Handbook of the History of Logic. Volume 07: Logic and the Modalities in the Twentieth Century

733 Pages·2006·5.54 MB·English

by  Dov M. Gabbay

Handbook of the History of Logic
Volume 7
Logic and the Modalities
in the Twentieth Century
Handbook of the History of Logic
Volume 7
Logic and the Modalities
in the Twentieth Century
ELSEVIER
NORTH
HOLLAND
Handbook of the History of Logic
Volume 7
Logic and the Modalities
in the Twentieth Century
Edited by
Dov M. Gabbay
Department of Computer Science
King’s College London
Strand, London, WC2R 2LS, UK
and
John Woods
Philosophy Department
University of British Columbia
Vancouver, BC Canada, V6T 1Z1
and
Department of Computer Science
King’s College London
Strand, London, WC2R 2LS, UK
and
Department of Philosophy
University of Lethbridge
Lethbridge, Canada
ELSEVIER
NORTH
HOLLAND
Amsterdam - Boston - Heidelberg - London - New York - Oxford - Paris
San Diego - San Francisco - Singapore - Sydney - Tokyo
Elsevier
Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands
The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK
First edition 2006
Copyright © 2006 Elsevier B.V. All rights reserved
No part of this publication may be reproduced, stored in a retrieval system
or transmitted in any form or by any means electronic, mechanical, photocopying,
recording or otherwise without the prior written permission of the publisher
Permissions may be sought directly from Elsevier’s Science & Technology Rights
Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333;
email: [email protected]. Alternatively you can submit your request online by
visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting
Obtaining permission to use Elsevier material
Notice
No responsibility is assumed by the publisher for any injury and/or damage to persons
or property as a matter of products liability, negligence or otherwise, or from any use
or operation of any methods, products, instructions or ideas contained in the material
herein. Because of rapid advances in the medical sciences, in particular, independent
verification of diagnoses and drug dosages should be made
Library of Congress Cataloging-in-Publication Data
A catalog record for this book is available from the Library of Congress
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
ISBN-13: 978-0-444-51622-0
ISBN-10: 0-444-51622-0
ISBN-13: 978-0-444-51596-4 (set)
ISBN-10: 0-444-51596-8 (set)
For information on all Elsevier publications
visit our website at books.elsevier.com
Printed and bound in The Netherlands
06 07 08 09 10 10 9 8 7 6 5 4 3 2 1
CONTENTS
Preface vii
List of Contributors xi
Mathematical Modal Logic: A View of its Evolution 1
Rob Goldblatt
Epistemic Logic 99
Paul Gochet and Pascal Gribomont
Deontic Logic 197
Paul McNamara
Relevant and Substructural Logics 289
Greg Restall
A. N. Prior’s Logic 399
Peter Øhrstrøm and Per F. V. Hasle
Modern Temporal Logic: The Philosophical Background 447
Peter Øhrstrøm and Per F. V. Hasle
The Gamut of Dynamic Logics 499
Jan van Eijck and Martin Stokhof
Situation Theory and Situation Semantics 601
Keith Devlin
Dialogue Logic 665
Erik Krabbe
Index 705
This Page is Intentionally Left Blank
PREFACE
In its traditional sense, a modal logic is one whose logical vocabulary contains
the modal expressions “possibly”, “necessarily” and “contingently”, construed as
sentence operators. If the first half of the twentieth century can lay fair claim
to having produced the deep and definitive accounts of classical logic, perhaps
the dominant achievement in the next quarter century was the attainment of a
firm semantic grip on a hefty plurality of modal systems, marked by the seminal
works of Hintikka, Kanger and Kripke. The semanticizing of modal sentences
— apart from the importance intrinsic to such an achievement — opened up an
importanttensionbetweenmodalandclassicallogics. Perhapsthemostsignificant
difference is that, whereas classical systems are extensional, modal setups are
intensional, a happenstance which various philosophers of logic have greeted with
suspicion and — in some cases — incredulity. Some of the skeptics — Quine and
Harman are two — raised doubts about whether a modal system could have the
bona fides of a genuine logic. This notwithstanding, the great burst of energy in
the modal research programmes since the 1950s has proved irresistible, and the
central semantic idea of accessibility relations on alternative possible worlds has
had a philosophical influence well beyond the confines of logic, especially in the
philosophy of language.
