Logic in Linguistics

196 Pages·1977·8.18 MB·English

by  Jens Allwood

CAMBRIDGE TEXTBOOKS IN LINGUISTICS
General Editors: J.BRF.SNAN, B.COMRIE, W.DRESSLER, R.HUDDLESTON,
R.LASS, D.LIGHTFOOT, J.LYONS, P.H.MATTHEWS, R.POSNER, S.ROMAINE,
N.V.SMITH, N.VINCENT
LOGIC IN LINGUISTICS
In this series:
P.H.MATTHEWS Morphology Second edition
R.A.HUDSON Sociolinguistics
B.COMRIE Aspect
R.M.KEMPSON Semantic Theory
T.BYNON Historical Linguistics
D.B.FRY The Physics of Speech
J.K.CHAMBERS and P.TRUDGILLZ)!tf/«:/o/og)>
A.J.ELLIOT Child Language
P.H.MATTHEWS Syntax
A.RADFORD Transformational Syntax
L.BAUER English Word-formation
S.C.LEVINSON Pragmatics
G.BROWN and G.YULE Discourse Analysis
R.HUDDLESTON Introduction to the Grammar of English
R.LASS Phonology
B.COMRIE Tense
A.CRUTTENDEN Intonation
W.KLEIN Second Language Acquisition
A.J. WOODS, P.FLETCHER and A.HUGHES Statistics in Language Studies
D. A.CRUSE Lexical Semantics
F.R.PALMER Mood and Modality
A.RADFORD Transformational Grammar
M.CARMAN Psycholinguistics
w.CROFT Typology and Universals
G.CORBETT Gender
H.GIEGERICH English Phonology
LOGIC IN LINGUISTICS
JENS ALLWOOD
LARS-GUNNAR ANDERSSON
OSTEN DAHL
CAMBRIDGE
UNIVERSITY PRESS
CAMBRIDGE university press
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore,
São Paulo, Delhi, Dubai, Tokyo, Mexico City
Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by
Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521291743
© Cambridge University Press 1977
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press.
First published in English translation 1977
Reprinted 1979,1981,1983,1985,1986,1987,1989,1991,1993,1995,1997
A catalogue record for this publication is available from the British Library
Library of Congress Cataloguing in Publication Data
ISBN 978-0-521-21496-4 Hardback
ISBN 978-0-521-29174-3 Paperback
Cambridge University Press has no responsibility for the persistence or
accuracy of URLs for external or third-party internet websites referred to in
this publication, and does not guarantee that any content on such websites is,
or will remain, accurate or appropriate. Information regarding prices, travel
timetables, and other factual information given in this work is correct at
the time of first printing but Cambridge University Press does not guarantee
the accuracy of such information thereafter.
CONTENTS
Preface viii Symbols and notational conventions, ix
i Logic for linguists page i
2 Set theory 3
2.1 Sets and elements 3
2.2 Relations between sets 5
2.3 Operations on sets 6
2.4 Relations and functions 9
3 Inference and logical analysis of sentences
IS
3.1 Inference
IS
3.2 Logical form 18
3.3 Sentences and propositions 20
3.4 Possible worlds and the truth-set of a proposition 22
3.5 Analytic and synthetic sentences 23
3.6 Simple and compound sentences 24
3.7 The depth of the logical analysis 25
4 Propositional logic 26
4.1 Connectives 26
4.2 The meaning of the logical connectives 30
4.2.1 Negation 30
4.2.2 Conjunction 32
4.2.3 Disjunction 34
4.2.4 Implication 37
4.2.5 Equivalence 40
4.3 i/ow /o indicate constituent structure 42
4.4 TAe syntax anrf semantics of propositional calculus 44
4.5 5y«ta* 45
4.6 Semantics 47
4.7 Tautologies and contradictions 5°
4.8 7VutA fa&fe* 5°
s Predicate logic page 58
Extending the logical analysis 58
5-2 Quantifiers 61
5.3 Summary of the syntax of predicate logic 71
5-4 The semantics of predicate logic 72
5-5 True in all interpretations 77
5.6 Summary of the semantics of predicate logic 83
5-7 A formal version of the semantics 84
5.8 Formal properties of relations 88
5.8.1 Reflexivity 88
5.8.2 Symmetry 89
5.8.3 Transitivity 89
5.84 Converse 90
5.8.5 Structure of the domain and co-domain of relations 90
