Proof Methods for Modal and Intuitionistic Logics

563 Pages·1983·13.794 MB·English

by  Melvin Fitting (auth.)

PROOF METHODS FOR MODAL AND INTUITIONISTIC LOGICS
SYNTHESE LIBRARY
STUDIES IN EPISTEMOLOGY,
LOGIC, METHODOLOGY, AND PHILOSOPHY OF SCIENCE
Managing Editor:
JAAKKO HINTIKKA, Florida State University, Tallahassee
Editors:
DONALD DAVIDSON, University of California
GABRIEL NUCHELMANS, University of Leyden
WESLEY C. SALMON, University ofP ittsburgh
VOLUME 169
MELVIN FITTING
Herbert H. Lehman College of the City University of New York
PROOF METHODS
FOR MODAL AND
INTUITIONISTIC LOGICS
SPRINGER-SCIENC.E..+.B. USINESS MEDIA, B.V .
''
Library of Congress Cataloging in Publication Data
Fitting, Melvin Chris.
Proof methods for modal and intuitionistic logics.
(Synthese library; v. 169)
Bibliography: p.
Includes index.
1. Proof theory. 2. Modality (Logic) 3. Intuitionistic
mathematics. 1. Title.
QA9.54.F57 1983 511.3 83-4409
ISBN 978-90-481-8381-4 ISBN 978-94-017-2794-5 (eBook)
DOI 10.1007/978-94-017-2794-5
Ali Rights Reserved
Copyright © 1983 by SpringerScience+BusinessMediaDordrecht
Originally pub1ished by D. Reide1 Pub1isbing Company, Dordrecbt, Holland in 1983
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
TABLE OF CONTENTS
INTRODUCTION 1
CHAPTER ONE I BACKGROUND
1. Propositional Formulas 11
2. Models 14
3. A Unifying Notation 22
4. Classical Semantic Tableaus 29
CHAPTER TWO I ANALYTIC MODAL TABLEAUS AND
CONSISTENCY PROPERTIES
1. Tableau Rules for K, K4, T, S4, D and D4 33
2. Uniform Modal Notation 41
3. Correctness 43
4. A Note on Completeness Proofs 46
5. Consistency Properties forK, K4, T, S4, D and D4 48
6. Tableau Completeness 60
CHAPTER THREE I LOGICAL CONSEQUENCE, COMPACTNESS,
INTERPOLATION, AND OTHER TOPICS
1. Introduction 64
2. Logical Consequence 65
3. Strong Tableau Completeness 70
4. Compactness Theorems 74
5. The Deduction Theorem 77
6. Gentzen Systems 81
7. Symmetric Gentzen Systems 88
8. The Craig Interpolation Lemma 93
9. Other Interpolation Theorems 99
10. The Beth Definability Theorem 105
11. Further Consequences of Interpolation Theorems 110
12. Decidability 115
v
vi TABLE OF CONTENTS
CHAPTER FOUR I AXIOM SYSTEMS AND NATURAL
DEDUCTION
1. Introduction 118
2. A Classical Propositional Axiom System 118
3. An Underlying Modal Logic 123
4. Results about the Logic U 127
5. The Logic K Axiomatized 130
6. Correctness and Completeness of the K Axiomatization 134
7. The Logics K4, T, S4, D and D4 Axiomatization 137
8. Finite Axiomatizability, Part I 141
9. Finite Axiomatizability, Part II 148
10. A Classical Natural Deduction System 153
11. Some Results about Natural Deduction 162
12. A Note on Modal Natural Deduction Rules 170
13. 1-style Natural Deduction Rules 171
14. 1-style Completeness and Correctness 176
15. A-style Natural Deduction Rules 184
16. A-style Completeness and Correctness 187
CHAPTER FIVE I NON-ANALYTIC LOGICS
1. Introduction 191
2. Synthetic KB, DB, B and S5 Tableaus 193
3. Semi-analytic KB, DB, B and S5 Tableaus 201
4. Consistency Properties for KB, DB, B and S5 205
5. Immediate Consequences 209
6. KB, DB, B and S5 Axiomatized 210
7. Natural Deduction for KB, DB, Band S5 213
8. Symmetric Gentzen Systems and Interpolation 214
9. Peculiar Non-peculiarities of S5 221
10. An Embedding of S5 into S4 222
11. Another S5 Tableau System 225
12. Finite Axiomatizability 229
13. S5 Interpolation Theorems 234
14. The Significance of G 241
15. Tableau Rules for G 245
16. G Tableau Correctness 249
17. G Consistency Properties 251
18. Consequences 255
vii
TABLE OF CONTENTS
CHAPTER SIX I NON-NORMAL LOGICS
1. Introduction 262