One of the byproducts of the modal groundswell is that there are a great many
more interpreted systems of the modals “possibly” and “necessarily” than there
are different meanings of these terms in ordinary English. It is easy to see that
they are ambiguous in English, that “possibly” encompasses the quite different
senses of logical, physical, causal, and practical possibility (ditto “necessarily”).
Butthesheerpluralityoflogicalsystemsinwhichthesetermsarecentrallyatissue
greatly exceeds this rather modest number of ordinary-language meanings. It is
notwhollyclearhowtounderstandthisproliferation. Onepossibilityisthatlogic
has a greater capacity to identify different concepts of possibility than do native
speakers of languages such as English. It is also possible that the multiplication
of heretofore unrecognized concepts of possibility is more a matter of the free
creation of the theoretical logician. Whatever is to be said for these and other
options, it is safe to say that, in having taken the modal turn, logic took on a
more experimental character than was evident in the classical heyday.1 Here, too,
1Intimations of the experimental proclivities of modal logicians are evident in Aristotle’s
modallogic,ofwhichthereareuptofivedistincttreatmentsoflogicalnecessity. Then,too,the
streamofsystemsproducedbyC.I.Lewisfrom1912intothe1930sencompassesvastlydifferent
axiomatizations.
viii Preface
it is not entirely clear what to make of this. Of the golden age of classical logic
it can be said with some confidence that logicians took themselves to be doing
one of two things. Taking the implication relation as an example, either they
were formalizing the pre-existing concept of implication or they were originating
a concept designed to facilitate some larger purpose, such as the construction
of logically precise languages adequate for science or capable of supporting the
reductiveburdensoflogicism. Partoftheanswermaybethatthemodeltheoretic
apparatus needed for the interpretation of modal systems is more complex, and
admits of greater recombinations of its elements, than do the standard models
of classical logic. Accordingly, it may be more natural for the modal theorist to
reconfigure a possible worlds semantics and wait for the kind of, e.g., implication
relation it embeds to “fall out”.
We should take care not to over-press the contrast between analyzing pre-
existing concepts and fashioning new ones. The distinction is present in Kant’s
pre-criticalwritings,andpersistsintheworksofhismaturity. Kantsawanalysisas
making concepts clear, a job for philosophy. Synthesis was the business of making
clear concepts, a task that falls to the mathematician. Since logic’s great classical
interlude arose from the contributions of philosophers and mathematicians alike,
we cannot say, especially in the aftermath of the paradox that dethroned intu-
itive (or “analytic”) set theory, that classical logic is synthesis-free. Far from it.
Whatismore,apartfromthelocaldisputeswithinclassicallogicitself,therewere
early rivals, such as intuitionism. Even so, classical logic has not been especially
pluralistic, whereas modal logic is vigorously so. It is moreover a rather pacific
andnon-antagonisticpluralism,whichattestsfurthertoitsreadinesstoviewlogic
as the exploration of mathematically interesting languages and model theoretic
structures, with a focus that is a good deal less analytical or philosophical than
most classical variations on classical semantics.
Italsobearsonthisissuethatthesemanticsofsomeofthe20thcentury’searliest
axiomsystems—Lewis’S2andS3forexample—muststretchthemselvesbeyond
ordinary recognition in order to keep up with the axioms. S2 and S3 cannot be
semanticized in a normal worlds approach; so nonnormal worlds were postulated
(more experimentation still), giving inadvertent anticipation of somewhat later
paraconsistent developments. A further case in point is the attempt by relevance
logicianstoimposerelevanceconstraintsontheimplicationrelation,soastoevade
the classical theorem that everything whatever is implied by an inconsistency. In
most relevant approaches, the disjunctive syllogism rule is demoted from a valid
to a merely admissible rule. On its face a rather slight adjustment, this actually
strips these logics of the truth-functional character of their classical predecessors.
