Handbook of Philosophical Logic Second Edition 10

HANDBOOK OF PHILOSOPHICAL LOGIC

2ND EDITION

HANDBOOK

OF PHILOSOPHICAL LOGIC

2nd Edition

Volume 10

edited by D.M. Gabbay and F. Guenthner

Volume 1 – ISBN 0-7923-7018-X Volume 2 – ISBN 0-7923-7126-7 Volume 3 – ISBN 0-7923-7160-7 Volume 4 – ISBN 1-4020-0139-8 Volume 5 – ISBN 1-4020-0235-1 Volume 6 – ISBN 1-4020-0583-0 Volume 7 – ISBN 1-4020-0599-7 Volume 8 – ISBN 1-4020-0665-9 Volume 9 – ISBN 1-4020-0699-3

HANDBOOK OF PHILOSOPHICAL LOGIC

2nd EDITION

VOLUME 10

Edited by

D.M. GABBAY

King’s College, London, U.K.

and

F. GUENTHNER

Centrum für Informations- und Sprachverarbeitung,

Ludwig-Maximilians-Universität München, Germany

A C.I.P. Catalogue record for this book is available from the Library of Congress.

Printed on acid-free paper

All Rights Reserved ©

No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording

or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. ISBN 978-90-481-6431-8 ISBN 978-94-017-4524-6 (eBook)

DOI 10.1007/978-94-017-4524-6

Originally published by Kluwer Academic Publisherd in 2003. 2003 Springer Science+Business Media Dordrecht

Editorial Preface

Dov M. Gabbay

CONTENTS

Modal Epistemic and Doxastic Logic

J ohn-J ules Ch. Meyer

Reference and Information Content: Names and

Descriptions

Nathan Salmon

Indexicals

Graeme Forbes

VII

1

39

87

Proposition al Attitudes

121

Rainer Bäuerle and M. J. Cresswell

Property Theories

143

George Bealer and Uwe Mönnich

Mass Expressions

249

Francis Jeffry Pelletier and Lenhart K. Schubert

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 first edition and there have been great changes in the landscape of philosophical logic since then.

The first edition has proved invaluable to generations of students and researchers in formal philosophy and language, as weIl as to consumers of logic in many applied areas. The main logic article in the Encyclopaedia Britannica 1999 has described the first edition as 'the best starting point for exploring any of the topics in logic'. We are confident that the second edition will prove to be just as good!

The first edition was the second handbook published for the logic commu-nity. It followed the North Holland one volume Handbook

0/

Mathematical Logic, published in 1977, edited by the late Jon Barwise. The four volume

Handbook

0/

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 artificial 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 organisa-tion on the one hand and to provide the theoretical basis for the computer program constructs on the other. The result was that the Handbook

0/

Philosophical Logic, which covered most of the areas needed from logic for these active communities, became their bible.

The increased demand for philosophicallogic from computer science and artificial intelligence and computational linguistics accelerated the devel-opment 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 com-puter 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 ofthe Handbook contributors became active in these application areas and took their place as time passed on, among the most famous leading figures 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 formallanguage and artificial in-telligence. It shows that the first edition is very close to the mark of what was needed. Two topics were not included in the first edition, even though

D. Gabbay and F. Guenthner (eds.),

Handbook of Philosophical Logic, Volume 10, vii-ix.

viii

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

• a chapter on non-monotonie logie

• a chapter on combinatory logie and A-calculus

We feIt at the time (1979) that non-monotonie logie was not ready for a chapter yet and that combinatory logie and A-calculus was too far re-moved.1 _{Non-monotonie logie is now a very major area of} philosophi-cal logie, alongside default logies, labelIed deductive systems, fibring log-ies, multi-dimensional, multimodal and substructural logies. Intensive re-examinations of fragments of classieal logie have produced fresh insights, including at time decision procedures and equivalence with non-classieal systems.

Perhaps the most impressive achievement of philosophieallogie as arising in the past decade has been the effective negotiation of research partnerships with fallacy theory, informallogie and argumentation theory, attested to by the Amsterdam Conference in Logie and Argumentation in 1995, and the two Bonn Conferences in Practieal Reasoning in 1996 and 1997.

These subjects are becoming more and more useful in agent theory and intelligent and reactive databases.

Finally, fifteen years after the start of the Handbook project, I would like to take this opportunity to put forward my current views about logie in computer science, computationallinguisties and artificial intelligence. In the early 1980s the perception of the role of logie in computer science was that of a specification and reasoning tool and that of a basis for possibly neat computer languages. The computer scientist was manipulating data structures and the use of logie was one of his options.

My own view at the time was that there was an opportunity for logie to playa key role in computer science and to exchange benefits with this rieh and important application area and thus enhance its own evolution. The relationship between logie and computer science was perceived as very much like the relationship of applied mathematies to physies and engineering. Ap-plied mathematies evolves through its use as an essential tool, and so we hoped for logie. Today my view has changed. As computer science and artificial 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-monotonie logie chapter. I wonder how the subjeet would have developed, if the AI research eommunity had had a theoretieal model, in the form of a chapter, to look at. Perhaps the area would have developed in a more streamlined way!

PREFACE TO THE SECOND EDITION ix such questions for centuries (unrestricted by the capabilities of any hard-ware).

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 effective 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 under lying 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 Temporal logie Modal logie. Multi-modal logies Algorithmie proof Non-monotonie reasoning Probabilistie and fuzzy logie Intuitionistie logie Set theory, higher-order logie, >.-ealculus, types 11 IT Natural language proeessing Expressive power of tense operators. Temporal indices. Sepa-ration of past from future generalised quantifiers Diseourse rep-resentation. Direct eom-putation on linguistie input Resolving ambigui-ties. Machine translation. Doeument dassifieation. Relevanee theory logieal analysis of language Quantifiers in logie Montague semanties. Situation semanties Program eontrol spee-ifteation, verifteation, eoneurreney Expressive power for re-eurrent events. Specifieation of tempo-ral eontrol. Decision prob-lems. Model eheeking. Action logie New logies. Generic theo-rem provers Loop ehecking. Non-monotonie decisions ab out loops. Faults in systems.

Real time sys-tems Constructive reasoning and proof theory about speeifi-eation design Non-well-founded sets Artiftcial in-telligenee Planning. Time depen-dent data. Event ealeulus. Persistenee through time-the Frame Problem. Tem-poral query language. temporal transactions. Belief revision. Inferential databases General theory of reasoning. Non-monotonie systems Intrinsie logical discipline for AI. Evolving and eom-munieating databases Expert sys-tems. Machine learning Intuitionistie logie is a better logical basis than classical logie Hereditary fi-nite predieates Logie pro-gramming Extension of Horn dause with time eapability. Event ealeulus. Temporal logie programming. Negation by failure and modality Proeedural ap-proach to logie Negation by failure. Dedue-tive databases Semantics for logie pro grams Horn dause logie is really intuitionistie. Extension of logie program-ming languages >.-ealeulus ex-tension to logie pro grams

Imperative vs. declar-ative lan-guages Temporal logie as a declarative programming language. The ehanging past in datahases. The imperative future Dynamie logie Types. Term rewrite sys-tems. Abstract interpretation Semanties for programming languages. Martin-Löf theories Semanties for programming languages. Abstract in-terpretation. Domain reeur-si on theory.

