easily. Consider then: Case A = OB . We are required to show that Ux OB i Ux (OB ). Well, for any B 2 0 and any x in W , we clearly have: U 1. OB i 8y(xRy U B ) by the de nition of truth in U Proof.

x

2.

U

3.

U B i U B y y

y

(Q ! B ) i by the U 8y(xR y&y 2 opt y B ) U x

de nition of truth in

by the induction hypothesis, y being any member of W

Hence: 4. (Right member of 1) i (right from 3 and the Lemma on Remember of 2) lations by elementary steps U U 5. from 1,2,4 by the transitivity x OB i x (Q ! B ) of ì' where 5 yields the desired result by the de nition of .

Case A = P B . The reasoning parallels that of the preceding case; we just make the necessary switches from O; ; !; 8; to P; ; ^; 9 and &, respectively. The proof of the Crucial Lemma is complete.

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Armed with the de nition of the KQ -model U and our three lemmas on it, we have fully justi ed the decisive step from 3 to 4 in our strategic argument given above. This completes the proof of the ìf' part of Case K = KQ and c(K) = OK. That case is thereby fully proved.

Remaining Cases. For the details of the remaining nine cases, see Aqvist [1987, Section 13.5.2{10]. In the present survey I only want to indicate the main novelty in each case, which appears in the proof of the ìf' part at the juncture where, given any c(K)-model U , we construct a corresponding K-model U with the `right' properties. Thus, in general, we lay down a de nition of this form: DEFINITION 50 (De nition of U ). Let U = hW ; R; Vi be any c(K)-model, so that R satis es the appropriate restriction. De ne U to be the structure hW; R ; opt, V i where: (i) R = the binary relation on W such that for all x; y in W : xR y i . (ii) opt = fy 2 W : for some x in W; xRyg = the converse domain of R in W . Having lled in the blank in (i) for each case, one goes on to state and prove a LEMMA 51 (Easy). As de ned, U is a K-model as well as the LEMMA 52 (On relations). Let U and U be as in the above De nition of U . Then, for all x; y in W : xRy i xR y and y 2 opt. Using this lemma on Relations, we give an inductive proof of the LEMMA 53 (Crucial). Ux A i Ux A for U and U as above, and for A and x as usual), just as in the case of K = KQ and c(K) =OK. Again, armed with the de nition of the K-model U and our three lemmas on it, we justify the decisive step from 3 to 4 in the strategic argument for the ìf' part. This will then complete the proof in each of the remaining nine cases. We now indicate how to ll in the blank in clause (i) of the De nition of U , for various cases.

Case K = MQ and c(K) = OM. Fill in the blank with this condition: (x = y or xRy).

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Case K = S4Q and c(K) = OS4. Fill in the blank with the same condi-

tion!

Case K = BQ and c(K) = (x = y or xRy or yRx)

OB.

Fill in the blank with the condition

Case K = S5Q and c(K) = OS5. Fill in the blank with this condition: For some natural number n 1 : xRn y where R is the relation on W de ned by: xRy i (x = y or xRy or yRx) and where Rn is the nth power of the relation R, de ned in the usual inductive way in terms of relative products. Thus, in the present case, R is de ned as the chain, or proper ancestral, of the relation R. Certain inductively provable additional

lemmata are then needed to establish the Easy Lemma and the Lemma on Relations in this case, which is more complicated than the preceding ones. The ve remaining + cases. De ne R just as in the corresponding case without +, i.e. the case where the axiomatic systems lack the schemata B3 and A3 and where the accessibility relations are not required to be opt-serial or serial in the relevant models. Our broad outline of the proof of the Translation Theorem for Monadic Deontic Logic is complete. V. FIRST STEPS IN DYADIC DEONTIC LOGIC 15 TWO NEW LANGUAGES AND A PROBLEM Consider the deontic language common to the Smiley{Hanson monadic systems and its set 0 of well formed sentences. Let us now think of O and P as dyadic (i.e. two-place) deontic connectives, expressing conditional obligation and permission, respectively. We then obtain a new deontic language, the set of sentences of which will be called 20 and is de ned as the smallest set S such that: (a) Every proposition letter is in S . (b) > and ? are in S . (c) If A is in S , then so is :A. (d) If A; B are in S , then so are (A ^ B ); (A _ B ); (A ! B ) and (A , B ). (e) If A; B are in S , then so are OB A and PB A. REMARK 54. Apart from the fact that clause (c) has been curtailed, the new thing about the present deontic language and its set 20 is of course embodied in clause (e). So, note that we write OB A and PB A, where

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many authors would write O(A=B ) and P (A=B ) and where still others would use (BOA) and (BP A); the former authors are obviously inspired by the notation familiar from probability theory, whereas the latter (e.g. Van Eck [1981]) stick to what might be labeled the standard binary connective notation (cf. clause (d) above). The motivation for our choice of notation will appear from what follows; it might be called a relative necessity or sententially indexed modality notation (cf. e.g. Chellas [1975, Section 5]). DEFINITION 55.

`Dyadic' De nitions of monadic deontic connectives: OA =df O> A P A =df P> A F A =df :P> A (alternatively: O> :A) Again, consider the alethic language common to the systems K and its set of well formed sentences. In this language Q was thought of as a propositional constant, i.e. as a zero-place connective, thus of degree 0. Now, think of Q as a monadic (i.e. one-place) prohairetic connective, so that QA might be read as òptimally A', ìdeally A', or what have you. We then obtain a new alethic language with an additional one-place connective Q; its set of sentences will be called 1 and is de ned as the smallest set S such that: (a) Every proposition letter is in S . (b)

> and ? are in S .

(c) If A is in S , then so are :A; A; A and QA. (d) As usual. As compared to the old alethic language with Q, the new one has just the nullary connectives > and ? (clause (b)), whereas Q reappears among the monadic ones (clause (c)).

15.1 Àlethic' De nitions of dyadic deontic connectives Q OB A PB A FB A

(the old propositional constant) =df Q> =df (QB ! A), =df (QB ^ A), =df (QB ! :A).

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We can now state the problem announced in the title to this section.

15.2 Problem Consider the two sets of sentences 1 and 20 . As usual, the new alethic language and the new deontic one are assumed to have the same set Prop of proposition letters. Corresponding to the de nitions in 1 of the dyadic connectives O and P , de ne a translation from 20 into 1 just as in the case of 0 and , except for the following fresh clauses: (vi) (OB A) = (QB ! A): (vii) (PB A) = (QB ^ A): Then, nd a system L of dyadic deontic logic and a system modal logic with our new monadic connective Q such that:

K of alethic

(i) The set of L-theses is a proper subset of 20 . (ii) The set of K-theses is a proper subset of 1 . (iii) The set of L-theses =the dyadic deontic fragment of K under as just de ned; i.e. we are to have for each sentence A in 20 : j L A i j K A. The word `proper' is inserted in requirements (i) and (ii) just in order to make sure that the logics L and K are consistent. We now consider, to start with, certain rather weak systems having the desired properties (i){(iii). 16 THE SYSTEMS Ody S4, Ody S5, S4Qmo AND S5Qmo The system Ody S4 is determined as follows. Rules of proof: (R1) (R2)

A; A ! B B A OB A

(modus ponens) (OB -necessitation)

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Axiom schemata: (a0) (a1) (a2) (a3) (a4)

All truth functional tautologies over 20 PB A , :OB :A; OB (A ! C ) ! (OB A ! OB C ); OB (OB A ! A); OB A ! OC OB A;

The axiomatic system Ody S5 results from Ody S4 by omitting schema a3 and by adding in its place the following new schema a5: (a5) PC OB A ! OB A; Note that a3 will be derivable in Ody S5 as a thesis schema. The notions of provability, consistency, derivability etc. are de ned for the systems Ody S4 and Ody S5 in the usual straightforward way. The axiomatic system S4Qmo is simply the familiar modal calculus S4 over the present alethic language whose set of sentences = 1 ; similarly, the system S5Qmo is S5 over that alethic language. See Section 12 above. We now turn to the semantics of the four systems just described. By an Ody S4-model we shall mean any ordered triple

U = hW ; R; Vi where: (i) W is a non-empty set and V is a function from Prop W into the set of truth-values f1,0g; thus, W and V are as usual. (ii) R is a function from the set of sentences 20 into the set of all binary relations on W , in symbols: R : 20 ! P (W W ). In other words, then: for each sentence B in 20 ; RB W W so that RB is a binary relation on W . Moreover, R is to satisfy the following two conditions, corresponding to axiom schemata a3 and a4: (3) For each B in 20 and any x; y in W : xRB y yRB y.

(4) For any B; C in 20 and x; y; z in W : xRC y & yRB z xRB z . Again, by an Ody S5-model we mean any Ody S4-model hW; R; V i where R satis es the following new restriction that corresponds to the schema a5:

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(5) For any B; C in 20 and x; y; z in W : xRC y & xRB z yRB z .

Truth conditions for dyadic deontic connectives: Let U = hW ; R; Vi be any or Ody S5-model, let x be any member of W , and let A be in 20 . The only change in the de nition of truth at x in U , given for our monadic deontic logics, will concern sentences of the new forms OB A and PB A, for which we adopt the following clauses: U O A i for every y in W such that xR y; U A. Ody S4-

x

B

U P A x B

B

y

i for some y in W such that xRB y; Uy A.

Notions of validity, satis ability and semantic entailment, which are relative to Ody S4 and Ody S5, are then introduced in the usual way. Furthermore, de ne an S4Qmo -model to be any ordered quadruple

U = hW ; Rx ; opt; Vi

with W; Rx ; V as in an S4Q -model so that Rx is re exive and transitive relation on W , and where: opt : 1 ! PW i.e. opt is a function which to each sentence B in 1 assigns a subset opt(B ) of W as its value. Truth condition for the monadic Q-connective: The old truth condition for the Kanger constant Q will have to be replaced by the following:

U QB i x 2 opt(B ) x

which then governs sentences of the form QB . Again, an S5Qmo -model is any S4Qmo -model where Rx has the additional property of being symmetric and, hence, an equivalence relation, on W . Validity etc. is de ned in the usual way relatively to S4Qmo and S5Qmo . THEOREM 56 (Soundness and completeness).

(i) For each A in 20 : A is provable in Ody S4 /Ody S5/ i A is valid in Ody S4 /Ody S5/. (ii) For each A in 1 : A is provable in S4Qmo /S5Qmo / i A is valid in S4Qmo /S5Qmo /. Proof.

See Aqvist [1987, Sections 15 and 16].

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THEOREM 57 (Translation). For each A in 20 : A is provable in Ody S4 /Ody S5/ i A is provable in S4Qmo /S5Qmo /. In other words, the set of Ody S4 /Ody S5/ theses = the dyadic deontic fragment (under ) of S4Qmo

/S5Qmo /.

See Aqvist [1987, Sections 15 and 16]. The demonstrations are a little bit tedious. Proof.

17 TWO NEW SYSTEMS: Ody S5N AND S5N Qmo At this juncture we observe that in the systems S4Qmo and S5Qmo there are no special axioms governing the monadic operator Q; correspondingly, there are no restrictions on the function opt in the models for these systems. So, the following is a natural expectation: if we start adding axioms for Q to these systems as well as matching restrictions on opt in their modellings, we should obtain a series of dyadic deontic logics as the deontic fragments of these extended alethic calculi with monadic Q; and to begin with, we are particularly interested in dyadic deontic logics that are in relevant respects similar to the systems DSDL1{DSDL3 proposed by Bengt Hansson [1969]. As it turns out, however, S4Qmo and S5Qmo are not quite t to serve as adequate bases for a development of dyadic deontic logic along those lines. But they come close to the basic system we are looking for; only one further step has to be taken. Consider the following alethic system S5N Qmo : its set of theses is identical to that of S5Qmo , but we want to be interpreted as what Scott [1970] calls universal necessity and what Kanger [1957] called ànalytic' necessity. This means simply that is to express truth at every world (point) in the set (space) W , unconditionally. Technically, we easily achieve this by de ning x an S5N Qmo -model as any structure U = hW ; R ; opt; Vi, where W , opt, V are x as usual and where R = W W (i.e. the universal binary relation on W ). So, in contrast to the case of S5Qmo -models, Rx is no longer an arbitrary equivalence relation on W , but is now identi ed with a particular equivalence relation on W , viz. W W ; and the set of S5N Qmo -models becomes a proper subset of the set of S5Qmo -models. As for the completeness of the system S5N Aqvist [1973, Qmo as just characterized, a proof can be extracted from ] Section 5 ; note in particular the treatment and the role of the operator u in that essay. Now, what is the dyadic deontic fragment under of S5N Qmo ? If you believe it to be Ody S5 (once again), then try to prove a translation theorem for Ody S5 and S5N Qmo along the familiar lines! You will nd that it doesn't work, because we will be unable to establish either the Easy Lemma or the Lemma on Relations. Instead, the correct answer to the above question is: the following system Ody S5N , which we are now going to describe quickly.

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Let us reconsider our dyadic deontic language with the set of sentences = 20 , and add a pair of one-place modal operators N and M to its stock of primitive logical connectives. N (M ) is to express universal necessity (possibility) in the sense indicated above. The set of sentences of the dyadic deontic language thus enriched will be called 20;N ; adjusting the formation rule (c) in the obvious way, we have that whenever A is in 20;N , so are NA and MA. As for the proof theory of Ody S5N , its set of theses is a proper subset of this new set of sentences 20;N . More precisely, it is determined by the following rules of inference and axiom schemata: (R1) (R200 ) (a0) (a1) (a2) (a6) (a7) (a8)

A; A ! B (modus ponens) B A (N -necessitation) NA All truth functional tautologies over 20;N PB A , :OB :A OB (A ! C ) ! (OB A ! OB C ) OB A ! NOB A NA ! OB A An appropriate set of S5-schemata for N and M (e.g. B1,B2,B5 and B6, with N; M respectively replacing ; ).

EXERCISE 58. Derive the schema PB A ! NPB A in Ody S5N as just described! Derive the system Ody S5 as a subsystem of Ody S5N ! Proceeding to the semantics for Ody S5N , we de ne an Ody S5N -model as any structure U = hW ; R; Vi, where W; V are as usual and where R : 20;N ! P (W W ) is a function from our fresh set of sentences 20;N into the set of all binary relations on W , satisfying the following condition that corresponds to schema (a6): (6) For each B in 20;N and any x; y; z in W : xRB y zRB y. EXERCISE 59. Show that in any Ody S5N -model R satis es the restrictions (4) and (5)! Clearly, we must supplement our earlier de nition of truth at x in U , where U is any Ody S5N -model, with new clauses governing the operators N and M :

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U NA i for each y in W; U A x y U U x MA i for some y in W; y A

The notions of validity, satis ability and semantic entailment, pertaining to Ody S5N , are then de ned in the usual way. THEOREM 60 (Soundness and completeness for Ody S5N ). For each A in 20;N :

j Ody S5N A Proof.

i

Ody S5N

A

See Aqvist [1987, Section 17.2].

17.1 Representation of Ody S5N in S5NQmo In this section we announce the result that the set of Ody S5N -theses is the dyadic deontic fragment under of S5N Qmo . Clearly, then, should be a translation from the new sentence-set 20;N into 1 , which is eected by adding the following clauses to our de nition of : (viii) (NA) = A; (ix) (MA) = A; THEOREM 61 (Translation for Ody S5N and S5N Qmo ). 2 For each A in 0;N :

j Ody S5N A Proof.

i

j S5NQmo A

See Aqvist [1987, Section 17.3.1].

VI. DEVELOPMENT OF DYADIC DEONTIC LOGIC THROUGH AXIOMATIC ADDITIONS TO THE SYSTEMS Ody S5N AND S5N Qmo 18 THE DYADIC CALCULI Ody S5N + i In this part we shall take Ody S5N as our basic and, in a certain sense, minimal system of dyadic deontic logic and form new calculi by adding to it one or more axiom schemata i from the following list 0{4:

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N (A , B ) ! (OA C , OB C ) OA A OA^B C ! OA (B ! C ) MA ! (OA B ! PA B ) PA B ! (OA (B ! C ) ! OA^B C )

0: 1: 2: 3: 4:

To start with, we consider the ve systems Ody S5N +i, for i = 0; 1; : : : ; 4; where Ody S5N +i is the calculus which results from Ody S5N by adding just the schema i to the latter. Turning quickly to semantics, we de ne, for i = 0; 1; : : : ; 4, an Ody S5N +i model as any Ody S5N -model U = hW ; R; Vi where R, in addition to meeting (6), satis es the condition i in the list 0{4 below of restrictions on R; where A; B are any members of 20;N ; x; y any members of W , and where, for each A in 20;N ,

k A kU = fy 2 W : Uy Ag (in other words, k A kU is to be the truth-set or extension in U

A):

0. 1. 2. 3. 4.

of the w

k A kU =k B kU RA = RB xRA y Uy A xRA y & Uy B xRA^B y k A kU = 6 ; 8x9y(xRA y) 9z (xRA z & Uz B ) (xRA^B y (xRA y & Uy B ))

THEOREM 62 (Soundness and completeness). For each i = 0; 1; : : : ; 4 and for each A in 20;N : j Ody S5N+i A i Ody S5N+i A: In other words, the sentences provable in Ody S 5N + i are exactly the sentences valid in that

system. Proof.

See Aqvist [1987, Section 18.0{1].

19 THE ALETHIC CALCULI S5N QMO + I Consider the following list 0{ 4 of axiom schemata that may be added to the system S5N Qmo ; they all govern our monadic operator Q:

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0: (A , B ) ! (QA , QB ); 1: QA ! A; 2: (QA ^ B ) ! Q(A ^ B ); 3: A ! QA; 4: (QA ^ B ) ! (Q(A ^ B ) ! (QA ^ B )): By the system S5N Qmo + i, for i = 0; 1; : : : ; 4, we mean the calculus which N results from S5Qmo by adding just the schema i to the latter. There are then ve systems of this sort to be considered. Moving on to semantics, we de ne, for i = 0; 1; : : : ; 4, an S5N Qmo + i model as any structure U = hW ; Rx ; opt,V i with W; V as usual, where 1 Rx = W W (just as in S5N Qmo -models) and where opt: ! P W satis es the condition i in the list 0{4 below of restrictions on opt (where A; B are any sentences in 1 ): 0. k A kU =k B kU opt(A) = opt(B ) 1. opt(A) k A kU 2. opt(A)\ k B kU opt(A ^ B ) 3. k A kU 6= ; opt(A) 6= ; 4. opt(A)\ k B kU 6= ; (opt(A ^ B ) opt(A)\ k B kU ) THEOREM 63 (Soundness and completeness). For each i = 0; 1; : : : ; 4 and each A in 1 :

j S5NQmo+ i A Proof.

i

S5N Qmo + i

A

See Aqvist [1987, Section 19.0].

20 WEAK REPRESENTATION OF Ody S5N + i IN S5N Qmo + i; IS FULL REPRESENTABILITY LOST? Bearing in mind that is now a translation from 20;N into 1 (with fresh clauses for N and M , see Section 17.1 above), we state the following result: THEOREM 64 (Weak translation for Ody S5N + i and S5N Qmo + i). For 2 each i = 0; 1; : : : ; 4 and each A in 0;N :

j Ody S5N +i A

only if

j S5NQmo+ i A:

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In other words, the set of Ody S5N +i theses the dyadic deontic fragment under of S5NQmo + i. Note that in this theorem `=' has been weakened to `' and ì' to ònly if'. Our task is to give an ordinary ònly if' part demonstration. The new thing here is to verify, for each i = 0; 1; : : : ; 4, that if A is an axiom under the new schema i, then its translation A is provable in S5N Qmo + i. And we easily accomplish this, appealing precisely to the schema i. Proof.

As to the converse result, i.e. the ìf' part, I have not been able to establish it; nor do I know whether it holds good or not. But I am inclined to believe that it does; if it does, however, its proof will be harder than any one so far met with in this essay | at least, so I believe. EXERCISE 65. Try to prove the converse of the weak translation theorem stated above! Explain why you got lost, or else: congratulations and many thanks! We like to add that, even if the full representability of Ody S5N +i in S5N Qmo + i should turn out not to hold, this does not entail that the enterprise of developing dyadic deontic logic and alethic modal logic with monadic Q in a parallel fashion is without considerable heuristic value. In fact, I think, the contrary will prove to be the case. 21 AN ATTEMPTED RECONSTRUCTION AND IDENTIFICATION OF THE HANSSON DYADIC SYSTEMS DSDL1, DSDL2, AND DSDL3: ALETHIC PRELIMINARIES The main idea proposed in Hansson [1969] is that the concept of validity in Von Wright-type deontic logic (Hansson is anxious to point out that he just deals with this type in his paper) can be semantically explained in terms of a preference relation ìs at least as ideal as' among possible worlds; this claim is to apply whether the Von Wright-type deontic logic be monadic or dyadic. Hansson himself thinks of possible worlds as Boolean valuations in the sense familiar from the teaching of elementary propositional calculus. We shall not follow him in making this identi cation, however, because our Kripkean semantical technique has already supplied us with an independent notion of (a set of) possible worlds. Again, given a preference relation (ordering, ranking) R on a set of possible worlds W , we are automatically equipped with the notion of the R-maximal (`best', òptimal', under R) elements of W and of various (perhaps all ) subsets of W . We could then give the following informal characterization of the function opt in our models U for alethic systems with monadic Q(S4Qmo etc.):

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opt(>) = the R-maximal elements of whole.

k > kU , i.e. of W

as a

Remember here that the set opt W in models for the alethic systems with the propositional constant Q is simply to be equated with this set opt(>). Moreover, in general, we should have for any sentence A in 1 : opt(A) = the R-maximal elements of k A kU , i.e. of the set of worlds in W where A is true (= the extension in U of the w A). In the light of these heuristic preliminaries we now supplement our alethic with a Hanssonian preference relation on W , consider some possible conditions on it, and see what happens when one interprets the sentences in 1 relatively to these new enriched structures. Later on, we are going to perform a similar operation on our dyadic deontic models and the sentence-set 20;N . DEFINITION 66 (Various sorts of <-supplemented alethic models). Let U = hW ; Rx , opt, <; V i be any structure where S5N Qmo -models

W 6= ; (as usual), Rx = W W (as has now become usual), opt: 1 ! P W (as usual), < W W (novelty), V : Prop W ! f1; 0g (as usual).

(i) (ii) (iii) (iv) (v)

If you like, call any ordered quintuple of this sort a minimal alethic Hmodel. Note that we place no further conditions on opt or on < in minimal alethic H -models. Furthermore, we say: (b)

U

is an alethic H -model i opt and < jointly satisfy the following condition (for each A in 1 ): Æ0. opt(A) = fx 2k A kU :(8y 2k A kU )x < yg:

(c)

U is an alethic H1 -model i <, in addition to meeting Æ0 jointly with

(d)

U

opt, satis es this condition Æ1: Æ1. < is re exive in W (i.e. for all x in W; x < x):

is an alethic H2 -model i <, in addition to meeting Æ0 and Æ1, satis es this condition Æ2 (for each A in 1 ): Æ2. If k A kU 6= ;, then fx 2k A kU :(8y 2k A kU )x < yg 6= ;.

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(e)

U is an alethic H3 -model i <, in addition to meeting Æ0, Æ1 and Æ2,

(f)

U

satis es this condition Æ3: Æ3. < is transitive in W .

is an alethic strong H3 -model i <, in addition to meeting Æ0, Æ1, Æ2 and Æ3, satis es this condition Æ4: Æ4. < is strongly connected (total, complete) in W ;

i.e. for all x; y in W : x < y or y < x (or both). LEMMA 67 (On <-supplemented alethic models). (b) In the inductive definition of truth at x in U , where U is an alethic H -models, the clause for Q

is equivalent to the following:

U QA i x

U A & 8y( U A x < y) x y

(b0 ) Let U be any alethic H -model so that opt and < jointly satisfy Æ0. Then opt satis es the three conditions 0, 1 and 2, given in Section 19 above. (c) Let U be any alethic H1 -model. Then opt satis es the three conditions just mentioned. (d) Let U be any alethic H2 -model. Then opt satis es these conditions as well as 3. (e{f) Let U be any alethic H3 -model or alethic strong H3 -model. Then opt satis es all the ve conditions 0{4. Proof.

See Aqvist [1987, Section 21].

THEOREM 68 (Soundness). The present result concerns axiomatic extensions of the alethic system S5NQmo ; we are thus dealing with subsets of 1 .

The result is presented in the right-most column of Table 3; in the head of that column, the locution `matching kind of validity' means, of course, the same as `truth at all points in every <-supplemented alethic model of the corresponding kind'. And the corresponding kinds of models are then to be found in the leftmost column, and the appropriate restrictions on < in the middle one. Proof.

See Aqvist [1987, Section 21].

REMARK 69. (i) Questions of completeness have been left open.

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Table 3. Kind of alethic Restriction(s) on model supple- such a model mented with <

< in

minimal H H-

none Æ0

H1 H2 -

Æ0 and Æ1 Æ0,Æ1 and Æ2

H3 -

Æ0,Æ1,Æ2 and Æ3

strong H3 -

Æ0,Æ1,Æ2,Æ3 and Æ4

Axiomatic extension of S5N Qmo , sound with respect to the matching kind of validity S5N Qmo S5N Qmo

itself + every i with i =

S5N Qmo

+ every i with i =

0; 1; 2 ditto

0; 1; 2; 3 S5N Qmo

+ every i with i = 0; 1; 2; 3; 4 ditto

(ii) Certain possible kinds of <-supplemented alethic models have been left out of consideration, e.g. alethic H -models where < satis es Æ1 and Æ3 (but not necessarily Æ2) as well as such where < satis es Æ3 and Æ4 (and hence Æ1; but not necessarily Æ2). 22 ATTEMPTED IDENTIFICATION OF HANSSON DYADIC SYSTEMS: <-SUPPLEMENTED DEONTIC MODELS In this section we supplement our dyadic deontic Ody S5N -models with a Hanssonian preference relation <, consider possible conditions on it, and see what happens when we interpret the sentences in the beautiful set 20;N relatively to these enriched structures. The exposition will parallel the one given in the last section. DEFINITION 70 (Various sorts of <-supplemented dyadic deontic models). Let U = hW ; R; <; Vi be any structure (i) W = 6 ; (as usual)

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(ii) R : 20;N (iii) (iv)

245

! P (W W ) (as has now become usual)

< W W (novelty) V : Prop W ! f1; 0g (as usual)

Then we say: (a)

U is a minimal deontic H -model i R meets condition (6) (see Section

(b)

U is a deontic H -model i R and < jointly satisfy the following con-

17 above).

dition 0 (for each A in 20;N and any x; y in W ):

0. xR y i U A & 8z ( U A y < z ): A

y

z

(c)

U is a deontic H1 -model i < meets the re exivity condition Æ1 as well

(d)

U is a deontic H2 -model i < meets the

(e)

U is a deontic H3 -model i < satis es the transitivity condition Æ3 as

(f)

U

as 0.

`Limit Condition' Æ2 as well as Æ1 and 0; where Æ2 now applies to any sentence in 20;N . well as Æ2, Æ1 and 0.

is a deontic strong H3 -model i < satis es the connectedness requirement Æ4 in addition to meeting Æ3, Æ2, Æ1 and 0.

LEMMA 71 (<-supplemented dyadic deontic models).

(b) Let U be any deontic H -model. Then, by 0, the inductive clauses for O and P in the de nition of truth at x in U become equivalent to the following:

U O A i x B

U P A x B

8y(( Uy B & 8z ( Uz B y < z )) Uy A)

i 9y(( Uy B & 8z ( Uz B y < z )) & Uy A):

(b') Let U be any deontic H -model. Then R satis es condition (6) (stated in Section 17 above) as well as the three conditions 0,1 and 2 (stated in Section 18 above). (c) Let U be any deontic H1 -model. Then R satis es those four conditions just mentioned.

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(d) Let U be any deontic H2 -model. Then R satis es those four conditions as well as 3. (e{f) Let U be any deontic H3 -model or a deontic strong H3 -model. Then R satis es (6) as well as all ve conditions 0{4.

THEOREM 72 (Soundness). The present result concerns axiomatic extensions of the dyadic deontic system Ody S5N ; we are thus dealing with subsets of 20;N . We state the result in the form of a table (Table 4) analogous to the one at the end of the last theorem (Theorem 68). Proof.

Straightforward and left as an exercise.

Table 4. Kind of dyadic Restriction(s) on < Axiomatic extension of deontic model and/or R in such a Ody S5N , sound with respect supplemented model to the matching kind of with < validity minimal H H-

(6)

0

H1 -

0 and Æ1

H2 -

0,Æ1 and Æ2

H3 -

0,Æ1,Æ2 and Æ3

strong H3 -

0,Æ1,Æ2,Æ3 and Æ4

Ody S5N

itself Ody S5N + every i with i = 0; 1; 2 Ody S5N + every i with i = 0; 1; 2 Ody S5N + every i with i = 0; 1; 2; 3 Ody S5N + every i with i = 0; 1; 2; 3; 4 Ody S5N + every i with i = 0; 1; 2; 3; 4

Use the soundness result for Ody S5N + every i, for relevant choices if i, together with the Lemma on <-supplemented dyadic deontic models. Proof.

