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First-Order Modal Logic, Most Decidedly, Is Not Just Propositional Modal Logic Plus Classical Quantifier Machinery

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First-Order Modal Logic, Most Decidedly, Is Not Just Propositional Modal Logic Plus Classical Quantifier Machinery SYNTHESE LIBRARY MELVIN FITTING AND RICHARD L. MENDELSOHN STUDIES IN EPISTEMOLOGY. Lehman College and the Graduate Cente" CUNY, New York. U.s.A. LOGIC. METHODOLOGY. AND PHILOSOPHY OF SCIENCE '0 Managing Editor: FIRST-ORDER JAAKKO HINTlKKA. Boston University MODAL LOGIC Editors: DIRK VAN DALEN. University of Utrecht. The Netherlands DONALD DAVIDSON. University of California. Berkeley THEO A.F. KUIPERS. University ofGroningen. The Netherlands PATRICK SUPPES. Stanford University. California JAN WOLENSKI. Jagiellonian University. Krak6w. Poland VOLUME 277 KLUWER ACADEMIC PUBLISHERS DORDRECHT I BOSTON I LONDON A C.J.P. Catalogue record for this book is available from the Library of Congress. ISBN 0-7923-5335-8 (PB) ISBN 0-7923-5334-X (HB) To Marsha, Robin, and losh, with love PubHshed by Kluwer Academic Publishers, RLM P.D. Box 17,3300 AA Dordrecht, The Netherlands. To Roma, who knows what I mean MP Sold and distributed in North, Central and South America by Kluwer Academic Publishers. 101 Philip Drive, NorweII, MA 02061, U.S.A. In all other countries, sold and distributed by KJuwer Academij; Publishers, P.D. Box 322, 3300 AH Dordrecht, The Netherlands. Printed on acid-free paper An Rights Reserved © 1998 KJuwer Academic Publishers No part of the material protected by this copyright notice may be reproduced or utilized in any fonn 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 Printed in the Netherlands. CONTENTS - , PREFACE xi CHAPTER ONE / PROPOSITIONAL MODAL LOGIC 1 1.1 What is a Modal? 2 1.2 Can There Be a Modal Logic? 3 I.3 What Are The Formulas? 5 1.4 Aristotle's Modal Square 7 1.5 Informal Interpretations 8 1.6 What Are the Models? II 1.7 Examples 14 1.8 Some Important Logics 18 1.9 Logical Consequence 21 1.1 0 Temporal Logic 24 I.I I Epistemic Logic 28 I.I 2 Historical Highlights 32 CHAPTER TWO / TABLEAU PROOF SYSTEMS 46 2.1 What Is a Proof 46 2.2 Tableaus 47 2.3 More Tableau Systems 51 2.4 Logical Consequence and Tableaus 55 2.5 Tableaus Work 57 CHAPTER THREE / AXIOM SYSTEMS 67 3.1 What Is an Axiomatic Proof 67 3.2 More Axiom Systems 71 3.3 Logical Consequence, Axiomatically 73 3.4 Axiom Systems Work Too 74 CHAPTER FOUR / QUANTIFIED MODAL LOGIC 81 4.1 First-Order Formulas 81 4.2 An Informal Introduction 83 4.3 Necessity De Re and De Dicto 85 4.4 Is Quantified Modal Logic Possible? 89 viii CONTENTS CONTENTS ix 4.5 What the Quantifiers Quantify Over 92 9.3 Predicate Abstraction 194 4.6 Constant Domain Models 95 9.4 Abstraction in the Concrete 195 4.7 Varying Domain Models 101 9.5 Reading Predicate Abstracts 200 4.8 Different Media, Same Message 105 4.9 Barcan and Converse Barcan Formulas 108 CHAPTER TEN / ABSTRACTION CONTINUED 204 10.1 Equality 204 CHAPTER FIVE / FIRST-ORDER TABLEAUS 116 10.2 Rigidity 210 5.1 Constant Domain Tableaus 116 10.3 A Dynamic Logic Example 216 5.2 Varying Domain Tableaus 118 10.4 Rigid Designators 217 5.3 Tableaus Still Work 121 10.5 Existence 220 10.6 Tableau Rules, Varying Domain 221 CHAPTER SIX '/ FIRST-ORDER AXIOM SYSTEMS 132 10.7 Tableau Rules, Constant Domain 227 6.1 -A Classical First-Order Axiom System 132 6.2 Varying Domain Modal Axiom Systems 135 CHAPTER ELEVEN / DESIGNATION 230 6.3 Constarit Domain Systems 136 11.1 The Formal Machinery 231 6.4 Miscellany 138 11.2 Designation and Existence 233 11.3 Existence and Designation 235 CHAPTER SEVEN / EQUALITY 140 11.4 Fiction 241 7.1 Classical Background 140 11.5 Tableau Rules 245 7.2 Frege's Puzzle 142 7.3 The Indiscemibility ofIdentica1s . 145 CHAPTER TWELVE / DEFINITE DESCRIPTIONS 248 12.1 Notation 7.4 The Formal Details 149 248 12.2 1\\'0 Theories of Descriptions 7.5 Tableau Equality Rules 150 249 12.3 The Semantics of Definite Descriptions 7.6 Tableau Soundness and Completeness 152 253 12.4 Some Examples 7.7 An Example 159 255 12.5 Hintikka's Schema and Variations 261 12.6 Varying Domain Tableaus CHAPTER EIGHT / EXISTENCE AND ACTUALIST QUANTIFICA- 265 12.7 Russell's Approach TION 163 273 12.8 Possibilist Quantifiers 8.1 To Be 163 275 8.2 Tableau Proofs 165 REFERENCES 8.3 The Paradox of NonBeing 167 277 8.4 Deflationists 169 INDEX 283 8.5 Parmenides' Principle 172 8.6 Inflationists 175 8.7 Unactualized Possibles 178 8.8 Barcan and Converse Barcan, Again 180 8.9 Using Validities in Tableaus 182 8.10 On Symmetry 185 CHAPTER NINE / TERMS AND PREDICATE ABSTRACTION 187 9.1 Why constants.should not be constant 187 o 'J ~rnnp 190 PREFACE After a rocky start in the first half of the twentieth century, modal logic hit its stride in the second half. The introduction of possible world semantics around the mid-century mark made the difference. Possible world semantics provided a technical device