模态逻辑 数理逻辑

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一本详细的关于模态逻辑的书 一本经典的研究计算机科学中的数理逻辑的必读物
A NEW INTRODUCTION TO MODAL LOGIC G.E.正 ughes Late professor of philosophy Victoria University of wellington M. J. Cresswell Professor of Philosophy Victoria Universiry of wellington London and New york First published 1996 by routledge 11 New fetter lane London ec4p ee Simultaneously published in the usa and Canada by routledge 29 West 35th Street, New York, NY 10001 Reprinted 1998 G 1996 M. J. Crcsswcll and the estate of G.E. hughes Typeset in Times by M. Cresswell Printed and bound in Great Britain by T J Intemational Ltd All rights reserved No part of this book may be reprinted or reproduced or utilized in any form or by any electronic 1. or other means, now known or hereafte vented, including photocopying and recording, or in any information storage or retrieval systcm, without permission in writing fro British library Cataloguing in Publication Data A catalogue record for this book is available from the British library Library of co loguing in publication data A catalogue record for this book ha quest SBN0-45-125995(bk) SBN0-415-12600-2(pbk) CONTENTS Preface Part One: Basic Modal propositional logic 1 The Basic notions 3 The language of Pc (3)Interpretation (4) Further operate terpretation of A,> and =(7)Validity 8)Testing for validity: () the truth-table method(10) Testing for validity:(ii)the Reductio method (11) Some valid wff of PC (13)Basic modal notions(13)The language of propositional modal logic(16) validity in propositional modal logic(17) Exercises -1(21)Notes(22) 2 The Systems K. T and D Systems of modal logic(23) The system K(24)Proofs of theorems(26) L and M (33)Validity and soundness(36) The system T (41)A definition of validity for T(43)The system D(43)A note on derived rules(45) Consistency(46) Constant wff (47) Exercises-2(48) Notes(49) 3 The Systems S4, S5, B, Triv and Ver Iterated modalities(51)The system S4(53) Modalities in S4(54)Validity for S4 (56)The systen S5(58) Modalities in S5(59)Validity for S5(60 The Brouwerian system(62) Validity for B(63) Some other systems(64) Collapsing into PC (64) Exercises-3(68) Notes (70) 4 Testing for validity 72 Semantic diagrams(73)Altematives in a diagram(80)S4 diagrams(85) S5-diagrams(91)Exercises-4(92)Notes (93) A NEW INTRODUCTION TO MODAL LOGK 5 Conjunctive Normal Form 94 Equivalence transformations(94)Conjunctive normal form(96)Moda functions and modal degree (97)S5 reduction theorem (98) MCNF theorem(101) Testing formulae in MCNF (103)The completeness of S5 (105) A decision procedure for S5-validity( 108 )Triv and Ver again(108) Exercises 5(110)Notes(110) 6 Completeness Maximal consistent scts of wff (113)Maximal consistent extensions( 114) Consistent sets of wff in modal systems(116)Canonical models(117)The completeness of K, T, B, S4 and S5(119) Triv and Ver again( 121) Exercises -6(122)Notes(123) Part TwO: Normal Modal systems 7 Canonical Models 127 Temporal interpretations of modal logic (127) Ending time (131) Convergence( 134)The frames of canonical models(136)A non-canonical system(139)Exercises- 7(141)Notes(142) 8 Finite Models 145 The finite model property (145) Establishing the finite model property (145)The completeness of Kw(150) Decidability(152)Systems without finite model property (153)Ey 8(156) Notes(156) 9 Incomplet 159 Frames and models(159)An incomplete modal system( 160)KH and KW (164)Completeness and the finite model property(165)General frames (166)What might we understand by incompleteness?(168)Er 9(169) Notes(l70) 10 Frames and Sys 172 Frames for T, S4, B and S5(172)Reflexiveness(176)Compactness (177)54.3.(79)First-order definability(181)Second-order logic(188) Exercises- 10(189)Notes(190) CONTENTS 11 Strict Implication 193 Historical preamble( 193) The'paradoxes of implication(194)Material and strict implication (195) The ' Lewis' systems(197) The system SI (198)Lemmon's basis for SI(199)The system S2(200) The system S3 (200) Validity in S2 and S3(201) Entailment(202)Exercises -11(205) Notes(206 12 Glimpses Beyond 210 Axiomatic PC (210) Natural deduction(211)Multiply modal logics(217) The expressive power of multi-modal logics(219)Propositional symbols 220) Dynamic logic(220) Neighbourbood semantics(221)Intermediate logics(224)'Syntactical approaches to modality (225) Probabilistic semantics(227)Algebraic semantics(229)Exercises -12(229)Notes (230) Part Three: Modal predicate logic 13 The Lower Predicate Calculus 235 Primitive symbols and formation rules of non-modal LPC(235) Interpretation(237)The Principle of replacement (240) Axiomatization (241)Some theorems of LPC (242) Modal LPC (243)Semantics for modal Lpc(243)Systems of modal predicate logic(244)Theorems of modal LPC (244)Validity and soundness (247) De re and de dicto( 250) Exercises- 13(254)Notes(255) 14 The Completeness of Modal LPC 256 Canonical models for Modal LPC (256)Completeness in modal LPC (262) Incompleteness (265)Other incompleteness results(270)The monadic modal LPC (271)Exercises-14(272)Notes(272) 15 Expanding domins 274 Validity without the barcan Formula (274)Undefined formulae(277) Canonical models without BF (280)Completeness(282)Incompleteness without the Barcan Formula(283)LPC +$4.4(S4.9)(283)Exercises 15(287) Notes(287 vIl A NEW INTRODUCTION TO MODAL LOGIC 16 Modality and Existence 289 Changing domains(289)The existence predicate(292)Axiomatization of systems with an existence predicate(293)Completeness for existence predicates (296) Incompleteness (302) Expanding languages (302 Possibilist quantification revisited ( 303)Krpke-style systems 304) Completeness of Kripke-style systems(306)Exercises-16(309)Notes (310) 17 Identity and Descriptions 312 Identity in LPC (312)Soundness and completeness(314)Definite descriptions (318) Descriptions and scope (323) Individual constants and function symbols(327) Exercises- 17(328)Notes(329) 18 Intensional Obiects 330 Contingent identity(330)Contingent identity systems(334)quantifying over all intensional objects(335)Intensional objects and descriptions( 342) Intensional predicates(344)Exercises-18(347)Notes(348) 19 Further issues 349 First-order nodal theories (349)Multiple indexing(350)Counterpart theory (353)Counterparts or intensional objects?