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REMARK 73. Our new <-supplemented deontic models obviously engender new `matching kinds of validity', such as deontic H -validity, deontic H1 validity etc. Are any of the axiomatic extensions of Ody S5N , which appear in the table above, complete with respect to the kind of validity associated with them by the table? If so, which? We shall try to answer some of these completeness questions elsewhere. Meanwhile, we make a rst attempt at an identi cation of the Hansson systems DSDL1, DSDL2 and DSDL3 and start with a few preliminary observations. (I) The language of these three systems is poorer than that of Ody S5N in the following respects: (a) it lacks the operators N and M of universal necessity and possibility; (b) its set of well formed sentences diers from 20;N in not allowing iterations and overlapping of dyadic deontic operators, nor any mixed formulas, e.g. of the type OB A ! A. Thus, in the language used by Hansson, there are, for one thing, no instances of any of our axiom schemata a3{a7. On the other hand, the set of sentences of Hansson's language should clearly be a subset of 20;N (inasmuch as we are at all able to discuss his systems in our present framework). We attempt the following characterization of it. Let be the smallest set which contains as its members all proposition letters in our set Prop as well as > and ?, and which is closed under :; ^; _; ! and ,. Thus, is simply the set of ws of the familiar propositional calculus with verum and falsum. We then de ne (= the set of sentences of our reconstructed Hansson language) to be the smallest set S such that: (a) If A; B are in , then OB A and PB A are in S .

(b) If C; D are in S , then so are :C; (C ^ D); (C _ D); (C ! D) and (C , D). Clearly, then, we have that 20 20;N . (II) As for the semantics of the Hansson dyadic system, we pointed out at the beginning of the previous section that we were going to use Kripkean possible worlds rather than Boolean valuations in the basic set on which < is de ned; this approach naturally leads to our present sort of deontic models. Now, as characteristic conditions on <, Hansson [1969, p. 395 f.] adopts the following:

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re exivity (Æ1). limitedness (Æ2; in addition to Æ1). transitivity (Æ3) and strong connectedness (Æ4) (in addition to Æ1 and Æ2).

For the system DSDL1: For the system DSDL2: For the system DSDL3:

As for the notion of truth as applied to sentences in of the forms OB A and PB A, Hansson proposes analogues of the two conditions stated under (b) in the Lemma on <-supplemented dyadic deontic models (we disregard here the fact that Hansson's own concept of <-maximility does not completely coincide with ours).

22.1 Semantic identi cation of DSDL1, DSDL2 and DSDL3

Let A be in 20;N , and let H (H ;strong H ) A mean that A is true at every world in every deontic H1 - (H2 -, strong H3 -)model. We then suggest the following characterization of the Hansson systems as subsets of 20;N : 1

(i) (ii) (iii)

2

3

= fA 2 : H Ag DSDL2 = fA 2 : H Ag DSDL3 = fA 2 : strong H Ag DSDL1

1

2

3

So, for i = 1; 2, DSDLi is the set of deontically Hi -valid sentences in , and DSDL3 is the set of deontically strong H3 -valid sentences in . What about axiomatic characterizations of the Hansson systems as deductive calculi? If such were available, our identi cation of them would be more satisfactory from the proof theoretical point of view. As things stand right now, however, we only have the following somewhat insuÆcient result:

22.2 Partial syntactic identi cation of DSDL1, DSDL2 and DSDL3 Let us introduce the following abbreviations: E F G

= the set of theses of = the set of theses of = the set of theses of

Ody S5N

+ every i with i = 0; 1; 2 + every i with i = 0; 1; 2; 3 Ody S5N + every i with i = 0; 1; 2; 3; 4 Ody S5N

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Furthermore, let E/ (F/,G/) = E(F,G) restricted to the set ; i.e.

\ etc. Then we have three groups of more or less obvious results:

E

I. Soundness results

II. Unsoundness results III. Incompleteness results

(i) E= DSDL1 (iv) F= 6DSDL1 (ii) F= DSDL2 (v) G= 6DSDL1 (iii) G= DSDL3 (vi) G= 6DSDL2

(vii) DSDL26E= (viii) DSDL36E= (ix) DSDL36F=

As for Group I, appeal to our last soundness theorem; as for Groups II and III, appeal in addition to deontic analogues of the Exercises on alethic H -, H1 -, and H2 -models studied in Aqvist [1987, Section 21]. Proof.

What makes the given syntactic identi cation only a partial one is, of course, the fact that as yet we do not know whether the converses of (i){ (iii) hold, so that we could strengthen `' to `='. Also, if it should turn out that G/ is not complete with respect to strong deontic H3 -validity among the members of , perhaps G/ is complete with respect to deontic H3 -validity in simpliciter? 23 ON THE COMPLETENESS PROBLEM FOR THE DYADIC DEONTIC LOGIC G As in the last section we let G be the (set of theses of the) system Qdy S5N + every i with i 0; 1; : : : ; 4. Thus G 20;N 0 its rules of proof are modus ponens (R1) and N -necessitation (R200 ), its axiom schemata are a0{a2, a6{ a8 (Section 17 above) as well as 0{4 (Section 18). The system G, as just characterized in purely axiomatic terms, is by far the most important dyadic deontic logic dealt with in the present chapter| recall that in Section 4 supra we de ned a strongly normal dyadic deontic logic as one containing precisely the system G. Obviously, then, the problem of nding an adequate semantical characterization relative to which G can be proved complete is an equally important one. In the original version of this chapter I conjectured that G was complete with respect to strong H3 validity in the sense of the last section; for a `possible', rather complicated proof of this conjectured result I referred the reader to Aqvist [1987, Section ] 23.3 .

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I am now convinced that the completeness problem for the system G has a much simpler solution than was suggested by my earlier conjectured proof. In the remainder of this section I shall then outline that simpler solution as rst presented in two recent papers by Aqvist [1996; 1997]. Again, in the three nal sections (Sections 24{26) of this chapter I shall present some further recent results on the crucial dyadic system G.

23.1 Improved semantics for G DEFINITION 74 (G-structures and truth at a point in a G-structure). By a G-structure we mean any ordered triple

U = hW; V; besti where (i) W = 6 ? (ii) V : Prop W (iii)

! f1; 0g best: 20;N ! P W .

We can now tell what it means for any sentence A in 20;N to be true at a point (\world") x(2 W ) in a G-structure U [in symbols: Ux A], starting out with obvious clauses like U p i V (p; x) = 1 (for any p in Prop)

x

U x

>

not Ux

?

and so on for molecular sentences having Boolean connectives as their main operator. We then handle sentences having modal and dyadic deontic operators as their main operator as follows: U NA i for each y in W : Uy A x U MA i for some y in W : Uy A x U OB A i for each y in best(B ) : Uy A x U P A i for some y in best(B ) : U A x

B

y

DEFINITION 75 (G-models, G-validity and G-satis ability). We now focus our attention on a special kind of G-structures called \G-models". By a G-model we mean any G-structure satisfying the following ve conditions

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251

on the function best, where, as usual, we let kAkU , or kAk for short, be the extension in U of A:

0 kAk = kB k only if best (A) = best (B ) 1 best (A) kAk 2 best (A) \ kB k best (A ^ B ) 3 kAjj 6= ? only if best (A) 6= ? 4 best (A) \ kB k 6= ? only if best (A ^ B ) best (A) \ kB k

for any sentences A; B in 20;N . The notions of G-validity and G-satis ability are then de ned in the usual way. REMARK 76. Conditions 0{4 are very much like the so-named conditions on the function opt in Section 19 above. But note that opt was de ned for all sentences in the alethic language of S5N Qmo , whereas best is de ned for those in the dyadic deontic language of Ody S5N and G. THEOREM 77 (Soundness and Completeness of the system G). Weak version:

for each A in 20;N : A is G-provable i A is G-valid.

Strong version: for each S 20;N : S is G-consistent i S is G-satis able. (sketchy). The soundness parts are unproblematic. So we concentrate on the ònly if' half of the strong version, from which the ìf' half of the weak one is immediate. Let S be any G-consistent set of sentences, and let S + be a maximal G-consistent extension of S , the existence of which is guaranteed by Lindenbaum's Lemma. Form the canonical G-structure generated by S + in the sense of the structure Proof.

US

+

= hW; V; besti

where (i) W = the set of maximal G-consistent sets x of sentences such that for all A in 20;N : if NA 2 S +, then A 2 x. (ii) V = the assignment de ned as follows: for each p in Prop and each x in W , V (p; x) = 1 i p 2 x. (iii) best = the function from 20;N into P (W ) de ned by setting, for all sentences B , best(B ) = fx 2 W : for all sentences A, if OB A 2 S + , then A 2 xg.

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Omitting details we can then prove that, as just de ned, the generated canonical G-structure U S satis es all lemmata needed for our desired completeness result. The most interesting one among them is the Veri cation Lemma, to the eect that U S is a G-model satisfying the ve conditions 0{4 on the function best. A proof to precisely that eect is easily obtained as follows: de ne our three-place relation R by the requirement +

+

(*) xRB y i y 2 best (B ) (for all x; y in W and all B in 20;N ) On the basis of this de nition, we quickly verify that our present truth conditions for OB A and PB A are equivalent to those given in Section 16 supra, that R meets condition (6) laid down in Section 17 supra, and that for each i = 0; 1; : : : ; 4, the condition i on best is equivalent to the restriction i on R (Section 18). Again, this result enables us to transform the proof given in Aqvist [1987, Section 18.1, pp. 161{165] into a proof that the canonical G-structure U S is indeed a G-model, as desired. +

24 AN INFINITE HIERARCHY OF EXTENSIONS OF G: THE DYADIC DEONTIC LOGICS Gm[m = 1; 2; : : : ] In this and the following sections of the present chapter we intend to shed some more light on the system G by studying an in nite hierarchy Gm[m = 1; 2; : : : ] of extensions of that system. The soundness and completeness of every system in that hierarchy is then asserted in a Theorem, for the proof of which we refer the reader to Aqvist [1996]. The main semantical technical device employed in our study of these extensions of G is this: in the models of each system Gm in the hierarchy, we work with a set

fopt1 ; opt2 ; : : : ; optm g which is to be a partition of the set W

of `possible worlds' into exactly m non-empty, pairwise disjoint and together exhaustive òptimality' classes, viewed as so many levels of perfection. Intuitively, we think of opt1 as the set of `best' [optimal] members of W as a whole, opt2 as the set of best members of W opt1 [the `second best' members of W ], opt3 as the set of best members of W (opt1 [ opt2 ) [the `third best' members of W ]; and so on. Now, we shall represent each level of perfection in the object-language of the systems by a so-called systematic frame constant. The truth conditions and axioms governing those constants can then be seen to play a highly important, characteristic role in our axiomatization. Thus, the primitive logical vocabulary of the systems Gm (m any positive integer) results from that of G by adding to the latter an in nite family

fQi gi = 1; 2; : : :

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253

of systematic frame constants, indexed by the set of positive integers. As just explained, the Qi are to represent dierent `levels of perfection' in the models of the system Gm. The set Sent of their well-formedsentences (formulas, ws) is then de ned in the straightforward way|we think of the Qi as zero-place connectives on a par with > and ?. We begin the presentation of the new dyadic deontic logics Gm by outlining their proof theory. The rules of proof modus ponens and N -necessitation are common to G and the Gm[m = 1; 2; : : : ]. In addition to the axiom schemata of G, each system Gm has the following: 5: Qi ! :Qj ; for all positive integers i; j with 1 < i 6= j < ! 6: PB Qi ! ((Q1 _ : : : _ Qi 1 ) ! :B ); for all i with 1 < i m 7: Q1 ! (OB A ! (B ! A)) 8: (Q1 ^ OB A ^ B ^ :A) ! PB (Q1 _ : : : _ Qi 1 ); for all i with 1 < i m 9: Q1 _ : : : _ Qm 10: MQ1 ^ : : : ^ MQm: Then, the axiomatic system Gm[m = 1; 2; : : : ] is determined by the axiom schemata 0{2, 6{8 and 0{10 (and the usual rules of proof). Moreover, we de ne the notions of provability, derivability, [in]consistency and maximal consistency for the systems Gm in the straightforward way. Turning next to the semantics of the logics Gm, we de ne, for any positive integer m, a Gm-structure as an ordered quintuple U = (W; V; fopti gi=1;2;:::; m; best) where (i) W 6= ? [W is a non-empty set of `possible worlds']. (ii) V : Prop ! pow(W ) [V is a valuation function which to each propositional variable assigns a subset of W ]. (iii) fopti gi=1;2;::: is an in nite sequence of subsets of W . (iv) m is the positive integer under consideration. (v) best: Sent ! pow(W ) [best is a function which to each sentence in the Gm-language assigns a subset of W , heuristically, the set of best worlds in the extension (truth-set) of the sentence under consideration]. The de nition of a sentence being true at a point in a Gm-structure remains as in the case of G, or G-structures, except for the following fresh clause governing our frame constants: U x Qi i x 2 opti (for all positive integers i):

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We now focus our attention on a special kind of Gm-structures called `Gm-models'. By a Gm-model we shall mean any Gm-structure U where fopti g, m and best satisfy the following additional conditions: Exactly m Non-empty levels of Perfection This condition requires the set fopt1 ; opt2 ; : : : ; optm g to be a partition of W in the sense that (a) opti \ optj = ?, for all positive integers i; j with 1 i 6= j m (b) opti [ : : : [ optm = W (c) opti 6= ?, for each i with 1 i m. (d) opti = ?, for each i with m < i < !. The second condition is one on our `choice' function best; it is intended to capture the intuitive meaning of that function:

0 : x 2 best(B ) i Ux B and for each y in W : i U y B; then x < y: Here, the weak preference relation <, ìs at least as good (ideal) as' is to be understood as follows. First of all, by clauses (a) and (b) in the condition Exactly m Non-empty levels of Perfection, we have that for each x in W there is exactly one positive integer i with 1 i m such that x 2 opti . We then de ne a `ranking' function r from W into the closed interval [1; m] of integers by setting r(x) = the i; with 1 < i < m; such that x 2 opti : Finally, we de ne < as the binary relation on W such that for all x; y in W : x < y i r(x) r(y): Armed with the notion of a Gm-model, we de ne, for m = 1; 2; : : : , those of Gm-validity and Gm-satis ability in the obvious way. THEOREM 78. (Soundness and completeness of the systems Gm[m = 1; 2; : : : ]) Weak version: For each A in Sent: A is Gm-provable i A is Gm-valid. Strong version: For each S Sent: S is Gm-consistent i S is Gm-

satis able.

See Aqvist [1996]. Also, we observe that a proof that the generated canonical Gm-structure U S satis es our condition 0 on the function best is easily extracted from the proof given in Aqvist [1993] and [1987, Ch ] VI, Section 23.3.6, pp. 187{191 that the three-place relation R satis es essentially the same condition (use the de nition (*) in the last section!). Proof.

+

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25 ON THE RELATION OF THE `CORE' SYSTEM G TO THE LOGICS Gm[m = 1; 2; : : : ] In the present section we deal with the system G and state a result to the eect that G is the intersection of all the logics Gm[m = 1; 2; : : : ] (identifying as usual a `logic' with the set of its theses, or sentences provable in it). Thus, the result answers the question how our crucial system G is related to the Gm. THEOREM 79. For each G-sentence A (i.e. member of 20;N ):

j G A i for each positive integer m; j Gm A:

See Aqvist [1997, Section 3]. Let us just outline the main structure of our proof as given in that paper. The non-trivial direction here is the right-to-left one, the contraposed version of which asserts the following: if A is not G-provable, then there exists a positive integer m such that A is not Gm-provable (either). We then argue as follows: 1. 6j G A hypothesis U 2. 6 G from 1 by the weak completeness of G U 3. 6 x , for some G-model U = from 2 by the de nition of Gvalidity. hW; V; besti and some x in W Let U = hW ; V ; best i be the ltration of U through the set of subsentences of A (in the sense rigorously de ned in my 1997 paper mentioned above), and let [x] be the equivalence class of x under a certain equivalence relation on W (also de ned in the paper). We then obtain: 4. 6 U[x] from 3 by the Filtration Lemma for G proved in Aqvist [1997]. We now observe that the ltration U is necessarily a nite G-model, so that there can be at most a nite number of levels of perfection compatible with and de nable on U . Again, this means that we can construct, for some positive integer m, a Gm-models U + = hW ; V ; fopt gi=1;2;:::; m; best i Proof.

i

with the property that 5. 6 U +

[x]

from 4 by the fact that the new items opti and m do not aect the truth-value of the G-sentence A

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and then argue:

6.

6 Gm

7.

6 j Gm A

from 5 by the de nition of Gmvalidity from 6 by the soundness of each system Gm

qvist [1997, Section 3] we then nish where 7 is our desired conclusion. In A the proof by providing a detailed careful justi cation of the two crucial steps 4 and 5 in the above overall argument. Among other things, our proof is seen to pro t from the fact that the system G is known to be deductively equivalent, under appropriate de nitions, to a certain logic PR of preference, as shown in Aqvist [1987, Appendix, x33]. 26 REPRESENTABILITY OF DYADIC DEONTIC LOGICS IN ALETHIC MODAL LOGICS WITH SYSTEMATIC FRAME CONSTANTS In this nal section of the present chapter we point out that the dyadic deontic logics Gm are representable in a hierarchy of alethic modal logics Hm[m = 1; 2; : : : ], which lack deontic operators in their primitive vocabulary, but which are such that we can de ne those operators in them, somewhat in the spirit of the well known Andersonian reduction. The detailed proof of this result in eect forms the bulk of Aqvist [1996]; here, we just present enough material so as to enable us to state that result in an intelligible way. Consider the result of banishing the dyadic deontic operators O and P from the primitive logical vocabulary of the systems Gm[m = 1; 2; : : : ]. Then, for any positive integer m, let the axiomatic system Hm of alethic modal logic with frame constants be determined by the rules of proof modus ponens and N -necessitation, and the axiom schemata a0 (= all tautologies over the present reduced language), a8 (= S5-schemata for N; M ), 5, 9 and 10; i.e. by those axiom schemata in Gm that do not contain occurrences of O or P . Clearly each system Gm is an extension of Hm. As to the semantics of the alethic modal logics Hm[m = 1; 2; : : : ], a Hm-structure will be the ordered quadruple that results from deleting the function best in a Gm-structure, and a Hm-model will be a Hm-structure satisfying (a){(d) in the requirement Exactly m Non-empty levels of Perfection (whereas the condition 0 on best vanishes altogether). We then have the following result: THEOREM 80. (Soundness and completeness of the systems Hm[m = 1; 2; : : : ])

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Weak version: For every sentence A: A is Hm-provable i A is Hmvalid. Strong version: For each set S of sentences: S is Hm-consistent i S is Hm-satis able. Proof.

See Aqvist [1996].

What is the interest of the just considered in nite hierarchy Hm of alethic modal logics? We take it to be this: although the operators (for conditional obligation) and P (for conditional permission) are not primitive in the language of the Hm, they can be de ned in those systems as follows:

Def O.

OB A = df [M (Q1 ^ B ) N ((Q1 ^ B ) A)]^ [(:M (Q1 ^ B ) ^ M (Q2 ^ B )) N (Q2 ^ B A)] ^ : : : ^ [(:M (Q1 ^ B ) ^ : : : ^ :M (Qm 1 ^ B ) ^ M (Qm ^ B )) N (Qm ^ B A)]:

Def P.

PB A = df M (Q1 ^ B ^ A)_ (:M (Q1 ^ B ) ^ M (Q2 ^ B ^ A)) _ : : : _ (:M (Q1 ^ B ) ^ : : : ^ :M (Qm 1 ^ B ) ^ M (Qm ^ B ^ A)):

We can then prove the following: THEOREM 81 (Deductive Equivalence for Hm and Gm). Let Hm+Def O+ Def P be the result of adding the de nitions Def O and Def P supra to the alethic system Hm. Then, for all m = 1; 2; : : : ; Hm + Def O + Def P is deductively equivalent to Gm in the sense that the following two conditions

are satis ed:

(i) Hm + Def O + Def P contains Gm. (ii) Each of Def O and Def P is provable in the form of an equivalence in Gm. Proof.

See Aqvist [1996].

An alternative, more `semantical' method of representing the dyadic deontic systems Gm in the alethic modal logics Hm is this: de ne recursively

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a certain translation from the set of sentences by the stipulations:

(p) (>) (?) (Qi ) (:A) (A ^ B )

Gm-sentences

into the set of

Hm-

= p; for each propositional variable p in Prop = > = ? = Qi ; for each positive integer i = :(A) = ((A) ^ (B ))

and similarly for Gm sentences having _; !; , as their principal sign.

(NA) = NA (MA) = MA where we have written A instead of (A) to the right. Finally, we have two characteristic clauses corresponding to Def O and Def P:

(OB A) = [M (Q1 ^ B ) N ((Q1 ^ B ) A)]^ [(:M (Q1 ^ B ) ^ M (Q2 ^ B )) N (Q2 ^ B A)] ^ : : : ^ [(:M (Q1 ^ B ) ^ : : : ^ :M ((Qm 1 ^ B ) ^ M (Qm ^ B )) N (Qm ^ B A)]

Similarly for (PB A): write it out as an m-termed disjunction! We then have the following result: THEOREM 82 (Translation for Gm and Hm). For each positive integer m, and for each Gm-sentence A:

j Gm A i j Hm A: Again, see Aqvist [1996]. The left-to-right direction is more or less immediate from the proof of the Deductive Equivalence theorem supra. The proof of the right-to-left is reminiscent of that of the Theorem on the relation of G to the Gm in respect of utilizing a relevant completeness result in the second step. Proof.

Combining the present Translation theorem with the Theorem on the relation of G to the Gm, we obtain the obvious COROLLARY 83. For any G-sentence A: j G A i for all m = 1; 2; : : : ; j Hm A. Proof.

Immediate from the two theorems just mentioned.

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ACKNOWLEDGEMENTS Research carried out under the auspices of the Bank of Sweden Tercentenary Foundation project RJ 79/23 on criminal intent and negligence.

Uppsala Universitet, Sweden. BIBLIOGRAPHY Within the boundaries of a reasonable-size bibliography it is now, i.e. in 1982, impossible to do justice to the existing literature on deontic logic, especially if one is to include such topics as the logic of imperatives, ethical and legal theory, and what have you. Laudable attempts have been made in the past, though: the bibliographies of Anderson [1956] (together with a Supplementary Bibliography (1966)), Rescher [1966], Von Wright [1968], Chellas [1969] and Fllesdal and Hilpinen [1971] are still of a manageable size. Really comprehensive bibliographies are: Berkemann and Strasser [1974] (around 800 items) and Conte and Di Bernardo [1977] (approaching 1500 items!). So, naturally, my own present attempt will be fairly selective; and it remains unclear to me exactly what principle of selection is being applied. Anyway, here we go: [Alchourron and Bulygin, 1971] C. E. Alchourron and E. Bulygin. Normative Systems. (Library of Exact Philosophy), Springer-Verlag, Vienna, New York, 1971. [Alchourron and Makinson, 1981] C. E. Alchourron and D. Makinson. Hierarchies of regulations and their logic. In R. Hilpinen, editor, New Studies in Deontic Logic, D. Reidel, Dordrecht, pp. 125{148, 1981. [al-Hibri, 1978] A. al-Hibri. Deontic Logic: A Comprehensive Appraisal and a New Proposal. Univ. Press of America, Washington DC. 1978. [Anderson, 1956] A. R. Anderson. The formal analysis of normative systems. In N. Rescher, editor, The Logic of Decision and Action, Univ. Pittsburgh, 1967, pp. 147{ 213. 1956. [Anderson, 1958] A. R. Anderson. A reduction of deontic logic to alethic modal logic. Mind 67:100{103, 1958. [Anderson, 1959] A. R. Anderson. On the logic of commitment. Philosophical Studies 10:23{27, 1959. [Anderson, 1962] A. R. Anderson. Logic, norms, and roles. Ratio 4:36{49, 1962. [Anderson, 1968] A. R. Anderson. A new square of opposition: Eubouliatic logic. In Akten des XIV. Internationalen Kongresses fur Philosophie (Wien, 2{9 Sept. 1968), Verlag Herder, Wien, pp. 271{284, 1968. [Anderson, 1971] A. R. Anderson. The logic of Hohfeldian propositions. Univ. Pittsburgh Law Review 33:29{38, 1971. [Apostel, 1960] L. Apostel. Game theory and the interpretation of deontic logic. Logique et Analyse 3:70{90, 1960. [ Aqvist, 1963] L. Aqvist. A note on commitment. Philosophical Studies 14:22-25, 1963. [ Aqvist, 1963a] L. Aqvist. Postulate sets and decision procedures for some systems of deontic logic. Theoria 29:154{175, 1963. [ Aqvist, 1965] L. Aqvist. Choice-oering and alternative-presenting disjunctive commands. Analysis 25:182{184, 1965. [ Aqvist, 1966] L. Aqvist. \Next" and \Ought", alternative foundations for Von Wright's tense-logic, with an application to deontic logic. Logique et Analyse 9:231{251, 1966. [ Aqvist, 1967] L. Aqvist. Good Samaritans, contrary-to-duty imperatives, and epistemic obligations. Noûs 1:361{379, 1967. [ Aqvist, 1969] L. Aqvist. Improved formulations of act-utilitarianism. Noûs 3:299{323, 1969.

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[Oppenheim, 1944] F. E. Oppenheim. Outline of a logical analysis of law. Philosophy of Science 11:142{160, 1944. [Powers, 1967] L. Powers. Some deontic logicians. Noûs 1:381{400, 1967. [Prawitz, 1968] D. Prawitz. A discussion note on utilitarianism. Theoria 34:76{84, 1968. [Prawitz, 1970] D. Prawitz. The alternatives to an action. Theoria 36:116{126, 1970. [Price, 1948] R. Price. A Review of the Principal Questions and DiÆculties in Morals. 1st ed. 1758. Edited by D.D. Raphael, Clarendon Press, Oxford, 1948. [Prior, 1954] A. N. Prior. The paradoxes of derived obligation. Mind 63:64{65, 1954. [Prior, 1955] A. N. Prior. Formal Logic. Clarendon Press, Oxford, 1955. [Prior, 1956] A. N. Prior. Review of G. Kalinowski: `Theorie des propositions normatives' (see above). J. Symbolic Logic 21:191{192, 1956. [Prior, 1958] A. N. Prior. Escapism: the logical basis of ethics. In A. Melden, editor, Essays in Moral Philosophy, Univ. Washington Press, Seattle, pp. 135{146, 1958. [Purtill, 1973] R. L. Purtill. Deontically perfect worlds and prima facie obligations. Philosophia 3:429{438, 1973. [Porn, 1970] I. Porn. The Logic of Power. Basil Blackwell, Oxford, 1970. [Porn, 1977] I. Porn. Action Theory and Social Science. Some Formal Models. D. Reidel, Dordrecht, 1977. [Quine, 1963] W. V. O. Quine. Carnap and logical truth. In P.A. Schilpp, editor, The Philosophy of Rudolph Carnap, Open Court, La Salle, Illinois, pp. 385{406, 1963. [Rescher, 1958] N. Rescher. An axiom system for deontic logic. Philosophical Studies 9:24{30, 1958. [Rescher, 1962] N. Rescher. Conditional permission in deontic logic. Philosophical Studies 13:1{6, 1962. [Rescher, 1966] N. Rescher. The Logic of Commands. Routledge & Kegan Paul, London, 1966. [Rescher and Robison, 1964] N. Rescher and J. Robison. Can one infer commands from commands? Analysis 24:176{179, 1964. [Robison, 1964] J. Robison. Who, What, Where, and When: A Note on Deontic Logic. Philosophical Studies 15:89{92, 1964. [Ross, 1944] A. Ross. Imperatives and logic. Theoria 7:53{71, 1941. Also in Philosophy of Science 11:30{46, 1944. [Ross, 1930] W. D. Ross. The Right and the Good. Clarendon Press, Oxford, 1930. [Ross, 1939] W. D. Ross. Foundations of Ethics. Clarendon Press, Oxford, 1939. [Scott, 1970] D. Scott. Advice on modal logic. In K. Lambert, editor, Philosophical Problems in Logic. Some Recent Developments, D. Reidel, Dordrecht, pp. 143{173, 1970. [Segerberg, 1971] K. Segerberg. Some logics of commitment and obligation. In R. Hilpinen, editor, Deontic Logic: Introductory and Systematic Readings, D. Reidel, Dordrecht, pp. 148{158, 1971. [Segerberg, 1971a] K. Segerberg. An Essay in Classical Modal Logic. Filoso ska foreningen, Uppsala, 1971. [Sellars, 1967] W. Sellars. Re ections on contrary-to-duty imperatives. Noûs 1:303{344, 1967. [Smiley, 1963] T. J. Smiley. Relative necessity. J. Symbolic Logic 28:113{134, 1963. [Smiley, 1963a] T. J. Smiley. The logical basis of ethics. Acta Philosophica Fennica 16:237{246, 1963. [Smullyan, 1978] R. M. Smullyan. What is the name of this book? | the riddle of Dracula and other logical puzzles. Prentice-Hall, Englewood Clis, 1978. [Spohn, 1975] W. Spohn. An analysis of Hansson's dyadic deontic logic. J. Philosophical Logic 4:237{252, 1975. [Stenius, 1963] E. Stenius. The principles of a logic of normative systems. Acta Philosophica Fennica 16:247{260, 1963. [Suppes, 1957] P. Suppes. Introduction to Logic. Van Nostrand, Princeton, 1957. [Thomason, 1981] R. H. Thomason. Deontic logic as founded on tense logic. In R. Hilpinen, editor, New Studies in Deontic Logic, D. Reidel, Dordrecht, pp. 165{176, 1981.

