Semantical Investigations in Heyting’s Intuitionistic Logic

304 Pages·1981·15.974 MB·English

by  Dov M. Gabbay (auth.)

SEMANTICAL INVESTIGATIONS IN
HEYTING'S INTUITIONIsTIC LOGIC
SYNTHESE LIBRARY
STUDIES IN EPISTEMOLOGY,
LOGIC, METHODOLOGY, AND PHILOSOPHY OF SCIENCE
Managing Editor:
JAAKKO HINTIKKA, Florida State University
Editors:
DON ALD DA VIDSON, University of Chicago
GABRIEL NUCHELMANS, University of Leyden
WESLEY C. SALMON, University of Arizona
VOLUME 148
DOV M. GABBAY
Professor of Logic, Bar-Ilan University
SEMANTICAL
INVESTIGATIONS
IN HEYTING'S
INTUITIONISTIC
LOGIC
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
Library of Congress Cataloging in Pnblication Data
Gabbay, Dov M. 1945-
Semantical investigations in Heyting's intuitionistlc
logic.
(Synthese library; v. 148)
Bibliography: p.
Includes index.
1. Intuitionistic rrtathematics. 2. Heyting, Arend,
1898- . I. Title.
QA9.47.G32 511'.2 80-27460
ISBN 978-90-481-8362-3 ISBN 978-94-017-2977-2 (eBook)
DOI 10.1007/978-94-017-2977-2
All Rights Reserved
Copyright © 1981 by Springer Science+Business Media Dordrecht
Originally published by D. Reidel Publishing Company, Dordrecht, Holland in 1981
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner
PREFACE
From the point of view of non-classical logics, Heyting's implication
is the smallest implication for which the deduction theorem holds.
This book studies properties of logical systems having some of the
classical connectives and implication in the neighbourhood of Heyt
ing's implication. I have not included anything on entailment, al
though it belongs to this neighbourhood, mainly because of the
appearance of the Anderson-Belnap book on entailment.
In the later chapters of this book, I have included material that
might be of interest to the intuitionist mathematician. Originally, I
intended to include more material in that spirit but I decided against
it. There is no coherent body of material to include that builds
naturally on the present book. There are some serious results on
topological models, second order Beth and Kripke models, theories of
types, etc., but it would require further research to be able to present
a general theory, possibly using sheaves. That would have postponed
pUblication for too long.
I would like to dedicate this book to my colleagues, Professors G.
Kreisel, M.O. Rabin and D. Scott. I have benefited greatly from
Professor Kreisel's criticism and suggestions. Professor Rabin's fun
damental results on decidability and undecidability provided the
powerful tools used in obtaining the majority of the results reported
in this book. Professor Scott's approach to non-classical logics and
especially his analysis of the Scott consequence relation makes it
possible to present Heyting's logic as a beautiful, integral part of
non-classical logics.
v
TABLE OF CONTENTS
v
PREFACE
1
INTRODUCTION
CHAPTER 1 / Logical Systems and Semantics 6
1. Scott and Tarski Systems 6
2. Scott Semantics 1 10
3. What is a Classical Connective? 12
4. Scott Semantics 2 17
Notes 19
CHAPTER 2 / Introducing HPC 20
1. Heyting's Implication 20
2. Heyting's Predicate Logic (HPC) 26
3. The Disjunction and Existence Properties 30
4. Heyting's Negation 35
5. Classical Predicate Calculus (CPC) 37
6. The System CD 40
7. The System MH* 41
Notes 42
CHAPTER 3 / The Kripke, Beth and Topological
Interpretations for HPC 43
1. Kripke Structures 43
2. Beth Interpretation 47
3. Structures with Constant Domains 50
4. Kripke Interpretation for MH 57
5. Topological Interpretation 59
Notes 62
CHAPTER 4 / Heyting's Propositional Calculus and
Extensions 63
1. hand c 63
2. Filtration 69
3. Incomparable Intermediate Logics 73
4. The Fragment without v 76
Notes 82
Vll
Vlll TABLE OF CONTENTS
CHAPTER 5 / Three Intermediate Logics 83
1. The Logic of Finite n-ary Trees 83
2. The Kreisel-Putnam System 95
3. An Intermediate Logic without f.m.p. 103
4. Exercises 105
Notes 107
CHAPTER 6 / Formulas in One Variable 108
1. Nishimura Lattice 108
2. The Disjunction Property for Yn 115
Note 120
CHAPTER 7 / Propositional Connectives 121
1. Negation 121
2. Strong Negation 124
3. Functional Completeness 127
4. New Intuitionistic Connectives 130
5. Modality 140
Notes 144
CHAPTER 8 / The Interpolation Theorem 145
1. General Consistency Theorem 145
2. Robinson's Theorem 146
3. Craig's Theorem 153
Notes 158
CHAPTER 9 / Second Order Propositional Calculus 159
1. The System 2h 159
2. The Completeness Theorem 160
3. Examples of Connectives in 2h 165
Notes 169
CHAPTER 10 / Modified Kripke Interpretation 170
1. Theory of Equality 170
2. Function Symbols and = 171
3. Some Model Theory 173
Note 177
CHAPTER II/Theories in HPC 1 178
1. Some Intuitionistic Theories 179
2. Some General Theorems 181
3. End Extensions 186
TABLE OF CONTENTS IX
CHAPTER 12 / Theories in HPC 2 193
1. Equality 193
2. The Apartness Relation 194
3. Linear Order 194
4. One Unary Function 196
5. Groups 197
6. Fields 199
7. A Fragment of Arithmetic 200
Note 202
CHAPTER 13 / Completeness of HPC with Respect to RE and
Post Structures 203
1. The Results 204
2. Positive Results 204
3. Negative Results 218
4. Tarski's Interpretation for HPC as a Calculus of Systems 220
5. Kreisel's Notion of Validity in Post Systems 221
6. Weak Validity in Post Systems 224
7. Validity in Post Systems 225
Notes 226
CHAPTER 14 / Undecidability Results 227
1. General Theorems 227
2. Undecidability Results 234
3. Undecidability of Classically Model Complete Theories 237
4. Undecidability of Theories Formulated with the Apartness
Relation 245
5. Undecidability of Some Finite Extensions of the Theory of
Abelian Groups 258
6. Undecidability of the Second Order Propositional
Calculus 262
W
N~s
CHAPTER 15 / Decidability Results 266
1. Decided Monadic Predicates 267
2. Unary Functions with Decided Equality 269
3. Monadic Fragment Again 272
4. Some Abelian Groups 277
5. Linear Order 279
m
N~s
x TABLE OF CONTENTS
REFERENCES 280
INDEX 285