2. Augmented Kripke Models 265
3. Semantic Tableaus 270
4. Consistency Properties 281
5. Deduction Theorems 286
6. Interpolation Theorems 292
7. Decidability 299
8. Axiom Systems 300
9. Natural Deduction Systems 306
10. Regular and Quasi-regular Logics 313
11. Logics between Regular and Normal 316
12. The Intermediate Logics Axiomatized 321
13. The Logic U 329
CHAPTER SEVEN I QUANTIFIERS
1. Introduction 332
2. First-order Languages 333
3. First-order Models 339
4. Interpretations 343
5. Uniform Notation 344
6. Tableau Rules 345
7. Tableau Correctness 350
8. Analytic Consistency Properties 353
9. Analytic Tableau Completeness 366
10. Natural Deduction and Axiom Systems 367
11. Compactness and Skolem-Lowenheim Theorems 372
12. Deduction Theorems 373
13. Interpolation Theorems 375
14. Dropping Monotonicity 380
15. Constant Domain Models 382
CHAPTER EIGHT I PREFIXED TABLEAU SYSTEMS
1. Introduction 386
2. Prefixed Propositional Tableaus 388
3. Correctness 398
4. A Systematic Tableau Procedure 401
5. Konig's Lemma 404
viii TABLE OF CONTENTS
6. Completeness 407
7. Decidability 410
8. Logical Consequence 416
9. The Fundamental Theorem 418
10. Quantifiers -Constant Domains 424
11. Correctness and Completeness 425
12. Constant Domain Logics Axiomatically 428
13. Quantifiers-Varying Domains 433
14. Concluding Remarks 436
CHAPTER NINE / INTUITIONISTIC LOGIC
1. Introduction 437
2. Propositional Intuitionistc Models 439
3. Independence of the Intuitionistic Connectives 444
4. Uniform Notation 450
5. Tableau Systems 453
6. Local and Global Satisfiability 462
7. Intuitionistic Consistency Properties 464
8. Decidability 468
9. First-order Intuitionistic Models 469
10. Uniform Notation, again 474
11. First-order Tableau Systems 476
12. First-order Intuitionistic Consistency Properties 481
13. Consequences 485
14. An Axiom System 487
15. Axiomatic Correctness and Completeness 491
16. A Natural Deduction System 495
17. Direct Consequences 500
18. Completeness of the Natural Deduction System 504
19. Gentzen Systems and Interpolation 508
20. Concluding Comments 518
BIBLIOGRAPHY 526
540
INDEX
SPECIAL NOTATION 555
INTRODUCTION
"Necessity is the mother of invention."
Part I: What is in this book - details.
There are several different types of formal proof
procedures that logicians have invented. The ones we
consider are: 1) tableau systems, 2) Gentzen sequent
calculi, 3) natural deduction systems, and 4) axiom
systems. We present proof procedures of each of these
types for the most common normal modal logics: S5, S4, B,
T, D, K, K4, D4, KB, DB, and also G, the logic that has
become important in applications of modal logic to the
proof theory of Peano arithmetic. Further, we present a
similar variety of proof procedures for an even larger
number of regular, non-normal modal logics (many introduced
by Lemmon). We also consider some quasi-regular logics,
including S2 and S3. Virtually all of these proof
procedures are studied in both propositional and
first-order versions (generally with and without the Barcan
formula). Finally, we present the full variety of proof
methods for Intuitionistic logic (and of course Classical
logic too).
We actually give two quite different kinds of tableau
systems for the logics we consider, two kinds of Gentzen
sequent calculi, and two kinds of natural deduction
systems. Each of the two tableau systems has its own uses;
each provides us with different information about the
logics involved. They complement each other more than they
overlap. Of the two Gentzen systems, one is of the
conventional sort, common in the literature. The other