PREFACE TO THE SECOND EDITION

Database theory Temporal databases and temporal transactions Database up-dates and action logie Abduction, rel-evanee Inferential databases. Non-monotonie eoding of databases Fuzzy and probabilistic data Database transactions. Inductive learning Complexity theory Complexity questions of deeision pro-eedures of the logies involved Ditto Ditto Ditto Ditto Ditto Ditto Agent theory An essential eomponent Possible ae-tions Agent's implementation rely on proof theory. Agent's rea-soning is non-monotonie Conneetion with decision theory Agents eon-struetive reasoning xi Special com-ments: A look to the future Temporal systems are beeoming more and more so-phistieated and extensively applied Multimodal logies are on the rise. Quantifieation and eontext beeoming very active A major area now. Impor-tant for formal-ising practieal reasoning Major area now Still a major eentral alterna-tive to classical logie More eentral than ever!

xii

Classical logic. Basic back- Pro gram syn- A basic tool Classical frag- ground lan- thesis

ments guage

LabelIed Extremely use- A unifying Annotated

deductive ful in modelling framework. logic programs

systems Context

theory.

Resource and Lambek calcu- Truth

substructural lus maintenance

logics systems

Fibring and Dynamic syn- Modules. Logics of space Combining

fea-combining tax Combining and time tures

logics languages

Fallacy theory

Logical Widely applied Dynamics here

Argumentation Game

seman-theory games tics gaining

ground

Object level! Extensively

metalevel used in AI Mechanisms: ditto Abduction, default relevance Connection with neural nets Time-action- ditto revision mod-els

PREFACE TO THE SECOND EDITION xiii

Relational Logical com- The workhorse The study of databases plexity classes of logic fragments is very active and promising.

Labelling Essential too!. The new

unify-allows for ing framework

context for logics

and contra!.

Linear logic Agents

have limited resources

Linked Agents are The notion of

databases. built up of self-fibring

al-Reactive various fibred lows for

self-databases mechanisms reference

Fallacies are really valid modes of rea-soning in the right context. Potentially ap- A dynamic plicable view of logic

On the rise in all areas of applied logic. Pramises a great future Important fea- Always central ture of agents in all areas Very important Becoming part far agents of the notion of

a logic Of great im-portance to the future. Just starting A new theory A new kind of of logical agent model

JOHN-JULES eH. MEYER

MODAL EPISTEMIC AND DOXASTIC LOGIC

1 INTRODUCTION

Knowledge has always been a topic central to philosophy (cf. e.g. [Glymour, 1992]). Since ancient times philosophers have been interested in the way knowledge comes to us and in what way it relates to reality, the world in which we live. As is the case with so many things, during this century also the topic of knowledge has become the subject of formal investigations. Questions arose such as what the logical properties of knowledge are, and in order to come up with answers to these, logics have been devised to study these quest ions in a formal setting. These logics are now generally called 'epistemic logics', i.e., logics pertaining to 'knowledge'. Mostly, also the notion of 'belief', which is sometimes thought of as a weaker form of knowledge (but this is debated among philosophers (cf. [Gettier, 1963; Pollock, 1986; Voorbraak, 1993]) is considered, and we will also incorporate this notion in our treatment. Sometimes logics of belief are referred to with the special term 'doxastic logics', but we will just use the term 'epistemic logic(s)' for logics of knowledge and belief. Jaakko Hintikka [1962] was the first who proposed a modal logic approach to knowledge and belief. We follow the tradition of most recent treatments of modal logics, including those of knowledge and belief, by adopting possible world semantics in the style of Kripke [1963].

Recently, that is, in the last decade or two, these logics have attracted new attention from computer scientists and researchers of artificial Intelli-gence (AI), since it appeared that the notions of knowledge and belief played an important role in understanding advanced systems of software. It was appreciated that the use of notions of knowledge and belief enabled one to describe systems that are complicated both in physical and in logical respect: for both so-called distributed systems as systems in which information is ma-nipulated in a non-trivial way, such as knowledge-based systems the concept of knowledge is important, and to give these systems a sound basis it ap-peared to be worth-while to use epistemic logic [Fagin et al., 1995; Fagin and Vardi, 1986; Fischer and Immerman, 1986; Halpern, 1986; Halpern, 1987; Halpern and Fagin, 1989; Halpern and Moses, 1984a; Halpern and Moses, 1985; Halpern and Moses, 1990; Halpern and Moses, 1992; Moses, 1988; Moses, 1992; Voorbraak, 1988].

In this chapter we will give an overview of the development of epistemic logic in recent years as well as its application in computer science and AI.

D. Gabbay and F. Guenthner (eds.),

Handbook of Philosophica/ Logic, Vo/ume 10. 1-38.

2 JOHN-JULES eH. MEYER

2 THE 'STANDARD' MODAL APPROACH TO KNOWLEDGE AND BELIEF

2.1 The Modal Language of Knowledge and Belief

The core (propositional) language that we will use consists of a fixed set of propositional atoms closed under the usual propositional connectives and modal operators K, M, Band N. (For ease of presentation we will treat both knowledge and belief in the same model. Of course, one might leave out one of the two, and then also simplify the models accordingly.)

Formally, this is given by the following definition:

DEFINITION 1. Let P be a denumerable set of (propositional) atoms. The language Leore consists of

1. pE Leore for all PEP;

2. if <p,

t/J

E Leore then also -'<p, <p 1\

t/J

and <p V

t/J

E Leore ;

3. if<p E Leore then also K<p, B<p E Leore •

We will furthermore use the abbreviations <p

-+

t/J

for -'<p V

t/J,

<p t+

t/J

for

(<p

-+

t/J)

1\

(t/J

-+

<p), ..L for p 1\ -,p for some fixed PEP, T for -,..L, M<p for

-,K -,<p and N<p for -,B-,<p.

The part of the language in which only the K- (and M-)modalities are used, will be denoted LK, while the language with only B-(and N-) modal-ities will be denoted LB. Formulas without any occurrence of a modal oper-ator will be called objective, and for a set q, ~ Leore the subset of objective

formulas will be referred to as Obj(q,). For any modal operator

0

(viz.