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[Thomason, 1981a] R. H. Thomason. Deontic logic and the role of freedom in moral deliberation. In R. Hilpinen, editor, New Studies in Deontic Logic, D. Reidel, Dordrecht, pp. 177{186,1981. [Tomberlin, 1981] J. E. Tomberlin. Contrary-to-duty imperatives and conditional obligations. Noûs 15:357{375, 1981. [Tomberlin and F. McGuinness, 1977] J. E. Tomberlin and McGuinness. \Because" and Good Samaritans. Critica 9:67{81, (Mexico), 1977. [Van Eck, 1981] J. E. Van Eck. A System of Temporally Relative Modal and Deontic Predicate Logic and its Philosophical Applications. Department of Philosophy, University of Groningen, The Netherlands, 1981. Also in Logique et Analyse, 25:249{290 and 25:339{381, 1982. [Van Fraassen, 1972] B. C. Van Fraassen. The logic of conditional obligation. J. Philosophical Logic 1:417{438, 1972. [Von Kutschera, 1973] F. Von Kutschera. Einfuhrung in die Logik der Normen, Werte und Entscheidungen. Alber, Freiburg, 1973. [Von Kutschera, 1974] F. Von Kutschera. Normative Praferenzen und bedingte Gebote. In H. Lenk, editor, Normenlogik, Verlag Dokumentation, Pullach bei Munchen, pp. 137{165, 1974. [Von Wright, 1951] G. H. Von Wright. Deontic logic. Mind 60:1{15, 1951. [Von Wright, 1951a] G. H. Von Wright. An Essay in Modal Logic. North-Holland, Amsterdam, 1951. [Von Wright, 1955] G. H. Von Wright. Om s.k. praktiska slutledningar (On so-called Practical Inferences). Tidskrift for Retsvidenskab 68:465{495, 1955. [Von Wright, 1956] G. H. Von Wright. A note on deontic logic and derived obligation. Mind 65:507{509, 1956. [Von Wright, 1963] G. H. Von Wright. Norm and Action. A Logical Inquiry. Routledge & Kegan Paul, London, 1963. [Von Wright, 1964] G. H. Von Wright. A new system of deontic logic. In Danish Yearbook of Philosophy 1:173{182, 1964. [Von Wright, 1965] G. H. Von Wright. A correction to a new system of deontic logic. Danish Yearbook of Philosophy 2:103{107, 1965. [Von Wright, 1967] G. H. Von Wright. The logic of action | A sketch. In N. Rescher, editor, The Logic of Decision and Action, Univ. Pittsburgh Press, Pittsburgh, pp. 121{136, 1967. [Von Wright, 1968] G. H. Von Wright. An Essay on Deontic Logic and the General Theory of Action. North-Holland, Amsterdam, 1968. [Von Wright, 1971] G. H. Von Wright. Deontic logic and the theory of conditions. In R. Hilpinen, editor, Deontic Logic: Introductory and Systematic Readings, D. Reidel, Dordrecht, pp. 159{177, 1971. [Von Wright, 1974] G. H. Von Wright. Handlungslogik. In H. Lenk, editor, Normenlogik, Verlag Dokumentation, Pullach bei Munchen, pp. 9{24, 1974. [Von Wright, 1977] G. H. Von Wright. Zur Einfuhrung. In A.G. Conte, R. Hilpinen and G.H. Von Wright, editors, Deontische Logik und Semantik, Athenaion, Wiesbaden, pp. 7{8, 1977. [Wedberg, 1951] A. Wedberg. Some problems in the logical analysis of legal science. Theoria 17:246{275, 1951. [Wedberg, 1969] A. Wedberg. Den klassiska deontiska konsekvensprincipens paradoxer: n agra preformella re exioner. In Logik, ratt och moral: Filoso ska studier tillagnade Manfred Moritz, Studentlitteratur, Lund, pp. 213{232, 1969. [Weinberger, 1970] O. Weinberger. Rechtslogik. Versuch einer Anwendung moderner Logik auf das juristische Denken. Springer-Verlag, Vienna, New York, 1970.

CARMO AND ANDREW J. I. JONES JOSE

DEONTIC LOGIC AND CONTRARY-TO-DUTIES

1 INTRODUCTION Deontic logic is concerned with the logical analysis of such normative notions as obligation, permission, right and prohibition. Although its origins lie in systematic legal and moral philosophy, deontic logic has begun to attract the interest of researchers in other areas, particularly computer science, management science and organisation theory. Among the application areas which have already received some attention in the literature are: issues of knowledge representation in the design of legal expert systems; the formal speci cation of aspects of computer systems, for instance in regard to security and access control policies, fault tolerance, and database integrity constraints; the formal characterisation of aspects of organisational structure, pertaining for example to the responsibilities and powers which agents are required or authorised to exercise. The \EON" workshop proceedings provide some illustrations of work in these areas (see [EON91; EON94; EON96]). Deontic logic is one of the formal tools needed in the design and speci cation of normative systems, where the latter are understood to be sets of agents (human or arti cial) whose interactions can fruitfully be regarded as norm-governed; the norms prescribe how the agents ideally should and should not behave, what they are permitted to do, and what they have a right to do. Importantly, the norms allow for the possibility that actual behaviour may at times deviate from the ideal, i.e. that violations of obligations, or of agents' rights, may occur. In [Jones and Sergot, 1992; Jones and Sergot, 1993] Jones and Sergot argue that it is precisely when the possibility of norm violation is kept open that deontic logic has a potentially useful role to play. If agents can always be assumed to behave in conformity to norm, the normative dimension ceases to be of interest: the actual does not depart from the ideal, so nothing is lost by merely describing what the agents in fact do. Thus, although it is correct to say that deontic logic deals with the logic of obligation, permission and other normative notions, a more insightful characterisation, Jones and Sergot suggest, views deontic logic as essentially concerned with representing and reasoning about the distinction between the actual and the ideal. Systems for which that distinction is relevant are genuinely normative systems, and their speci cation will ordinarily include \secondary" norms which indicate what is to be done in circumstances in which actual behaviour has deviated from the ideal. The methodological guidelines proposed in [Jones and Sergot, 1992; Jones and Sergot, 1993] strongly suggest D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, Volume 8, 265{343.


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that \secondary" norms of this kind ( rst dubbed \contrary-to-duty" in [Chisholm, 1963]) will be a prominent feature of normative systems, and thus that any adequate deontic logic must accommodate them. However, the analysis of contrary-to-duty obligation sentences has proved to be a task of some considerable complexity. And it is this issue | at the very core of deontic logic | which this chapter addresses. The plan is as follows: we rst (Section 2) describe Standard Deontic Logic, and a number of its defects, including problems regarding the representation of conditional obligation sentences. In the course of Section 3 we examine a number of dierent theories which have attempted to accommodate contrary-to-duty obligation sentences (CTDs), and in the course of this examination we identify several criteria | eight in all | which, we argue, an adequate treatment of Chisholm's puzzle about CTDs should meet. Some of these criteria are not tied to Chisholm's problem, but apply quite generally to the analysis of CTDs. Section 4 presents a revised, and in parts considerably modi ed version of the [Carmo and Jones, 1997] theory of CTDs; its application to a number of CTD \scenarios" is investigated in some detail in Section 5, and this provides a further impression of the broad range of representational and reasoning issues which a CTD theory must address. Section 6 examines some possible counter-examples to the proposed analysis, thereby relating its treatment of CTD problems to other well-known issues in deontic logic, concerning | in particular | the closure of deontic operators under logical consequence, and the representation of con icts of obligations. Section 7 oers further observations on alternative approaches based on temporal logic, the logic of action, and preference orderings, respectively. The overall aim of the chapter is to supply a rather detailed overview of a group of problems at the heart of deontic logic, and a guide to existing attempts to solve them. 2 DEONTIC LOGIC: THE STANDARD APPROACH

2.1 Standard Deontic Logic The standard approach to deontic logic takes it to be a branch of modal logic, interpreting the necessity operator as expressing ethical/legal necessity, i.e. as meaning \it is obligatory that", and denoting it by O ; accordingly, the dual possibility operator = : : is interpreted as expressing \it is permitted that" (and is denoted by P), and the impossibility modal contruction : is interpreted as expressing \it is forbidden that" (and is often denoted by F). Axiomatically, the weakest deontic logic (called standard deontic logic, SDL for short) is then obtained by replacing the modal necessity schema (T) ( A ! A: unacceptable for a deontic interpretation, since what is


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obligatory may fail to be the case) by the (D) schema (which requires that what is obligatory is permitted). Thus, following the Chellas classi cation [Chellas, 1980], SDL is the weakest normal modal system of type KD; that is, its theorems can be characterized as the smallest set of formulas that includes all instances of the following axiom schemas, and that is closed under the O-necessitation rule and Modus Ponens (MP). Axiom schemas (PC) (K) (D)

All instances of tautologies (PC stands for Propositional Calculus) O (A ! B ) ! (O A ! O B ) (O A ! PA)

Rules O -necessitation: OAA

Modus Ponens (MP):

A; A!B B

We here employ capital letters (A; B; C; : : :) to stand for arbitrary formulas (well-formed sentences of the underlying propositional modal logic), and we use lower case letters (p; q; : : :) for arbitrary atomic sentences, and ? and > to denote, respectively, a contradiction and a tautology; parentheses will be omitted following the usual precedence rules for the operators; the Boolean connectives will be denoted by :, ^, _, ! and $; in the meta-language we denote such connectives by \not", \and", \or", \if ... then ..." (or \implies") and \i" (if and only if), and in the meta-language we also avail ourselves of the universal and the existential quanti ers (these do not appear in the object language: we are concerned only with propositional modal logics). Moreover, as usual, we will use ` A (respectively 6` A) to denote that A is a theorem (respectively, A is not a theorem) of the underlying logical system; and, following the traditional philosophical/logical approach to deduction, we say (cf. [Chellas, 1980; Hughes and Cresswell, 1984]) that A is deducible from a set of hypotheses , written ` A (or simply A1 ; : : : An ` A if is nite), i A belongs to the smallest set of formulas that contains and the theorems and that is closed under (MP).1 1 In this way we get a Boolean, compact, deductive system (see e.g. [Bull and Segerberg, 2001]). Non-Boolean axiomatic approaches to deduction, where nontautological rules may also be applied to the hypotheses, and not only to the theorems, may be found in some works in the eld of mathematical logic, such as [Hamilton, 1978; Mendelson, 1979].


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Semantically, the models M of SDL are standard models [Chellas, 1980]: = (W; R; V ), where W is a non-empty set (the set of worlds), R is a binary relation on W and V is an assignment to each atomic sentence of a set of worlds; informally, V (p) denotes the set of worlds where p is true. In order to validate the schema (D) we require that the accessibility relation R is serial, i.e. (8w)(9v)wRv (using w; v; : : : to denote worlds and, as usual, writing wRv instead of hw; vi 2 R). The deontic interpretation of the accessibility relation is as follows: wRv i v is a deontic alternative to, or an ideal version of, w. The truth of a formula A in a world w of a model M is denoted by M j=w A and is de ned as usual: for instance, M j=w O A i (8v) (if wRv then M j=v A); thus, informally, O A is true in a world w i A is true in all ideal versions of w. A formula A is true in a model M, written M j= A, i A is true in all the worlds of the model M; and a formula A is valid, written j= A, i A is true in all models.

M

2.2 SDL and its problems It is widely accepted that SDL is not adequate as a basic deontic logic. In fact, few systems of logic have been as heavily criticised as SDL; SDL gives rise to a set of \paradoxes" (theorems of SDL that many have deemed to be counter-intuitive) and there are some deontic concepts and constructions which apparently cannot be expressed in SDL in a consistent manner. Some of the main examples will be given below. We have essentially two aims here: rst, without any claims to originality, we comment on the reasons underlying the so-called paradoxes; secondly, we indicate which of these problems have a counterpart in other areas of applied modal logic (e.g., epistemic, doxastic and action logics), and which seem to be particular to deontic logic. A rst group of paradoxes has its origin in the closure of the O-operator under logical consequence (that is, in the fact that SDL, like any normal modal logic, contains the (RM)-rule: \if ` A ! B then ` O A ! O B "). Some well known examples are:

Ross paradox: (` O A ! O (A _ B ))

\If it is obligatory to mail the letter, then it is obligatory to mail the letter or to burn it"

The question of the signi cance of this paradox has been the subject of considerable dispute. Whereas some claim that the second obligation (the one in the consequent) is a counter-intuitive consequence of the rst, since it seems to leave open to the agent a choice to mail or to burn the letter, others maintain that the consequent does not leave a choice of this kind, because burning the letter is clearly not a way of meeting the obligation expressed by the antecedent. Given


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the perspective on deontic logic advocated in Section 1, however, we should also look at the problem from the point of view of violation: supposing that A is obligatory and that A is not the case, how many obligations have been violated? If we accept the Ross theorem, then not only has the obligation that A been violated, but | in addition | for each state of aairs B which actually fails to obtain, an obligation that A _ B has also been violated. This is a peculiar result; it contrasts, of course, with how things look from a ful lment perspective; for if the obligation that A is ful lled, then so are all the other obligations which can be derived by application of the Ross theorem. (We are assuming, as is natural, that within SDL violation of an obligation O C is to be expressed as the conjunction O C ^ :C .)

Free Choice Permission paradox: This paradox has to do with the fact that (in SDL) 6` P(A _ B ) ! (PA ^ PB ), whereas | ordinarily | if it is permitted that A or B this would be understood to imply that A is permitted and B is permitted. We include this paradox in this group, since the reason why we cannot add P(A _ B ) ! (PA ^ PB ), as a new axiom, to SDL is the fact that, by the (RM)-rule, ` PA ! P(A _ B ), which together with P(A _ B ) ! (PA ^ PB ) would imply PA ! (PA ^ PB ); so permission to go to the cinema would imply permission to kill the President! Moreover, from any permission we could then deduce P?, which is inconsistent with the fact that ` O >. However, in common with some other researchers, we think that this \paradox" is a pseudo-problem: if what we want to express is that both A and B are permitted, then we should simply represent that formally by PA ^ PB (instead of by P(A _ B )).

Good Samaritan paradox:

\If it is obligatory that Mary helps John who has had an accident, then it is obligatory that John has an accident" On the assumption that \Mary helps John who has had an accident" is represented as the conjunction \Mary helps John and John has had an accident", then the antecedent of the above conditional takes the form \O (A ^ B )". Since, tautologically, a conjunction implies each of its conjuncts, the (RM)-rule yields the SDL theorem: ` O (A ^ B ) ! O B . In our view, the formal concepts needed to deal with problems about contrary-to-duty obligations can also provide an appropriate analysis of the Good Samaritan problem. So we return to this issue below, in Section 6.

Deontic/epistemic paradox:

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\If it is obligatory that Mr. X knows that his wife commits adultery, then it is obligatory that X's wife commits adultery" We here assume, as is usual, that the (T)-schema holds for the epistemic operator. So this problem is again a result of the fact that, in SDL, any logical consequence of that which is obligatory is itself obligatory. We note that the closure of the necessity operator under logical consequence is also a source of problems for other applications of modal logic, for instance epistemic and doxastic logics, where the assumption that every agent knows (believes) every logical consequence of what he knows (believes) is an extreme idealisation. In the logic of action, too, it is surely not acceptable to suppose that an agent brings about all the logical consequences of that which he brings about (cf. [Elgesem, 1993]).2 A second problem of SDL has to the with the O -necessitation rule itself, according to which any tautology (more generally, any theorem) is obligatory, which is incompatible with the idea that obligations should be possible to ful ll and possible to violate. Similar problems occur with this rule in the epistemic and doxastic logics, where it requires that an agent knows (or believes) all theorems (called in [Hintikka, 1975] the \logical omniscience problem"), and in the logic of action, where it is in general supposed that that which can be brought about must be avoidable (see, e.g., [Elgesem, 1993; Santos and Carmo, 1996]). A third problem of SDL is that, because of the (D)-schema, it is not possible to express consistently a con ict of obligations, even though, as a matter of fact, normative systems may indeed contain con icting obligations. We shall return to this issue later in this chapter. But, again, we note at this point that this is not a problem only of deontic logic: similar problems may appear, for instance, in the logic of belief. However, it is fair to say that, for most deontic logicians, the problem of how to represent conditional obligation sentences has been their principal reason for seeking an alternative to SDL. Let us denote by O (B=A) the \conditional obligation of B , given A"; so O (B=A) is intended to mean that \it is obligatory that B , if A is the case". In SDL there are two possible ways to represent such sentences: (option1) and (option2)

O (B=A)= A ! O B df

O (B=A)= O (A ! B ) df

2 In [Konolige and Pollack, 1993] it is argued that this problem, called the \sideeect problem" in [Bratman, 1987], is even worse for the logic of intentions - a logic which, it has been suggested, has very close similarities to deontic logic (see [Porn, 1977; Jones, 1991]).


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Note rst what these two options have in common. With both of them we get (within SDL): (UN) (SA)

` O B $ O (B=>) ` O (B=A) ! O (B=A ^ C )

The rst theorem is generally seen as a good property, and has been accepted by many authors on the grounds that an unconditional obligation is a particular (limiting) case of a conditional obligation, where the condition is a logical truth. Here, however, we shall adopt the opposite view, in line with the opinion expressed by Carlos Alchourron in [1993, pp. 62], who argued that (UN) was one of the wrong steps followed by almost all researchers in deontic logic. (SA) is known as the \principle of strengthening of the antecedent"; it is problematic, since it appears to make the expression of defeasible (conditional or unconditional3) obligations impossible; but of course it is a commonplace feature of obligations that they are subject to exceptions. Consider, for instance, a conditional obligation to the eect that, if your aged mother is sick, then you should help her. Such a conditional obligation might well leave room for exceptions, just as penguins might be the exception to the generalisation that birds y; supposing for example that your young child has been injured in a car accident, and urgently needs you at the hospital, the obligation to help your sick, aged mother may well be deemed to have been defeated, or overturned. But again this problem (which has some connections with the problem of how to deal with con icting obligations) is not a speci c issue of deontic logic; the problem of how to deal with defeasible conditionals appears in many other areas and has been a source of intensive research. So far we have not yet found a problem that sets deontic logic apart from other branches of modal logic. But we here return to the point emphasised in the introduction and suggest that the issue of how to represent contrary-to-duty obligation sentences (CTDs) | obligations which come into force when some other obligation is violated | seems to be a speci c problem of deontic logic. It has sometimes been proposed, however, that CTD obligations may be seen as handling exceptions to (primary) obligations. Although we accept that there may be some connections between the problem of how to deal with CTDs and the problems concerning allowable exceptions and default reasoning, it should be stressed that there are also crucially important dierences (cf. [Prakken and Sergot, 1994]): when a CTD obligation comes into force because of some violation, we do not want then to say that the violated obligation has been defeated; it has not been overturned, it has been violated! We need to be able to integrate in

3 Note that combining the two previous theorems we get ` O B ! O (B=A), and so an unconditional obligation remains obligatory under any condition.

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a single logical framework the ability to make deductions at two dierent levels: on the level of what ideally should be the case, and on the level of what actually should be the case, given the circumstances (where, of course, the circumstances might include the fact that what has happened deviates from the ideal). The simultaneous speci cation of both ideal behavior and of what to do when actual behavior deviates from the ideal is a central task of deontic logic. 3 CONTRARY-TO-DUTIES

3.1 Chisholm's CTD-paradox and SDL Consider the following set of four sentences, formulated by Chisholm in 1963 [Chisholm, 1963]: EXAMPLE 1. (a) It ought to be that a certain man go to help his neighbours. (b) It ought to be that if he goes he tell them he is coming. (c) If he does not go, he ought not to tell them he is coming. (d) He does not go. There is widespread agreement in the literature that, from the intuitive point of view, this set is consistent, and its members are logically independent of each other; and there is a good deal of disagreement in the literature as regards which further requirements an adequate formal representation of the Chisholm set should meet. We start by discussing whether the Chisholm set can be represented in SDL in a way that meets this set of two minimum requirements, leaving the discussion of other further requirements to later. It is straightforward to represent sentences (a) and (d) in SDL; the question is how to represent (b) and (c), since they express conditional obligations. Let us leave that open for the moment, and represent them by the use of our binary conditional obligation operator above; we then get (using \tell" and \help" in an obvious way as abbreviations of the sentences concerned): (a) O help

(or O (help/>), since in SDL ` O help $ O (help/>))

(b) O (tell / help) (c) O (:tell / :help) (d) :help


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In regard to the representation of conditional obligations in SDL, recall the two alternatives: (option1) and (option2)

O (B=A)= A ! O B df

O (B=A)= O (A ! B ) df

With (option1) we get the following results: (FD)

` :A ! O (B=A) ` A ^ O (B=A) ! O B

Given that an expression of the form O (B=A) is intended to mean that, in circumstances A, B is obligatory, the rst of these two results is clearly problematic. From the fact that it is not raining we should not be able to deduce that, in circumstances where it is raining, it is obligatory that the President be assassinated. The other theorem has to do with the fundamental issue of how we can detach new (unconditional) obligations from conditional obligations, and it states a kind of \factual detachment" principle, allowing the deduction of the actual obligations of the agent, that is, the obligations which arise given the actual facts of the situation. With (option2) we get the following results: (DD)

` O :A ! O (B=A) ` O A ^ O (B=A) ! O B

According to the rst theorem everything is obligatory on the condition that some forbidden fact is the case: thus (option 2) clearly does not allow us to express CTDs. The second theorem represents a kind of \deontic detachment" principle, allowing the deduction of the ideal obligations of the agent, i.e. the further obligations which arise if he behaves in a way which conforms with some existing set of obligations. The surface structures of lines (b) and (c) in the original Chisholm set might be taken to indicate that, within SDL, (option 2) should be chosen for (b), and (option 1) for (c), giving: (a) O help (b) O (help ! tell) (c) :help ! O :tell (d) :help

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This was Chisholm's choice, and he rightly went on to point out that this formalisation yields an inconsistency, since O tell is derivable from (a) and (b), whilst O :tell is derivable from (c) and (d) (and an inconsistency follows by the (D)-schema). If, alternatively, we use (option 1) for both lines (c) and (b), then the resulting set is consistent, but logical independence is lost, since (b) will then be a consequence of (d). Likewise, if (option 2) were adopted for both (b) and (c), then (c) would be a consequence of (a) by the (RM)-rule. So, in SDL, the conclusion is that the Chisholm set cannot be represented in a way which satis es both of the two minimum requirements of consistency and logical independence.

3.2 Some further requirements on the representation of CTDs A number of deontic logicians have argued that the problems raised by CTDs involve in an essential way either a temporal dimension or actions. We shall have a good deal more to say about these lines of approach later on (especially in Section 7), but for the moment we just want to register agreement with Prakken and Sergot [Prakken and Sergot, 1994; Prakken and Sergot, 1996], who have indicated that there are examples of CTD scenarios where it is far from obvious how considerations of the temporal or action dimensions might be applicable. Consider: EXAMPLE 2. (a) There ought to be no dog. (b) If there is no dog, there ought to be no warning sign. (c) If there is a dog, there ought to be a warning sign. (d) There is a dog. EXAMPLE 3. (a) There must be no fence. (b) (c) If there is a fence, then it must be a white fence. (d) There is a fence. Examples of these kinds suggest that a treatment of CTD's which is tied to temporal or action aspects will not be suÆciently general in its scope. A further question which existing treatments of CTD's raise is this: are lines (b) and (c) of the Chisholm set to be assigned fundamentally dierent


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logical forms? The theory we develop below gives a negative answer to this question, and supplies a uniform treatment of deontic conditionals. Our view is that, in the absence of strong arguments to the contrary, the surface forms of (b) and (c) should be deemed to be merely stylistic variants of essentially the same type of underlying logical structure. In particular, we reject the position taken in [Prakken and Sergot, 1994; Prakken and Sergot, 1996], where it is argued that (b) and (c) should be given distinct logical representations just because (c), unlike (b), is a contrary-to-duty conditional, expressing as it does the obligation which comes into force when the obligation expressed by line (a) is violated. Prakken and Sergot's approach makes the assignment of logical form to deontic conditionals a highly context-dependent matter, with the consequence that any insertion or deletion of a norm may require that some revision then has to be made to the formalisation of some other norm in the set; (e.g., deleting line (a) of the Chisholm set would require, on their approach, a change in the formalisation of line (c)). Likewise, the form initially assigned to a given sentence might have to be revised in virtue of what turns out to be derivable from other sentences; suppose, for instance that \if A then it is obligatory that B " is in the initial set, and is assumed not to be a CTD; if it then transpires that \it is obligatory that not A" is derivable from other members of the initial set, then the conditional becomes a CTD and its logical form has to be changed accordingly. This change may, in turn, have further repercussions regarding what can be derived . . . and so on. Now with a small initial set, such as Chisholm's, it will of course be relatively easy to see where changes need to be made; but with a large corpus of norms it is not diÆcult to imagine that the problem could become intractable. The disadvantages which accrue from this kind of context-dependence of logical form are so great, in our opinion, that any approach to the analysis of CTDs which manages to avoid it is - other things being equal - to be preferred. Thus, we have so far identi ed the following requirements that an adequate formalisation of the Chisholm set should meet: (i) consistency; (ii) logical independence of the members; (iii) applicability to (at least apparently) timeless and actionless CTDexamples; (iv) analogous logical structures for the two conditional sentences, (b) and (c). One important group of deontic logics that satisfy these requirements employs a primitive dyadic conditional obligation operator O ( / ), where

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O (B=A) is read \it is obligatory that B , given that A". These logics usually take the unconditional obligation O B to be equivalent to O (B=>),

and they represent the Chisholm set as follows: (a) O (help / >) (b) O (tell / help) (c) O (:tell / :help)

(d) :help

Following [Lower and Belzer, 1983] we can distinguish between two main \families" of dyadic deontic logics, according to the kind of detachment principles they support: one supports the \factual detachment" principle (FD), and we call it the \FD-family";4 the other supports the \deontic detachment" principle (DD), and we call it the \DD-family".5 Returning again to the Chisholm set, it is clear that (as for its proposed representation within SDL) acceptance of both (FD) and (DD) would permit the derivation of O :tell (by (FD) on lines (c) and (d)) and O tell (by (DD) on lines (a) and (b)). If the (D)-schema is accepted, then the situation arising from adoption of both (FD) and (DD) would of course be one of logical inconsistency. But even if the (D)-schema is not accepted, so that the conjunction O tell ^ O :tell is not deemed to be logically inconsistent, the derivation from the Chisholm set of a con ict of obligations of the type expressed by this conjunction is surely unacceptable from the intuitive point of view. The situation described by the Chisholm set does not present the agent concerned with a moral dilemma, on our view. Requirement (i), above, should be understood as one to the eect that a conjunction of the form O A ^ O :A should not be derivable from the formal representation of the set, regardless of whether that conjunction is deemed logically inconsistent. Of course, neither the FD-family nor the DD-family accepts both (FD) and (DD). Nevertheless, it might be suggested that a fully adequate representation of the Chisholm set should be able to capture, in a way which generates neither inconsistency nor a moral dilemma, both the fact that | given the circumstances, and particularly the occurrence of the violation of the obligation expressed by line (a) | the agent's actual obligation is 4 In general the logics in this family have a semantics based on minimal models (proposed, independently, by Dana Scott and Montague, and popularised by [Chellas, 1980]). As representatives of this family [Lower and Belzer, 1983] mention [Mott, 1973; al-Hibri, 1978; Chellas, 1974]; however, as regards [Chellas, 1974] it is not entirely clear whether Chellas commits himself to acceptance of (FD). 5 [Lewis, 1974] presents an overview of several members of the DD-family. These logics introduce, in the semantics, a preference relation between the worlds, that orders the worlds according to their ideality; then O (B=A) is true at a world i there is some world where A ^ B is true and that is more ideal than any world where A ^ :B is true. See also, below, Sections 7.1 and 7.3.


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not to tell his neighbours he is coming, and the fact that | under ideal circumstances, in the absence of violation of the obligation expressed by line (a) | the agent's obligation would be to help his neighbours and to tell them he is coming. Accepting these suggestions, we oer three further requirements which we believe an adequate representation of the Chisholm set should meet: (v) capacity to derive actual obligations; (vi) capacity to derive ideal obligations;6 (vii) capacity to represent the fact that a violation of an obligation has occurred. Neither the FD-family nor the DD-family meet both (v) and (vi).7 We will return later to the issues raised by (vii).