(357)Notes(358) Axioms, Rules and Systems 359 Axioms for normal systems(359)Some normal systems(361)Non normal systems(363)Modal predicate logic(365) Table 1: Normal Moda Systems(367) Table II: No l Modal Systems (368 Solutions to selected Exercises 369 Bibliography 384 Index 398 PREFACE Modal logic is the logic of necessity and possibility, of"mist be' anday e,. By this is meant that it t only truth and falsi wlat is or is not so as things actually stand, but considers what would be so if things were different. If we think af how things are as the actual world hell we may think of how things ught have been as how things are in an ternative non-actual but possible, state of affairs -or possible world Logic is concenel with truth and falsity, In modal logic we are conceme with truth or falsity u other possible works as well as the real one. In this sense a proposition will be necessary n a world if it is true in all worlds sible relative to that world and possiblc in a world if it is ti in at least one workl possible reletive to that workl. all this is explained the first chapter of this book. Our ain in this book is to introduce readers to nodal logic, and we assullle that to begin with the reader knows nothing of inodlal logic. We have attempted ta make the book self contained so that it could even be tackled by someone who had not studie any logic at all. However,we anticipate that most readers will already know a little about the(non-modal) propositional and predicate calculi, and will be able to use tlus knowlelge as a foundation for understanding Unocal logic This book is untended as a replacenent for our earlier two books An Tnurolnction to Molal Logic(Hughes ad Cresswell, 1968, IML)and A Cormpaniorl to Modal Logic(Hughes and Cresswell, 1984, CML) and we shall here say a little about the relation between it and the earlier books Part I covers most of the ground covered in IML with two important changes. First, as in CML, we take the systeln K as basic rather than T. Second as also in CML, we have (u Chapter 6) used the inethod of canonical odels to prove completeness. We have retained(in Chapter 5) the nethod of nodal conjunctive normal forrs to prove the completeness of s5, but wlule (in Chapter 4)we have retained from IML the nethod of semantic diagrams for testing formulae, we llave omitted the completeness proofs based on this nethod Part II covers a range of topics 1l mocal propositional logic, mlost of wluch are also discussed in CML. In the present work we have attcmpptel to be particilarly sensilive to its iole as an introcliction. thlis, to take one A NEW NTRODUCTION TO MODAL LOGIC example, our approach to fuite models is one that we believe is easier to follow than the more standard method of filtrations which we used in CML Although this part of the book may be seen as more of interest to specialists we have tried to present its topics in a way which should be easily accessible to the reader who has followed Part I. Part Ill of imi contained a survey of modal logic as it was in 1968. A comparable survey would be impossible today but we have attempted, in Chapter 1l of the present book, to give an outline of the more important developments in the earlier history of modal logic. Readers who need more may be referred to IML Part Il was the most difficult to write. Modal predicate logic is rightly regarded as the most philosophically important branch of modal logic, and although this is a book on formal logic not philosophical logic, we have attempted to discuss topics which have a bearing on such important philosophical questions as what to say about things which exist in one world but not another or about things claimed to be identical but not necessarily sO.Unfortunately, the semantics of odal predicate logic is extremely complicated, and while we have tried to make our discussions as approachable as we can, we are conscious of the burden imposed on the reader. All we can say is that we have attempted to set out all technical naterial so that with patience a reader should be able to follow every proof without requiring more than is in this book George Hughes died on 4 March 1994. At the time of his death we had completed the first five chapters. Chapter 6 and most of Part Il has been clapted from CML, and we had discussed nany issues in that area. In preparing flie manuscript I have endeavoured, to the best of my ability, to write it As a joint work and present it in a style as close as I can to what would have emerged had George Hughes lived to see its completion. It is n Part II that I have felt the greatest lack of his collaboration, and I am grateful especially to Rob Goldblatt and Edwin Mares here in Wellington who have looked at and commented on nany passages. We would also thank various readers, and colleagues froin around the world, whose'wish lists'we liave not al ways been able to take as much note of as they would ur depArtment secretary, Debbie Luyinda, put the initial manuscript into the computer, so that we could then play with it, and we would thank her for this, at times frustrating, work. Wellington, Ne January 1995

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模态逻辑 数理逻辑

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