Modal Logic ALEXANDER CHAGROV. Tver State University. and MICHAEL ZAKHARYASCHEV

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1 Modal Logic ALEXANDER CHAGROV Tver State University and MICHAEL ZAKHARYASCHEV Moscow State University and Institute of Applied Mathematics Russian Academy of Sciences CLARENDON PRESS OXFORD 1997

2 CONTENTS I INTRODUCTION Classical logic 3 Syntax and semantics 3 Semantic tableaux 6 Classical calculus 9 Basic properties of Cl 15 Exercises 19 Notes 21 Intuitionistic logic 23 Motivation 23 Kripke frames and models 25 Truth-preserving operations 28 Hintikka systems 35 Intuitionistic frames and formulas 40 Intuitionistic calculus -^.. 45 Embeddings of Cl into Int 46 Basic properties of Int 49 Realizability logic and Medvedev's logic 52 Exercises \ 54 Notes ' 56 3 Modal logics Possible world semantics Modal frames and models Truth-preserving operations \ Hintikka systems Modal frames and formulas Calculus K Basic properties of K A few more modal logics Embeddings of Int into S4, Grz and GL Other types of modal logics Exercises Notes - 105

3 xii CONTENTS 4 From logics to classes of logics Superintuitionistic logics Modal logics "The roads we take" Exercises and open problems Notes 125 II KRIPKE SEMANTICS 5 Canonical models and filtration The Henkin construction Completeness theorems The filtration method Diego's theorem Selective nitration Kripke semantics for quasi-normal logics Exercises Notes Incompleteness 6.1 Logics that are not finitely approximable 6.2 Logics that are not canonical and elementary 6.3 Logics that are not compact and complete 6.4 A calculus that is not Kripke complete 6.5 More Kripke incomplete calculi 6.6 Complete logics without countable characteristic frames 6.7 Exercises and open problems 6.8 Notes III ADEQUATE SEMANTICS 7 Algebraic semantics 7.1 Algebraic preliminaries 7.2 The Tarski-Lindenbaum construction 7.3 Pseudo-Boolean algebras 7.4 Filters in pseudo-boolean algebras 7.5 Modal algebras and matrices 7.6 Varieties of algebras and matrices 7.7 Operations on algebras and matrices 7.8 Internal characterization of varieties 7.9 Exercises 7.10 Notes

4 CONTENTS xiii Relational semantics 235 General frames 235 The Stone and Jonsson-Tarski theorems 241 From modal to intuitionistic frames and back Descriptive frames 250 Truth-preserving operations on general frames 258 Points of finite depth in refined finitely generated frames 267 Universal frames of finite rank 272 Exercises and open problems 279 Notes Canonical formulas Subreduction Cofinal subreduction and closed domain condition Characterizing transitive refutation frames Canonical formulas for K4 and Int Quasi-normal canonical formulas Modal companions of superintuitionistic logics Exercises and open problems Notes 332 IV PROPERTIES OF LOGICS 10 Kripke completeness The method of canonical models revised P-persistence and elementarity " Sahlqvist's theorem Logics of finite width The degree of Kripke incompleteness of logics NExtK Exercises and open problems \ Notes Finite approximability Uniform logics Si-logics with essentially negative axioms and modal logics with DO-axioms " Subframe and cofinal subframe logics Quasi-normal subframe and cofinal subframe logics The method of inserting points The method of removing points Exercises and open problems Notes Tabularity Finite axiomatizability of tabular logics 417

5 xiv CONTENTS 12.2 Immediate predecessors of tabular logics 12.3 Pretabular logics 12.4 Some remarks on local tabularity 12.5 Exercises and open problems 12.6 Notes 13 Post completeness 13.1 m-reducibility 13.2 O-reducibility, Post completeness and general Post completeness 13.3 Exercises and open problems 13.4 Notes 14 Interpolation 14.1 Interpolation theorems for certain modal systems 14.2 Semantic criteria of the interpolation property 14.3 Interpolation in logics above LC and S Interpolation in Extlnt and NExtS Interpolation in extensions of GL 14.6 Exercises and open problems 14.7 Notes 15 The disjunction property and Hallden completeness 15.1 Semantic equivalents of the disjunction property 15.2 The disjunction property and the canonicalfprmulas 15.3 Maximal si-logics with the disjunction property 15.4 Hallden completeness 15.5 Exercises and open problems 15.6 Notes \ V ALGORITHMIC PROBLEMS 16 The decidability of logics 16.1 Algorithmic preliminaries 16.2 Proving decidability 16.3 Logics containing K4.3, 16.4 Undecidable calculi and formulas above K Undecidable calculus and formula in Extlnt 16.6 The undecidability of the semantical consequence problem on finite frames 16.7 Admissible and derivable rules 16.8 Exercises and open problems 16.9 Notes

6 CONTENTS xv 17 The decidability of logics' properties A trivial solution Decidable properties of calculi Undecidable properties of modal calculi Undecidable properties of si-calculi Exercises and open problems Notes Complexity problems Complexity function. Kuznetsov's construction Logics that are not polynomially approximable Polynomially approximable logics Extremely complex logics of finite width and depth Algorithmic problems and complexity classes Exercises and open problems Notes 564 Bibliography 567 Index 597

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