3.3 The \pragmatic oddity" One of the logics that ful lls all these requirements is the one proposed in [Jones and Porn, 1985]. Jones and Porn adopt a completely dierent approach from that of the dyadic deontic logics, and de ne a deontic logic where non-normal obligation operators are obtained as Boolean combinations of normal modal operators, following a strategy that had already been used in the eld of action logic by Kanger and by Porn. Taking as its point of departure the observation that SDL fails in its attempt to capture CTDs because | from the semantical point of view SDL considers only the ideal versions of each world, Jones and Porn propose, in addition to SDL's accessibility relation, a second accessibility relation which picks out the sub-ideal versions of a given world (and they further require that each world is either an ideal or a sub-ideal version of itself). Then they introduce into the logical language two modal necessity operators,8Note that the notation employed here for the operators diers in most cases from that used in [Jones and Porn, 1985]. here denoted by ! and ! . The rst of these is just the obligation operator of SDL, so that an

6 s

i

Some might call them prima facie obligations. However, we avoid using this term here since its meaning in the literature seems to us to be far from clear. Furthermore, [Prakken and Sergot, 1997] provide good reasons for supposing that the term is most at home in the discussion of defeasibility, rather than CTDs. 7 In [Jones, 1993] it is argued that a further problem of the (FD)-family is that they reject the \principle of strengthening of the antecedent" (SA) whilst at the same time accepting unrestricted factual detachment. The problem is that one of the reasons for rejecting (SA) is that one wants to be able to represent conjunctions of the form O (B=A) ^ O (:B=A ^ C ), without getting logical inconsistency or moral dilemma of the form O B ^ O :B , even in circumstances in which both A and C are true. 8`


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expression of the form ! A is true at a given world w i A is true at all of the ideal versions of w. By contrast, ! A is true at a given world w i A is true at all of the sub-ideal versions of w. (A sub-ideal version w1 , of w, is informally seen as a version of w in which at least one of the obligations in force at w is violated.) The duals of ! and ! are, respectively, ! and !. Finally they introduce both a deontic necessity operator ! , de ned as follows:

i

s

i

s

! A= ! A ^ ! A df

i

i

s

s

and an actual-obligation operator Ought , de ned by:

Ought A= ! A ^ ! :A df

i

s

(the second conjunct guarantees that ` :Ought

>).

The Chisholm set is then represented in [Jones and Porn, 1985] as follows: (a) Ought help

! (:help ! Ought :tell) (c)

(b) ! (help ! Ought tell) (d) :help

The set, on this representation, is consistent and its members are logically independent of each other. Lines (c) and (d) imply Ought :tell (note that ! is a \success" operator, i.e. it satis es the (T)-schema), and lines (a) and (b) imply ! Ought tell. Furthermore, the conjunction of (a) and (d) may be taken as expressing the fact that the unconditional obligation to help the neighbours has been violated; and, had it been the case that \(d') help", rather than (d), were true, then from (b) one could have deduced the actual obligation to tell, Ought tell. Apparently, all is well! However, [Prakken and Sergot, 1994; Prakken and Sergot, 1996] point out that the Jones and Porn treatment of Chisholm, in common with a number of others, generates what they call the \pragmatic oddity": line (a), together with the derived actual obligation Ought :tell, require that, in all ideal versions of the given world, the agent concerned goes to help his neighbours but does not tell them he is coming | a result which appears highly counterintuitive. Prakken and Sergot correctly point out that, for a number of cases, a reasonable temporal interpretation is available which enables the pragmatic oddity to be avoided. For instance, perhaps the obligation expressed in line (a) would ordinarily be understood as an obligation to go to help the

i


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neighbours no later than a particular time, t. Then, if line (d) were to be true after time t, the accessible deontically ideal worlds would be characterised in such a way that, after time t, these worlds would require that the agent does not tell his neighbours he is coming (but they would not, of course, also require that he goes to help, since it would then be too late). However, as we indicated above, Prakken and Sergot also point out that there are instances of the Chisholm set which may be interpreted in such a way that the temporal dimension is completely absent ([Jones, 1993, pp. 153-4] makes a similar point). Example 2, above, is one such case: a very ordinary way of understanding that set takes each sentence to be true at one and the same moment of time, and | without any insertion of temporal quali cations concerning when there ought to be no dog, or when there ought to be a warning sign | allows the conclusion to be drawn that there ought, in the circumstances, to be a warning sign, without thereby generating the pragmatic oddity, i.e., without forcing the further conclusion that, in all ideal versions of the given situation, there is no dog but there is a sign warning of one. Unfortunately, Prakken and Sergot oer little by way of explanation of the pragmatic oddity: they say little about what it is that creates the sense of oddity. In [Carmo and Jones, 1997] we suggest an explanation which exploits a parallel between examples of type Example 2, which on the [Jones and Porn, 1985] analysis exhibit the pragmatic oddity simpliciter, and examples like Example 3 above (also due to Prakken and Sergot) which, by virtue of some assumed logical truth, are inconsistent when formalised in the style of [Jones and Porn, 1985] (according to which, in all ideal versions of the given world, there is a white fence and no fence at all!). The suggested parallel is as follows: as represented in the language of [Jones and Porn, 1985], Example 2 exhibits the pragmatic oddity because an inconsistency would be generated were one to add to the example the further constraint that it ought not to be the case that there is both no dog and a sign warning of one. The sense of oddity arises because there is an interpretation of Example 2 according to which it remains consistent even if supplemented with that further constraint; and the problem with the [Jones and Porn, 1985] approach is that it fails to capture that interpretation. Thus we add another requirement which an adequate representation of CTDs should satisfy: (viii) capacity to avoid the pragmatic oddity (interpreted according to the previous diagnosis).

3.4 Two attempts to resolve the \pragmatic oddity" In [Prakken and Sergot, 1994; Prakken and Sergot, 1996], Prakken and Sergot argue that the proper response to the problems raised by Example 2 |

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and in particular the problem of pragmatic oddity | is to assign distinct logical forms to primary obligations, on the one hand, and CTD obligations, on the other. For CTD obligations, they relativise an obligation operator to a speci c \context of violation"; more precisely, an expression of the form OA B is intended to be read as \there is a secondary obligation that B given that, or presupposing, the sub-ideal context A", or \given that A, which is a violation of some primary obligation, there is a secondary, compromise obligation that B " [Prakken and Sergot, 1996, section 5]. They emphasise that expressions of the form OA B are not to be read as conditional primary obligations. \The expression OA B : : : represents a particular kind of obligation. There is no meaningful sense . . . in which the obligation O B can be detached from the expression OA B " [loc.cit.]. Their representation of Example 2 takes the form: (a) O :dog

(b) :dog ! O :sign

(c) dog ! Odog sign

(d) dog We shall not pursue the Prakken and Sergot 94 treatment of CTDs here (although we return to their work brie y in Section 7). SuÆce it to say that their approach (in [Prakken and Sergot, 1994; Prakken and Sergot, 1996]) rejects the fourth of our requirements for a satisfactory theory in this area. For them, the choice of logical form for an apparently conditional deontic sentence will itself be dependent on which other norms are contained in, or derivable from, the set of norms being formalised.9 In [Carmo and Jones, 1995]10 we attempted a dierent kind of approach to the problem of the pragmatic oddity, distinguishing between \ideal obligations" (line (a) in Examples 1, 2 and 3, for instance) and \actual obligations", which indicate what is to be done given the (perhaps less-than-ideal) circumstances. The operator Oa , for representing actual obligations, was de ned in the same way as the Ought -operator of [Jones and Porn, 1985], described above. As regards ideal obligations, the basic model-theoretic 9 There are also some diÆculties in understanding how OA B should be interpreted, particularly since Prakken and Sergot insist that A (in OA B ) necessarily represents a context of violation. For instance, the formula (PA ^ OA B ) ! O B is valid, on their account (where P is the permission operator), but not trivially so. As we see it, intuitively the conjunction in the antecedent of this conditional (given their reading of OA B ) could only be false, so it should imply anything. Furthermore, what can they possibly mean by the claim that O B is an abbreviation of O> B ? Are we to suppose that it is obligatory that B only if the tautology represents a context of violation? 10 We there adapt the logic proposed in [Carmo and Jones, 1994; Carmo and Jones, 1996] for the analysis of deontic integrity constraints.


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idea was to distinguish between ideal versions of a given world (the fundamental feature of SDL), and ideal worlds themselves. Accordingly, we divided the set of possible worlds W into two mutually exclusive sub-sets, the set of ideal worlds and the set of sub-ideal worlds; importantly for our purposes, we allowed that a world w1 could be an ideal version of a given (sub-ideal) world w without also itself being an ideal world. And we xed truth conditions for expressions of the form Oi B (\it ought ideally to be the case that B ") in terms of the truth of B in all ideal worlds and falsity in some sub-ideal world.11 We represented Example 2 in the following way: (a) Oi :dog

! (dog ! O (c)

(b) ! (:dog ! Oa :sign) a sign)

(d) dog

All the requirements (i){(viii) are met by this analysis. In particular, the pragmatic oddity disappears because the conjunction Oi :dog Ôa sign, which is clearly derivable from (a){(d), does not imply that, in all ideal versions of the given world, there is no dog but a sign warning of one. What the conjunction does say, essentially, is that in all ideal worlds there is no dog, but in all ideal versions of the given (clearly sub-ideal) world there is a warning sign. The proposal worked well for this and a number of other examples, and we have de ned a complete axiomatization for the logic. However, as we now see things, this approach suered from a defect similar to the one we have criticised in relation to [Prakken and Sergot, 1994; Prakken and Sergot, 1996]: the assignment of logical form for some of the norms in the set is dependent on the other norms in it. In [Prakken and Sergot, 1994; Prakken and Sergot, 1996] this was re ected in the use of dierent obligation operators for representing the deontic conditionals expressed by lines (b) and (c); in [Carmo and Jones, 1995] it is re ected in the use of dierent obligation operators for representing line (a) and lines (b) and (c). So, in order to capture the general issue motivating the adoption of adequacy requirement (iv), it should be reformulated as follows: (iv) the assignment of logical form to each of the norms in the set should be independent of the other norms in it. 11 The second conjunct simply guarantees the violability of ideal obligations (i.e. j= :Oi >). [Carmo and Jones, 1995] contains discussion of possible connections between

the notions of ideal/sub-ideal world and ideal/sub-ideal versions of a world, but we omit those details here.


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A related observation is that problems appear within the [Carmo and Jones, 1995] approach if we add to the Chisholm set other norms that interact in some signi cant way with the norms in the original set. In particular, serious diÆculties arise as soon as a \second-level" of CTDs is considered. Suppose, for instance, that lines (e) and (f), below, are added to Example 2: (e) If there is a dog and no warning sign, there ought to be a high fence. (f) There is no warning sign. The [Carmo and Jones, 1995] representation of this extended set is:

(a) Oi :dog (b) ! (:dog ! Oa :sign) (c) ! (dog ! Oa sign) (d) dog (e) ! (dog ^ :sign ! Oa fence) (f) :sign And the pragmatic oddity now re-appears, since the conjunction Oa sign ^ Oa fence is derivable. So, in all ideal versions of the given world, there is a sign and a fence. (If this does not seem \odd", imagine that the sign says \Beware of the unfenced dog": it may well be forbidden to have both a sign of that kind and a fence. Thus the pragmatic oddity, in the sense of our proposed diagnosis, re-emerges.) The problem of \further levels" of CTDs would force the [Carmo and Jones, 1995] approach to allow the possibility of an in nity of obligation operators: the need to associate (in some way or other) a context to each obligation operator seems to re-appear.

4 CONTRARY-TO-DUTIES: A NEW APPROACH On one very common interpretation of the set (a){(f) above, the actual obligation which applies in the circumstances is the obligation to put up a fence, and it applies because the other two obligations (not to have a dog, and to put up a sign if there is a dog) have been violated. As we have emphasised above, it would be incorrect to say that the obligations not to have a dog, and to put up a sign if there is a dog, have been defeated, or overturned; they have been violated, and any proper representation of the situation must register the fact that, because of these violations, the obligation which becomes actual is the obligation to erect a fence. But how are these points to be articulated in a formal theory? To that question we now turn.


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4.1 Motivation Consider again Example 2, particularly lines (a), (c) and (d). The norms governing, or in force in, the situation are that there ought to be no dog, and that if there is a dog there ought to be a warning sign; and the relevant fact is that there is a dog. So what is the actual obligation, of the agent concerned, in these circumstances? To erect a warning sign? But why not insist on getting rid of the dog, rather than on erecting a warning sign12 ? We wish to suggest that the answer to such questions turns on the status assigned to the fact that there is a dog | in the following sense: so long as there is a dog, but this, for one reason or another, is not deemed to be a xed, unalterable feature of the situation, then the actual obligation which applies is that there ought to be no dog. However, as soon as, for one reason or another, the fact that there is a dog is deemed xed, i.e., it is seen as a necessary, unavoidable feature of the situation, so that | in consequence | the practical possibility of satisfying the obligation that there ought to be no dog has to all intents and purposes been eliminated, then the actual obligation which applies is that there ought to be a warning sign.13 What do we mean when we say that for some reason or another a fact of the situation | in this case that there is a dog | may be deemed a xed, necessary, unalterable feature of that situation? Well, there are various ways in which this \ xity" might arise; those who proposed temporal solutions to the problems associated with CTDs focussed on one of these ways. If books shall be returned by date due, then if you still have the books after the date due there is no way that obligation can be met. It is too late! It is unalterably the case that the books are not returned by the date due, and consequently the possibility of satisfying the obligation to return the books by date due has been eliminated. But temporal reasons, although very common, are not the only reasons why things become xed, in the sense of necessity or unalterability we here seek to explicate; for instance, it is not for temporal reasons that the deed of killing, once done, cannot be undone. What explains xity in this case is not temporal necessity, but rather causal necessity. Nor need temporal considerations have any role to play in explaining why the presence of a dog may be, to all intents and purposes, an unalterable feature of the situation; it may, for some reason, be practically impossible in the situation to remove the dog; perhaps, for instance, its owner stubbornly refuses to remove it, and nobody else dares attempt the feat. The presence of the dog is a xed fact: the dog remains unless the intervention of some agent leads to its 12 Remember that, in keeping with our analysis of the pragmatic oddity, we seek an answer to these questions which is compatible with a further assumption to the eect that there ought not to be both no dog and a warning sign. 13 Some remarks in a similar spirit are to be found in [Hansson, 1971, section XIII: \on the interpretation of circumstances"].

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removal, and no agent is prepared to perform the required action. From the practical point of view | from the point of view of deciding which obligation actually applies to the situation | the key feature is that the possibility of satisfaction of the requirement that there be no dog is eectively eliminated. As a further illustration, consider next the example of the \considerate assassin".14 EXAMPLE 4. (a) You should not kill Mr. X. (b) (c) But if you kill Mr. X, you should oer him a cigarette. When does the assassin have an actual obligation to oer Mr. X a cigarette? After killing him? But then it is too late! One intuitively acceptable interpretation is that the assassin's actual obligation to oer Mr. X a cigarette arises when he rmly decides that he is going to kill Mr. X. It is then that it becomes a settled or xed fact that Mr. X will be killed, and then that the assassin's actual obligation is to oer a cigarette. Notice that the examples indicate that two dierent notions of necessity | and their associated notions of possibility | need to be considered. Mr. X, once killed, cannot be oered a cigarette because nobody has the ability or the opportunity to make oers to the dead, just as nobody has the ability or opportunity to return a book by date due if the date due has passed. On the other hand, the dog-owner may have both the ability and the opportunity to remove the dog, and the assassin may have both the ability and the opportunity to refrain from killing Mr. X; but once each has made a rm and de nite decision (to keep the dog and to kill Mr. X, respectively), then to all intents and purposes the persistent presence of the dog and the future performance of the assassination become xed features of the respective situations; so questions about which actual obligations arise in these situations have to be answered in the light of the fact that alternatives in which there is no dog, or no assassination, are not actually available. Now it may well be that the judge, at the assassin's trial, insists that the assassin should never have decided to commit the murder, just as it may be that the manager of the housing estate refuses to accept that the dog owner was entitled to decide that he would keep his dog. Furthermore, it is a well known feature of, for instance, disputes in legal cases, that the parties to the dispute may disagree about what an agent was able to do, or 14 This example can also be found in [Prakken and Sergot, 1996] (using \the witness" instead of \Mr. X"). In fact [Prakken and Sergot, 1996] provides an excellent survey of the principal examples of CTDs, and we use them in Section 5 to test the adequacy of our proposal.


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what he had the opportunity to do. But the existence of disagreements of these kinds is perfectly compatible with the approach to CTD scenarios we develop below. For it will not be the task of our logical system to determine the reasons which justify the classi cation of some fact as settled. Rather, what the system will do is this: rst, it will specify the role of assumptions about two types of xity in reasoning about actual and ideal obligations; and, second, it will show which actual/ideal obligations can be derived from a given set of norms when some facts are taken to be xed in the one sense or the other.15

4.2 The new theory and its fundamental semantic features We now present the basic features of a modal-logical language designed to capture the approach to CTDs described above in a way that conforms to the constraints, or requirements, (i){(viii). We shall then show how the new language may be applied to the Chisholm set, and to the analysis of some other problematic CTD \scenarios". We adopt the following approach to the formal representation of these scenarios: their deontic component (the obligation norms which they explicitly contain) will be represented throughout in terms of a dyadic, conditional obligation operator O ( / ); their factual component will be represented by means of either unmodalised sentences, or modalised sentences in two categories. These two categories correspond to the two notions of necessity (and their associated dual notions of possibility) which we shall employ to articulate the ideas regarding xity, or unalterability, of facts alluded to in the previous subsection. From the deontic and factual components taken together, some further obligation sentences may be derivable. The derived obligation sentences are of two types, pertaining to actual obligations and ideal obligations, respectively. There is an intimate conceptual connection between these two notions of derived obligation, on the one hand, and the two notions of necessity/possibility used in characterising the factual component, on the other. Consider rst the dyadic conditional obligation operator. How do we wish to interpret a sentence of the kind \if there is a dog then there shall be a warning sign"? On our view, this sentence is to be understood as saying that in any context in which the presence of a dog is a xed, unalterable fact, it is obligatory to have a warning sign, if this is possible. We think of a context as a set of worlds | the set of relevant worlds for the situation at hand. So the above sentence is to be understood as saying that, for any context in which there is a dog (i.e., for any context in which there is a dog in each world of that context), if it is possible to have a warning sign then 15 Our thanks to Layman Allen for raising a question at the Sesimbra EON96 Workshop related to this point.


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it is obligatory to have a warning sign.16 In order to capture this idea we introduce in our models a function ob:}(W ) ! }(}(W )) which picks out, for each context, the propositions which represent that which is obligatory in that context. That is, kB k 2 ob(X ) (where kB k denotes the truth set of B in the model in question) if and only if the proposition expressed by B represents something obligatory in context X . Accordingly, we say that a sentence O (B=A) is true in a model if and only if, in any context X where A is true and B is possible (i.e. in any context having A true in each of its worlds and B true in at least one of its worlds), it is obligatory that B .17 On the basis of this operator we could now derive the obligations that were applicable in each context, assuming that our language contained a means of representing contexts. The question is: what are the types of contexts that we need to be able to talk about in our formal language, given that we want to be able to derive sentences of two kinds, describing actual obligations and ideal obligations, respectively? We answer this question in terms of the two notions of necessity. The rst of these will be denoted by ! , and its dual possibility notion denoted by ! . Intuitively, ! is intended to capture that which | in a particular situation | is actually xed, or unalterable, given (among other factors) what the agents concerned have decided to do and not to do. In, for instance, the \dog scenario" (Example 2), if the agents concerned have rmly decided that the dog is not going to be removed, then the sentence ! dog is true of that situation (the presence of the dog is actually an unalterable fact). On the other hand, if some actual possibility existed for getting rid of the dog, then the situation would be appropriately described by ! :dog (i.e. : ! dog). Which actual obligations arise in the dog scenario will depend, in particular, on whether or not ! :dog is true. We must emphasise one important dierence between this notion of necessity/possibility and the second one we employ. For the reasons discussed in the previous subsection, we do not exclude, a priori, that a sentence of the form ! A might be true even though the agents concerned have the ability and the opportunity to see to it that :A. That is, we shall want to consider scenarios where, despite their abilities and opportunities for action, the agents have rmly resolved not to see to it that :A, and where | given that this is what the agents have decided | there is (for all intents and purposes) no way that A could be false.

16 We are here using the term \obligatory" in a weak sense; in a strict sense, for a sentence B to be obligatory in a context X we would also claim that there must exist at least one world in X where B is false (i.e., we would insist, for the strict sense, that obligations must be violable). However, our actual and ideal obligations, to be de ned next, will be considered in this strict sense. 17 We also add the further requirement that the conjunction of A and B is not contradictory, in order to avoid some \absurd" vacuous conditional obligations, and as one of the conditions needed to secure the result that if O (B=A) is true then kB k 2 ob(kAk).


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In order to capture the semantics of the necessity operator ! , our semantical models will contain a function, av, which picks out (for any given world w) a set of worlds av(w) | the set of worlds which are the actual versions of w (the open alternatives for the current world w), those which constitute the context that it is actually relevant to take into account in determining which obligations are actually in force, or actually apply, at w. Accordingly, a sentence of the form ! A will be said to be true at a given world w if and only if A is true at all of the worlds contained in av(w). Given the way the function ob is understood, the set of propositions ob(av(w)) will be the set of propositions which represent that which is obligatory in the context av(w) (that is to say, in the context of the alternatives that are actually open at w). In line with this, we shall say that a sentence of the form Oa A (read as \it is actually obligatory that A", or \it actually ought to be the case that A") is true at a world w only if the proposition expressed by A is one of those propositions picked out by ob for the argument av(w). In addition, the truth of Oa A at w will require that there is at least one world in av(w) where the sentence A is false; the reason for this second requirement is that that which is actually obligatory might actually fail to obtain. The second of the two notions of necessity will be denoted by , and its dual possibility notion denoted by . Intuitively, is intended to capture that which | in a particular situation | is not only actually xed, but would still be xed even if dierent decisions had been made, by the agents concerned, regarding how they were going to behave. For instance, certain features of the situation will be such that it is beyond the power of the agents to change them | they may lack the ability, or the opportunity, or both. Of such features it is appropriate to say that they are xed in the sense that they could not have been avoided by the agents concerned, no matter what they had done. It is not even potentially possible for the agents to alter them. In the original Chisholm scenario, for example, if the bridge that leads to the man's neighbours' house has been destroyed by a storm, and the man is unable to repair it, then clearly it is a necessary feature of the situation, in this second sense of necessity, that he does not help his neighbours. This is to be understood in contrast to the situation in which it is potentially possible for the agent to go to his neighbours' house to help them, but the agent has made a de nite decision, from which he will not budge, not to go. His will is rm, and thus in all actual relevant alternatives open to the agent he does not go to help them (and so actually he ought not to tell them he is coming). But given that he has the ability and opportunity to go, it is potentially possible | in the sense expressed by the operator | that he does so18, and so we want then be able to derive

18 So the best short readings for these two pairs of operators we can oer are the following:


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that his ideal obligation was to go and to tell them he was coming. To articulate the semantics of the second pair of notions of necessity and possibility, we introduce into the models a function, pv, which picks out (for any given world w) a set of worlds pv(w) | the set of worlds which are the potential versions of w. These worlds will constitute the context it is relevant to take into account in determining which ideal obligations hold at w (\what should have been done"). A sentence of the form A will be said to be true at a given world w if and only if A is true at all of the worlds contained in pv(w). Furthermore, given the way the function ob is understood, the set of propositions ob(pv(w)) will be the set of propositions which represent that which is obligatory in the context pv(w). Thus we shall say that a sentence of the form Oi A (read as \it is ideally obligatory that A", or \it ideally ought to be the case that A") is true at a world w only if the proposition expressed by A is one of those propositions picked out by ob for the argument pv(w). In addition, the truth of Oi A at w will require that there is at least one world in pv(w) where the sentence A is false | since that which is ideally obligatory might potentially fail to obtain. Finally, we de ne the notion of violation in terms of the notion of ideal obligation, as follows:

viol(A)= Oi A ^ :A df

This choice is in accordance with the intuitive idea that ideal obligations express what should have been done, and ts in well with our treatment of the pragmatic oddity and other features of CTD scenarios, as will become clearer when we analyse a number of examples in some detail. Brie y, the main points may already be explained as follows: in, for instance, the dog scenario, if it is a xed fact that there is a dog (i.e., if ! dog is true), but it is actually possible that a sign may be erected and potentially possible that there is no dog, then we shall be able to derive that it is actually obligatory that a sign is erected and ideally obligatory that there is no dog. The pragmatic oddity will be avoided because it will not, in these circumstances, be possible to derive an actual obligation that there be no dog. Nevertheless, we still of course want to say that an obligation (that there be no dog) has been violated, and this result is secured if violation is characterised as above. As the formal analysis of this example will show, ! A : it is actually possible that A

A ! A A

it is potentially possible that A

: :

it is not actually possible that :A

: it is not potentially possible that :A (In a number of cases, the natural reading of statements about potential possibility will be in the past tense.)


289

we shall also be able to derive a second violation in this situation, if no sign has been erected. The semantic models described next will be subject to various constraints, designed to achieve a particular pattern of relationships between the dyadic obligation operator, the two types of necessity/possibility operators, and the operators for actual and ideal obligations.

4.3 Syntax and semantics of the formal language Syntax

Alphabet: a set of (natural language) terms (dog, fence, sign, . . . ) for atomic sentences

: , ^ , _ , ! , $ (sentential connectives) ( , ) (parentheses) ! (dual: ! = : ! :) (dual: = : :) O (/) (dyadic deontic operator) Oa (monadic deontic operator - for actual obligation) Oi (monadic deontic operator - for ideal obligation) Rules for construction of well-formed sentences: as usual viol(A)= Oi A ^ :A

df

df

df

Semantics

Models:

M = hW; av; pv; ob; V i, where: 1) W = 6 ; 2) V - a function assigning a truth set to each atomic sentence 3) av : W ! }(W ) (alternatively: Ra W W and av(w) = fw1 : wRa w1 g) such that:


290

3-a) av(w) 6= ; 4) pv : W ! }(W ) (alternatively: Rp W W and pv(w) = fw1 : wRp w1 g) such that: 4-a) av(w) pv(w) 4-b) w 2 pv(w) 5) ob : }(W ) ! }(}(W )) such that (where X; Y; Z designate arbitrary sets of members of W ): 5-a) ; 2= ob(X ) 5-b) if Y \ X = Z \ X , then (Y 2 ob(X ) i Z 2 ob(X )) 5-c) if Y; Z 2 ob(X ); then Y \ Z 2 ob(X ) 5-d) if Y X and Y 2 ob(X ) and X Z , then ((Z X ) [ Y ) 2 ob(Z )

Truth in a world w in a model M = hW; av; pv; ob; V i is characterised as follows (where kAk = kAkM = fw 2 W : M j=w Ag):

M j=w p

i

w 2 V (p)

. . . (the usual truth conditions for the connectives :, ^, _, ! and $)

M j=w ! A i av(w) kAk M j=w A i pv(w) kAk M j=w O (B=A)19 i kAk \ kB k 6= ; and (8X )(if X kAk and X \ kB k 6= ;, then kB k 2 ob(X )) M j=w Oa A i kAk 2 ob(av(w)) and av(w) \ k:Ak = 6 ; (i.e. i kAk 2 ob(av(w)) and av(w) \ (W kAk) 6= ;) M j=w Oi A i kAk 2 ob(pv(w)) and pv(w) \ k:Ak = 6 ;

19 An alternative would be to de ne the dyadic obligation operator in the strict sense referred to in footnote 16, in which case M j=w O (B=A) i kAk \ kB k 6= ; and kAk \ k:Bk 6= ; and (8X )(if X kAk and X \kBk 6= ; and X \k:Bk 6= ;, then kBk 2 ob(X )). In that case we would require that if Y 2 ob(X ) then X \ (W Y ) 6= ;; and the truth in a world w of Oa A (respectively Oi A) would be de ned as follows: M j=w Oa A i kAk 2 ob(av(w)) (resp. M j=w Oi A i kAk 2 ob(pv(w))). With both approaches we get exactly the same semantics for Oa A and Oi A.


291

(Note that the de nition of M j=w O (B=A) entails that if M j=w O (B=A), then kB k 2 ob(kAk).) A sentence A is said to be true in a model M = hW; av; pv; ob; V i, written M j= A, i kAkM = W ; and A is said to be valid, written j= A, i M j= A in all models M. Some comments about the conditions: i) As would be expected, the set av(w) is required to be a subset of pv(w), for any w (condition 4-a)), so that actual possibility entails potential possibility. Conditions 3-a) and 4-b) are also obvious. In [Carmo and Jones, 1997]20 we required that w 2 av(w) (which implies 3-a) and, together with 4-a), also implies 4-b)). Although in most scenarios it makes sense to say that the actual world is always an actual alternative to itself, we sometimes need to be able to describe situations where A is not yet the case, but nevertheless in all relevant future alternatives open to the agent, A is the case. Consider, for instance, the scenario of the \considerate assassin" (Example 4): there may be situations where, although the assassin has not yet killed Mr. X, in all the relevant future alternatives open to the assassin Mr. X is going to be killed by him (because the assassin has so decided); the natural way to represent this situation in our logic is: :kill ^ ! kill. But of course this can only be expressed consistently if we do not require that, for all w; w 2 av(w). There exist other conditions that it may seem natural to impose on av and pv, such as the transitivity of both the actual and potential relations. But, for simplicity, we consider here only those conditions which appear to have a direct bearing on the analysis of the key examples of CTD scenarios in Section 5.

ii) Condition 5-a) means that we do not accept that a contradiction might be obligatory. iii) Condition 5-b) means that if, from the point of view of a context X , two propositions Y and Z are indistinguishable, then one of them is obligatory i the other is (this corresponds to a kind of \contextual" RE-rule). It is also appropriate at this point to state some of the main consequences of condition 5-b), and to attach numbered labels to them, in order to facilitate later discussion of a possible weakening of 5-b).

