Table Of ContentHANDBOOK OF
PHILOSOPHICAL
LOGIC
Volume 111:
Alternatives in Classical Logic
Eäited by
D. GABBAY
Department 0/ Computing, Imperial College, London, England
and
F. GUENTHNER
FNS, University o[ Tuebingen, West Germany
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
Library of Congress Cataloging-in-Publication Data
Main entry under title:
Alternatives in c1assicallogic.
(Handbook of philosophical logic ; v. 3) (Synthese
Iibrary ; v. 166)
Inc1udes bibliographies and indexes.
1. Logic - Addresses, essays, lectures. 1. Gabbay,
Dov M., 1945 - . H. Guenthner, Franz. III. Series.
IV. Series: Synthese Iibrary ; v. 166.
BC71.A56 1985 160 85-25692
ISBN 978-94-010-8801-5 ISBN 978-94-009-5203-4 (eBook)
DOI 10.1007/978-94-009-5203-4
All Rights Reserved
© 1986 by Springer Science+Business Media Dordrecht
Originally published by D. Reidel Publishing Company in 1986
Softcover reprint of the hardcover 1st edition 1986
No part Qf the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
inc1uding photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner
CONTENTS TO VOLUME III
ACKNOWLEDGEMENTS ~
PREFACE ~
A NOT EON NOT A T IO N xi
III.I. STEPHEN BLAMEY / Partial Logic
III.2. ALA SDA IR U R QU H ART / Many-valued Logic 71
II1.3. J. M I C H A E L DUN N / Relevance Logic and Entailment 117
I1I.4. D IRK V AND ALE N / Intuitionistic Logic 225
111.5. W A L TE R FE L S CH E R / Dialogues as a Foundation for
Intuitionistic Logic 341
111.6. ERMANNO BENCIVENGA / Free Logics 373
IIL7. MARIA LUISA DALLA CHIARA / Quantum Logic 427
III.8. GO RAN SUN D H 0 L M / Proof Theory and Meaning 471
NAME INDEX 507
SUBJECT INDEX 513
TAB LEO F CON TEN T S TO VOL U M E S I , II, AND IV 521
ACKNOWLEDGEMENTS
The preparation of the Handbook of Philosophical Logic was generously
supported by the Lady Davis Fund at Bar-Han University, Ramat-Gan, Israel
and the Werner-Reimers-Stiftung, Bad Homburg, West Germany, which
provided us with the chance of discussing the chapters in the Handbook
at various workshops with the contributors. It is a great pleasure to acknowl
edge the assistance of these institutions during the preparation of this col
lection. We benefitted further from the editorial and personal advice and
help from the publisher.
Most important of all, we would like to express our thanks to all the
contributors to the Handbook for their unlimited goodwill, their professional
counsel, as well as their friendly support. They have made the preparation
of this collection a stimulating and gratifying enterprise.
D. GABBA Y (Imperial College, London)
F. GUENTHNER (University of Tuebingen)
PREFACE
This volume presents a number of systems of logic which can be considered
as alternatives to classical logic. The notion of what counts as an alternative
is a somewhat problematic one.
There are extreme views on the matter of what is the 'correct' logical
system and whether one logical system (e.g. classical logic) can represent
(or contain) all the others. The choice of the systems presented in this volume
was guided by the following criteria for including a logic as an alternative:
(i) the departure from classical logic in accepting or rejecting certain theorems
of classical logic following intuitions arising from significant application
areas and/or from human reasoning; (ii) the alternative logic is well-established
and well-understood mathematically and is widely applied in other disciplines
such as mathematics, physics, computer science, philosophy, psychology, or
linguistics. A number of other alternatives had to be omitted for the present
volume (e.g. recent attempts to formulate so-called 'non-monotonic' reason
ing systems). Perhaps these can be included in future extensions of the
Handbook of Philosophical Logic.