More intensionality still.
Anattractionofthetraditionalalethicmodalsistheeasewithwhichtheybear
new interpretations in the breakthrough work of Hintikka, von Wright and others
in epistemic and deontic logic. These were significant developments twice-over.
On the one hand, the new logics of knowledge and belief, and of obligation and
permission, were able to retain much of the syntactic and semantic machinery of
Preface ix
theiralethicpredecessors,showingthatalltheselogicsaretoadegreevariationsof
eachother. Ontheotherhand,however,thelogicsofknowledgeandobligationhad
taken yet another step away from classical logic. Not only are these newer logics
intensional and more experimentally oriented than their classical vis-a`-vis, but
there is now the looming presence of agents operating in time. We say “looming”
rather than “overt” inasmuch as neither agents nor times are much developed, if
at all, in the semantics of these particular systems.
Theephemeralpresenceofagentsandtimesisimportantinanotherrespect. It
indicates that logic was developing in ways that would satisfy a broader interpre-
tationoftheadjective“modal”. IfoneweretoconsultTheNewOxfordDictionary
of English, corrected reprint, 2001, it would be seen that in the entry for modal
logic the first reference is not to a logic of possibility and necessity, but rather to
alogicinwhichsentencesaresubjectto“somequalification”. If, then, wewereto
accept the trichotomy of the basic modes of language introduced by the linguist
C.W. Morris — the trichotomony between syntax, semantics and pragmatics —
we would be reminded that a pragmatic approach to language is one that takes
expressly into account the roles of language-users and the contexts in which they
operate. By these lights, the developments in the 1950s and 1960s within the
epistemic and deontic adaptations of alethic modal logic mark the transition of
logic from a purely syntacto-semantic enterprise, to a “Morrisean” enterprise in
which agents, times and situations have a load-bearing role. This was the prag-
matic turn in logic.2 In the broad sense of “modal”, a pragmatic logic counts
as modal, a happenstance that the Editors have allowed themselves to be guided
by in organizing the present volume. In addition to chapters on the traditional
modal logics, their epistemic and deontic variations and relevant logic, there are
chapters on systems in which the times of utterance are taken note of, in which
temporal change is tracked, in which an utterer’s situation is taken into account,
in which interpersonal utterance is acknowledged, and in which agents compete
with one another in the furtherance of their interests. In each case, the sentences
ofthelogicaremodifiedbytheseotherfactors—time,change,agents,situations,
dialogue roles, and procedural strategies. Modal logics in the broad sense reflect
another change in logic’s conception of itself. In the classical heyday, logicians
were preoccupied with the analysis of properties (such as implication and logical
truth) of abstractly linguistic constructions or of linguistic artifacts in relation to
abstractly set theoretic structures. If such logicians gave any thought to the ins-
and-outs of human reasoning in the here and now, it was much the received view
that the classical laws were also norms of reasoning, albeit in a highly idealized
form. Even so, the attention to reasoning was at best an afterthought.3
With the rise of modern modal logic, the emphasis began to shift. Under press
2Theword“pragmatic”invitesconfusion. Initslogico-epistemologicalsense,itistheQuinean
doctrine that no principle of logic is immune from overthrow. In its linguistic meaning, it is a
logicthattakesexpressnoteoftheroleoflinguisticagents. Thesecondsenseisintendedhere.
3Thepretensionsofso-callednaturaldeductionsystemstobemore“natural”thanaxiomatic
systemsreflectedrathermoreadistrustoftheepistemicprivilegethatlogicianssoughttoextend
totheiraxiomsthantoaburninginterestingettingreasoningonthegroundright.