20 Where we use va, vp and pi, instead of the more suggestive names, av, pv and ob, used in this chapter.

292


Since Y \ X = (Y \ X ) \ X , we get as particular cases of 5-b): 5-b1) if Y 2 ob(X ), then Y \ X 2 ob(X ) 5-b2) if Y \ X 2 ob(X ), then Y 2 ob(X ) On the other hand, using 5-a) and 5-b1) we get the condition (which in turn implies 5-a)): 5-ab) if Y 2 ob(X ), then Y \ X 6= ; iv) Condition 5-c) requires that the conjunction of two obligatory propositions within a context X is also obligatory in that context. A natural extension of condition 5-c) would be to require the closure of ob under arbitrary intersections (and not only under nite intersections): T 5-c+) if ob(X ) and 6= ;, then ( ) 2 ob(X ) T (where is any set of subsets of W , ( ) is de ned as: T ( ) = fw(2 W ) : (8X 2 )(w 2 X )g) T If we impose this stronger condition then we would get ( ob(X )) 2 ob(X ),Tif ob(X ) is non-empty; note also that, by 5-b1), ifTob(X ) = 6 ; then ( ob(X )) X ; if ob(X ) = ; then, by de nition, ( ob(X )) = W . However, for reasons to be explained, what we shall in fact propose later is a weakening of condition 5-c. v) Condition 5-d) states that if a subset Y of X is an obligatory proposition in a context X , then in a bigger context Z it is obligatory to be either in Y or else in that part of Z which is not in X . Taking into account condition 5-b), it may be shown that each of the following conditions is equivalent to 5-d) - they can sometimes be used to simplify proofs: 5-bd1) if Y X and Y 2 ob(X ) and X Z , then ((W X ) [ Y ) 2 ob(Z ) 5-bd2) if Y 2 ob(X ) and X Z , then ((Z X ) [ Y ) 2 ob(Z ) 5-bd3) if Y 2 ob(X ) and X Z , then ((W X ) [ Y ) 2 ob(Z ) 5-bd4) if Y 2 ob(X ) and X Z , then ((W X ) [ (X \ Y )) 2 ob(Z ) Using conditions 5-b), 5-c) and 5-d), we can also prove that if Z 2 ob(X ) and Z 2 ob(Y ), then Z 2 ob(X [ Y ).

4.4 Syntactic/axiomatic characterisation of the modal operators In what follows we introduce the axioms and rules for the various modal operators. For some of the axioms and theorems we introduce special labels


293

| if there is no standard label | in order to facilitate reference to them later on.

Characterisation of

:

is a normal modal operator of type KT. !: Characterisation of ! is a normal modal operator of type KD. 2. !: Relationship between and !A ! )) 3. A ! (axiom schema ( ! 1.

Characterisation of O : 4. :O (?=A) (the schema :N for O : (O :N )) 5. O (B=A) ^ O (C=A) ! O (B ^ C=A) (the schema C for O : (O

C ))

6. Restricted principle of strengthening of the antecedent - 1: O (B=A) ! O (B=A ^ B ) (SA1) 7. the RE-rule with respect to (w.r.t.) the antecedent: if ` (A $ B ) then ` O (C=A) $ O (C=B ) 8. the \contextual RE-rule" w.r.t. the consequent: if ` C ! (A $ B ) then ` O (A=C ) $ O (B=C )

Relationship between O and : 9. O (B=A) ! O (B=A)

(O ! O ) 10. Restricted principle of strengthening of the antecedent - 2: (A ^ B ^ C) ^ O (C=B) ! O (C=A ^ B) (SA2)

Characterisation of Oa / Oi : 11. Oa A ^ Oa B ! Oa (A ^ B ) Oi A ^ Oi B ! Oi (A ^ B )

(Oa

(Oi

Relationships between Oa (respectively:

C) C)

Oi ) and ! (resp.:

):


294

(:O ) (:O ) ! (A $ B ) ! (O A $ O B ) 13. ($ O ) (A $ B) ! (O A $ O B) ($ O ) ! (resp.: ): Relationships between O ; O (resp.: O ) and 14. Restricted factual detachment: ! A ^ !B ^ ! :B ! O B O (B=A) ^ (O F D) O (B=A) ^ A ^ B ^ :B ! O B (O F D) ! (A ^ B ) ^ ! (A ^:B ) ! O (A ! B ) 15. O (B=A) ^ (O ! O !) O (B=A) ^ (A ^ B ) ^ (A ^:B ) ! O (A ! B ) (O ! O !) 12. ! A ! (:Oa A ^ :Oa :A) A ! (:Oi A ^ :Oi :A) a

a

i

i

a

a

i

a

i

i

a

a

i

i

a

a

i

i

Notes:

i) Simply using Propositional Calculus, we can deduce from 3. ( the following useful theorem: ` !A ! A (! ! )

!! )

ii) From the contextual RE-rule (8.) it follows that if ` A $ B then ` O (A=C ) $ O (B=C ); thus O is classical w.r.t. to both of its arguments. iii) Some comments are in order regarding axiom 9. It re ects the fact that norms which comprise the deontic component of a CTD scenario are themselves taken to be xed, in the sense that they are features of the scenario which the agents concerned have neither the ability nor the opportunity to change. We maintain that this is a reasonable assumption to make, given that our concern is not with the dynamics of normative systems, but with the determination of which ideal and actual obligations may be derived from a xed set of norms, given the facts of the case. But it might be asked whether our analysis of the given norms (i.e., of the dyadic deontic conditionals) is compatible with the obvious fact that norms are man-made and may be subject to change. Our answer is that there is no incompatibility here: were we to switch our interest from normative statics to normative dynamics, one natural move | from the semantic point of view | would be to treat the models of our current semantics as the worlds of a semantical framework for the investigation of normative dynamics.


295

iv) Axiom 10. embodies only a weak restriction on strengthening of the antecedent and it is clear that our dyadic deontic conditionals are not \exception-allowing" default conditionals. Any attempt to re-de ne them as default conditionals | and thus to eliminate 10. from the class of theorems | would have to be acompanied by elimination of the factual detachment principles expressed in 14. (cf. footnote 7, above). The factual detachment principles would then need to be replaced by a theory of default reasoning, de ning the conditions under which sentences about actual and ideal obligations could be drawn as default conclusions from any given deontic and factual scenario. In our view, changes of this kind in the underlying logic of deontic conditionals would have no direct bearing on how the CTD problems themselves are to be handled. v) The counterpart to axiom 12. ( rst part) also held in the system DL of [Jones and Porn, 1985], where their modality Ought represented actual obligation. The point captured by this axiom is as follows: if it is not actually possible that A is false, then A is not actually obligatory (for if it is guaranteed that A is true, any such obligation has been discharged); in addition, :A is not actually obligatory (for that which it is not actually possible to realise cannot be actually obligatory). However, the fact that it is not actually possible that :A does not rule out the possibility that, ideally, it ought not to be the case that A. vi) Using the axiom schemas ($Oa ) and ($Oi ), we can prove the RErules: if ` (A $ B ) then ` Oa A $ Oa B and if ` (A $ B ) then ` Oi A $ Oi B Thus both Oa and Oi are classical operators. vii) Using the schemas (:Oa ) and (:Oi ), we can deduce the following theorems: ` :Oa > (Oa :N ) ` :Oi > (Oi :N ) ` :Oa ? (Oa OD) ` :Oi ? (Oi OD) And any classical operator with (OD) and (C ) also has the schema (D) as a theorem; a revision of the logic which avoids (D) will be discussed in Section 6. It is important that neither Oa nor Oi validates the schema (M) | the converse of (C ) | otherwise we would get the (RM )-rule, to be discussed further in Section 6.1.


296

viii) We write just (F D), intead of (Oa F D) or (Oi F D), whenever it is clear from the context to which of the two obligation operators we are referring. RESULT 1. The previous axiomatisation is sound.21 Hint to the proof: 1.-4. 5. 6. 7.-8. 9.-10. 11.-13. 14. 15.

Besides the relevant truth conditions, simply use (respectively) the condition: 4-b), 3-a), 4-a), and 5-a) (for this last case recall that M j=w O (B=A) implies kB k 2 ob(kAk)). Use 5-c) and 5-ab) (plus the relevant truth condition); condition 5-ab) is needed only to prove that kA ^ B ^ C k = 6 ;. Trivial (simply use the relevant truth condition). It is trivial to prove that these rules preserve truth in a model (and so also preserve validity, as desired). For the contextual RE-rule use condition 5-b). Simply use the relevant truth condition. (w.r.t. 10., note that M j=w (A ^ B ^ C ) implies kA ^ B ^ C k = 6 ;.) Besides the relevant truth conditions, use (respectively): 5-c), 5-ab), and 5-b). Simply use the relevant truth conditions. Using conditions 5-b) and 5-d), prove that if M j=w O (B=A) and Z \ kAk \ kB k 6= ;, then ((W kAk) [ kB k) 2 ob(Z ); and then apply that result with Z = av(w) (resp. pv(w)).

End-proof The next result provides a list of some other useful theorems and rules concerning the obligation operators. RESULT 2. i) ii) iiii) iv) v)

` O (B=A) $ O (A ^ B=A) ` :O (A=?) ` :O (:A=A) ` :O (B=A) ! :O (B=A) ( :O ! :O ) ` (A ^ B ) ^ O (A ! B=>) ! O (B=A) if ` A ! B then ` (A ^ C ) ^ O (C=B ) ! O (C=A)

21 Some completeness results are also available, but are not reproduced in this chapter.


vi)

` ! A ! (Oa B ! Oa (A ^ B )) ` A ! (Oi B ! Oi (A ^ B ))

297

(Oa ! Oa ^) (Oi ! Oi ^)

vii) Restricted deontic detachment:22 ` Oa A ^ O (B=A) ^ ! (A ^ B ) ! Oa (A ^ B ) ` Oi A ^ O (B=A) ^ (A ^ B ) ! Oi (A ^ B )

(Oa DD) (Oi DD)

Hint to the proof: i) From the contextual RE-rule, since ` A ! (B $ A ^ B ). ii) From the contextual RE-rule and (O and ` A ! (? $ :A)).

:N ) (since ` ? ! (? $ A)

! O ). iv) Since ` (A ^ B ) ! (A ^ > ^ (A ! B )), using (SA2) we get ` (A ^ B ) ^ O (A ! B=>) ! O (A ! B=A ^ >), and the desired theorem follows by using the RE-rule (w.r.t. the antecedent) and the

iiii) From ( O

contextual RE-rule (since ` A ! ((A ! B ) $ B )).

v) If ` A ! B then ` (A ^ C ) ! (A ^ B ^ C ); by (SA2), ` (A ^ C ) ^ O (C=B ) ! O (C=A ^ B ); the theorem follows by the RE-rule w.r.t. the antecedent (since ` A ! B implies ` A ^ B $ A). vi) From ($ Oa ), since ` ! A ! ! (B $ A ^ B ). (Analogously for Oi .)

vii) Using axioms (O ! Oa !) and (Oa C ), we deduce ` ! (A ^ :B ) ! (Oa A ^ O (B=A) ^ ! (A ^ B ) ! Oa (A ^ B )); by axiom ($ Oa ), we deduce ` : ! (A ^ :B ) ! (Oa A ! Oa (A ^ B )) (since ` : ! (A ^:B ) ! ! (A $ A ^ B )); thus ` Oa A ^ O (B=A) ^ ! (A ^ B ) ! Oa (A ^ B ). (Analogously for Oi .)

End-proof As can be seen in the next section, the theorems that play the dominant 22 We note, however, that we cannot detach Oa B (Oi B ) from the antecedent of these theorems. In fact, even the \weaker" formulas below are not valid: Oa A ^ O (B=A) ^ ! (A ^ B ) ^ ! (A ^ :B ) ! Oa B Oi A ^ O (B=A) ^ (A ^ B ) ^ (A ^ :B ) ! Oi B . Although it may seem, at rst sight, that the failure of these implications represents a weakness of the logic, the analysis of some examples, to follow, indicates that it is | on the contrary | an advantage.


298

role in determining what may, or may not, be derived from a chosen representation of a CTD scenario are:

the factual detachment axioms (FD); axioms (O

! Oa !) and (O ! Oi !);

the deontic detachment theorems (DD); axioms (:Oa ) and (:Oi );

axioms ($ Oa ) and ($ Oi ); and the theorems (Oa

! Oa ^) and (Oi ! Oi ^).

Besides these theorems, we will also make extensive use of the T -normality of ; the D-normality of ! ; axiom ( ! ! ) and theorem ( ! ! ).

5 THE ANALYSIS OF SOME CTD SCENARIOS We shall focus on six scenarios which exhibit CTD structures: Scenario 1 is the Chisholm set | Example 1 above; Scenario 2 is the extended version of Example 2 | \the dog scenario" involving \contrary-to-contrary-to-duties"; Scenario 3 is \the white fence" - Example 3 above; Scenario 4 is \the gentle murderer"; Scenario 5 is \the considerate assassin" | Example 4 above; Scenario 6 is the so-called \Reykjavic scenario"23. The scenarios illustrate the range of problems which a theory of CTD should be able to handle; and they enable us to exhibit the expressive and deductive capabilities of our logical system.

23 This is also discussed in [Belzer, 1987] and [McCarty, 1994], and is one of the many examples considered in [Prakken and Sergot, 1994; Prakken and Sergot, 1996].


299

SCENARIO 1. The Chisholm Set

The deontic component (d1) It ought to be that a certain man go to help his neighbours (d2) It ought to be that if he goes he tell them he is coming (d3) If he does not go, he ought not to tell them he is coming

Logical representation (d1) O (help / >) (d2) O (tell / help) (d3) O (: tell / : help)

Assumptions regarding the representation of the facts In this example we assume the following obvious hypotheses regarding the representation of the facts:

(a) help ! ! help (b) tell ! ! tell

(thus, by ( ! D), ! :help ! :help) (analogously, ! :tell ! :tell)

Moreover, we also assume that

(c) (help ^ :tell)! ! :tell

(but not :tell ! ! :tell)

since when the agent concerned has helped his neighbours but did not tell them he was coming, it makes no sense to consider any actual alternative where he tells them he is coming. On the other hand, although in some of the cases it might be reasonable to accept both :help ! ! :help and :tell ! ! :tell, we shall not adopt either of these assumptions. (Nevertheless, by ( T ), we have, for any sentence A, ` A ! A:)

CASE 1.1.

Factual component (f1) X (the agent concerned) decides not to go to help his neighbour (and, of course, he has not yet gone to help them).


300

(f2) But it is potentially possible for X to help and to tell and potentially possible for X to help and not to tell. (f3) X has not in fact told that he is coming, although it is still actually possible that he does tell and actually possible that he does not tell.

Logical representation (f1) ! :help (f2) (help ^ tell) ^ (help ^ :tell) (f3) :tell ^ ! tell ^ ! :tell

(Note that, without assumption (a), we would represent (f1) as :help ^ ! :help, since we have not adopted the T -schema for ! .)

Conclusions

In virtue of the factual detachment axioms (FD), we may derive the following: viol(help) ^ Oa :tell that is to say, X violates his obligation to help his neighbours, and is actually (given the circumstances) under an obligation not to tell them he is coming. It is also possible to derive the conclusion that X violates his obligation to \help and tell", since it is also possible to deduce, in virtue of the deontic detachment theorem (DD), that Oi (help ^ tell). We remark here (cf. footnote 22 regarding Result 2-vii)) that it is not also possible to conclude that X has violated an obligation to tell simpliciter, since that particular obligation would not come into eect until X's helping was a xed fact. This result seems to accord well with intuition. Note also that the pragmatic oddity, as we have diagnosed it, is avoided, for we could consistently add to the deontic component above an obligation to the eect that X ought not both go to help and not say that he is coming, i.e., an obligation of the form O (:help _ tell / >). CASE 1.2.

Factual component (f1) X has helped his neighbours and told them he was coming. (f2) But it was potentially possible that X did not help his neighbours.


Logical representation (f1) help ^ tell

301

(by assumptions (a) and (b), this implies ! (help ^ tell)) (f2) :help

Conclusions We may derive the following (using (FD) and (DD)): Oi help ^ Oi (help^tell) ^ help ^ tell So, X has met his ideal obligations, and no actual obligation arises (as can be seen by taking into account (:Oa )). CASE 1.3.

Factual component (f1) X has helped his neighbours and he did not tell them he was coming. (f2) It was potentially possible that he both helped and told, as well as that he did not help them.

Logical representation (f1) help ^ :tell

(by assumptions (a) and (c), this implies ! (help ^ :tell)) (f2) (help ^ tell) ^ :help

Conclusions

We may derive the following (using (FD) and (DD)): Oi help ^ help ^ viol(help ^ tell) So, X meets his ideal obligation to help, but violates his obligation to help and tell; and no actual obligations are derivable. CASE 1.4.

Factual component (f1) X has helped his neighbours, although it was potentially possible that he did not help them. (f2) X did not tell his neighbours he was coming, since that was potentially impossible for X to do (imagine that there were no available means of communication).


302

Logical representation (f1) help ^ (f2)

:help

:tell

(recall that, by (a), help ! ! help)

Conclusions We may derive the following (using (FD)):

Oi help ^ help

So, X has not violated any obligation: the obligation sentence Oi (help ^ tell) cannot be derived, because it would be impossible to satisfy such an obligation; furthermore, X has met his ideal obligation to help. No actual obligations are derivable. CASE 1.5.

Factual component (f1) It is not potentially possible for X to help his neighbours (for some reason or other | perhaps, for instance, there are no available means for X to travel to his neighbours' house); however, X tells his neighbours he is coming, but he might not have told them so.

Logical representation (f1)

:help ^ tell ^ :tell

Conclusions We may derive the following (again using (FD)):

viol(:tell)

SCENARIO 2. The dog example { extended with a second-level CTD


303

The deontic component (d1) There ought to be no dog (d2) If there is no dog, there ought not to be a warning sign (d3) If there is a dog, there ought to be a warning sign (d4) If there is a dog and no warning sign, there ought to be a high fence

Logical representation (d1) O (:dog / >) (d2) O (:sign / :dog) (d3) O (sign / dog) (d4) O (fence / dog ^ :sign)

Assumptions regarding the representation of the facts We here adopt no speci c hypotheses regarding the representation of the facts. CASE 2.1.

Factual component (f1) There is a dog, and it is actually possible to keep it or to get rid of it (and there is no information regarding the possibility of having, or not, a sign or a fence).

Logical representation

(f1) dog ^ ! dog ^ ! :dog

Conclusions

We may derive the following regarding violation and actual obligation: viol(:dog) ^ Oa :dog So, there is a violation of the obligation not to have a dog, and there is an actual obligation to get rid of it.


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CASE 2.2.

Factual component (f1) There is a dog and it is not actually possible that there is not a dog (i.e., the presence of the dog is actually xed), but it was potentially possible that there was no dog.

Logical representation (f1) dog ^ ! dog ^ :dog Conclusions

We may derive the following regarding violation and actual obligation: viol(:dog) There is a violation of the prohibition against having a dog, and no actual obligations may be derived - although it is possible to derive ! sign ^ ! :sign ! Oa sign

CASE 2.3.

Factual component (f1) There is a dog and that fact is actually xed (i.e., it is not actually possible that there is not a dog), but there might potentially have been no dog. (f2) There is no sign and it is not actually possible that there is a sign, but there might potentially have been both a dog and a sign. (f3) There is not a fence, and both the presence and the absence of a fence are actual possibilities.

Logical representation (f1) dog ^ ! dog ^ :dog (f2) :sign ^ ! :sign ^ (dog ^ sign) (f3) :fence ^ ! :fence ^ ! fence Conclusions

We may derive, in particular, the following violations and actual obligation: viol(:dog) ^ viol(dog!sign) ^ viol(dog^:sign!fence) ^ Oa fence The ideal obligation that there be no dog is obtained by applying (FD). By application of the axiom schema (O !Oi !), we derive that: (i) it is ideally obligatory that if there is a dog then there is a sign


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(note that, from (f1) and (f2) we get dog^:sign, and so (dog^ :sign); the application of (O !Oi !) is then trivial, taking into account (d3) and (f2)); and (ii) it is ideally obligatory that if there is a dog and no sign then there is a fence

(for instance, from (f1), (f2) and (f3), plus the normality of ! , deduce ! (dog^:sign^fence) and ! (dog^:sign^:fence), which imply (by ( ! ! )) (dog ^:sign^fence) and (dog^:sign^ :fence), and then use (O !Oi !) and (d4)).

Axiom (FD) enables us to derive that, given the circumstances, the actual obligation is to put up a fence. SCENARIO 3.The white fence case - Example 3

The deontic component (d1) There must be no fence (d2) But, if there is a fence it must be white

Logical representation (d1) O (:fence / >) (d2) O (white-fence / fence)

Assumptions regarding the representation of the facts (white-fence ! fence)

CASE 3.1.

Factual component (f1) There is no fence, and it is still actually possible not to erect a fence and actually possible to erect a fence, white or not.

Logical representation (f1) :fence ^ ! :fence ^ ! white-fence ^ ! (fence ^ :white-fence)

! white-fence we can derive (Note that, by the assumption made, from ! (fence ^ white-fence).)


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Conclusions We may derive the following regarding ideal/actual obligations:

Oi :fence ^ Oi (fence ! white-fence) ^ :fence ^ Oa :fence ^ Oa (fence ! white-fence)

There is an ideal obligation not to have a fence and an ideal obligation that if there is a fence then it must be white; neither of these ideal obligations has been violated; and both persist as actual obligations. CASE 3.2

Factual component (f1) There is a white fence, and it is actually xed that there will be a fence, possibly white or of another colour. (f2) But it was potentially possible not to have a fence.

Logical representation (f1) white-fence fence) (f2)

:fence

^

! fence ^ ! white-fence ^ ! (fence ^ :white-

Conclusions We may derive, in particular, the following violation, and ideal and actual obligations:

viol(:fence) ^ Oi (fence!white-fence) ^ Oa (fence!white-fence) ^ Oa white- fence

The ideal obligation not to have a fence has been violated and does not persist as an actual obligation, since it can no longer actually be ful lled; the obligation to the eect that if there is a fence then it must be white has not been violated, and it persists as an actual obligation. We may also derive the actual obligation to have a white fence.


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SCENARIO 4. The gentle murderer

The deontic component (d1) You should not kill Mr. X (d2) But, if you kill Mr. X, you should do it gently

Logical representation (d1) O (:kill / >) (d2) O (kill-gently / kill)

Assumptions regarding the representation of the facts We make the following assumptions:

! kill) (thus:(kill-gently ! (kill-gently ! kill)) (kill-gently ! kill) and ! kill (b) kill ! ! kill-gently (c) kill-gently ! ! :kill-gently (d) kill ^ :kill-gently ! (a)

In the cases discussed below, assumptions (b), (c) and (d) are relevant only in regard to determining that no actual obligations arise. Note that the dierence between Scenarios 4 and 3 is that the counterparts to assumptions (b), (c) and (d) cannot be adopted for Scenario 3. CASE 4.1

Factual component (f1) The assassin has killed Mr. X, but gently. (f2) It was potentially possible for the assassin not to kill and potentially possible that he killed \non-gently".

Logical representation (f1) kill-gently (f2) :kill ^

(kill ^ :kill-gently) (Note that ` kill-gently ! kill-gently.)


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Conclusions

We may derive the following (using (FD) and (O !Oi !)): viol(:kill) ^ Oi (kill ! kill- gently) ^ kill-gently The assassin has violated his obligation not to kill and has ful lled his obligation to kill Mr. X gently if he was going to kill him; no actual obligation arises; (using, for instance, (f1), (a) and (b) we can derive ! kill: so, it is actually impossible to ful l the obligation not to kill Mr. X; also we cannot derive an actual obligation to kill gently, since that cannot actually be violated, taking into account (c)).

CASE 4.2.

Factual component (f1) The assassin has killed Mr. X and not gently. (f2) It was potentially possible for the assassin not to kill and potentially possible for him to kill gently.

Logical representation (f1) kill ^ :kill-gently (f2) :kill ^ kill-gently Conclusions

We may derive the following: viol(:kill) ^ viol(kill ! kill- gently) The assassin has violated both his obligation not to kill and his obligation to kill gently if he does kill; no actual obligation arises; (use (f1), (b) and (d)). CASE 4.3.

Factual component (f1) It has been proved (in court) that the \assassin" has killed Mr. X, but gently (and this is admitted by the \assassin"). (f2) The \assassin" argues in court that he had no other choice, i.e. that it was potentially impossible for him not to kill Mr. X, because a real assassin had told him that he would kill his son if he (the \assassin") did not kill Mr. X. The prosecution argues that it was potentially possible for the \assassin" not to kill Mr. X (for instance because he could ask the police for protection for his son).


309

Logical representation of the facts according to the point of view of the defence (f1) kill-gently (f2)

kill ^ :kill-gently

Conclusions Since it was impossible not to kill Mr. X the ideal obligation not to kill cannot be derived; using (FD) we can derive the ideal obligation to kill gently, and this obligation was ful lled.

Logical representation of the facts according to the point of view of the prosecution (f1) kill-gently (f2) :kill ^

(kill ^ :kill-gently)

Conclusions As in case 4.1, we can derive viol(:kill). So, the prosecution argues: the \assassin" should be considered guilty.

Conclusions Of course it may well be suggested that the tactic of the defence here is most unwise. Perhaps they should rst accept the prosecution's claim that it was potentially possible for the \assassin" to refrain from killing, and thus that the \assassin" was ideally obliged not to kill; but then the defence should point out that obviously the \assassin" acted under duress, because of the threat to his son, and thus that the option of not killing was not one which the \assassin" could reasonably be expected to choose. Again we note here that the logic does not determine the status of the facts; but its language is capable of representing the opposing views concerning their status - in this case, concerning what is taken to be potentially possible. The logic's task is to show which conclusions regarding obligations and violations follow from a given set of norms, once a particular proposal has been made as to the status of the facts.


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SCENARIO 5. The considerate assassin

The deontic component (d1) You should not kill Mr. X (d2) But, if if you kill Mr. X, you should oer him a cigarette

Logical representation (d1) O (:kill / >) (d2) O (oer / kill)

Assumptions regarding the representation of the facts (a) kill ! ! kill (b) oer ! ! oer (c) kill ^ :oer ! ! :oer

CASE 5.1.

Factual component (f1) The assassin has not yet killed Mr. X and has not oered him a cigarette. (f2) It is still actually possible for the assassin to kill Mr. X and to oer him a cigarette or to kill and not oer a cigarette or not to kill and oer him a cigarette or not to kill and not oer him a cigarette.

Logical representation (f1) :kill ^ :oer (f2) ! (kill ^ oer) ^ ! (kill ^ :oer) ^ ! (:kill ^ oer) ^ ! (:kill ^ :oer) Conclusions We may derive the following (using (FD), (O !Oa !), and (O ! Oi !)): Oi :kill ^ Oi (kill ! oer) ^ :kill ^ :oer ^ Oa :kill ^ Oa (kill ! oer)

The assassin has not violated his ideal obligations not to kill Mr. X and to oer Mr. X a cigarette if he was going to kill him, and these obligations persist as actual obligations.


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CASE 5.2

Factual component (f1) The assassin has not yet killed Mr. X but he has rmly decided to kill him. (f2) It is actually possible for the assassin to oer or not a cigarette.

Logical representation (f1) :kill ^ ! kill (f2) ! oer ^ ! :oer Conclusions

We may derive, in particular: Oi :kill ^ Oa (kill ^ oer) ^ Oa oer Given that it is actually a xed fact that the assassin is going to kill Mr. X, his actual obligation is to oer Mr. X a cigarette. Of course, his ideal obligation is not to kill. Note that from Oa oer (by (O !Oa !)) we can derive Oa (kill!oer); and the obligation Oa (kill ^ oer) can then be obtained by (f1) and (Oa !Oa ^). We shall comment on this conclusion in Section 6, below, but note at this point that Oa kill is not derivable. SCENARIO 6. The Reykjavik scenario

The deontic component

Consider the following instructions given to oÆcials accompanying Reagan and Gorbachov at the Reykjavik meeting: (d1) The secret shall be told neither to Reagan nor to Gorbachov (d2) But if the secret is told to Reagan it shall also be told to Gorbachov (d3) And if the secret is told to Gorbachov it shall also be told to Reagan

Logical representation

Suppose that `r' represents `Reagan knows the secret' and that `g' represents `Gorbachov knows the secret' (d1) O (:r ^ :g=>) (d2) O (g=r) (d3) O (r=g)


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Comments on the logical representation of the deontic component (a) Sentence (d1) ought to be represented in the form indicated and not as O (:r=>) ^ O (:g=>). The reason is that there is no \absolute" obligation according to which, e.g., Reagan is not to know the secret (in just any context in which that would be possible); what is \absolutely" obligatory is that neither of them knows the secret. In fact, if the above alternative formulation of (d1) were to be employed, then in the situation where, for instance, the secret has been told to Gorbachov but not to Reagan, it would be possible to derive con icting actual obligations, and that derivation would be intuitively correct. It is a merit of the logic proposed that it would detect such a con ict. (A similar point is made in [Prakken and Sergot, 1994; Prakken and Sergot, 1996] in their discussion of this scenario.) (b) A possible alternative representation of (d2)+(d3) would be (d2+3): O (g ^ r=g _ r). (Although it would then be possible to generate the same conclusions as from (d2)+(d3), dierent patterns of derivation would be involved.)

Assumptions regarding the representation of the facts We make the following assumptions: (a) (b) (c) (d) (e)

r! ! r g! ! g ! (r ^ g) (which gives: ! r and ! g, as well as :r ! ! :r :g ! ! :g

(r ^ g))

CASE 6.1.