Chapter 1 deals with partial logics, that is, systems where sentences do not
always have to be either true or false, and where terms do not always have to
denote. These systems are thus, in general, geared towards reasoning in
partially specified models. Logics of this type have arisen mainly from philo
sophical and linguistic considerations; various applications in theoretical
computer science have also been envisaged.
Chapter 2 deals with many-valued logics, one of the first alternatives to
classical logic. Its characteristic is - as the name suggests - to allow for
sentences to take many possible truth values. The chapter surveys the most
well-known versions of many-valued logic from 3-valued to fuzzy-valued
systems.
Chapter 3 covers relevance and entailment logics, another main con
tender to classical logic for capturing intuitively correct human reasoning.
Many classically valid theorems and rules of inference are not accepted in
relevance logic and alternative systems are suggested.
Chapter 4 deals with intuitionistic logic, a logic that arose in connection
with the foundations of mathematics. The chapter gives a detailed account
of intuitionistic proof theory and semantics as well as some discussion of
issues in intuitionistic mathematics. Recently, intuitionistically-based systems
ix
D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, Vol. III, ix-x.
© 1986 by D. Reidel Publishing Company.
x
have begun to play a useful and important role in theoretical computer
science as well.
Chapter 5 is a supplement to the previous chapter and contains a proof
theoretical study of intuitionistic logic from the point of view of dialogue
games.
Chapter 6 presents systems of logic known as free logics. These logics
agree with classical logic in the propositional part, but differ in the way they
deal with non-denoting terms and quantifiers at the predicate logic level.
Chapter 7 covers quantum logic, i.e. the logical system that has arisen in
connection with logical problems arising in certain physical systems.
Finally, Chapter 8 concludes the volume with a survey of the ways proof
theoretical rules can be used to determine the nature of the logical con
nectives.
D.M.GABBAY
F. GUENTHNER
A NOTE ON NOTATION
Writings in the field of philosophical logic abound with logical symbols and
every writer seems to have avowed or non-avowed (strong) preferences for
one or the other system of notation. It had at one point been our intention
to adopt a completely uniform set of symbols and notational conventions
for the Handbook. For various reasons, we have left the choice of minor
bits of notation (e.g. sentential symbols and the like) to the particular
authors, unifying only whenever confusion might arise. The reader is invited
to translate whatever seems to go against his or her notational taste into his
or her favorite variant.
CHAPTER II1.1
PARTIAL LOGIC
by STEPHEN BLAMEY
Introduction 2
1. A sketch of simple partial logic 2
1.1. Classical semantics as partial semantics 2
1.2. Partial semantics as monotonic semantics 7
1.3. Comparisons with supervaluations 12
2. Some motivations and applications 15
2.1. Presupposition 15
2.2. Conditional assertion 18
2.3. Sortal incorrectness 19
2.4. Semantic paradox 21
2.5. Stage-by-stage evaluation 22
2.6. Enrichments of simple partial logic 25
2.7. Situation semantics 27
3. Fregean themes 30
3.1. Reference failure 30
3.2. Functional dependence 33
4. Non-classical sentence connectives 36
4.1. Interjunction and transplication 36
4.2. Non-monotonic connectives 40
5. Partial logic as classical logic 44
5.1. Partial truth languages 44
5.2. Natural negation 47
6. First-order partial semantics 49
6.1. Languages and models 49
6.2. Monotonicity and compatibility 52
6.3. A parenthesis on description terms 55
6.4. Semantic consequence 58
7. First-order partial theories 61
7.1. Logicallaws 61
7.2. Model-existence theorems 66
References 68
D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, Vol. III, 1-70.
© 1986 by D. Reidel Publishing Company.
2 STEPHEN BLAMEY
INTRODUCTION
There is a confusing diversity of philosophical motivations and practical
applications for logics in which sentences do not have to be either 'true' or
'false', or in which singular terms do not have to 'denote' anything, or both.