with intuitive appeal, and almost overnight the subject became something people felt they understood, rightly or wrongly. Today there are many books that deal with modal logic. But with very few ex­ ceptions, treatments are almost entirely of propositionai modal logic. By now this is a well-worked area, and a standard part of philosophical training. First­ order modal logic, on the other hand, is under-represented in the literature and under-developed in fact. This book aims at correcting the situation. To make our book self-contained, we begin with propositional modal 10- gic, and our presentation here is typical of the entire book. Our basic approach is semantic, using possible world, or Kripke, models. Proof-theoretically, our machinery is semantic tableaus, which are easy and intuitive to use. (We also include a treatment of propositional axiom systems, though axiom systems do not continue throughout the book.) PhilosophicalIy, we discuss the issues that motivate formal modal logics, and consider what bearing technical de­ velopments have on welI-known philosophical problems. This three-pronged approach, Kripke semantics, tableaus, philosophical discussion, reappears as each new topic is introduced. Classically, first-order issues like constant and function symbols, equality, quantification, and definite descriptions, have straightforward formal treat­ ments that have been standard items for a long time. Modally, each of these items needs rethinking. First-order modal logic, most decidedly, is not just propositional modal logic plus classical quantifier machinery. The. situation is much subtler than that. Indeed, many of the philosophical problems attending the development of modal logic have arisen because of the myopia of looking at it through classical lenses. For this reason we initially develop first-order modal logic without constants and functions. This arrangement separates the philosophical problems due to names and definite descriptions from those of quantifiers, and facilitates the development and understanding of first-order modal notions. Successive chapters are -arranged along the following lines. First, as we I said, we introduce the quantifiers, and address the difference between actual­ is! quantification (where quantifiers range over what actually exists) and pos­ i sibiIist quantification (where they range over what might exist). Semantically, I I xii PREFACE CHAPTER ONE this is the difference between varying domain models and constant domain models. (If you don't know what this means, you will.) Next, we introduce equality. We discuss Frege's well-known morning star/evening star puzzle in PROPOSITIONAL MODAL LOGIC this modal context, while presenting a formal treatment of equality. We then return to the matter of actualist and possibilist quantification and show how to embed possibilist quantification in an actualist framework and conversely. By this point in the exposition we are dealing both with a defined notion of For analytic p~ilosophy, formalization is a fundamental tool for clarifying existence as well as a primitive notion, and the time is ripe for a full-fledged language, leadl.ng ~o rn:tter understanding of thoughts expressed through lan­ discussion of the logical problems with existence and, in particular, of the guage. Formahzatton mvolves abstraction and idealization. This is true in idea that there are things that could exist but don 'I. Only after all of this has the sciences as well as in philosophy. Consider physics as a representative been examined are constants and function names introduced, along with the exa~ple. N.ewton's laws of .motion formalize certain b,asic aspects of the idea of predicate abstraction and rigidity. ?hyslcal umve:se. Mathematical abstractions are introduced that strip away . Predicate abstraction is perhaps the most central notion of our treatmenl. Irrelevant detatls of the real universe, but which lead to a better understand, This is -a syntactic mechanism for abstracting a predicate from a formula, ing of its "deep" structure. Later Einstein and others proposed better models in effect providi.ng a scoping mechanism for constants and function symbols than ~at of ~ewton, reflecting deeper understanding made possible by the similar to that provided for variables by quantifiers. Using predicate abstrac­ expenence gamed through years of working with Newton's model. tion we are able to deepen the unilerstanding of singular terms and reference, "Model" is the key word here. A mathematicai model of the universe is enabling distinctions that were either not visible or only dimly visible clas­ not the univ.erse. It is. ~ aid to the understanding-an approximately correct, sically. Since the variable binding of predicate abstracts behaves similarly to t~ough parttal, descnptlOn of the universe. As the Newton/Einstein progres­ the variable binding of quantifiers, analogues to some of the issues.
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