K, M, B, N, but also for any other modal operator that we shall intro-duce in the sequel), we use the convention that Oq, = {O<p I <p E q,}. For ease of notation, we will also use this convention for

0

= -'.

The operator K will denote 'it is known that' and B will denote 'it is

believed that', while the dual operators M and N will denote possibility on the basis of knowledge and belief, respectively.

This will be the core language that we will use for the moment. Later on, we will extend this language to accommodate for further epistemic modali-ties.

2.2 K ripke-style Semantics

The core language Leore is given an interpretation on the basis of

Kripke-models, as is usual for a modallanguage. We will use tt and

ff

for the truth values.

DEFINITION 2. A Kripke model is a structure M ofthe form (S, 7r,R, T),

MODAL EPISTEMIC AND DOXASTIC LOGIC 3

• S is a non-empty set (the set of states or possible worlds);

• 7l" : S

-+

(P

-+

{tt,

In)

is a truth assignment function to the atoms

per state;

• R ~ S x S is the knowledge accessibility relation;

• T ~ S x S is the belief accessibility relation.

The intuition behind the accessibility relation is the following: if R(s, t) holds, then the state

t

is considered possible on the basis of the available knowledge in state s. In some way, having a number of available alternatives tl, t2, t3, ... for which R(s, ti) holds, expresses the knower's uncertainty about the true nature of the state s jhe is in. Something similar holds for the relation T with respect to belief.

The interpretation of the language is now given by the following. DEFINITION 3. Let

M

=

(S,7l",R,T)

be a given model and

sES.

Then:

• M, s

F

p Hf

7l"(s)(P)

= tt, for pE Pj

• M,

s

F

-'<p Hf not M, s

F

<pj

• M, s

F

<p 1\ 'Ij! iff

M, s

F<P

and

M, s

F

'lj!j

• M, s

F

<p V 'Ij! iff

M, s

F<P

or

M,

S

F

'lj!j

• M, S

F

K<p iff M, t

F

<p for all t with R(s, t)j

• M, s

F

B<p iff M, t

F

<p for all t with

T(s, t)

A formula <p is valid in a model M =

(S,7l",R,T),

denoted M

F

<p, if M, s

F

<p for every

sES.

A formula<p is valid with respect to a set MOD

of models, denoted MOD

F <p,

if

M

F

<p for every model

M

E MOD. If MOD is the set of all Kripke models of the above form, we generally write

F<P

instead of MOD

F

<po

One may now verify that the following holds: PROPOSITION 4.

1.

F

K(<p

-+

'Ij!)

-+

(K<p

-+

K'Ij!), and likewise for Bi 2. If

F

<p then

F

K<p and

F

B<p.

This proposition says that by modelling knowledge and belief by a (nor-mal) modallogic, the notions knowledge and belief are closed under logical implication. Furthermore, validities are always known and believed. Al-though perhaps defensible as properties of idealized notions of knowledge and belief, these properties are not always desirable: they give rise to the problem of logical omniscience, to which we will return in a later section.

4 JOHN-JULES eH. MEYER

2.3 The Systems S4, S5, KD45 and Their Models

In the previous section we encountered some first properties of knowledge and belief. The properties obtained thus far, however, are very weak indeed (although in some sense already controversial). In this section we explore further properties, which are obtained by considering subdasses of Kripke models for oUf core language.

We consider now Kripke models with constraints on the accessibility re-lations Rand T. A relation T ~ S

x

S is serial if for all sES there is a

t E S with R(s, t). A relation T ~ S x S is Euclidean if for s, t, u E S it holds that R(s, t) and R(s, u) implies R(t, u).

DEFINITION 5. A Kripke model for S4-knowledge and KD45-belief is a Kripke model M

=

(S, 7f, R, T) such that R is reflexive and transitive, and

T is serial, transitive and Eudidean. We denote the dass of these models by S4/KD45. A Kripke-model for S5-knowledge and KD45-belief is a Kripke model for S4-knowledge and KD45-belief such that R is Euclidean as weIl (and hence an equivalence relation). The dass of these models are denoted S5/KD45.

Regarding these models we obtain a number of additional interesting properties of knowledge and belief.

PROPOSITION 6. 1. S4/KD45

F=

K<p -t <p; 2. S4/KD45

F=

K<p -t KK<p; 3. S4/KD45

F=

-,B~;

4·

S4/KD45

F=

B<p -t BB<p; 5. S4/KD45

F=

-,B<p -t B-,B<p;

additionally for S5/KD45 we have: 6. S5/KD45

F=

-,K<p -t K-,K<p;

Items 1 states that knowledge is true; items 2 and 4 are caIled the 'posi-tive introspection' properties; items 5 and 6 are the 'negative introspection' properties. (The one for knowledge, item 6, is rather controversial, philo-sophically.) Item 3 states that beliefs are consistent.

The deductive system (for knowledge) consisting of propositional logic together with the axioms

(P) any axiomatisation of propositional logic

MODAL EPISTEMIC AND DOXASTIC LOGIC

(T) K<p -t <Pi (4) K<p -t KK<p,

and rules modus ponens (MP) and (NK )

lcp

is called 84. The system 85 is obtained by adding the axiom: (5) --,K<p-tK--,K<p

5

Since axiom (4) is derivable from (K), (T) and (5) in the context of the rules (NK ) and modus ponens, the system 85 can alternatively and more

concisely be defined as the system consisting ofaxioms (K), (T), (5) and rules (N

K)

and (MP). The system KD45 for belief consists of the axioms:

(P) any axiomatisation of propositionallogic (K) B(<p -t 'ljJ) -t (B<p -t B'ljJ);

(D) --,BJ..;

(4) B<p -t BB<p; (5) --,B<p -t B--,B<p

together with the modus ponens rule and the rule (NB)

-ltp

We denote derivability of a theorem <p in a deductive systems 8 by 8 I- <po

Let 84K

+

KD45B stand for the deductive system resulting from

tak-ing the system 84 for the K-modality and the system KD45 for the B-modality, and likewise for the system 85K

+

KD45B. One can now obtain

completeness results along the lines of those in [Hughes and Cresswell, 1968; Hughes and Cresswell, 1984; Chellas, 1980; Meyer and van der Hoek, 1995b], as follows:

THEOREM 7. FOT any <p E Leore we have that

S4jKD45

F

<p -<===} 84K

+

KD45B I- <p and

S5jKD45

F

<p -<===} 85K

+

KD45B I- <p

U ntil now there is no relation between the modalities of knowledge and belief whatsoever. To achieve a meaningful relation between these two n(}-tions we may put a relation between the relan(}-tions Rand T.