And a correspondingly wide variety of formal approaches has grown up for
treating this kind of departure from traditional classical logic. In this chapter
we shall pursue one simple approach, which is guided by an emphasis on
functors rather than the sentences or singular terms which they take as
arguments and yield as compounds. We shall pull together bits and pieces
from various places, suggest a few novelties, and aim to present an apparatus
of semantics to handle truth-value gaps and denotation failures in a uniform
way.
Two ·complementary themes are followed through. The first is a con
straint on the interpretation of functors, in the light of which truth-value
less sentences and denotationless singular terms can be seen to emerge as
semantically 'undefined'. This constraint, 'mono tonicity' , is familiar from
the theory of computation and the theory of inductive definitions. The
exercise we set ourselves is to pursue partial logic as 'monotonic logic'. The
second theme is then fully to exploit the fact that, within this constraint,
functors can yield undefined compounds even when all their arguments
are defined. In particular, it is fruitful to adopt connectives which make
sentence functors of this kind.
Some of the following material is some of the material in my D.Phil.
thesis. It was written under the supervision of Dana Scott, whose nose for
monotonicity sniffed out some sense in my funny connectives. But this
essay was planned jointly with Albert Visser; and he picks up partial logic
again in his Chapter IV.14 of the Handbook.
1. A SKETCH OF SIMPLE PARTIAL LOGIC
1.1. Qassical Semantics as Partial Semantics
In classical logic sentences are either true (T) or false (1) and the interpreta
tion of the standard sentence connectives can be given in the following way:
-,~ is T iff ~ is1
-,~ is 1 iff ~ is T
~1\tJ; is T iff ~ is T and tJ; is T
~1\tJ; is 1 iff ~ is1 or tJ; is 1
IlI.1: PAR TIAL LOGIC 3
qJvljJ is T iff qJ is T or IjJ is T
qJvljJ is 1 iff qJ is 1 and IjJ is 1
qJ-+1jJ is T iff qJ is 1 or IjJ is T
qJ-+1jJ is 1 iff qJ is T and IjJ is 1
qJ +-+-1jJ is T iff (qJ is T and IjJ is T)
or (qJ isl and IjJ is 1)
qJ +-+-1jJ is 1 iff (qJ is T and IjJ is 1)
or (qJ is 1 and IjJ is T).
For simple partial logic we shall adopt precisely these classical T/1-conditions;
only we give up the assumption that all sentences have to be classified either
as T or as 1. This leaves room for the classification neither-T-nor-1. At present
we are concerned merely to highlight a parallel with classical semantics,
and under the parallel we can think of the third classification as a 'truth
value gap'. This thought is taken a little further in Sections 1.2 and 3. But
the point, if any, of seeing the third classification as different in philosophical
kind from T and 1, will of course depend on what particular motivation we
consider for adopting the forms of partial logic. (See, especially, Sections 2
and 5.)
To interpret universal and existential quantifiers over a given domain D,
we shall again exploit the fact that the classical interpretation leaves room
for a gap between T and 1 when we write out T-conditions and 1-conditions
separately. Assuming that a language has - or can be extended so as to
have - a name ii for each object a in D:
VXqJ(x) is T iff qJ(ii) is T for any a inD
Vxcp(x) is 1 iff cp(ii) is 1 for some a in D
3xcp(x) is T iff cp(ii) is T for some a in D
3xqJ(x) is 1 iff cp(ii) is 1 for any a in D.
Most treatments of classical logic stipulate that the domain be non-empty.
We shall not be so restrictive: D may be empty.
These T/1-conditions for '<Ix and 3x of course presuppose a semantic
account of predicate/singular-term composition. And this mode of com
position deserves some attention, since it is the most familiar place to locate
the cause of a sentence's being neither 'true' nor 'false'. It has been con
sidered to give rise to a truth-value gap in two different ways: either (i)
because a term t may lack a denotation and may, for this reason, make a
Description:This volume presents a number of systems of logic which can be considered as alternatives to classical logic. The notion of what counts as an alternative is a somewhat problematic one. There are extreme views on the matter of what is the 'correct' logical system and whether one logical system (e. g.