Factual component (f1) The secret has not yet been told to either of them

Logical representation (f1) :r ^ :g Conclusions Oi (:r ^ :g) ^ :r ^ :g ^ Oa (:r ^ :g)

The obligation to tell the secret to neither of them has not been violated, and persists as an actual obligation. Application of (FD) is instrumental in the derivation of these conclusions.


313

CASE 6.2

Factual component (f1) Reagan knows the secret but Gorbachov does not (f2) But it was potentially possible that neither of them knew

Logical representation (f1) r ^ :g (f2) (:r ^ :g) Conclusions viol(:r ^ :g) ^ Oa g

(it suÆces to use (FD)) We may also derive Oa (r ^ g), by direct application of (Oa ! Oa ^), on the basis of the prior derivation of Oa g. On the other hand, if we employ (d2+3) instead of (d2) and (d3), we get Oa (r ^ g) by direct application of (FD), and then on the basis of that result we can use ($ Oa ) to obtain Oa g.

CASE 6.3

Factual component (f1) The secret has been told at the same time to Reagan and Gorbachov

Logical representation (f1) r ^ g ^ (:r ^ :g) Conclusions viol(:r ^ :g)

There is violation of the obligation to tell neither of them, and no actual obligations are now derivable.

CASE 6.4

Factual component (f1) One, and only one, of Reagan and Gorbachev has been told the secret (f2) It might potentially have been the case that neither of them was told


314

Logical representation (f1) (r _ g) ^ :(r ^ g) (f2) (:r ^ :g) Conclusions viol(:r^:g) ^ Oa (r^g)

Note that the derivation of Oa (r ^ g) requires the use of (FD), and then the application of (Oa ! Oa ^); the derivation could be made directly by application of (FD) if (d2+3) were used to represent the second and third lines of the deontic component. 6 EVALUATING THE PROPOSED APPROACH

6.1 Closure under implication Our ideal/actual obligation operators are not closed under the (RM)-rule. Since we have (:N ) (for both Oi and Oa ), and since the operators are classical, closure under the (RM)-rule would yield the result that any ideal or actual obligation implies a contradiction. But even if Oi and Oa had been de ned in such a way that (:N ) were not valid, there are still reasons for not wanting the (RM)-rule; as we remarked in Section 2.2, from the point of view of violation the Ross problem | which acceptance of the (RM)-rule would generate | does seem to be genuine. So, given the focus on violation in our approach, the Ross problem is to be avoided. This was a main reason for not supplementing our models with respect to òb'. If we had imposed that \if YZ and Y2 ob(X), then Z2 ob(X)", then although we would not get the (RM)-rule in its full generality (because of the second conjunct in the truth conditions for actual and ideal obligation sentences), we would still get weaker versions of it, such as: \if j= A ! B then j= :B ! (Oi A ! Oi B )",24 that are strong enough to generate the Ross problem. Nevertheless, some would maintain25 that at least weaker versions of the (RM)-rule are needed, since if, for instance, it is forbidden to kill, it seems strange that we cannot also derive that it is forbidden to strangle. The claim seems to be this: for a particular class of pairs of sentences (A; B ), the conditional relation between A and B is such that it should license the derivation of \if B is forbidden then A is also forbidden". Suppose that we introduce a connective ) to express the kind of conditional

24 Using condition 5-b), we would even get the stronger result \j= ( (A ! B ) ^ :B) ! (Oi A ! Oi B)" (and \j= ( ! (A ! B) ^ ! :B) ! (Oa A ! Oa B)"

with respect to actual obligations). 25 We are grateful to Henry Prakken for this criticism.


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relation concerned (leaving its semantics open for the moment). Instances of this conditional relationship (taken from the scenarios analysed in Section 5) would be (white-fence ) fence) and (kill-gently ) kill), besides the previously mentioned (strangle ) kill). Using this connective, one possible way of interpreting the claim, in terms of our approach, might be to require the validity of the following sentences:

A ! (O :B ! O :A) ! A ! (O :B ! O :A) ($a) (A ) B ) ^ ($i) (A ) B ) ^

i

i

a

a

It would not be a diÆcult matter to de ne appropriate truth conditions for ) and to relate them to the semantics of òb' in such a way as to secure the validity of these two sentences. But we do not pursue this line here, since we think that, in terms of our approach, there are reasons for supposing that they should not be deemed valid. To see why, recall again the gentle murderer scenario, supposing that \A" is \kill-gently" and \B " is \kill". (In what follows we will concentrate on ($i) although a similar argument could be raised against ($a).) Consider the case where it was potentially possible to kill or not to kill, and potentially possible to kill gently or not to kill gently. According to ($i) we would derive that there is an ideal obligation not to kill gently. Is this an acceptable derivation? We think not. In our opinion what we should derive | as we do in our logic | is that Oi :kill Ôi (kill ! kill-gently). Of course, if the assassin kills gently he violates the prohibition to kill, and that is indeed secured by our logic: assuming the obvious hypothesis that (kill- gently ! kill) then we may derive

() Oi :kill ! (kill-gently ! viol(kill)) The fact that we can derive () goes a long way towards accommodating the point behind Prakken's criticism, we believe.

$ Oa) and condition 5-b)

6.2 Axiom (

First possible counter-example: Suppose that Mr. X has an actual obligation to help his friend move on Saturday, and suppose that Mr. X ( rmly) decides that he will help his friend move on Saturday if and only if he will borrow his brother's convertible on Saturday. Thus we will have Oa help ^ ! (help$borrow) and, using axiom ($Oa ), we derive Oa borrow. However, it has been suggested26 that Mr. X might have an actual obligation to help his friend, but not an actual obligation to borrow the convertible. We disagree: the

26 We are grateful to Donald Nute for this criticism.

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central point, as we see it, is that the truth of ! (help$borrow) means that it is not actually possible that either Mr. X helps his friend and does not borrow the convertible or Mr. X borrows the convertible and does not help his friend. Given that actual impossibility, there can be an actual obligation to help the friend if and only if there is an actual obligation to borrow the convertible. The fact that the decision to help a friend involves a commitment to another person, whereas the decision to borrow the car perhaps does not, is irrelevant to the conception of actual obligation we wish to explicate. Second possible counter-example: Suppose that Mr. X has an actual obligation to go to the cinema on Saturday, and suppose that Mr. X decides that he will go to the cinema on Saturday if and only if his friend, Mr. Y, also goes to the cinema on Saturday. Thus we will have Oa go-X ^ ! (go-X$go-Y) and we derive Oa go-Y. Does this mean that Mr. Y has an actual obligation to go to the cinema on Saturday? Clearly not! Examples of this sort indicate that there are cases where the formal language needs to be extended to include means of indexing decisions to particular agents, and means of relativising obligations to particular agents. Then it could be made explicit that X has taken the decision, that X bears the obligation to go to the cinema, and that X also bears the obligation to secure Y's presence. Third possible counter-example: Recall Case 5.2, from \the considerate assassin" scenario, in Section 5. The facts were: (f1) \The assassin has not yet killed Mr. X but he has rmly decided to kill him" and (f2) \It is actually open for the assassin to oer or not a cigarette"; from these facts, and the norms, and some background assumptions, we concluded not only that the assassin has an actual obligation to oer Mr. X a cigarette (Oa oer), but also (using the theorem (Oa !Oa ^), which follows from the axiom ($Oa )) that the assassin has an actual obligation to kill Mr. X and to oer him a cigarette (Oa (kill ^ oer)). This result may seem odd, just because it seems odd to say that someone has an actual obligation to kill (and whatever). But there is a good reason why this result should indeed be forthcoming. For remember that it is a xed fact that the assassin will kill Mr. X | it is assumed that it is actually impossible not to kill; thus, whatever actual obligations now come into force do so in the context of that assumption. Importantly, Oa kill cannot be derived from Oa (kill ^ oer), and clearly there cannot be an actual obligation to kill, since it must be actually possible that that which is actually obligatory fails to obtain. There is a connection between the last point and the Good Samaritan paradox of SDL mentioned in Section 2.2. In our view, the proper response to the Good Samaritan scenario is as follows: it is a xed fact of the situation that a man X has been robbed. The actual obligation to help him thus arises


317

in the context of that xed fact, and so it is appropriate to represent the content of the actual obligation in terms of a conjunction: \X is robbed and X is helped". In our system, in contrast to SDL, it does not follow that there is an obligation to rob; and we have a good explanation for why it should not follow, for that which is actually xed cannot actually be otherwise, and thus cannot be the object of an actual obligation.

6.3 Violations, and ideal and actual obligations Violations of conditional obligations A sentence of form viol(B ) is true if we can deduce from a set of deontic norms, and a set of facts, both that ideally it ought to be the case that B (Oi B ) and that B is not the case (:B ). However, it might be suggested that a normgiver often wants to express explicitly in the object language sentences of the form \violation(deontic norm) ! Sanction". The question is how \violation(O (B=A))" could be represented in our system?27 Our answer is that \violation(O (B=A))" can be characterized as: O (B=A) ^

(A ^ B) ^ A ^ :B

The explanation is as follows:

Suppose O (B=A). Two factual situations are then possible:

A is the case. Then (by (FD) and (:O )) we derive an ideal obligation O B i B ^ :B . And in order to get viol(B ) we must have :B (which implies :B ). Thus a violation occurs if B ^ :B , which is equivalent, given that A, to (A ^ B ) ^ A ^ :B . Second possibility: : A is the case. In this case (using axiom (O ! O !)) we derive from O (B=A) an ideal obligation of the form O (A ! B ) if (A ^ B ) ^ (A ^ :B ). And in order to get viol(A ! B ) we must have A ^ :B (which implies (A ^ :B )). Thus a violation occurs if (A ^ B ) ^ A ^ :B .

First possibility:

i

i

i

i

In the absence of other conditional obligations, no other ideal obligations can be deduced from O (B=A). The only other way we could deduce an ideal obligation from O (B=A), would be if we already had Oi A and (A ^ B ), in which case we could deduce,

27 We are grateful to Henry Prakken for posing us this question.

318


by (DD), the further obligation Oi (A ^ B ). We omit further details here, but it may be shown that this case too provides no diÆculties for the proposed characterisation of \violation(O (B=A))". Violation vs non-ful llment Consider the following new case of the Reykjavik scenario (Scenario 6 in Section 5):

Factual component

Suppose that the following facts hold at three dierent points in time (t1 < t2 < t3 ): (t1 {f1) Neither Gorbachov nor Reagan knew the secret (which oÆcial 007 knows), and oÆcial 007 tells the secret to Reagan (t2 {f1) OÆcial 007 has not yet told the secret to Gorbachov (t3 {f1) OÆcial 007 tells the secret to Gorbachov

Logical representation of the facts (t1 {f1) r ^ :g ^ (:r ^ :g) (t2 {f1) :g ^ r (t3 {f1) g ^ :g ^ r Conclusions At time t1 : viol(:r ^ :g) ^ Oa g At time t2 : Oi g ^ :g ^ Oa g At time t3 : Oi g ^ g

The natural reading of these conclusions is as follows: at time t1 , OÆcial 007 has violated his obligation to tell neither of them the secret, and gets an actual obligation to tell the secret to Gorbachov; at time t2 , OÆcial 007 has an ideal obligation to tell the secret to Gorbachov, which he has not yet ful lled, and which persists as an actual obligation; at time t3 , OÆcial 007 has ful lled his ideal obligation to tell the secret to Gorbachov, and has no actual obligation. However, according to our de nition of violation, we derive that at time t2 007 has violated his ideal obligation Oi g. Is this an intuitively correct interpretation of the situation at t2 ? Surely, we can say that at time t2 there exists a violation of the deontic norm (d2 ) (O (g=r)), in the sense described above. But should we conclude that at t2 the obligation Oi g has been violated, or rather that it has not yet been ful lled? This suggests that in some cases where there is a clear temporal dimension involved it may be natural to distinguish between the violation of


319

an obligation, on the one hand, and the situation in which an obligation has not been ful lled, on the other. The way we have de ned violation (Oi A ^ :A) would perhaps more appropriately be said to correspond to the latter, weaker notion, since that de nition does not rule out the actual possibility of ful lling the (ideal) obligation concerned (Oi A ^ :A ^ ! A); whereas violation in the full sense of the obligation Oi A implies that falsity of A is an actual necessity. Analogously, we may wish to distinguish between dierent degrees of ful llment of an ideal obligation. We do not pursue this point here, but it seems clear that our formal language as it stands is expressive enough to capture some of the distinctions concerned. Relationship between ideal and actual obligations In our approach, there is no direct logical connection between the notions of actual and ideal obligation. However, a reasonable question to raise is this: should it not be supposed that an ideal obligation which it is still actually possible to ful ll and actually possible to violate entails an actual obligation to the same eect? An aÆrmative answer would require the validity of the following sentence:

Oi A ^ ! :A ^ ! A ! Oa A

(Oi

! Oa )

This result would be secured were the following model condition to be adopted: 5-e) if Y

X and Z 2 ob(X ) and Y \ Z 6= ;, then Z 2 ob(Y )

In [Carmo and Jones, 1997], we conjecture that a condition of this kind could be adopted without untoward consequences. However, the situation is not quite so simple; 5-e) may con ict with the other conditions on our models.28 A deeper analysis shows that the problem depends fundamentally on the combination of 5-d) and 5-e), together with 5-c), for reasons we now explain. Consider the graphical description in Figure 1 and suppose Y 2 ob(X ). Then, by condition 5-d), ((Z X ) [ Y ) 2 ob(Z ): if a subset Y of X is an obligatory proposition in a context X , then in a bigger context Z it should be the case that either we are not in X or we are in Y . With respect to the set S, 5-d) does not require ((S X )[Y ) 2 ob(S ), contrary to what we would obtain if we also adopted condition 5-e) | taking also into account 5-b). 28 The following counter-example is due to Bjrn Kjos-Hanssen: Suppose O (A=>) is true in a model where W = f1 ; 2 ; 3 g and kAk = f1 g; then it can be shown that the conditions 5-b) and 5-d), together with the truth of O (A=>), imply that if 1 2 X , then ob(X ) = fZ : 1 2 Z g. Thus, supposing Y = f2 ; 3 g, if 5-e) is also assumed, then we would get (from ob(W )) that f1 ; 2 g 2 ob(Y ) and f1 ; 3 g 2 ob(Y ); and, from 5-b), we would get f2 g 2 ob(Y ) and f3 g 2 ob(Y ), and a contradiction would follow from 5-c) and 5-a).


320

That is: S =

Z X Y

S

X=

Y = Figure 1.

But ((S X ) [ Y ) 2 ob(S ) means that (S X ) 2 ob(S )! Is this acceptable? And, if the deontic norms also require that X 2 ob(S ), we would derive a con ict of obligations. The following case of the dog scenario illustrates the point.

Factual component

Suppose that the facts are as follows: (f1) there is no dog and there is a sign warning of one; and it was also potentially possibly to have no dog and no sign, or to have a dog and no sign, or to have a dog and a sign (f2) the agent is rmly decided to have a sign (for instance, because he wants to frighten possible robbers), and it is still actually possibly to have, or not, a dog

Logical representation { (f1) :dog ^ sign ^ (:dog ^ :sign) ^ (dog ^ :sign) ^ (dog ^ sign) (f2) ! sign ^ ! dog ^ ! :dog

From these facts, as our logic stands, we can derive that the ideal obligation not to have a dog has not been violated, and that there was a violation of the ideal obligation not to have a sign if there is no dog; with respect to the actual obligations, we derive the actual obligation not to have a dog. If we now adopt condition 5-e), and introduce (Oi !Oa ) as a new axiom schema, then, since we have Oi (:dog!:sign), ! (:dog!:sign) and ! :(:dog! :sign), we would derive also Oa (:dog!:sign); and, since ! sign ! ! ((:dog!:sign) $ dog)), we derive Oa dog, con icting with Oa :dog. So the question is, in this situation which conclusion should follow, from the intuitive point of view: simply the actual obligation not to have a dog, or a con ict of obligations? If intuition indicates that a con ict of obligations should not be derivable from this scenario, then 5-e) and (Oi !Oa ) must not be adopted. Then


321

there will be no direct logical connection between ideal and actual obligations - each of them will be derived, independently, directly from the deontic norms, taking into account the relevant context; nevertheless, whenever an ideal obligation Oi A is deduced from a deontic norm O (A=B ) by factual detachment (FD), we may still also deduce ! :A ^ ! A ! Oa A. If, on the contrary, intuition indicates there does indeed exist a con ict of obligations in the previous situation, then we should adopt condition 5e), and we should weaken condition 5-c) so that con icting obligations can be expressed without logical contradiction. For reasons to be explained in the next sub-section, we think it is necessary to weaken 5-c). Although we do not here commit ourselves to acceptance of 5-e), we note in passing that its adoption would have some further interesting consequences, besides providing a direct link between ideal and actual obligations. First consequence: relationships between O (B=A) and O (A ! B=>), and the \pragmatic oddity"

Recall Result 2-iv) in subsection 4.4): ` (A ^ B ) ! (O (A ! B=>) ! O (B=A)). Adopting condition 5-e) (plus 5-b1) and 5-d)), the formula below would also become valid:29 O (B=A) ! O (A ! B=>)

(O

! O !)

Thus, to represent a deontic conditional by O (B=A) rather than by O (A ! B=>) would become almost a question of taste (the former is only slightly more general). Moreover, were (O ! O !) to be adopted as a new axiom, we could conclude that the \pragmatic oddity", as we have diagnosed it, does not introduce anything really new to the orginal Chisholm set, since O (B=A) ! O (:(A ^ :B )=>) | and so, for instance, O (:sign / :dog) ! O (:(:dog ^ sign) / >)). Second consequence: rede nition of M j=w O (B=A) With the condition 5-e) (plus 5-ab)), the condition for M j=w O (B=A) becomes equivalent to the following, simpler, condition:

M j=w O (B=A)

i

kB k 2 ob(kAk)

29 It is trivial to see that M j=w O (B=A) implies kA ! B k 6= ;. On the other hand, let Z be such that Z \ kA ! B k 6= ;. We have M j=w O (B=A) implies kB k 2 ob(kAk); by condition 5-bd3), this implies kA ! B k 2 ob(kAk [ Z ); thus, by condition 5-e), kA ! Bk 2 ob(Z ).


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6.4 Unrelated deontic norms, contrary-to-duties and con icting obligations Consider again the Case 5.2 of the \considerate assassin" example (Scenario 5 of Section 5):

Factual situation (f1) The assassin has not yet killed Mr. X but has rmly decided to kill him (f2) It is actually possible for the assassin to oer, or not, to Mr. X a cigarette and suppose now that there is also another deontic norm applicable to the situation saying that it is forbidden to oer cigarettes. Thus we have: (d1) (d2) (d3) (f1) (f2)

O (:kill / >) O (oer / kill) O (:oer / >)

! kill ! oer ^ ! :oer

:kill ^

In this case we get, using (FD): Oa oer (from (f1), (f2) and (d2)) and Oa :oer (from (f2) and (d3)), and thus a contradiction (since Oa veri-

es schemas (C) and (OD)). Is this a problem? We think not. There is a reasonable interpretation of this situation according to which it does contain a con ict of obligations; the logical system we have described was not designed to solve con icts of obligations, but it should certainly be able to detect them when | as here | they arise. The key feature of this scenario is as follows: (d2) was designed to be \a CTD w.r.t. (d1)", i.e. to describe the obligations in force in a context of violation of the obligation speci ed by (d1); but (d1) and the new sentence (d3) express unrelated, or independent, deontic norms; (d2) cannot be seen to be \a CTD w.r.t. (d3)", and so the obligation not to oer a cigarette \transports down"30 to the context ( ! kill) of violation of (d1), and a con ict is obtained according to (d2). \Now one may ask how this con ict should be resolved and, of course, one plausible option is to regard (d2) as an exception to (d3) and to formalize this with a suitable nonmonotonic defeat mechanisms. However, it is important to note that this is a separate issue, which has nothing to do with the CTD aspects of the example". This remark is quoted from [Prakken and Sergot, 1994, pp.310-311] (replacing their sentences (2) and (3) by our

30 Using the terminology of [Prakken and Sergot, 1994; Prakken and Sergot, 1996], where a similar result is obtained for this case.


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(d2) and (d3)); and we fully agree with it, although we leave entirely open the question of how con icts of obligations might be resolved, since that is a matter falling outside the logical analysis of CTDs themselves. The above example indicates that semantical condition 5-c) needs to be weakened; consider also the following observation,31 that in our logic any two \unrelated deontic norms" on the same \antecedent" will give rise to a contradiction, in the following sense: Suppose that both O (B=A) and O (C=A) are true in a model M, and suppose that kAk \ kB k \ k:C k 6= ; and kAk \ kC k \ k:B k = 6 ;. Then, de ning X = (kAk\kB k\k:C k) [ (kAk\kC k\k:B k), we get kB k 2 ob(X ) and kC k 2 ob(X ); and, from condition 5-c), it follows that (kB k \ kC k) 2 ob(X ), contradicting 5-ab). Our response is to replace 5-c) by 5-c ): 5-c ) if Y; Z 2 ob(X ) and Y

\ Z \ X 6= ;, then Y \ Z 2 ob(X )

It is easy to see that a weakning of this kind does not aect the theorems used in our analysis of the CTD scenarios. This move is in keeping with our belief (a) that sets of norms, as human artifacts, may indeed be imperfectly designed and thus contain the possiblity for generating con icting obligations; and (b) that it is the task of the logic to identify such con icts when they arise, and to supply con ict-free representations of those CTD scenarios which are, from the intuitive point of view, normatively consistent.

6.5 Axiomatisation revisited

Given the weakening of condition 5-c), in order to obtain a sound axiomatisation we need to replace the axioms (O C ); (Oa C ) and (Oi C ) by the weaker schemas:

(A ^ B ^ C) ^ O (B=A) ^ O (C=A) ! O (B ^ C=A) (O ! (A ^ B ) ^ O A ^ O B ! O (A ^ B ) 110 . (O C ) (A ^ B) ^ O A ^ O B ! O (A ^ B) (O C ) 50 .

a

a

i

i

a

i

C )

a

i

All the other axioms remain unchanged. Moreover, it is easy to see that all the theorems stated in Result 2 (of Subsection 4.4) are retained. On the other hand, if we were also to adopt condition 5-e), then the axiom schemas (O ! Oa !) and (O ! Oi !) would be replaced by the new axiom (O ! O !) (since the latter, in conjunction with (FD), allows 31 We are grateful to Henry Prakken for this criticism.

324


the derivation of the former pair), and we would also add (Oi ! Oa ) as a new axiom. Obviously, it is then possible to deduce some new theorems32. 7 A FURTHER LOOK AT SOME OTHER APPROACHES

7.1 Temporal approaches A number of researchers have maintained that the problems raised by the Chisholm set essentially involve a temporal dimension, and that previously proposed solutions fail in as much as they do not capture this dimension. In this section we review some aspects of temporal deontic approaches and compare them with ours. Temporal approaches to the semantics of deontic notions are generally based on tree-structures, representing branching time with the same past and open to the future. Following [Chellas, 1980, section 6.3], we can think of time as an ordered set T of moments (or instants), and | in order to regard the possible worlds as time-stretched | de ne them as functions from T into an otherwise unspeci ed set of momentary world-states S ; we use the term history to denote this speci c interpretation of a possible world, and, in the models, we denote by H the set of the possible histories. Thus we can de ne the tree-like structures as a tuple hT; <; S; H; V i, where: - T and S are non-empty sets; - < is a strict linear order on T (i.e. < is irre exive, transitive and (8i1 ; i2 2 T ) (i1 < i2 or i2 < i1 or i1 = i2 )); - H is a non-empty subset of the set of all functions from T into S , satisfying the following restriction (where h i h0 i h(i1) = h0 (i1 ) at every instant i1 i): () (8h; h0 2 H ) (8i 2 T ) (if h(i) = h0 (i) then h i h0 ). Constraint () is intended to give the tree-like form to the temporal structures, allowing branching to the future, but not to the past. On top of these structures, temporal deontic logics typically de ne one necessity modal operator plus obligation operators (either monadic or dyadic33 ). However, a main dierence appears in the way the temporal dimension is syntactically re ected in the formal language. One family of logics indexes

32 For instance, we can prove that ` ((A _ C ) ^ B ) ^ O (B=A) ^ O (B=C ) ! O (B=A _ C ) as follows: by (O ! O !) we deduce ` O (B=A) ^ O (B=C ) ! O (A ! B=>) ^ O (C ! B=>); and the new theorem then follows by (O C ) and Result 2-iv). 33 We are excluding from this comparison the analysis of other deontic operators, such as permission operators, or the obligation-related operators proposed in [Brown, 1996].


325

the modal and deontic operators with temporal terms; as representatives of the \indexed" family, we may mention [van Eck, 1981] and [Lower and Belzer, 1983].34 Another family introduces temporal operators that can be iterated with the modal and deontic operators; as representatives of the \non-indexed" family we may mention [Chellas, 1980, sections 6.3 and 6.4], [ Aqvist and Hoepelman, 1981], [Thomason, 1981; Thomason, 2001] and [Brown, 1996]. Although the indexed family is more expressive and less abstract than the other family, this dierence is not essential in regard to deontic aspects. So we shall introduce a uniform setting where we can abstract from such dierences and concentrate on the main distinctions they provide regarding the deontic notions. In order to see how the truth-value of a sentence can be evaluated within the temporal framework, we rst need to extend the tree structures with a valuation of the atomic sentences. Tree-like models are then tuples hT; <; S; H; V i, where: - V assigns to each atomic sentence p a subset of H T , satisfying: () (8i 2 T ) (8h; h0 2 H ) (if h(i) = h0 (i) then (hh; ii 2 V (p) i hh0 ; ii 2 V (p)) Informally, hh; ii world-state h(i).

2 V (p)

i p is true at the time instant i in the

Constraint () is intended to capture the idea that the truth-value of an atomic sentence is a function solely of the actual current world state, and independent of its past and future. We are here assuming that the atomic sentences take the simple form of propositional variables, thus avoiding most of the complications introduced by a rst order component, such as the one considered in [van Eck, 1981]. With respect to truth in a model, we can follow Prior's \Ockhamist" approach to indeterminist time [Prior, 1967], and say that a sentence A is true in M(= hT ; <; S ; H; Vi) i A is true at all pairs hh; ii (belonging to H T ). Thus the basic semantic unit for the truth analysis is the pair hh; ii: intuitively i denotes the current instant; although the intuition behind h is less obvious, we may see h as xing the current and past states and pointing out a possible future (that we may call actual, or prima facie, following Prior). So kAkM = fhh; ii(2 H T ) : M j=hh;ii Ag and kpkM = V (p); as usual, we write kAk, leaving M implicit. Many temporal operators can be de ned in terms of this semantical framework, and the various \non-indexed" temporal deontic logics vary a good deal with respect to the kinds of temporal operators they consider. 34 In [Lower and Belzer, 1983] the dyadic O -operator is not time-indexed, but all the other modal/ deontic operators are.


326

Just to give one example, we can de ne a \sometime in the (actual) future" operator as follows (corresponding past operators are de ned similarly):

M j=hh;ii FA

i (9i < i1 )M j=hh;i i A (dual: G; for past: P and H) 1

On the other hand, as we have mentioned, the \indexed" temporal deontic logics do not adopt any temporal operator; instead, they allow a direct reference to the underlying time structure in the proper formal (object) language. More precisely, they include time variables and possibly other time terms35 t; t1 ; : : : which refer to the time instants through a function v, added to the temporal models, that applies each term into an element of T . Their languages may be seen as a Boolean combination of two sublanguages: the sublanguage of time and the modal/deontic sublanguage, where the latter is built from atomic sentences of the form pt (for t a time term), with time-indexed modal and deontic operators, and sentential connectives. Interestingly, if we ignore this time indexing, we can see that the modal/deontic sublanguages of the indexed and the non-indexed temporal deontic logics are analogous. So, if we could \separate" the time index from the modal/deontic operators, we would get a uniform semantical and logical setting for analysing the modal/deontic component of both types of temporal deontic logics, with obvious advantages for the purposes of comparison. This can be achieved by means of the temporal realisation operator of [Rescher and Urquhart, 1971]: (t), for t a time term (Rt in [Rescher and Urquhart, 1971]), semantically de ned as follows:

M j=hh;ii (t)A

i

M j=hh;v(t)i A

Intuitively, (t)A signi es that A is true at time t, and using this operator we can translate each modal/deontic indexed sentence into a non-indexed sentence, with the same meaning, by replacing each pt by (t)p, and each modal/deontic operator #t by (t)#. For instance, the sentence t < t1 ! Ot1 pt , valid in [van Eck, 1981]'s logic, is translated to t < t1 ! (t1 )O (t)p. Turning now to the modal/deontic component of temporal deontic logics, we start by noting that they all introduce a necessity modality, expressing some kind of \inevitability". However, a main division appears here between the temporal deontic logics | independently of whether or not they are indexed | regarding which of the following two types of necessity operator they adopt; for instance [ Aqvist and Hoepelman, 1981], [Thomason, 1981; ] [ Thomason, 2001 , Lower and Belzer, 1983] and [Brown, 1996] adopt the rst, whilst [Chellas, 1980] and [van Eck, 1981] opt for the second: 35 For particular known numerical time structures T , it is usually considered that T is directly \included" in the formal language, allowing the existence of constants and functions, and supposing that terms (of sort T ) can be written in the formal language as in the semantics (e.g. in the form t + 1). The same assumption will be made here whenever appropriate.