DEFINITION 8. A KL(S4jKD45)-model (KL(S5jKD45)-model) is a Kripke model for S4-knowledge (S5-knowledge) and KD45-belief M = (5, 7r, R, T), satisfying the following additional constraints:

6 JOHN-JULES eH. MEYER

• T~R;

• for all s, t, U ES: R(s, t) & T(t, u)

=>

T(s, u)

The classes ofKL(S4jKD45)-models and KL(S5jKD45)-models are denoted

KC(S4jKV45) and KC(S5jKV45), respectively.

The extended deductive system of S4K

+

KD45B (S5K

+

KD45B) to-gether with the axioms

(K ---+B) K<p ---+ B<p

(KB) B<p ---+ KB<p

is denoted KL(S4jKD45) (and KL(S5jKD45), respectively)

The system KL(S5jKD45) was first proposed by Kraus and Lehmann [1986] (although they also included a number of other modalities that we will encounter later). They also noted that adding the 'intuitive' axiom (BK) B<p ---+ BK<p ("an agent believes to know what it believes")

would lead to a collapse of knowledge and belief, since then the formula

K<p t+ B<p becomes derivable. Van der Hoek [1991a; 1993] has investi-gated an alternative system which allows the addition of formula (BK) as an axiom without collapsing knowledge and belief. This system results from KL(S5jKD45) by replacing the negative introspection axiom (5) for knowl-edge by the positive introspection axiom (4) for knowlknowl-edge, and adding the properties • BB<p ---+ B<p • B-,B<p ---+ -,B<p • B<p ---+ BK<p • -,B<p ---+ K -,B<p • -,K<p ---+ B-,K <p

This system is so to speak 'as close as' KL(S5jKD45) together with (BK) as one can get without collapsing K and B, provided one is willing to sac-rifice the negative introspection axiom for knowledge. However, also other solutions to obtaining a sensible logic for knowledge and belief in the spirit of Kraus and Lehmann's logic supporting (BK) are possible. Voorbraak [Voorbraak, 1991; Voorbraak, 1993] argues that negative introspection for knowledge is incompatible with the principle that knowledge implies ('ra-tional') belief, and the latter is dropped rather than the former as in the solution of Van der Hoek. Furthermore, in [Voorbraak, 1993] the notion of rational belief in the sense of justified true belief and its associated logic are discussed extensively, and the latter is identified as a logic in between S4 and S5, namely S4.2, consisting of the axioms and rules of S4 together with the axiom M K <p ---+ KM <po

MODAL EPISTEMIC AND DOXASTIC LOGIC

3 DESCRlBING KNOWLEDGE AND IGNORANCE BY MEANS OF EPISTEMIC LOGIC

7

Epistemie logie is partieularly suited for reasoning ab out what an agent knows and what it doesn't know. Sometimes describing the agent's ignorance is as important as giving a description of its knowledge. By using epistemie logic one can express the an agent doesn't know a formula <p, viz. by the formula --,K <po On the other hand, standard epistemie logie does not allow one to infer what is not known from the facts that are known. E.g. from K p 1\ K q it does not follow that --,K r, for different propositional atoms p, q and r. In general, this would also be undesirable: it might be the case that by asserting Kp 1\ Kq incomplete information about the knowledge of an agent is given, just as in ordinary propositional logic the assertion p 1\ q

might also be partial information of what is actually the case (e.g. p, q and r hold!), so that it would be wrong to conclude --,r from p 1\ q. Of course, if

it would be indieated somehow that p 1\ q is all that is true of the atoms in our language, we would be able to conclude --,r. One way of indieating this is the well-known Closed World Assumption (CWA) [Reiter, 1978].

In our example regarding knowledge we can do something similar and indieate in some way that Kp 1\ Kq, or equivalently K(p 1\ q), is all that is known. In the literature there have been proposed several approaches to doing this. In this chapter we will discuss two of these, viz.s(}-called minimal 85, whieh was in essence firstly proposed by Halpern and Moses [Halpern and Moses, 1984b], and Levesque's 'All I know' logie [Levesque, 1990] which has its roots in Moore's autoepistemie logic (AEL) [Moore, 1984; Moore, 1985a]. Abasie difference between these two approaches is that in the minimal S5 approach the 'minimality of knowing' is obtained on the 'meta-level' by considering a (non-monotonie) entailment relation, while in the 'All I know' logic of Levesque a special operator in the object language of the logic itself is added to express that its argument is all that is known.

3.1 Minimal Epistemic Logic and Honesty

The basic idea behind minimal epistemic logie is the consideration that not all models of a certain theory (or formula) are equally good if one wants to model what exactly is known by the agent. This results in a preference of 'models that contain less knowledge' over those in whieh more is known. Considering such preferences on models falls in with a well-known stream of approaches within the area of nonmonotonie reasoning (see Makinson's chapter in a later volume of this Handbook), where models are ordered according to preference and mostly the minimal models according to some criterion (or ordering) are preferred.

8 JOHN-JULES eH. MEYER Simple S5-models

We have seen that the S5-notion of knowledge corresponds to Kripke models in which the accessibility relations are equivalence relations. Ignoring the belief modalities and their accessibility relations for the purpose of this section, we thus consider models of the form M = (S, 7r, R) where R is an equivalence relation. The dass of these models will be denoted by S5. The standard completeness result for S5 thus becomes

THEOREM 9. For any 'P E

LK,

S5

F

'P {:} S5 f- 'P

However, without loss of generality we can simplify the above models. We call a relation R ~ S x S universal, if R(s,

t)

holds for all s,

t

E S. DEFINITION 10.

1. A reduced S5-model is a Kripke-model M

=

(S, 7r, R) where R is universal. We sometimes denote such a model simply by M

=

(S,7r).

This dass of models is denoted RS5.

2. A simple S5-model is a Kripke-model M

= (S),

where the set S is a set of dassical truth assignment functions P -+

{tt,ff}.

This dass of models is denoted SS5. The dass of simple S5-models of a set formulas q, ~

LK

is denoted by

Modss

5(q,).

We now have also the following completeness results: THEOREM 11. For any 'P E

LK,

1. RS5

F

'P {:} S5 f-'P

2. SS5

F

'P {:} S5 f- 'P

Simple S5 models are basically just sets (offunctions), and this gives us a simple idea of defining an ordering on such models, viz. simply taking the set-indusion ordering.

DEFINITION 12. Simple S5-models are ordered by: let

MI

=

(SI) and

M 2 = (S2) be two simple S5-models. Then

MI

-< M2 Hf S2

c

Sr, where

c

stands for strict set-indusion. The subset of -<-minimal models in a set MOD of models is denoted Min-<.(MOD).

One can show that for simple S5-models this ordering corresponds to an ordering on the knowledge contained in these models.

DEFINITION 13. Let M

=

(S) be a simple S5-model. Then the knowledge ofmodel M if defined as: Knowl(M)

=

{'P

I

M

F

'P}.

