Table Of Content

Lattices of intermediate and cylindric modal logics Nick Bezhanishvili Lattices of intermediate and cylindric modal logics ILLC Dissertation Series DS-2006-02 For further information about ILLC-publications, please contact Institute for Logic, Language and Computation Universiteit van Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam phone: +31-20-525 6051 fax: +31-20-525 5206 e-mail: [email protected] homepage: http://www.illc.uva.nl/ Lattices of intermediate and cylindric modal logics Academisch Proefschrift ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam op gezag van de Rector Magniflcus prof.mr. P.F. van der Heijden ten overstaan van een door het college voor promoties ingestelde commissie, in het openbaar te verdedigen in de Aula der Universiteit op vrijdag 17 maart 2006, te 12.00 uur door Nikoloz Bezhanishvili geboren te Tbilisi, Georgi˜e. Promotor: Prof.dr. D.H.J. de Jongh Co-promotor: Dr. Y. Venema Faculteit der Natuurwetenschappen, Wiskunde en Informatica Copyright (cid:176)c 2006 by Nick Bezhanishvili Printed and bound by PrintPartners Ipskamp ISBN: 90{5776{147{5 Contents 1 Introduction 1 I Lattices of intermediate logics 9 2 Algebraic semantics for intuitionistic logic 11 2.1 Intuitionistic logic and intermediate logics . . . . . . . . . . . . . 11 2.1.1 Syntax and semantics . . . . . . . . . . . . . . . . . . . . . 11 2.1.2 Basic properties of intermediate logics . . . . . . . . . . . 17 2.2 Heyting algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.1 Lattices, distributive lattices and Heyting algebras . . . . . 19 2.2.2 Algebraic completeness of IPC and its extensions . . . . . 23 2.2.3 Heyting algebras and Kripke frames . . . . . . . . . . . . . 26 2.3 Duality for Heyting algebras . . . . . . . . . . . . . . . . . . . . . 28 2.3.1 Descriptive frames . . . . . . . . . . . . . . . . . . . . . . 28 2.3.2 Subdirectly irreducible Heyting algebras . . . . . . . . . . 32 2.3.3 Order-topological duality . . . . . . . . . . . . . . . . . . . 33 2.3.4 Duality of categories . . . . . . . . . . . . . . . . . . . . . 36 2.3.5 Properties of logics and algebras . . . . . . . . . . . . . . . 37 3 Universal models and frame-based formulas 39 3.1 Finitely generated Heyting algebras . . . . . . . . . . . . . . . . . 39 3.2 Free Heyting algebras and n-universal models . . . . . . . . . . . 46 3.2.1 n-universal models . . . . . . . . . . . . . . . . . . . . . . 46 3.2.2 Free Heyting algebras . . . . . . . . . . . . . . . . . . . . . 49 3.3 The Jankov-de Jongh and subframe formulas . . . . . . . . . . . . 56 3.3.1 Formulas characterizing point generated subsets . . . . . . 56 3.3.2 The Jankov-de Jongh theorem . . . . . . . . . . . . . . . . 58 3.3.3 Subframes, subframe and coflnal subframe formulas . . . . 59 v vi Contents 3.4 Frame-based formulas . . . . . . . . . . . . . . . . . . . . . . . . . 65 4 The logic of the Rieger-Nishimura ladder 79 4.1 n-conservative extensions, linear and vertical sums . . . . . . . . . 80 4.1.1 The Rieger-Nishimura lattice and ladder . . . . . . . . . . 80 4.1.2 n-conservative extensions and the n-scheme logics . . . . . 83 4.1.3 Sums of Heyting algebras and descriptive frames . . . . . . 85 4.2 Finite frames of RN . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.3 The Kuznetsov-Gerciu logic . . . . . . . . . . . . . . . . . . . . . 93 4.4 The flnite model property in extensions of RN . . . . . . . . . . . 98 4.5 The flnite model property in extensions of KG . . . . . . . . . . . 105 4.5.1 Extensions of KG without the flnite model property . . . 105 4.5.2 The pre-flnite model property . . . . . . . . . . . . . . . . 111 4.5.3 The axiomatization of RN . . . . . . . . . . . . . . . . . . 114 4.6 Locally tabular extensions of RN and KG . . . . . . . . . . . . . 117 II Lattices of cylindric modal logics 121 5 Cylindric modal logic and cylindric algebras 123 5.1 Modal Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 5.1.1 Modal algebras . . . . . . . . . . . . . . . . . . . . . . . . 126 5.1.2 Jo¶nsson-Tarski representation . . . . . . . . . . . . . . . . 127 5.2 Many-dimensional modal logics . . . . . . . . . . . . . . . . . . . 130 5.2.1 Basic deflnitions . . . . . . . . . . . . . . . . . . . . . . . . 130 5.2.2 Products of modal logics . . . . . . . . . . . . . . . . . . . 131 5.3 Cylindric modal logics . . . . . . . . . . . . . . . . . . . . . . . . 132 5.3.1 S5£S5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 5.3.2 Cylindric modal logic with the diagonal . . . . . . . . . . . 135 5.3.3 Product cylindric modal logic . . . . . . . . . . . . . . . . 137 5.3.4 Connection with FOL . . . . . . . . . . . . . . . . . . . . 139 5.4 Cylindric algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 5.4.1 Df -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 140 2 5.4.2 Topological representation . . . . . . . . . . . . . . . . . . 141 5.4.3 CA -algebras . . . . . . . . . . . . . . . . . . . . . . . . . 143 2 5.4.4 Representable cylindric algebras . . . . . . . . . . . . . . . 145 6 Normal extensions of S52 149 6.1 The flnite model property of S52 . . . . . . . . . . . . . . . . . . 149 6.2 Locally tabular extensions of S52 . . . . . . . . . . . . . . . . . . 155 6.3 Classiflcation of normal extensions of S52 . . . . . . . . . . . . . . 160 6.4 Tabular and pre tabular extension of S52 . . . . . . . . . . . . . . 161 Contents vii 7 Normal extensions of CML 167 2 7.1 Finite CML -frames . . . . . . . . . . . . . . . . . . . . . . . . . 167 2 7.1.1 The flnite model property . . . . . . . . . . . . . . . . . . 167 7.1.2 The Jankov-Fine formulas . . . . . . . . . . . . . . . . . . 170 7.1.3 The cardinality of ⁄(CML ) . . . . . . . . . . . . . . . . . 172 2 7.2 Locally tabular extensions of CML . . . . . . . . . . . . . . . . 173 2 7.3 Tabular and pre-tabular extensions of CML . . . . . . . . . . . 178 2 8 Axiomatization and computational complexity 187 8.1 Finite axiomatization . . . . . . . . . . . . . . . . . . . . . . . . . 188 8.2 The poly-size model property . . . . . . . . . . . . . . . . . . . . 195 8.3 Logics without the linear-size model property . . . . . . . . . . . 199 8.4 NP-completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 Bibliography 209 Index 219