Table Of ContentINTRODUCTORY MODAL LOGIC
Introductory
Modal Logic
KENNETH KONYNDYK
University of Notre Drone Press
Notre Drone, Indiana 46556
University of Notre Dame Press
Notre Dame, Indiana, 46556
www.undpress.nd.edu
All Rights Reserved
Published in the United States of America
Copyright © 1986 by University of Notre Dame
Reprinted in 1993, 1996, 2002, 2008
Library of Congress Cataloging-in-Publication Data
Konyndyk, Kenneth.
Introductory modal logic.
Bibliography: p.
1.Modality (Logic) I. Title.
BC199.M6K66 1986 160 85-41007
ISBN 13: 978-0-268-01159-8 (pbk.)
ISBN 13: 978-0-268-02407-9 (hardback)
This book is printed on acid-free paper.
∞
CONTENTS
Preface ix
Introduction 1
1 Propositional Logic 3
1. 1 Symbolism 4
1.2 Rule of Replacement and Equivalences 5
1.3 Rules of Inference 6
1.4 Proofs 9
1.5 Exercises 9
2 Propositional Modal Logic 11
2.1 The Modal Concepts 12
2.1.a An Explanation of Necessity 12
"Ungiueupable" 14
"Unable to be rationally rejected" 15
"Self-evident" 15
A priori 15
2.1.b The Possible Worlds Picture 16
2.1.c A Map of the Modal Concepts 17
2.1.d Symbols and Definitions 18
2.1.e Exercises 19
2.1.f Symbolizing Conditionals 20
2.1.g The Necessity of the Consequence and the
Necessity of the Consequent 21
2.1.h Exercises 23
2.2 The Logic of Modality 23
2.2.a The Need for a Logic of Modality 23
2.2.b Logical Relationships 25
2.2.c Systems of Propositional Modal Logic 30
2.3 The System T 31
Exercises 32
V
vi CONTENTS
2.3.a Rules 32
The necessity elimination rule 33
Necessity introduction subproofs 34
The necessity introduction subproof rule 34
The necessity introduction rule 35
The T-reiteration rule 36
2.3.b Hints 36
2.3.c Exercises 38
2.3.d More Rules 39
Possibility introduction 39
Possibility elimination 39
Modal modus ponens 40
Exercises 40
2.3.e Decision Procedure 41
2.3.f Counterexamples 41
2.3.g Exercises 44
2.3.h Iterated Modalities and Reduction of
Modalities 46
2.4 The System S4 49
2.4.a Rules 49
2.4.b Exercises 50
2.4.c Reduction of Modalities in S4 50
2.5 The System S5 51
2.5.a Rules for S5 52
2.5.b Exercises 53
2.5.c Reduction Laws 53
2.5.d Exercises 55
2.6 Philosophical Matters 55
2.6.a Differences between Systems 55
2.6.b Some Assumptions Implicit in Our
Presentation 60
2.6.c Is There Only One Correct System of
Modal Logic? 64
3 Quantification 69
3.1 Symbols 69
3.2 Interpretation 70
3.3 Equivalences 71
3.4 Rules 71
Universal Instantiation 72
Universal Generalization 72
Existential Generalization 72
CONTENTS vii
Existential Instantiation 72
Exercises 75
4 Quantified Modal Logic 77
4.1 Motivation 77
4.2 'l\vo distinctions 78
4.3 Symbolization 84
4.4 Some Objections to Quantified Modal Logic 85
4.5 Essential Properties and Possible Worlds 88
4.6 Considerations Bearing on Quantified Modal Logic 90
4. 7 The Systems 92
4.8 A Kripke System 92
Exercises 94
4.9 Objections to This System 95
4.10 Kripke's Revised S5 97
The necessity introduction rule 98
Possibility elimination 99
4.11 Reflections on Kripke's Revision 101
4.12 The Actualistic Version of Possible Worlds 103
4.13 Actualism and Quantification 105
Exercises 108
4.14 Actualistic Quantified Modal Logic 108
4.14.a Rules 109
4.14.b Exercises 112
4.15 Existence and Modality 113
4.16 The Problem of Transworld Identity 115
4.1 7 Superiority of Actualism 117
Appendix I 119
Appendix II 121
Appendix III 125
Notes 127
Bibliography 131
PREFACE
This text is designed as an introduction to modal logic for a stu
dent who has studied elementary symbolic logic. It presents the sym
bolism, natural deduction proof procedures, and problems of modal
logic, using the systems which have come to be known as T (or M),
S4, and S5.
Since elementary symbolic logic texts differ with respect to sym
bolism, proof techniques, and rules, this book begins with a brief
specification of symbols, rules, and proof technique to be used here.
Propositional modal logic and quantified modal logic are devel
oped separately. The chapter treating propositional modal logic can
be studied immediately after ordinary propositional logic. Although
it introduces some metatheoretical considerations, it does not pre
suppose any prior acquaintance with metatheory. I consider some
of the philosophical motivation for modal logic and do not try to pre
tend futilely that modal logic is or should be philosophically neutral,
preferring instead to point out areas of philosophical controversy over
modal logic.
Quantified modal logic raises even more philosophical questions.
Which system is preferable depends quite directly on how certain
metaphysical questions are answered. Indeed, the point of quantified
modal logic is to represent the logical structure of certain correct
metaphysical views. Accordingly, I develop the chapter on quanti
fied modal logic so as to make clear where metaphysical choices have
to be made, and I indicate what I think are the correct choices. There
are plenty of references to discussions of the philosophical issues at
stake. But I develop this chapter so a student can get a good grasp
of quantified modal logic without being overwhelmed by the philo
sophical difficulties.
My presentation owes a great deal to the work of Saul Kripke and
to the interpretations of Alvin Plantinga and Arthur N. Prior, but
I am most deeply indebted to my colleagues Alvin Plantinga and
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