MODAL EPISTEMIC AND DOXASTIC LOGIC 9

PROPOSITION 14. Let

Mi

=

(Si)

and M 2 = (S2) be two simple 85-models. Then:

Thus, the less objective knowledge is contained in a model, the more preferred it iso So if we want to minimalize this knowledge we should con-centrate on the -<-minimal models.

Minimal

85

N ow we are ready to define an entailment of the form cP

r-

H M 'Ij; expressing that 'Ij; can be concluded on the basis of only knowing the (finite) set of formulas cP.

DEFINITION 15.

cP r-HM 'Ij; {::} IM all

M

E Min-«Modss5

(cP»,M

F

'Ij;

The logic thus obtained is often referred to as minimal 85.

REMARK 16. In Halpern and Moses' original approach [Halpern and Moses, 1984b] it is required that the premise <p determines exactly one minimal (or

preferred) model. In this case the formula <p is called honest, and only for

honest formulas <p the entailment relation <p

r-

H M 'Ij; is defined. In the more general approach above, which is also taken in e.g. [Lin and Shoham, 1992; Schwarz and Truszczynski, 1994; Engelfriet, 1996], we also allow multiple minimal models and consider as consequences the formulas that hold in all of these. (In nonmonotonic reasoning this is called the skeptical ap-proach (cf. [Brewka, 1991].) The use of a preference relation on models to select models (viz. the minimal ones) makes minimal 85 a so-called

prel-erentiallogic in the sense of Shoham [Shoham, 1987], which is nowadays a

prominent way to describe nonmonotonie logics, cf. Makinsons' chapter on Nonmonotonic Logic in a later volume ofthis Handbook. That minimal 85 is nonmonotonic can easily be seen from the examples below.

EXAMPLE 17. Let p, q and r be distinct atoms. 1. p r-HM KpA ---,Kq 2. pAq 'rfHMKpA---,Kq 3. pAqr-HMKpAKq 4. pVq r-HM K(pVq) A---,KpA---,Kq 5. KpV Kq r-HM ---,(KpAKq) 6. ---,Kp -t q r-HM KpV Kq

10 JOHN-JULES eH. MEYER

Note that the fifth example above involves a dis honest premiss, so that this example falls outside the scope üf Halpern and Moses' approach ([Halpern and Moses, 19S4b]). Here it is important to consider a11 minimal models (the skeptical approach).

3.2 AEL and (All I Know'

Autoepistemic logic (AEL) and its derivative 'The Logic of All I Know' take a slightly different approach to the concept of 'only knowing'. The minimization of knowledge in AEL is done differently, and in the Logic of All I Know an explicit operator 0 with the meaning 'only knowing' appears in the object language rather than leaving this aspect implicit on the (meta-) inferentiallevel as in minimal 85.

AEL: the basic definitions

AEL is built on the doxastic fragment CB of our core language. (Actually,

mostly in the AEL literature the symbol L is used for belief, but we will use B für this, as before.) However, the language is given a different semantics, which appears to be non-modal. AEL along with its semantics was first developed by Moore [1984; 1985al. The results along with the proofs in this subsection can be found there.

DEFINITION 18. An AE-model is a pair [w, Xl, where w is a propositional valuation P ~

{tt,!!}

and X<;;;; CB. Formulas in CB are now interpreted by means of AE-models as fo11ows:

• [w,Xl

F

p iff w(p)

=

tt

for PEP;

• [w,

Xl

F ''P

iff not

[w,

Xl

F 'P;

• [w, Xl

F 'P

t\ 1jJ iff [w, Xl

F 'P

and [w, Xl

F

1jJ; • [w, Xl

F 'P

V 1jJ iff [w, Xl

F'P

or [w, Xl

F

1jJ; • [w,Xl

F

B'P iff

'P

E X;

• [w,Xl

F

Nip iff 'ip ~ X

An AE-model of the form [., Xl that satisfies a formula 'P is called an X-interpretation of

'P.

A formula is caUed AE-satisfiable if it has an AE-model. A set <P <;;;; CB AE-implies a formula

'1/),

denoted <P FAE 1jJ, if aU AE-models of<p (Le. all AE-models

[w,

Yl

such that

[w,

Yl

F 'P,

for a11

'P

E <p) are AE-models of 1jJ. A set <P <;;;; CB X-implies a formula 1jJ, denoted <P

FX

1jJ,

if a11 X-interpretations of<p (Le. all X-interpretations [w,Xl such that

MODAL EPISTEMIC AND DOXASTIC LOGIC 11

The intuition behind an AE-model

[w,X]

is that

w

describes the world the agent lives in, while X represents the agent's beliefs. AE-implication in general is not a very interesting concept, witness the following proposition. PROPOSITION 19.

ep

F

AE 1/J {::::::::} ep Fpl 1/J

where Fpl denotes the ordinary entailment in classical proposition al logic.

However,

F

x is an interesting notion with which can express a kind of 'minimal' knowledge (or rather belief) that is contained in some premiss set. To this end we define the notion of an AE-extension (in the literature often called AE-expansion).

DEFINITION 20. Let ep ~ 'cB be given. A set X ~ 'cB is called an

AE-extension of ep Hf X

=

{<pIep FX <p}.

An AE-extension of a set ep of formulas can be viewed as a set of conse-quences that is (minimally) contained in ep. In general, a set ep may have one or multiple AE-extensions, or none at all. Also, in general we do not have something like epl ~ ep2 implying that Xl ~ X2 for AE-extensions

Xl of epl and X2 of ep2. This means that in the notion of an AE-extension

some kind of nonmonotonicity is involved. (See Makinson's chapter in a later volume of this Handbook.)

AE-extensions enjoy a number of ni ce properties, of which we mention the following. We use the abbreviations Bep = {B<p I <p E ep} and -,B(fi

=

{-,B<p I <p ~ ep}, for ep ~ 'cB. Furthermore, let Conpl(ep) stand for the (classical) propositional closure ofthe set ep.

THEOREM 21. ep is an AE-extension 01 lJ! iff ep

=

Conpl(lJ! U Bep U -,B(fi) Universal 85-models

Interestingly, AE-extensions can be described by special S5-models, thus establishing a relation between AEL and 'standard' epistemic logic after all.

DEFINITION 22. Let M = (S) be a simple S5-model. The universal

85-model M derived /rom M is a Kripke model M

=

(U, R), where U is the uni verse of all classical propositional valuation functions, i.e. U

=

{1fI1f :

p -+ {tt, If}}, and R is the accessibility relation U x S, i.e. (s, t) E R Hf tE S.

We can now also interpret our language 'cB on universal models in the obvious way:

Let M

=

(U, R) be the universal model derived from the simple S5-model M = (S). For s E U, M, s

F

B<p Hf M, t

F

<p for all t with (s, t) E R, i.e. for all t E

S.