M j=hh;ii (dual

) A

A A

i

327

(8h0 2 H )(if h i h0 then M j=hh0 ;ii A)

M j=hh;ii A i (8h0 2 H )(if h i h0 then M j=hh0 ;ii A) where h i h0 i h(i1 ) = h0 (i1 ), at every i1 < i (dual: )

Informally, means \for all histories, with the same past and present", and means \for all histories, with the same past". Obviously, A! A is a valid sentence within this semantics. A similar division also appears in the way the deontic components are A

36

A

de ned, as we explain below. Starting with the analysis of the unary obligation operator O , with the exception of [Brown, 1996], all the mentioned works employ a normal logic for O , which can be obtained as follows: enrich the temporal models M = hT; <; S; H; V; vi with a new component (\best" histories) bh : H T ! }(H ), and de ne

M j=hh;ii O A

i

(8h0 2 bh(hh; ii))M j=hh0 ;ii A

Obviously, the function bh must satisfy some conditions, besides the nonemptiness of each bh(hh; ii) required by all of these researchers. Importantly, we can distinguish between (a) those approaches | the ones that employ | that can be said to interpret bh intuitively as giving the \best" histories that are still open, given the past and present (which are xed); they adopt the following conditions (or conditions that imply them):

A

bh(hh; ii) fh0 : h i h0 g if h i h0 and i1 i then bh(hh; i1i) = bh(hh0 ; i1i) and (b) those approaches | the ones that opt for | that can be said to

interpret bh intuitively as giving the \best" histories, that were open, given the ( xed) past; they adopt:

bh(hh; ii) fh0 : h i h0 g if h i h0 and i1 < i then bh(hh; i1i) = bh(hh0 ; i1i)

36 The symbol

is normally used for both of them, but since we want to distinguish them - and since the symbol has already been used for potential necessity | we use A and . We also note that, for an integer-like time, we can express as Y A X, where Y and X denote the previous and next time operators, semantically de ned as follows: M j=hh;ii YA i M j=hh;i 1i A and M j=hh;ii XA i M j=hh;i+1i A.


328

J

(In what follows we shall use whenever we want to stress that we are referring to the unary obligation operator under this latter class of approaches.) Some comparison can already be made between these temporal deontic approaches and our own. In fact, it is not diÆcult to see that the operators and have some similarities with the operators for, respectively, potential and actual necessity, and ! , in those contexts where they are assigned a speci cally temporal reading. More generally, it is easy to J nd connections between each pair of operators ( , ) and ( , O ), and our pairs ( , Oi ) and ( ! , Oa ). However, there are also some relevant dierences. We compare them both at a technical level and from the point of view of their motivation, starting with the former. As regards the relationships between each necessity operator and its associated obligation operator, we note that | as for our corresponding operators | what is obligatory must be possible to ful l; that is, the following sentences are valid:

A

A

A A A !

OA ! J

A

However, in contrast to ours, these logics do not require that what is obligatory must also be violable. On the contrary, all necessary truths (in their sense of necessary) are (vacuously) obligatory; that is, the following sentences are valid:

A ! OA A ! A A

J

Thus, since A $ A is valid if A does not include reference to the future, if one wanted to require that an obligation must be both possible to ful l and violable, then in order for O A to be true the sentence A would have to be characterised in such a way that it refers to the future; in the non-indexed logics this would imply that A can never be a propositional variable and must include future operators (such as F or X); in the indexed case, Ot pt1 could be true only if t1 > t; (for a general sentence Ot A one would need to require that the \temporality of" A is greater than t; this notion is from trivial: see e.g. [van Eck, 1981]). On our approach, by contrast, we can simply write Oa p,37 abstracting both from the exact time A

37 The possibility of violation can be described in our logic by the valid sentence Oa A ! ! :A, where for cases where there is a meaningful temporal interpretation,

we can see ! :A as saying that there is a future instant within some of the alternatives open to the agent where A is not the case. Note that, in contrast to what happens with A , our operator ! does not verify the (T )-schema.


329

instant t where this obligation appears (as in the non-indexed logics), and from the description of the other aspects related with the time of occurrence of the \events" encoded in p. J Consider now ( , ) and ( , Oi ). There seems to be a main technical dierence between and . For the latter, the potential alternatives (using our terminology) to the actual current world-state correspond to the alternatives that were open immediately before the current instant, i.e., at tc 1, supposing that tc denotes the current instant and assuming here, and in what follows, an integer-like time. On the other hand, the potential alternatives to the actual current world-state, as described by our operator , include the alternatives that were open at some relevant time before the current time instant, but not necessarily the one immediately before. Even in cases where it is appropriate to give a temporal interpretation to our operators, it is necessary to consider only two time instants, on our view: the current instant tc (the one relevant to determining the actual obligations), and some previous time instant t, relative to which an assessment is made of the potential possibilities which were open to the agent | an assessment which, in turn, will determine the agent's ideal obligations at tc and his violations at tc . J As regards , it appears that for the analysis of violations only the instant tc 1, immediately preceding tc , is relevant. The informal idea seems to be that one can violate only those obligations that require that something be done immediately. But then there is a problem if the obligation was in fact incurred at some time t prior to tc 1: how is the persistence of the obligation to be represented? As is noted in [Prakken and Sergot, 1997], the issue of the persistence of obligations through time has been poorly studied within the temporal approaches to deontic logic, despite its apparently close connections with CTD-problems. One of the main exceptions is [Brown, 1996], where there is some discussion of persistence conditions according to which an obligation persists until it is (de nitely) ful lled or violated. In [Thomason, 2001] we also nd a condition related with the persistence of obligations, a condition that more or less corresponds (in our temporal semantics) to: if i i1 and h1 2 bh(hh; ii), then bh(hh1 ; i1 i) = bh(hh; ii) \ fh0 : h i h1 g. However, on the one hand, this condition seems too strong, since it seems not to allow the appearance of other obligations between i and i1 ; and, on the other hand, since it only refers to h1 2 bh(hh; ii), it says nothing about how an obligation persists over time when some other obligation is violated. Besides these technical similarities and dierences, there is one fundamental dierence between the motivation underlying our approach and the one underlying the temporal approaches. This dierence has to do with the reasons why a history h1 , which was considered an alternative to the actual history h at some time instant i 1 (so that h1 i h), is no longer taken to be an alternative to h at i (i.e., not h1 i h). For the temporal

1

330


approaches, the essential reason why, at i; h1 is no longer an alternative to h, is that a state h(i) makes h1 impossible (that is, h1 (i) 6= h(i)). On the other hand, in our approach, in cases for which a temporal interpretation can meaningfully be assigned to our necessity operators, we would say that our worlds roughly correspond to the pairs hh; ii; and although av(hh; ii) would be seen as a subset of fhh1 ; j i : h1 i h and j > ig (satisfying the condition that if j; k > i then hh1 ; j i 2 av(hh; ii) i hh1 ; ki 2 av(hh; ii)), it need not necessarily be equal to the set fhh1; j i : h1 i h and j > ig; thus a pair hh1 ; j i may belong to av(hh; i 1i), but may fail to belong to av(hh; ii), even if h1 i h and j > i. The essential reason for this is that the agent may have decided to exclude from the worlds that are actual alternatives to hh; ii all the worlds provided by the history h1 , even if this history is still potentially possible to follow. This fundamental dierence between our theory and the temporal approaches is the key to understanding why we are able to accommodate such a CTD-scenario as the gentle murderer, which remains problematic for the temporal approach. We next note that there is a signi cant dierence between our dyadic obligation operator and those proposed in, e.g., [ Aqvist and Hoepelman, 1981] and [van Eck, 1981], both of which see the unary O as a limiting case of the dyadic O , in the sense that O A is an abbreviation of O (A=>). As we have indicated above, we reject a move of this kind; we view unary obligations (of types Oi and Oa ) as derivable from the dyadic, in ways that depend on the context, and on matters pertaining to the satis ability and violability of obligation. The rejection of the idea that O A is an abbreviation of O (A=>) is also a feature of the temporal-based approach in [Lower and Belzer, 1983], to which we now turn. Loewer and Belzer's 3D-logic involves a combination of a temporal approach with a preference-based approach. Brie y, the basic idea of [Lower and Belzer, 1983] appears to be the following: there is a hypothetical time instant zero, at which some deontic norms are assumed to come into force; these are expressed by the use of a dyadic O -operator, which is not timeindexed. Actual obligations, expressed by means of a time-indexed unary operator Ot , evolve from this moment on in ways determined by the deontic norms and by the facts which are taken to be settled at each time t (settledness being represented by means of an operator of type t ). They also use a notion of \ethical suÆciency", and introduce two versions of sentences of the form R(A; B ), one time-indexed and one not, intended to express the idea that B is ethically suÆcient for A. (We omit here the details of their semantical analysis of R.) Although they have little to say regarding ideal obligations and their violations, they do oer a suggestion to the eect that ideal obligation of A can be de ned as an abbreviation of O (A=>); in our opinion this would lead to diÆculties in cases where at the instant zero some of the deontic norms cannot be potentially satis ed. Consider, for instance, a case of Scenario

A


331

3 in which it is potentially impossible not to have a fence - suppose that the fence is installed in such a way that the agent concerned has neither the ability nor the opportunity to remove it; were Oi A to be de ned as an abbreviation of O (A=>), then Oi :fence would be derived, instead of | as in our logic | Oi white-fence. The semantics of the dyadic O is de ned a la Lewis, imposing a ranking on the members of H . This ranking is then connected with the model component (\best" histories) bh : H T ! }(H ). In their own words, the intuitive idea is the following [Lower and Belzer, 1983, pp. 308]: \Our basic idea for connecting the ranking with F (here denoted by \bh"), that is connecting conditional obligations with actual obligations is this: At the rst instant of time t0 , we will assume that some of the histories which are ideal according to can be achieved. But as time proceeds, events and the actions of men may render the ideal histories unattainable. Still at every moment the actual obligation is to bring about one among those best histories that remain". The factual and deontic detachment principles that are validated by their semantics are as follows: O (B=A) ^ R(B=A) ^ t A ^ B ^ ! Ot B t

O (B=A) ^ R(B=A) ^ O A ^ B ^ ! O B A

A

t

A

t

t

Ignoring the \ethical suÆciency" operator R, their factual detachment principle strongly resembles a \temporal version" of ours (although, as in the previous logics, they do not require that an obligation must be violable). On the other hand, their deontic detachment principle is stronger than ours, and, for the reasons that we have previously mentioned, we suspect that it is too strong, particularly if one also wants to address the problem of violations of ideal obligations: recall our conclusions in Case 1.1, Section 5 above. Finally, some further comments about the representation of CTD-scenarios within the temporal deontic approaches, as compared to our own. Of the temporal approaches here analysed that explicitly discuss the representation of CTD-scenarios ([ Aqvist and Hoepelman, 1981], [van Eck, 1981] and [Lower and Belzer, 1983]), the one most similar to ours is [Lower and Belzer, 1983]. In fact, it is the only one of the three that respects our requirement (iv) on the representation of CTD scenarios, and | like us | Loewer and Belzer make a distinction between the \deontic norms" applicable to a situation (expressed through the dyadic O ) and the speci c obligations that can be deduced from them given the facts of the case. However, it appears that they opt to represent the rst sentence of the Chisholm set by a time-indexed actual obligation rather than as a deontic norm expressed in terms of the dyadic O -operator. An important dierence remains: in our approach, given the \deontic norms", in order to derive which violations may have occurred and what

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the actual obligations are, we need only consider what it was potentially and actually possible to do. No speci c reference to any time instant is needed, in contrast to what happens in [Lower and Belzer, 1983] and other indexed temporal approaches, where sometimes one is forced to introduce time instants in the representation of cases in a way that seems more or less arti cial, since their speci c values are irrelevant. Moreover, as we have emphasised, our potential and actual necessitation operators are not tied to temporal settledness, and so can be used to represent cases where a temporal dimension is absent. In short, we think that our theory oers patterns of representation which are more abstract and simpler than those supplied by the typical temporal deontic approaches. Of course, whenever it is essential to state the exact time of realisation of some obligatory state of aairs | for instance because there is an obligation to do something by a speci c deadline | then an indexed temporal logic seems to be necessary. The simplicity of the logic will always depend on the degree of abstraction that we want to achieve, and when we choose, for instance, to use a deontic propositional language we abstract from many features | the temporality of some state of aairs being just one of these. But this abstraction is justi able precisely on the grounds of the simplicity that it provides for illustrating the essential features underlying CTD-reasoning.

7.2 Action-based approaches In the work of Casta~neda, and of those in uenced by him, it has frequently been maintained that the problems that beset deontic logic can be solved if proper recognition is given to the role of the concept of action. It is perhaps fair to say that Casta~neda's own work is not always easy to penetrate, but fortunately in this case two papers by James Tomberlin [1983; 1986] supply both an outline introduction to Casta~neda's system of deontic logic, and an appraisal of (respectively) his treatments of the Chisholm set and the Good Samaritan. And, judging by his replies | published in the same sources | Casta~neda accepted that Tomberlin provided a faithful account of his position. We focus here on just those aspects which seem essential in regard to the analysis of the Chisholm set. \Central to Casta~neda's enterprise, we encounter his pivotal distinction between practitions and propositions" [Tomberlin, 1983, pp. 204]. Propositions are understood in the usual way, whereas practitions are understood, roughly, as the semantical content of such sentences as commands, orders, requests, entreaties. Within the scope of deontic operators, such as \it is obligatory that . . . ", both types of components may occur: propositions indicate the circumstance, or condition, of a deontic judgment, whereas practitions indicate \. . . the target or focus of the deontic operator; components of this sort are actions practically considered. . . " [Tomberlin, 1983, pp. 235]. However, deontic operators are assumed to apply to practitions


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only, and thus a mixed scope formula, within the scope of a deontic operator, will have a practition as its content. Deontic judgments are said to belong to dierent families, where each family determines a particular sense or type of obligation, permission, and so on. This type is indicated in the formal language by an index on the deontic operator. Let us turn immediately to the representation of the Chisholm set, stating in due course those principles of Casta~neda's logic which are relevant to an assessment of that representation. Suppose that the type of obligation concerned is denoted by \s", and that we use p; q; : : : to stand for propositions, and p, q, . . . to stand for practitions. Then line 1 of the Chisholm set is assigned the form: 1. Os p whereas line 4 expresses a proposition and is accordingly represented by: 4. :p As regards line 3 (the CTD), Casta~neda insists that it must be understood as specifying what is obligatory in those circumstances in which the obligation expressed by line 1 is violated. He insists, then, on factual detachment, requiring that 1 and 3 must together imply 5. Os :q Accordingly, 3 is assigned the following form, where conditional:

! is the material

3. :p ! Os :q Line 2 is interpreted as an obligation sentence with a mixed component within the scope of the deontic operator: 2. Os (p ! q) The crucial point to note now is that lines 1 and 2 do not imply 6. Os q because the embedded antecedent in line 2 is a proposition, whereas the scope formula in line 1 is a practition. So, the claim is, in essence: distinguish properly action components and propositional components, and incorporate that distinction in the representation of the set, and the inconsistency which Chisholm's formalisation contained will disappear. Although Casta~neda's deontic logic contains the theorem (CasK)

Os (p ! q ) ! (Os p ! Os q )


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its application is of course restricted to instances where both the component sentences in the antecedent express practitions. This representation of the Chisholm set, then, is consistent. Before moving on to Tomberlin's criticisms, note that it is already clear that Casta~neda's solution generates the pragmatic oddity, in virtue of lines 1 and 5. Outlining the basic feature of Casta~neda's semantics of obligation sentences, Tomberlin says that a sentence of the form Os p \. . . is true at a world w if and only if the practition p belongs to every world v such that v is deontically compossible with w : : :" [Tomberlin, 1983, pp. 237]. Then in all the worlds which are deontically compossible with the given world in which the four sentences of the Chisholm set are true, the agent concerned helps his neighbours (by line 1) but does not tell them he is coming (by line 5). Note that (TC1)

(p ! Os q) $ Os (p ! q)

is a theorem of Casta~neda's system. It follows immediately that line 4 implies line 2, and thus that the requirement of logical independence is not met. Tomberlin and Casta~neda are among the very few deontic logicians who do not accept this requirement; we do not discuss their reasons for this rejection here | the reader is referred to [Tomberlin, 1983] and Casta~neda's reply | but choose rather to focus on Tomberlin's criticism of Casta~neda's treatment of the Chisholm set, which also begins from the observation that (TC1) is a theorem. What troubles Tomberlin is that, in virtue of (TC1), line 4 also implies 7. Os (p ! :q) Putting lines 2 and 7 together we have: it is obligatory that if X goes to help his neighbours then he tells them he is coming, and it is obligatory that if X goes to help his neighbours then he does not tell them he is coming. This is clearly an intuitively unacceptable consequence | it would not be implied by a proper representation of the Chisholm set! So what has gone wrong? Tomberlin's view, in brief, is that line 2 is not the correct way to represent the second line of the Chisholm set. By virtue of (TC1), which he seems inclined to accept, line 2 is equivalent to a conditional obligation and thus, ignoring the negation signs, has the same logical structure as line 3. But Tomberlin is of the opinion that the second sentence of the Chisholm set should be understood as expressing an obligation simpliciter, of the form: 20 . Os (p ! q)


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But then, of course, as he immediately notes, the original Chisholm \paradox" re-emerges, since line 1 and line 20 together imply line 6, by (CasK). Tomberlin advocates an alternative line of solution to the Chisholm problem, within the framework of Casta~neda's deontic logic. \Biting the bullet", as he puts it, one should accept that another aspect of Casta~neda's deontic logic | that deontic operators can be relativised to dierent senses of obligation - has a key role to play in the representation of the Chisholm set. As Tomberlin sees it, the four sentences of the Chisholm set can be assumed to be both true and mutually consistent if and only if the sense or type of obligation pertaining to the third line is dierent from the sense or type of obligation expressed by the rst and second lines. So, the rst two lines are obligation sentences whose truth conditions refer to deontically perfect worlds (he calls these absolute obligations), whereas the third line is an obligation sentence whose truth conditions refer to worlds which | like the actual world in which line 4 is true - are deontically imperfect. At this point, naturally, we begin to experience a gentle sense of deja vu. Tomberlin is here treading one of the paths we investigated in Section 3, above, and which | as we there tried to indicate | leads only to further troubles. How would he cope with second-level CTDs of the kind exhibited by Scenario 2 (the dog-sign-fence example)? By introducing a third sense of obligation, i.e., a second sense of imperfection or sub-ideality? Clearly, this proliferation of senses of the fundamental deontic notions is to be avoided. To embrace it is to admit defeat. Despite the diÆculties that arise for Tomberlin's own positive proposal, it does seem clear that his criticism of Casta~neda casts considerable doubt on whether the proposition/practition distinction has any role to play in solving Chisholm' s puzzle. Our conclusion is that Casta~neda failed to supply an analysis of deontic conditionals that both copes with the pragmatic oddity and generates suitable deontic and factual detachment principles. And we should add, with Prakken and Sergot, the observation that such CTD examples as Scenario 3 (the white fence) are apparently devoid of any action component whatsoever. Nevertheless, we think that there is at least the following to be said for the action approach: the two notions of necessity to which we assigned a crucial role in the analysis of CTD scenarios are intimately connected to praxiological concepts, in particular decision, ability and opportunity. A more elaborate development of our logical system should make these connections explicit. But, even when that is done, we very much doubt that the resulting picture will provide con rmation of Casta~neda's account of the role of action concepts. It is suggested in [Hilpinen, 1993, pp. 89] that one interesting way to understand Casta~neda's distinction is \. . . to consider it from the standpoint of dynamic deontic logic. In (propositional) dynamic logic, the non-logical

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expressions are divided into action terms and propositions, and the deontic operators behave in the same way as in Casta~neda's deontic logic . . . : they transform action terms into deontic propositions or statements . . . the way in which the distinction between action terms and propositions is made in the semantics of dynamic logic corresponds nicely to Casta~neda's conception of propositions as descriptions of the circumstances under which an action is considered or performed." Existing work (e.g., [Meyer, 1988]) falls short of providing convincing evidence that an approach to deontic logic based on dynamic logic holds the key to solving CTD problems. One diÆculty, as Hilpinen himself notes [Hilpinen, 1993, pp. 94, footnote 4], is that the fourth line of the Chisholm set appears to have \. . . no plausible representation in Meyer's dynamic deontic logic". Hilpinen's suspicion is con rmed in a later paper by Meyer, Wieringa and Dignum [1997], in which they oer a representation of what they call the òught-to-do' version of the Chisholm set in a deontic logic (named PDeL) based on dynamic logic. Concerning their representation they make the following remark: \. . . the fourth premise of the set . . . cannot be represented in PDeL. In some sense, statements of actions in PDeL and the underlying dynamic logic are of a hypothetical nature: ìf one (would) perform the action, the following holds'. The implication implicit in a formula []' is therefore more like a conditional in conditional logic. As such, it is not really important what actually happens. Here and in the sequel we shall just ignore the fourth assertion in the formal representation" [Meyer, Wieringa and Dignum, 1997, x 1.6.3]. (The formula []' says that execution of action leads to some state(s) where ' holds.) However, in the context of analysing CTD scenarios, what actually happens is | as we have seen | of paramount importance. Meyer, Wieringa and Dignum further maintain that it is necessary to distinguish explicitly between the logic of òught-to-do' and the logic of òught-to-be', and accordingly they also oer an analysis of the òught-to-be' version of the Chisholm set. We refer the reader to their paper for the details [Meyer, Wieringa and Dignum, 1997, x 1.5], but note that their òught-tobe' logic relies essentially on the availability of an inde nite number of distinct obligation operators, each of which is relativised to a particular `frame of reference', as they put it. They do not oer precise criteria for determining when one has moved from one frame of reference to another, but they take it for granted that the rst three lines of the Chisholm set involve reference to three distinct frames of reference, one for each of the obligations contained. Not surprisingly, problems of inconsistency are thereby avoided, but at the cost of introducing an inde nite number of obligation operators, and in the absence of clear guidelines determining how many operators the representation of a given scenario will need.


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7.3 Preference-based approaches \SDL cannot distinguish between various grades or levels of non-ideality; in the semantics of SDL worlds are either ideal or non-ideal. Yet the expression ìf you kill, kill gently' says that some non-ideal worlds are more ideal than other non-ideal worlds; it says: presupposing that one kills, then in those non-ideal worlds that best measure up to the deontically perfect worlds, one kills gently. In formalising CTD reasoning the key problem is formalisation of what is meant by `best measure up' " [Prakken and Sergot, 1997, p. 244]. This passage is quoted from the most detailed study currently available of the treatment of CTD-phenomena within a preference-based approach to the semantics of obligation sentences. The passage captures very succinctly the reason why a number of researchers in deontic logic have accepted the idea that an appropriate semantics for obligation sentences calls for an ordering on possible worlds, in terms of preference or relative goodness. As the authors note, the idea stems from Bengt Hansson [1971], with the later work of David Lewis [1974] providing a more comprehensive investigation of the semantical framework involved. Prakken and Sergot's paper, although in parts rather dense | it is best read as a sequel to [Prakken and Sergot, 1996] | explores in considerable depth the Hansson{Lewis analysis of dyadic deontic logic, assesses its shortcomings in regard to the treatment of CTDs, and oers a remedy for them which draws on techniques deriving from the study of default reasoning. We shall not attempt to give a summary of their nal position, but con ne ourselves to a few observations. A signi cant feature of the Prakken and Sergot account of the Hansson{ Lewis approach is that they add a notion of alethic necessity, the function of which is in part to clarify the nature of detachment properties. As was pointed out in Section 3.2, above, the characteristic feature of the DD-family of dyadic deontic logics (to which the Hansson{Lewis approach belongs) is that it validates the deontic detachment principle, but does not validate the factual detachment property. Expressed in terms of the notation used by Prakken and Sergot in [Prakken and Sergot, 1997], this means that (DD)

O [B ]A ! (O B ! O A)

is valid, whereas (FD)

O [B ]A ! (B ! O A)

is not. (Formulas of the form O [B ]A express what Prakken and Sergot call contextual obligations, and they correspond to formulas of the form O (A=B ) in the Lewis notation. The formula O A is an abbreviation of O [>]A, where > is any tautology.) However, since

O) B ! OB

(

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is valid in the Prakken and Sergot extended version of Hansson-Lewis, (DD) immediately yields (SFD) O [B ]A ! ( B ! O A) Prakken and Sergot call this principle `strong factual detachment', and they say of the alethic necessity operator that it expresses the notion òbjectively settled'. They do not say much by way of further characterisation of òbjectively', except that they want to distinguish it from necessity in a subjective sense, \. . . such as when an agent decides to regard it as settled for him that there will be a fence" [Prakken and Sergot, 1997, p. 241]. They have the further interesting comment to make about the import of (SFD) for CTD contexts: \For CTD obligations this form of strong factual detachment seems very appropriate, but it must be read with extreme care. As long as it is possible to avoid violation of a primary obligation O :B a CTD obligation O [B ]A remains restricted to the context; it is only if the violation of O :B is unavoidable, if B holds, that the CTD obligation comes into full eect, pertains to the context >" [loc.cit.]. Note, however, that if B holds, then O B holds, and thus | since the (D)-schema is valid in their system | :O :B holds. That is, going back to the dog-and-sign scenario, one could detach the obligation to put up a sign only in circumstances in which there was no longer an obligation not to have a dog! These observations bring to the fore one of the most basic dierences between our approach and that adopted by Prakken and Sergot: there is a quite fundamental disagreement between us regarding what an adequate theory of CTD scenarios should be expected to achieve. As we see matters, it is of paramount importance that the theory can show which actual obligations are derivable in circumstances of violation of some primary obligation, and can do so without also requiring that the sentence expressing the primary obligation must be false. A deontic logic which cannot show what actually ought to be done in circumstances of violation is, in our view, of limited interest. Prakken and Sergot's theory, by contrast, belongs to a tradition in deontic logic which, it seems, takes the detachment of actual obligations to be a matter of no particular importance. The notion of `settledness' Prakken and Sergot employ is peculiar, at least with respect to its relation to the concept of obligation. How can that which is settled, unalterable, be obligatory? Surely that which is genuinely obligatory must be violable! Here again is a basic point of contrast between our approach and that of Prakken and Sergot. The principles (:Oa ) and (:Oi ) | Section 4.4, above | express, we believe, the correct connection between settledness and obligation concepts, and they also play a key role, as we saw in Section 5, in determining the consequences which can be drawn from various CTD scenarios. Returning to Prakken and Sergot's discussion of the Hansson{Lewis framework, it should be noted that a prime reason for their dissatisfaction with


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that framework resides in the fact that it fails to capture a reading of the èxtended' considerate assassin scenario according to which that scenario is inconsistent. (This scenario was discussed above, Section 6.4, in response to some critical questions from Prakken. It extends the considerate assassin scenario by adding the further requirement that it is forbidden to oer cigarettes. See [Prakken and Sergot, 1997, x 6.1].) The underlying problem, as they diagnose it, is that the Hansson{Lewis semantics \. . . allows for the possibility of sub-ideal worlds but has very little to say about what they are like and nothing to say about how they compare with ideal worlds" [Prakken and Sergot, 1997, p. 250]. Their attempted solution involves a rather complex extension of the Hansson{Lewis framework, adapting techniques from the study of default reasoning in order to provide a means of ranking sub-ideal worlds with respect to the degree to which they measure up to ideal worlds. We shall not here attempt to summarise these complexities. But from the point of view of comparison with our own theory, we observe in particular that Prakken and Sergot chose to impose on their investigation two constraints which we feel | for reasons discussed, in particular, in Section 6, above | are best rejected: they insist on retaining the (D)-schema for obligation sentences, and they refuse to abandon the consequential closure principle expressed by (A ! C ) ! (O [B ]A ! O [B ]C ) (However, one of the factors which further complicates their account is that they also introduce a notion they call èxplicit obligation', which is not closed under consequence.) Our rejection of the (D)-schema and consequential closure, our insistence on the importance of (a restricted form of) factual detachment, our exploitation of a distinction between actual and ideal obligations, and our characterisation of the relationship between settledness and obligation | all of these factors mark fundamental dierences between our theory and that of Prakken and Sergot. But perhaps none of these constitutes the most fundamental dierence. For we have not found it necessary at all to resort to the use of an explicit preference ordering in the semantics, in order to capture an adequate representation of the various CTD scenarios. Recall Lewis' claim in [Lewis, 1974, pp. 3]: \A mere division of worlds into the ideal and the less-than-ideal will not meet our needs. We must use more complicated value structures that somehow bear information about comparisons or gradations of value." The treatment of the Chisholm scenario in [Jones and Porn, 1985] deliberately attempted to indicate that Lewis was wrong: the semantics used a \mere division" into two types of worlds, de ned two accessibility relations pertaining to them, and de ned some simple relations between these relations. But it imposed no ranking, no ordering, on the possible worlds. Yet it supplied the basis for an analysis of the Chisholm

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set that met all the standard adequacy criteria . . . until Prakken and Sergot identi ed the \pragmatic oddity". The question then became: is a preference ordering in the semantics essential in order to cope with the pragmatic oddity, and to cope with a broader class of CTD-phenomena, including but not limited to those exposed by the Chisholm set? In contrast to Prakken and Sergot, our answer to the last question is negative; the alternative strategy is to focus on another of the notions mentioned by Hansson which, as we have seen, also plays a role for both Loewer and Belzer and Prakken and Sergot: the notion of settledness or xity of the facts. Our basic conjecture is that, properly characterised, and appropriately connected to the notions of actual and ideal obligation, the concept(s) of settledness provide the fundamental key to unravelling the tangled knot of CTD-problems. Future research, we trust, will facilitate comparison of the preferencebased approach and the approach which has formed the core of this chapter, with a view to furthering our understanding of the Contrary-to-Duty, and thus of normative reasoning itself. POSTSCRIPT (SEPTEMBER, 2001) The writing of this chapter was completed in 1999. Of relevant material that has been published since that time, we would like in particular to mention Makinson and van der Torre's work on \input/output" logics, which the authors claim to be applicable to the treatment of CTD-problems; see Makinson and van der Torre [2000; 2001]. ACKNOWLEDGEMENTS Some of the research here reported was carried out within the Portuguese research projects no. PCSH/C/OGE/1038/95-MAGO, and PCEX/P/MAT/ 46/96- ACL, and the ESPRIT Basic Research Working Group 8319 MODELAGE (\A Common Formal Model of Cooperating Intelligent Agents"). Andrew Jones also wishes to thank the PRAXIS XXI Programme of the Portuguese Research Council for its support in the Spring of 1996. Finally, the authors are very grateful to Henry Prakken, Donald Nute, Bjrn Kjos-Hanssen and Marek Sergot for their criticisms and comments. At the time of completion of the writing of this chapter, Jose Carmo was at the Department of Mathematics, Instituto Superior Technico, Lisbon, Portugal; and Andrew Jones was at the Department of Philosophy and Norwegian Research Centre for Computers and Law, Univerity of Oslo, Norway. Jose Carmo

Department of Mathematics, University of Madeira, Portugal. Andrew Jones

Department of Computer Science, King's College, London, UK.


