دانلود کتاب Lattices of intermediate and cylindric modal logics [PhD Thesis]

فهرست مطالب :

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 cofinal subframe formulas . . . . 59 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 finite model property in extensions of RN . . . . . . . . . . . 98 4.5 The finite model property in extensions of KG . . . . . . . . . . . 105 4.5.1 Extensions of KG without the finite model property . . . 105 4.5.2 The pre-finite 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 Jonsson-Tarski representation . . . . . . . . . . . . . . . . 127 5.2 Many-dimensional modal logics . . . . . . . . . . . . . . . . . . . 130 5.2.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . 130 5.2.2 Products of modal logics . . . . . . . . . . . . . . . . . . . 131 5.3 Cylindric modal logics . . . . . . . . . . . . . . . . . . . . . . . . 132 5.3.1 S5 x 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 Df2-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 140 5.4.2 Topological representation . . . . . . . . . . . . . . . . . . 141 5.4.3 CA2-algebras . . . . . . . . . . . . . . . . . . . . . . . . . 143 5.4.4 Representable cylindric algebras . . . . . . . . . . . . . . . 145 6 Normal extensions of S52 149 6.1 The finite model property of S52 . . . . . . . . . . . . . . . . . . 149 6.2 Locally tabular extensions of S52 . . . . . . . . . . . . . . . . . . 155 6.3 Classification of normal extensions of S52 . . . . . . . . . . . . . . 160 6.4 Tabular and pre tabular extension of S52 . . . . . . . . . . . . . . 161 7 Normal extensions of CML2 167 7.1 Finite CML2-frames . . . . . . . . . . . . . . . . . . . . . . . . . 167 7.1.1 The finite model property . . . . . . . . . . . . . . . . . . 167 7.1.2 The Jankov-Fine formulas . . . . . . . . . . . . . . . . . . 170 7.1.3 The cardinality of Λ(CML2) . . . . . . . . . . . . . . . . . 172 7.2 Locally tabular extensions of CML2 . . . . . . . . . . . . . . . . 173 7.3 Tabular and pre-tabular extensions of CML2 . . . . . . . . . . . 178 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