The interpretation regarding the other clauses of the language are as usual.

12 JOHN-JULES eH. MEYER

One can now prove ([Vreeswijk, 1991; Meyer and van der Hoek, 1995bl) that:

THEOREM 23. Let lvi

=

(S) be a simple S5-model, and let M be the universal S5-model derived /rom M. Then: Knowl(M) is an AE-extension of cP ijJ S

=

{8

I

M, 8

F

cp}

By enriching the modal language by a modal operator B* (and dual operator N*) referring in a universal S5-model M

=

(U, R) to the states of

U

that are not in S, we can formulate further properties relating to AE-extensions. In this way we get what we call an enriched universal S5-model. DEFINITION 24. Let M

=

(U,

R) be a universal S5-model derived from simple S5-model M

=

(S). The enriched universal model derived from M is a structure (U, R, R*), where R*

=

U x (U\S), i.e. (8, t) E R* iff t

rt

S. To emphasize the role of the set S, we may also write M

=

(U, S, S*), where S*

=

U\S. The dass of (enriched) universal S5-models is denoted by U.

We extend the language

C

B with the operator B* to refer to the states

outside S, and denote this language by CB,B*.

DEFINITION 25. Let M

=

(U, R, R*) the enriched universal S5-model de-rived from

M.

Then, for

s

EU,

• M, s

F

B*rp iff M, t

F

rp for all t with (s, t) E R*, i.e. for all t

rt

S. The other clauses remain the same.

Now we can state the following result [Vreeswijk, 1991]:

THEOREM 26. Let cP ~ CB. Then cP has an AE-extension iff Bcp !\B*-,cp

is satisfiable in an (enriched) universal S5-model.

If one calls an enriched universal S5-model M

=

(U, S, S*) such that all sES satisfy cP and all 8 E S* don't, a preferred belief model of CP, we can also put this in other words, as e.g. in [Schwarz and Truszczyriski, 1994]: cP has an AE-extension if it has a preferred belief model. This view renders AEL a preferentiallogic in the style of Shoham [1987] again.

The logic of 'all I know'

Once we have extended the language of AEL with the B* operator, we can even go a step further and also introduce an operator

0

denoting 'only knowing' as done first by Levesque [1990]. Here we can simply introduce Orp as an abbreviation for Brp !\ B*-,rp.

It is easy to check that this gives us the following nice interpretation for the 0 operator:

PROPOSITION 27. Let M

=

(U, S, S*) the enriched universal S5-model derived /rom M. Then, for 8 EU,

MODAL EPISTEMIC AND DOXASTIC LOGIC 13

• M, s

F=

O<p ijj, for alt t, t E S ~ M, t

F=

<p.

With Theorem 26 we thus obtain a direct link between AEL and the 'only knowing' operator:

COROLLARY 28. Let <p E LE. Then <p has an AE-extension ijj O<p is satisfiable in an (enriched) universal S5-model.

Moreover, by enriching the language even further we can obtain a com-plete axiomatisation of the logic of only knowing: we extend the language with the operators D and

<>.

The operators are interpreted on a univer-sal S5-model as the necessity and possibility operators, respectively, on the whole uni verse U of such models:

DEFINITION 29. Let M = (U, S, S*) the enriched universal S5-model de-rived from

M.

Then, for s EU,

• M, s

F=

D<p Hf M, t

F=

<p for all t E U;

• M, s

F=

<><p Hf M, t

F=

<p for some t EU.

The logic of only knowing is captured by the system OK.

DEFINITION 30. The system OK consists of the following axioms: 1. All (or sufficiently many) propositional tautologies

2. D(<p -+ 'IjJ) -+ (D<p -+ D'IjJ) 3. D<p -+ <p

4. ,D<p -+ D,D<p

5. <><p ++ ,D,<p

6. <><p, for every satisfiable objective formula <p

7. D<p ++ (B<p 1\ B*<p)

8. <p -+ D<p, for every basic subjective1 _{formula <p} 9. B(<p -+ 'IjJ) -+ (B<p -+ B'IjJ)

10. B*(<p -+ 'IjJ) -+ (B*<p -+ B*'IjJ)

11. <p -+ B<p, for every basic subjective formula <p 12. <p -+ B*<p, for every basic subjective formula <p

13. ,B~ -+ (B<p -+ <p), for every basic subjective formula <p

1 A basic subjective formula is a formula of the form ®tp, where ® stands for olle of hte modalities 0,0, B, B* ,N and N* (thus excluding 0).

14 JOHN-JULES eH. MEYER

14. ,B* ~

-+

(B*<p

-+

<p), for every basic subjective formula <p

15. ,(N<p 1\ N*<p), for every uniquely satisfiable objective formula <p

16. D<p B (B<p 1\ B*,<p)

and the rules: modus ponens (MP) and (ND)

J'cp

THEOREM 31. For any <p we have that U

F

<p -<===} OK I-<p

Proof. Cf. [Levesque, 1990; Vreeswijk, 1991; Boutilier, 1994a; Boutilier,

1994b]. •

REMARK 32. The logic OK is very much related to the (propositional frag-ment of the) logic of 'All I know' as originally proposed by Levesque [1990]. A minor difIerence is that instead of our axiom 7, Levesque employs as an ax-iom B<p

-+

,B*<p, for any objective <p that is falsifiable. But this axiom fol-lows from axiom 7, by the observation that (h<p {:} ,O<p {:} ,(B<pI\B*<p) {:} (B<p

-+

,B*<p). Another difIerence is that Levesque takes both Band B*

to be weak-S5- (or KD45-) modalities, while we assume these to be K45-modalities, which also explain the extra antecedents regarding consistency in our axioms 13 and 14. A more important difIerence is the inclusion in the logic OK of the axiom 15, which essentially states that in a universal model Rn R*

= 0.

It can be shown (cf. [Humberstone, 1983]) that this property cannot be expressed by just using the modal language without reference to additional means like referring to uniquely satisfiable objective formulas. (In other words, this property cannot be expressed on the level of frames, but needs properties of the valuation function on models.) The interested reader is referred to Humberstone [1983], where the issue of inac-cessible worlds is studied in a more general setting. There it is shown that by considering two modalities 01 and O2 each satisfying the axioms and rules of system K, together with the schema

where W stands for strings of any (including zero) length of occurrences of the duals <>1 and <>2 of 01 and O2 , respectively, and S stands for strings of any (including zero) length of occurrences of 01 and O2 , one obtains a complete logic with respect to Kripke models with two accessibility relations that are each other's complement. Finally, we remark that Boutilier [1994a; 1994b] has shown that in the context oftwo modalities 01 and O2 , of which the former satisfies the system S4 and the latter that of K it is sufficient

MODAL EPISTEMIC AND DOXASTIC LOGIC 15 to add the following single 'pseudo-instance' (2) of (1) to obtain a complete system with respect to models with two accessibility relations of which one is reflexive and transitive and the other is its complement:

where Dip ++ (01 ip 1\ D2ip) and Oip is its dual -,o-'ip. In the context of our universal S5-models, (2) is equivalent with the right-to-left half of our axiom 7, as can be seen by the following derivations:

Assuming (2), we derive: Bip 1\ B*ip

*S5

O(Bip 1\ B*ip)

*

D(ip V ip)

*

Dip, while assuming 7 we derive: O(Bip 1\ B*'1j;)

*

O(B(ip V '1j;) 1\ B*(ip V

'1j;))

*

OD(ip V '1j;)

*S5

D(ip V '1j;). Although proposed for entirely different purposes, Boutilier's logic CT40 [Boutilier, 1994a; Boutilier, 1994b] can thus be regarded as a kind of S4-variant of Levesque's logic of 'All I know'.