6 Deduction 96
6.1 The deductive system 96
6.2 Deduction rules in everyday conversation 104
7 Modal logic 108
7-1 Modal operators 108
7-2 Strict implication no
7-3 Other modalities in
7-4 Problems connected tvith scope and identity in modal logic 114
7.4.1 ^De dicto'- de re* ambiguities 114
7.4.2 Specificity 116
74.3 Opacity 117
7.4-4. Cross-world identification 119
7-5 Counterfactual sentences 120
7.6 Tense logic and reference points 121
8 Intensional logic and categorial grammar 125
8.1 Intensions and extensions 125
8.2 Intension 127
8.3 The Fregean principle 130
8.4 The Fregean principle and categorial grammar 132
8.5 Categories, intensions and types 136
9 Further extensions 148
9.1 Second-order predicate logic and predicate operators 148
9.2 Presuppositions and definite descriptions 149
93 Pragmatic analysis of presuppositions 153
94 The abstraction- or lambda-operator 155
io Logic for linguists ? page 158
10.1 General 158
10.2 The concept of meaning 158
10.3 The role of formal languages in analysing natural language 164
10.4 The limitations of classical logic 168
References 172
Answers to exercises 175
Index 181
PREFACE
This book has grown out of an introduction to logic for linguists in
Swedish published by Studentlitteratur, Lund, 1971, a German transla-
tion of which appeared in 1973 (Niemeyer Verlag). All the chapters of
the Swedish book have been revised and several additions (including
the whole of chapter 8) have been made. The goal is still the same: to
give linguists and others interested in language an orientation in logic
which will enable them to understand how logical concepts are used (or
could be used) in linguistic theory, in particular semantics.
We have benefited from suggestions and critical remarks made by
several people. In addition to several generations of students at the
Department of Linguistics in Goteborg, who have had to act as our
guinea-pigs, we want to thank Claes Aberg, who has read the book in
manuscript at various stages of its development, Michael Grabski, who
translated the Swedish book into German and who suggested a number
of changes, many of which have been incorporated into the present
version, and John Lyons, an editor of this series. We are also grateful to
Peter Hinst for a devastating review of the German version in Zeitschrift
fur Germanistische Ltnguistik, 3: 3 (1975), with which we partly agree.
Were it not for our stubbornness, all these people might have managed
to exert a still more positive influence on this book. For kind assistance
with typing and editorial matters, we want to thank Pierre Javanaud,
Ann-Mari Ranstrand and the staff of the Cambridge University Press.
Although the book is the result of team-work, the main responsibility
for the respective chapters has been distributed as follows: chapters 1,
3, 4, 8 and 9.3-Jens Allwood; chapters 5 and 6 - Lars-Gunnar
Andersson; and chapters 2, 7, 10 and the rest of chapter 9 - Osten Dahl.
Vlll
Symbols and notational conventions
Symbol Name Explained
e element sign ('is a member of) page 3
£ ' is not a member of 3
{} curly brackets (denoting sets) 3-4
0 empty set 4
1 universal set 5
c proper inclusion ('is a proper subset of) 6
£ inclusion ('is a subset of) 6
= set identity 6
fl intersection 7
U union 7
— difference 8
C complement 9
<> angle brackets (denoting n-tuples) 9
t 'true' 28
f 'false' 28
I Sheffer's stroke 29
~ (—r, —) negation 30
& (•, A) conjunction 32
V disjunction (inclusive) 34
© disjunction (exclusive) 35
->•(=') implication (material) 37
— (*->> c) equivalence 41
V universal quantifier 62
3 existential quantifier 65
R converse of R 90
possibility operator 109
necessity operator 109
strict implication ('fishhook') in
IX