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[Jones, 1993] A. J. I. Jones. Towards a formal theory of defeasible deontic conditionals. Annals of Mathematics and Arti cial Intelligence, 9, 151{166, 1993. [Jones and Porn, 1985] A. J. I. Jones and I. Porn. Ideality, sub-ideality and deontic logic. Synthese, 65, 275{290, 1985. [Jones and Sergot, 1992] A. J. I. Jones and M. J. Sergot. Deontic logic in the representation of law: towards a methodology. Arti cial Intelligence and Law, 1, 45{64, 1992. [Jones and Sergot, 1993] A. J. I. Jones and M. J. Sergot. On the characterisation of law and computer systems: the normative systems perspective. In Deontic Logic in Computer Science - Normative System Speci cation, J.-J. Ch. Meyer and R. J. Wieringa, eds. pp. 275{307. John Wiley and Sons, Chichester, UK, 1993. [Konolige and Pollack, 1993] K. Konolige and M. E. Pollack. A Representationalist theory of intention. In Proceedings of IJCAI'93, pp. 390{395, 1993. [Lewis, 1973] D. Lewis. Counterfactuals. Blackwell, Oxford, 1973. [Lewis, 1974] D. Lewis. Semantic Analysis for Dyadic Deontic Logic. In Logical Theory and Semantic Analysis. Stenlund, ed. pp. 1{14. D. Reidel, Dordrecht, 1974. [Lower and Belzer, 1983] B. Lower and M. Belzer. Dyadic deontic detachment. Synthese, 54, 295{318, 1983. [Makinson and van der Torre, 2000] D. Makinson and L. van der Torre. Input/output logics. Journal of Philosophical Logic, 29, 383{408, 2000. [Makinson and van der Torre, 2001] D. Makinson and L. van der Torre. Constraints for input/output logics. Journal of Philosophical Logic, 30, 155{185, 2001. [McCarty, 1994] L. T. McCarty. Defeasible deontic reasoning. Fundamenta Informaticae, 21, 125{148, 1994. [Mendelson, 1979] E. Mendelson. Introduction to Mathematical Logic. Van Nostrand, 2nd edn. 1979. [Meyer, 1988] J.-J. Ch. Meyer. A dierent approach to deontic logic: deontic logic viewed as a variant of dynamic logic. Notre Dame Journal of Formal Logic, 29, 109{136, 1988. [Meyer, Wieringa and Dignum, 1997] J.-J. Ch. Meyer, R. J. Wieringa and F. P. M. Dignum. The role of deontic logic in the speci cation of information systems. In Logics for Databases and Information Systems, J. Chomicki and G. Saake, eds. Ch. 1, Kluwer Academic Publishers, Boston/Dordrecht/London, 1997. [Mott, 1973] P. L. Mott. On Chisholm's Paradox. Journal of Philosophical Logic, 2, 197{211, 1973. [Porn, 1977] I. Porn. Action Theory and Social Science: Some Formal Models. Synthese Library 120, D. Reidel, Dordrecht, Holland, 1977. [Prior, 1967] A. Prior. Past, Present and Future. Clarendon Press, Oxford, 1967. [Prakken and Sergot, 1994] H. Prakken and M. J. Sergot. Contrary-to-duty imperatives, defeasibility and violability. In Proceedings of the Second International Workshop on Deontic Logic in Computer Science (EON'94), A. J. I. Jones and M. J. Sergot, eds. pp. 296{318. Complex 1/94 NRCCL, Tano A. S., Oslo, 1994. [Prakken and Sergot, 1996] H. Prakken and M. J. Sergot. Contrary-to-duty obligations and defeasible deontic reasoning. Studia Logica, 57, 91{115, 1996.


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[Prakken and Sergot, 1997] H. Prakken and M. J. Sergot. Dyadic deontic logic and contrary-to-duty obligations. In Defeasible Deontic Logic. D. Nute, ed. Synthese Library, pp. 223{262, 1997 [Rescher and Urquhart, 1971] N. Rescher and A. Urquhart. Temporal Logic. SpringerVerlag, 1971. [Santos and Carmo, 1996] F. Santos and J. Carmo. Indirect action, in uence and responsibility. In Deontic Logic, Agency and Normative Systems (Proceedings of the Third International Workshop on Deontic Logic in Computer Science), M. Brown and J. Carmo, eds. pp. 194{215. Springer, Workshops in Computing Series, 1996. [Thomason, 1981] R. H. Thomason. Deontic logic as founded on tense logic. In New Studies in Deontic Logic. R. Hilpinen, ed. pp. 165{176. D. Reidel, Dordrecht, 1981. [Thomason, 2001] R. H. Thomason. Combinations of tense and modality. In Handbook of Philosophical Logic, 2nd edition, Volume 6, D. M. Gabbay and F. Guenthner, eds. D. Reidel, Dordrecht, 2001. [Tomberlin, 1983] J. E. Tomberlin. Contrary-to-duty imperatives and Casta~neda's system of deontic logic. In Agent, Language, and the Structure of the World, J. E. Tomberlin, ed. pp. 231{249, Hackett Publishing Co., Indianapolis, USA, 1983. (A reply by Casta~neda appears at pp. 441{448 of the same volume.) [Tomberlin, 1986] J. E. Tomberlin. Good samaritans and Casta~neda's system of deontic logic. In Hector-Neri Casta~neda, J. E. Tomberlin, ed. pp. 255{272. D. Reidel, Dordrecht, Holland, 1986. (A reply by Casta~neda appears at pp. 373{375 of the same volume.) [van Eck, 1981] J. van Eck. A System of Temporally Relative Modal and Deontic Predicate Logic and its Philosophical Applications, Ph. D. thesis, Rijksuniversiteit te Gronigen, 1981.

INDEX

absolute vs. relative, see normative concept absurdity, see Falsum accessibility relation, 231 Ackermann, W., 170 act-utilitarianism, 198, 201 action logic of, 154, 197, 204 action dimension of CTDs, 274, 332{336 actual/potential possibility/necessity !, , ! , , 285{ 288, 290, 291, 293{295, 297, 298, 317, 319{323, 326, 328{332, 337{340 adequacy, see translation adequacy criteria (for a theory of commitment), 186 adjunction, 177, 181, 229 al-Hibri, A., 190 alethic modal logic, 149, 151, 152, 156, 188 with a monadic Q connective, 152, 235, 236, 241 with a propositional constant, 149, 151, 179, 185, 217 alethic models, see models almost re exive relation, 209 almost symmetric relation, 209, 210 alphabet, 166, see also vocabulary, 205, 217 alternative relation, see accessibility relation to an action ancestral, see chain (of a relation) Anderson, A. R., 148{150, 152, 153, 156{158, 160, 179, 184{187, 189, 204, 259

332 2

Andersonian de nitions of deontic operators, 151 reduction of deontic logic to alethic logic, 179 answer, 1 Aqvist, L., 147, 149, 152, 159, 178, 189, 190, 202, 203, 216, 230, 236, 238{240, 243, 249, 250, 252, 254{257 Aqvist, L., 5 Aqvist, L., 147 Aqvist, L., 15, 24, 31, 33, 49, 50 Aristotle, 3, 150 assignment (function), 214, 221, 251 autonomy, 164 auxiliary symbols, 161 axiom, single vs. schema, 155 axiomatic point of view, 150 axiomatic systems, see theories, proof theories Bailhache, P., 154 Belnap's Analysis, 10 Belnap's Hauptsatz, 22 Belnap, N. D., 2, 4, 5, 7{10, 34, 37, 40, 41, 49{51 Bentham, J., 150, 154 Bergstrom, L., 199, 202 Berkemann, J., 259 bestness, see optimality betterness, 193 bi-intuitionistic logic, 118 biconditionality, 164 Bolinger, D., 7

346

INDEX

completeness theorem, 152, 211, 212, 214, 221, 223, 249 strong, 221, 222, 256 weak, 211, 221, 255 completeness-claim, 14, 15 compounding, 18 ca-complete, 55 compounds, 52 ca-derivation, 55 conclusive answer, 31 canonical model, 221 conclusiveness, 32 Carnap, R., 3 conditional obligation, 148 Casta~neda thorem, 202 conditional proof (rule of), see deCasta~neda, H.-N., 200{203 duction theorem, impliceteris paribus proviso, 198, 199 cation introduction, 215 chain (proper ancestral) of a relacon ict of obligations, 266, 270, tion, 231 276, 320, 322, 323 Chellas, B. F., 185, 188, 200, 203, connective 232, 259 degree of, 162, 166, 168, 206 Chisholm's Paradox, 151, 153, 190, dyadic (binary), 148 273{277, 279, 281, 282, main, 168 285, 287, 298, 299, 301, monadic, 148{152 302, 321, 324, 332{336, zero-place, 232, 253 339, 340 consequential closure, 266{270, 314, theorem on, 192, 197 315, 339 Chisholm's paradox, 272, 282, 300, considerate assassin example, 284, 332, 334 291, 310, 311, 316, 322, Chisholm, R. M., 148, 151, 153, 339 190, 192, 197, 198, 204 consistency, 166, 172, 190, 191, choice-oering vs. alternative-presenting 193, 195, 198, 207, 218, (sense of disjunctive obli234, 253 cations and permissions), constant, see variable vs. constant, see free choice permission 152, 155, 161, 165, 166, circumstance, 148, 201 186, 187, 203, 219, 232 closure, 158, 159 Conte, A. G., 259 co-permissibility, see accessibility contingency relation alethic, 157 Cohen, F., 3 deontic, 157 coherence principal, 30 contradiction, see falsum coincidence lemma, 214, 216, 222 correction, 28 commanding vs. commands, 148 corrections-accumulating sequence, commitment, 148, 149, 181, 183{ 49 187, 189, 190, 197, 198 correctness, see truth, soundness theorem paradox of, 148, 151, 153, 179, Cresswell, M., 44 180, 197 Cresswell, M. J., 218 completeness, 8, 16, 89 Bolzano criterion, 153, 173{175, 181, 182 Bolzano, B., 153, 173{175 brackets, see auxiliary symbols Bromberger, S., 41

INDEX

crucial lemma, 229 EON workshops, 285 EON workshops, 265 Dacey, R., 6 Dahlquist, T., 189 Danielsson, S., 152, 185, 188, 198, 200 decision theory, 179 deducibility, see derivability deduction theorem, 196 deductive completeness, 10 deductive equivalence, 257, 258 defeasible obligations, 271, 282 de nite descriptions, 205 de nitional enrichment of a language, 164, 165, 168, 204 extensions of a theory, 153, 157 de nitions theory of, 165 deliberative question, 43 denotational semantics of proofs, 110 deontic contingency, 157 deontic detachment, 273, 276, 297, 298, 331, 335, 337 deontic logic, 147 dyadic, 148, 151, 153 monadic, 148{152, 155, 169 natural vs. formal, 152, 153, 172, 205 reduction to alethic modal logic, see Andersonian relationship of act-utilitarianism to, 198, 201 von Wright-type, 148, 149, 154, 155, 159, 179, 241, see Hansson dyadic systems, Smiley{Hanson monadic systems deontic models, see models deontic paraadoxex, see paradox

347

deontic/epistemic paradox, 269, 270 Deontik (Mally's), 150 depreviation of counterintuitive force, 183 derivability, 172, 191, 210, 218, 234, 253 derived obligation, see commitment detachment, principle of, 156, see

Modus ponens

di Bernardo, G., 259 dilemma, see paradox Jephta, 198, 200, 201, 203 of commitment and detachment, 150 DIntKt, 103 direct answer, 15 direct answers, 1, 29 discharging of hypotheses, see implication introduction, negation introduction, disjunction elimination, existential instantiation disjunction elimination, 215 disjunction introduction, 163, 205 display logic, 81 distinctness-claims, 15 DKt, 89 dog example, 274, 279{286, 288, 302, 304, 320, 335, 338 doxastic logic, 270 duty, con ict of, 151 dyadic deontic logic, 276, 285, 286, 289, 290, 292, 294{298, 317{323, 330, 331, 337{ 340 dynamic logic, 335, 336 easy lemma, 228, 230, 231, 236 eectiveness, 8 eectivity, 16 Egli, U., 4, 10 elementary questions, 12 elementary-like questions, 17 epistemic analysis of questions, 24

348

INDEX

equivalence (material), 163 equivalence relation, 235, 236 erotetic, 2 erotetic logic, 2, 51 ethics (ethical theory), 150, 201, 204, 259 Euathlus, 147 Euclidean relation, 209{211, 220 existential generalisation, 177, 181, 215, 229 existential instantiation, 215 expressive completeness, 9 expressive incompleteness, 9 expressive resources of deontic languages, 150 extension, see de nitional, truthset factual detachment, 273, 276, 295, 321, 331, 333, 335{339 faithfulness, see representation fallacy of many questions, 35 falsum, 155, 163, 192, 205, 247 Ficht, H., 4 Finn, V., 38 Fisher, M., 154 xed/unalterable/settled facts, 283{ 288, 294, 316, 330, 338{ 340 forbiddance, see prohibition formalization, 153, see translation, 161, 169, 170, 183, 184, 190, 191, 197, 204 formula(s), see sentences formulas-as-types for temporal logics, 101 fragment (generated by de nitions), see representability, representation deontic, 150, 151, 223{225, 233, 236, 241 of normative English, 153, 167, 168, 170, 171, 175, 179, 183

free choice permission, 178, 269 full answer, 31 function from sentences into binary relations, 168, 193, 208, 219, 232 Fllesdal, D., 150, 158, 174, 175, 178, 185, 190, 197, 259 Gardenfors, P., 156 gaggle theory, 129 general erotetic logic, 51, 52 gentle murderer example, 307{309, 315, 330 Gentzen system, 62 Gentzen systems for normal modal logics, 63 Gentzen terms, 62 genuine vs. spurious deontic sentences, 154 Gestalt, 162 GMA (`Give Me an Answer'), 34 good samaritan, 269, 316 Gornstein, I., 4 Grewendorf, G., 41 Groenendijk, J., 2, 7, 32, 37, 44 grouping, ambiguity of, 169 GS5, 80 guarding, 28 guarding a risky question, 28 Halonen, I., 42 Hamblin's Postulate, 37, 43, 44 Hamblin's Postulates, 5 Hamblin, C., 2, 3, 5, 34 Hanson, W. H., 150, 151, 154, 207 Hansson systems of dyadic deontic logic, 152, 185, 193, 236, 241, 244, 247 partial syntactic identi cation, 248 semantic identi cation, 248 Harrah, D., 1, 9, 21, 35, 52, 53, 57 Harris, S., 32 Hedenûs, I., 154

INDEX

Henkin technique in modal logic, 151, 225 Henkin, L. A., 151, 225 Hi_z, H., 4, 23, 38 Higginbotham, J., 4, 36, 37, 44, 51 higher-arity sequent systems, 75 higher-dimensional sequent systems, 74 higher-level sequent systems, 73 Hilbert, D., 170 Hilpinen, R., 150, 158, 174, 175, 178, 185, 190, 197, 259 HIntKt, 135 Hintikka deontic consequence, 159 Hintikka's development, 31 Hintikka, J., 5, 25, 31, 32, 40, 42, 150, 159, 199, 200, 203 historical possibility (necessity), 149, 217 Hoepelman, J., 203 Hoh ed, W. N., 204 how questions, 43 Hughes, G. E., 218 Hull, R., 37 hypersequents, 80 hypothesis, see discharging ideal/actual obligations, Oi A=Oa A, 271, 273, 278, 283{286, 288{297, 314{317, 319{ 323, 328{331, 338{340 identity, 171, 203 imperative vs. descriptive interpretation of deontic sentences, 154 imperatives contrary-to-duty, 148, 153, 190, 192, 195, 197, 203 logic of, 148, 259 implication introduction, 196, see conditional proof (rule of) implying, 44 improved formalization, 183, 184

349

incompleteness, 249 independence, 195 individual acts, 203 inductive de nition, see recursive de nition inferential equivalence, see deductive equivalence information-retrieval, 51 internal vs. external deontic sentences, 154 interpretation, 148, 149, 154, 163, 183, see assignment, models, truth, imperative vs. descriptive, internal vs. external, modal vs. factual, genuine vs. spurious interrogative, 1 interrogative model of inquiry, 32 interrogative operators, 26 interrogatives, 26 introduction rules, 86 introduction schemata, 68 IQW (Ìt Is the Question Whether'), 38 iteration, 247 Jephta, see dilemma Kalinowski, G., 159 Kalish, D., 176, 181 Kamp, J. A. W., 178 Kampe de Feriet, J., 50 Kanger, H., 204 Kanger, S., 150, 151, 175, 186, 201, 203, 204, 217, 235, 236 Karttunen, L., 7 Keenan, E., 37 Kiefer, F., 7, 32 Knuuttila, S., 150, 200 Koura, A., 41 Kracht's algorithm, 96

350

Kripke, S., 34, 150, 151, 218, 241, 247 Kt, 97 Kubinski, T., 3, 10, 15, 17, 23, 24, 44, 46 LS5,

66 language, see alphabet, vocabulary of alethic modal logic with Q connective, 151, 152 of dyadic deontic logic, 151, 152, 159, 186, 187 of monadic deontic logic, 152, 161, 169, 173, 175 of normative English, 153, 167 legal theory, 149, 259 Lehnert, W., 4, 44 Leibniz, G. W., 150 lemma on relations, 230, 231, 236 on G, 255 Lemmon, E. J., 151, 218, 222 Lenzen, W., 147, 203 Leszko, R., 48 Lewis, D., 33, 152, 178, 185 Lewis, S., 33 limit assumption (limitedness), 248 Lindahl, L., 150, 204 Lindenbaum lemma, 213, 221, 251 Lindenbaum, A., 213, 221, 251 logic, see alethic modal logic, deontic logic, predicate logic, preference theory, action logic of action, 270, 332, 334{336 logic of intention, 270 logical analysis, 149 connectives, 206 consequence, 159, 172, 176{ 178 dictionary, 163 validity (truth), 153, 163, 171{ 174, 176, 181

INDEX

logics, see theories LS4, 69 main connective, see principal sign Makinson lemma, 213, 215, 221 Makinson, D., 151, 156, 203, 213, 215, 218, 222 Mally, E., 150, 158 Materna, P., 37 matrix method, 225 maximal (complete) consistent set, see saturated sets maximality under a preference relation, see optimality Mayo, B., 43 McGuinness, F., 200 mixed sentences, 51 MMB (`Make Me Believe'), 33 MMK (or `Make Me Know') approach, 24 modal logic, see alethic modal logic modal vs. factual interpretation of normative sentences, 154 modality, see predicate vs modality model-theoretical semantics, 150, 183, 203 models <-supplemented deontic, 244 <-supplemented dyadic deontic, 245 classi cation of, 209 classi cation of alethic, 219 deontic, 242 modus ponens, 156, 196, 206, 212, 215, 217, 233, 237, 249, 253, 256 modus tollens, 182, 189 monadic vs. dyadic, see deontic logic, normative concept Montague, R., 150, 176, 181, 200, 203 Moritz, M., 156

INDEX

morphology, see alphabet, vocabulary Mott, P. L., 190 multiple-sequent systems, 78 natural deontic logic, 153, 171 natural vs. restricted sentence-set (over a vocabulary), 153, 171, 172 necessitation (inference rule of), 159 N , 159, 249, 253 O, 155, 156, 206, 213 OB , 233 negation introduction, 215 negative requirements, see unprovables nominally true, 20 non-normal operators, 277 normal deontic logic, 155, 157 normative concept absolute vs. relative, 148 monadic vs. dyadic, 148 unconditional vs. conditional, 148 notation (for dyadic normative concepts) probabilistic, 186 sententially indexed modality, 232 standard binary connective, 232 Nute, D., 178 obedience vs. disobedience, 178 obligation prima facie vs. actual, 198, 199 conditional, see commitment expressions of, 148, 178 Oppenheim, F., 149 opt-serial relation, 219, 220, 231 optimal obligation and permission, 217 optimality (bestness), 217

351

or, 18 other-things-being-equal, see ceteris paribus proviso ought-implies-can, 198, 199 overrule, 199 Porn, I., 204 Paradox Chisholm, 151 Good Samaritan, 173, 198 Morning Star, 201, 203 Protagoras, 147 Ross, 153 partial answer, 44 partial answers, 1 penalty, 148, 185 permission, 148, see commitment, deontic logic (dyadic), 152, 185 expressions of, 151, 154, 163, 178, 186 Picard, C., 50 Polish notation, 169 positive law, 148 possible worlds semantics, see modeltheoretical semantics power-relation, see chain (proper ancestral) of a relation Powers, L., 185, 188, 190 pragmatic oddity, 278{279, 288, 300, 321, 334, 335 Prawitz, D., 202 predicament, see duty, con ict of predicate logic, 126 predicate logic, temporally relative, 203 predicate vs. modality, 159 preference logic of, 217, 256 relation, 152, 202, 217, 241, 242, 254 theory, 152, 217 preference-based semantics, 276, 331, 337{340

352

INDEX

prescription, 148 presumptions, 56 presupposes, 20 presupposition, 20, 28 Price, R., 199 principal sign, 168 Prior, A. N., 2, 3, 148, 151, 153, 156, 157, 159, 179, 184, 200, 203 Prior, M., 2, 3 prohairetic, 217, 218, 232 prohibition expressions of, 148, 151, 154, 157 proof theory, 205, 206, 217, 237, 253 proof-theoretical point of view, see axiomatic point of view propositional constant, 148, 149, 151, 152, 162, 165, 179, 217, 242 letter, 152, 155, 156, 161, 164, 167, 169, 216 parameter, 167 variable, 162, 165 von Wright-type deontic logic, 148, 154, 155, 159 Protagoras, 147, 203 provability, 172, 173, 186, 207, 218, 234, 253 prudence, safety, risk in embouliatic logic, 179 Purtill, R. L., 199

Q-connective

monadic, 151, 235 zero-place, 232, 253 QIE (quanti ed imperative-epistemic) logic, 25 quanti cation, 203 queriables, 12 question trees, 48 question-types, 1 questions, 1

questions as context descriptions, 34 questions as hyper-complete entities, 38 questions as incomplete entities, 37 questions as intensional entities, 35 questions as their presuppositions, 34 Quine, W. V. O., 174 quotation marks, 168 raising, 44 rational reconstruction, 149 reading, 162 real answer, 16 really true, 20 recursive de nition, 162, 179, 208, 257 reductio ad absurdum, 178 reduction of questions, 46 relative product, 231 relevant replies, 53, 54 reply, 1 representability, 217, 256 representation, 217, 238 faithful, 174 left-to-right faithful, 174 right-to-left faithful, 174 requirement of consistency andnonredundancy, 190 Rescher, N., 148, 150, 178, 185, 259 residuation, 81 Reykjavik example, 311{314, 318 riskiness, 28 risky, 22, 28 Robison, J., 204 Rosetus, R., 200 Ross paradox, 268, 269, 314, 315 Ross, A., 153, 165, 175 Ross, Sir D., 199 rule of proof (inference), 156, 161

INDEX

Russell, B., 147

S 5, 65

SA reduction of questions, 5 safe, 22, 28 sanction, 148, 151, 185 satis ability, 210, 219, 235, 238 saturated sets, 151, 212 lemma on, 212 saturation Lemma for canonical models, 214, 216, 222, 223 Schleichert, H., 4 Schliechert, H., 10 Schmidt-Radefeldt, J., 44, 48 Scott, D., 151, 159, 218, 222, 236 Segerberg, K., 156, 185 Sellars, w., 190 semantic entailment, 238 semantic tableaux, 151 semantics, see model theoretical semantics sentences basic, 154, 161, 163, 170 set of all well formed, 154, 157, 161, 231, 232, 247, 253 sequences of questions, 48 sequencing, 48 serial relation, 219, 231 set-of-answers methodology, 4, 5 Smiley, T. J., 150, 151 Smiley{Hanson monadic systems, 150{153, 155, 159, 161, 163, 169, 173, 176, 179, 180, 204{206, 210, 216 Smullyan, R. M., 147, 203 soundness theorem, 194, 195, 210, 211, 249 speech act theory, 150 Spohn, W., 185, 197 square of opposition, 149 Stahl, G., 3, 6, 7, 19, 49 standard deontic logic, 266{270, 272{274, 277, 317

353

Steel, T., 2, 4, 5, 9, 10, 40, 41, 49, 51 Stenius, E., 154, 156, 157 Stokhof, M., 2, 7, 32, 37, 44 Strasser, P., 259 strict implication, 185 strict preference, see betterness strong cut-elimination, 90 strong factual detachment, 337, 338 strong normality (of deontic logics), 152, 156{159 strong vs. weak permission, 179 structural rules, 87 stylistic elegance, 167 stylistic variant, 176, 181 substitution, 155, 156, 165, 167 substructure property, 70 suÆcient reply, 56, 57 Suppes, P., 164, 166 suppressing, 44 symbolization, see formalization symmetric relation, 219, 220, 235 Szaniawski, K., 6 temporal deontic logic, 203 resources of expression, 150, 153, 157 temporal dimension of CTDs, 274, 279, 283, 324{331 temporally dependent (relative) possibility (necessity), 199, 200 theories, 148, 153, 162 thesishood, see provability Thomason, R. H., 203 Tichy, P., 5, 7, 9, 35, 37, 51 TMT (`Tell Me Truly'), 33 Todt, G., 44, 48 Tomberlin, J. E., 190, 200 transitive relation, 209, 219, 220, 235, 243 translation, 153, 161, 169, 173, 223, 236, 258

354

INDEX

from natural language into formal, 153 fully adequate, 174, 175, 177, 181 left-to-right adequacy, 173, 174, 197, 204, 205 right-to-left adequacy, 173, 174, 177, 229 translation of hypersequents, 125 translation of multiple-sequent systems, 124 translation theorem for monadic deontic logic (Smiley), 225 for dyadic deontic logic ( Aqvist), 152, 236 for monadic deontic logic (Smiley), 151, 152, 225 true, 20 truth, 20 condition (de nition), 183, 193, 208, 222, 235, 237, 248 function, 155, 162 set, 164, 194, 234, 239, 253 truth value analysis of interrogatives, 39 truth-or-falsity of deontic sentences, 154 turn rules, 82 typed -calculus t , 105 types of answer, 21 types of elementary-like question, 23 types of question, 40 types of reply, 43 undsoundness, 249

ungetrenntes ganzes, 170

universal generalization, 223 universal instantiation, 215 universal necessity and possibility, 159, 236, 237, 247 univocity, 16 unprovables, 158

validity, 149, 153, 163, 171{174, 188, 210, 219, 235, 238, 250 valuation (function), see assignment van Eck, J., 150, 153, 159, 188, 190, 197{200, 202, 203, 232 van Fraassen, B. C., 152, 185, 188, 200 variable vs. constant, 149 vectored sentences, 52 veri cation lemma, 214, 216, 222, 223, 252 verum, 155, 163, 187, 205, 247 violation, 265, 269, 271, 277, 282, 288, 314, 315, 317{319, 322, 329{331, 333, 338 vocabulary, 252, 256 von Kutschera, F., 152, 185, 260 von Wright, G. H., 148, 149, 153, 154, 156, 157, 161, 178, 179, 184{187, 190, 194, 200, 203, 204, 259 Wachowicz, K., 24 Wedberg, A., 149 what, 40 Wheatley, J., 43 whether-questions, 12 which-questions, 12 white fence example, 274, 305, 306, 315, 331, 335 who, 40 why, 40 why questions, 41, 42 Wisniewski, A., 2, 4, 10, 24, 44, 45, 47

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