As we remarked already, although Halpern and Moses [1984b] and Levesque [1990] had more or less the same philosophical considerations for proposing their logics, they have turned out to be quite different. As Schwarz and Truszczynski [1994] remark: AEL is more adequate for de-scribing belief sets (although the name autoepistemic suggest otherwise), while minimal S5 seems to be more appropriate for dealing with knowledge sets of an agent, which is illustrated succinctly by the following example:

The set {Kp}, for propositional atom p, has a minimal simple S5-model, whereas its analogue in AEL {Bp} has no AE-extension. The former de-scribes the knowledge of an agent containing Kp as well as p, whereas the latter cannot really be the basis of the beliefs of an agent, since the belief Bp is not really justified if one only believes Bp and lacks the property Bp -+ p, as is the case with belief.

3.3 Schwarz and Tr-uszczynski's Logic

0/

Minimal Knowledge

Having noticed the differences in minimal S5 and AEL (as well as some other related approaches such as that of Lin and Shoham [1992]), Schwarz and Truszczynski have proposed yet another logic for minimal knowledge.

In particular, they wanted to devise a logic for minimizing knowledge (as Halpern and Moses wanted) rather than belief (as AEL turned out to be), while retaining one of the in their opinion nice properties of AEL, namely conservativity with respect to introducing explicit definitions: the introduc-tion of some formulas of the form q ++ ip, for some atom not occurring in formula ip, should not affect the theorems not involving q. This property does not hold for Halpern and Moses' minimal S5, as is illustrated by the following example.

Consider <P = {p V -,p}, where p is a propositional atom. Then <P

r--

H M

-,Kp. Now consider <P'

=

<P U {q ++ Kp}, for some atom q f::- p. The set <P' has two minimal simple S5-models: one consisting of all truth assignment

16 JOTIN-JULES eH. MEYER

functions that assign ff to q, and one that contains all truth assignment functions that assign tt to both p and q. Thus, in the sceptical approach, considering all minimal models, we obtain that <1>' ffHM ,Kp.

The approach of Schwarz and Truszczynski is further based on two key conditions:

• The 'epistemic state' of a possible world is determined not only by the objective facts that hold in that world, but also by what is known / believed in this world. Extending a collection of possible worlds should preserve their epistemic states.

• An agent regards its beliefs as knowledge, that is, it is unaware of any possible world, compatible with its initial assumptions, in which some of its beliefs no longer hold.

Schwan,; and Truszcllynski now first define abi-modal logic, involving both the not ions of knowledge (K) and belief (B).

DEFINITION 33. An ST-model is a structure M

=

(S, 7f, T), where S is a set of possible worlds; 7f is a truth assignment function per possible world; and T ~ S such that T

f=.

0.

Formulas are interpreted on ST-models as follows:

DEFINITION 34. Let M

=

(S,7f,T) be aST-model and sES. Then:

• M,

s

FST

K<p Hf

M,

t

FST

<p for all t ES, if s

f/-

T; • M, S

FS'I'

K<p iff M,

t

FST

<p for all

t

E T, if SET;

• M,

s

FS'I'

B<p iff

M,

t

FST

<p for all tE T;

• The rest of the clauses as usual.

REMARK 35. Note the slightly nonstandard interpretation of the knowl-edge operator K: it depends on the world of evaluation how it is evaluated: for worlds within the 'belief' cluster T it is evaluated the same as belief, while for the remaining worlds it is evaluated as usual taking the whole set of worlds into account.

It appears that this bimodallogic can be reduced to a unimodal one since the belief operator can be expressed in terms of knowledge:

PROPOSITION 36. For any ST-model M

=

(S, 7f, T) and any formula <p it holds that: M

F

ST B<p B ,K ,K <p.

This means that one can just use the K-operator as primitive and B<p

as an abbreviation for ,K ,K<p. Conceptually, this me ans that the belief operator expresses in this logic something like 'considering it possible to know'.

MODAL EPISTEMIC AND DOXASTIC LOGIC 17 Note that any reduced S5-model

M

=

(S,7I") can be viewed as an ST-model (S,7I",S). Moreover, in these models it holds that K<p {:} B<p {:}

-,K -,K<p, Le. -,K<p ++ K -,K<p, which gives us just the negative introspection property for knowledge again!

Now the crucial definition of the adapted preference relation on ST-models, as proposed in [Schwarz and Truszczynski, 1994] is the following: DEFINITION 37.

• Let

N

=

(S,7I"N,T) and M

=

(T,7I"M, V) be two ST-models such that 7I"N(t)

=

7I"M(t) for all T E T. We say that

N

extends

M

if for some objective formula 'IjJ it holds that

M

FST 'IjJ, but for some tE S\T,

N,t

'F

'IjJ.

• We define that

N

is preferred over

M,

denoted

N

-<ST

M,

Hf

N

extends

M.

• Let M

=

(S,7I") be a reduced S5-model. We say that M is a preferred ST-model of<p if

M

FST <P, and for every ST-model

N

such that

N

-<ST

M,

it holds that

N

'F

<P.

• We may define an entailment relation r-ST as follows:

<P r-ST 'IjJ {:} fM alt preferred ST - models M of<p M FST 'IjJ

EXAMPLE 38. Let p and q be distinct atoms. • Kp r-ST p /\ Kp /\ -,Kq

• -,Kp --t q r-ST q

Note the difference between the latter item and with Example 17, last item. Whereas the formula -,Kp --t q in minimal S5 has two minimal models, viz. one in which all states satisfy q and one in which all states satisfy p, only the former is a preferred ST-model. So he re we see that the ST-approach is more in line with AEL, whereas in the first item above it is more in line with minimal S5. So we observe that this approach has some features of both. It can be shown that the resulting logic meets the desidemta above. Moreover, it can be viewed as a nonmonotonie modal logic (cf. [Marek and Truszczynski, 1993]). More precisely it is related to the logic 84F, which is the logic 84 together with the axiom

(F) (M<p /\ M K 'IjJ) --t K(M<p V 'IjJ) in the following sense:

A reduced S5-model M is a preferred ST-model for <P Hf the theory of M is an S4F-expansion W of <P, Le. W satisfies the equation W = Cons4F(<P U { -,K 'IjJ

I

'IjJ

rt

w}) (see [Schwarz and Truszczynski, 1994]).

18 JOHN-JULES eH. MEYER

4 THE LOGICAL OMNISCIENCE PROBLEM AND ALTERKATIVE APPROACHES

4.1 The Problem of Logical Omniscience

The modal approach to knowledge and belief, as based on (normal) Kripke models gives us a number of properties, 'for free' so to speak, whieh are not always eonsidered desirable. So, for instanee, the following assertions are valid in any normal modal logie, viz. a logie that satisfies the axioms (P), (K) and the rules (MP) and (NO) of Seetion 2.3. We use the modal operator D as generie for either knowledge or belief.

• F Dip t\ D(ip -+ 'I,b) -+ D'I,b LOI

• F ip::::}F Dip L02

• F ip -+ 'I,b ::::}F Dip -+ D'I,b L03

• F ip ++ 'I,b ::::}F Dip ++ D'I,b L04

• F (Dip t\ D'I,b) -+ D(ip t\ 'I,b) L05

• F Dip -+ D (ip V 'I,b) L06

• F ,(Dip t\ D,ip) L07

These properties are inherent to the 'standard' possible world semanties as we have treated. One may view this as a kind of idealization of the eoneepts of knowledge and belief of an agent / knower, whieh might be eonsidered eonvenient. On the other hand, when regarding more praetieal situations, such as e.g. an agent with limited reasoning resourees, like some partieular robot, and like indeed we humans all are in some way or another, it seems unrealistie to assume that this agent believes (knows) all the eon-sequenees of its beliefs (the things it knows, respeetively), as is stated in property L03 above. Maybe also the other properties above are not very realistie in this ease. This is ealled the problem

0/

logical omniscience in the literature. Similar problems oeeur in other modal logies. For instanee, the logieal omniscience problem is related to some of the so-called paradoxes in

deontic logic [Jones and Carmo, 2002].

Of course, if we would want to deny all of the above properties we are faced with the problem whether there is any logieal prineiple left that we want to retain, apart from perhaps the introspection property K ip -+ K K ip.

This presents us with a kind of paradox: in the extreme the knowledge and belief operators do not respeet any logical prineiple. For instanee, when denying even the prineiple of replacing logieal equivalents, we should not be surprised that a modal logie, or any logie whatsoever for that matter, eannot help us to describe the knowledge and beliefs of an agent. Then it is

MODAL EPISTEMIC AND DOXASTIC LOGIC 19 perhaps better to resort to cognitive science right away rather than trying to employ logic.

Another way of looking at this, is that the properties above specify to what extent we may consider our agents as ideal reasoners/knowers/believers. Instead of denying all the properties we could focus on a few of them which we want to deny while retaining the other ones, where it depends on the applications which properties are denied. As will be clear from the above, if one wants to do this, one has to resort to 'nonstandard' approaches, since the standard possible world semantics leaves no room for doing this. In the following subsection we will explore some of these nonstandard models.

4.2 'Nonstandard' Approaches

Generally, we can distinguish between two approaches to overcome (in-stances of) logical omniscience, a syntactical one and a semantical one. In the former approach one uses syntactical elements in the model to avoid logical omniscience, whereas in the latter one uses 'non-standard' models in the sense that the possible world semantics is changed as compared to normal Kripke-style semantics, e.g. by treating some worlds difIerently or considering accessibility relations between sets

01

worlds rather than worlds. Admittedly, the line between the syntactical and the semantical approaches cannot be drawn very strictly since the inclusion of a syntax-based function in an otherwise standard possible world model has indeed both a syntactical and a semantical flavour.

The 'purely' syntactical approach

Of course, the most radical approach of avoiding logical omniscience, pro-posed in [Fagin et al., 1995], would be to consider models of the form (S, a),

where S is a set of possible worlds and a : S -t (CE -t

{tt, If})

is a func-tion that assigns truth per world to every proposifunc-tion. In order to keep the standard logical behaviour with respect to the classical operators outside the scope of modal operators, we stipulate that the function a respects the clauses:

• a(s)(--,ip) = tt ifI a(s)(ip) =

11;

• a(s)(ip 1\ 'Ij;)

=

tt ifI a(s)(ip)

=

tt & a(s)('Ij;)

=

tt; • and similarly for the other classical connectives.

Clearly with respect to these models alllogical omniscience can be avoided, since any formula L01- L07 can be made false at some state in some model. But the price is high. One may wonder whether this is a logic of belief at all. In any case, as Fagin et al. state, this way of modelling beliefs does

20 JOHN-JULES eH. MEYER

not explain anything about beliefs: it is a representation method for belief rather than true modelling.

Impossible world semantics

Perhaps the next to most radical form of using syntactical elements is called

'impossible world' or Rantala semantics [Rantala, 1982]. Here we distinguish between standard possible worlds where every formula is evaluated as usual, but we also consider so-called impossible worlds, where the evaluation of formulas is completely free! Formally, we define Rantala models as foHows. DEFINITION 39. A Rantala model is a structure M ofthe form (S,a, T, S*),

where

• S is a non-empty set, the set of (possible and impossible) worlds;

• S* ~ S is the set of impossible worlds;

• a : (S\S*

-+

(P

-+

{tt,

if}»

u

(S*

-+

(CE

-+

{tt,

if}»,

a nmction assigning truth to atoms on possible worlds, and truth to arbitrary

formulas on impossible worlds; is a truth assignment function to the atoms per state;

• T

~

S

x

S

is the belief accessibility relation.

The interpretation of formulas on Rantala models is for possible worlds

s E S\S* the same as in standard Kripke models, but for an impossible world s E S* it is given directly by a:

M,s

F<P

Hf a(s)(<p)

=

tt,

if sE S*.

Impossible world semantics resolves all logical omniscience as weH: also in this semantics none of LO 1-L07 are valid.

Sieve semantics

Instead of going so far as introducing impossible worlds into the frame-work on can also stick to standard possible worlds and introduce some non-standard element by means of a special (syntax-based) function. This is called sieve semanties. As the term suggests, it sieves the true interpreta-tions of belief formulas (which are determined in a standard way) by means of an operator that constraints the syntax of the formulas to be believed. Since this operator can be loosely interpreted as 'being aware' of the formu-las concerned, this is also called the awareness approach, and the operator is usually denoted by A; A<p expressing that the agent is aware of the formula