This is a PhD Thesis of Sumit Sourabh written under supervision of Prof. Dr. Yde Venema.
The focus and main contribution of this thesis is the understanding of the mechanisms
underlying correspondence and canonicity, and their application to the
development of a uniform correspondence and canonicity theory for a wide family
of non-classical logics which includes but is not limited to regular distributive
modal logics, and bi-intuitionistic modal mu-calculus. Uniformity is the critical
feature of our results, and it is made possible by our methodology, which crucially
relies on algebraic and order-topological notions and tools.
Author(s): Sumit Sourabh
Publisher: University of Amsterdam
Year: 2015
Language: English
Pages: 258
City: Amsterdam
1 Introduction 1
1.1 Outline of chapters . . . . . . . 13
2 Sahlqvist correspondence and canonicity 17
2.1 Modal logic . . . . . . . . . 17
2.2 Correspondence . . . . . . . . . 22
2.3 Syntactic classes . . . . . . . . 23
2.4 Algorithmic strategies . . . . . . . 26
2.5 Duality and Canonicity . . . . . . . 31
3 Basic algebraic modal correspondence 33
3.1 Preliminaries . . . . . . . . . 34
3.1.1 Meaning Function . . . . . . . 34
3.1.2 Definite implications . . . . . . 35
3.2 Algebraic correspondence . . . . . . . 36
3.2.1 The general reduction strategy . . . . . 37
3.2.2 Uniform and Closed formulas . . . . . 38
3.2.3 Very simple Sahlqvist implications . . . . 40
3.2.4 Sahlqvist implications . . . . . . 44
3.3 Conclusions . . . . . . . . . 49
4 Algorithmic correspondence and canonicity for regular modal logic 51
4.1 Preliminaries . . . . . . . . . 52
4.1.1 Regular modal logics . . . . . . 53
4.1.2 Kripke frames with impossible worlds and their complex algebras . . . . . . . . . 55
4.1.3 Algebraic semantics . . . . . . . 56
4.1.4 The distributive setting . . . . . . 57
4.1.5 Canonical extension . . . . . . . 58
4.1.6 Adjoints and residuals . . . . . . 59
4.2 Algebraic-algorithmic correspondence . . . . . 61
4.2.1 The basic calculus for correspondence . . . 64
4.3 ALBA on regular BDL and HA expansions . . . . 66
4.3.1 The expanded language L+ . . . . . 66
4.3.2 The algorithm ALBAr . . . . . . 68
4.3.3 Soundness and canonicity of ALBAr . . . . 71
4.4 Sahlqvist and Inductive DLR- and HAR- inequalities . . 73
4.5 Applications to Lemmon's logics . . . . . . 78
4.5.1 Standard translation . . . . . . 79
4.5.2 Strong completeness and elementarity of E2-E5 . . 80
4.6 Conclusions . . . . . . . . . 84
5 Algorithmic correspondence for intuitionistic modal mu-calculus 87
5.1 Preliminaries . . . . . . . . . 88
5.1.1 The bi-intuitionistic modal mu-language and its semantics 88
5.1.2 Perfect modal bi-Heyting algebras . . . . 91
5.2 ALBA for bi-intuitionistic modal mu-calculus: setting the stage . . . . . . 92
5.2.1 Preservation and distribution properties of extremal fixed points . . . . . . . . . 92
5.2.2 Approximation rules and their soundness . . . 96
5.2.3 Adjunction rules and their soundness . . . 97
5.2.4 Recursive Ackermann rules . . . . . 98
5.2.5 From semantic to syntactic rules . . . . . 99
5.3 Recursive mu-inequalities . . . . . . . 99
5.3.1 Recursive mu-inequalities . . . . . 100
5.3.2 General syntactic shapes and a comparison with existing Sahlqvist-type classes . . . . . . 102
5.4 Inner formulas and their normal forms . . . . . 105
5.4.1 Inner formulas . . . . . . . 105
5.4.2 Towards syntactic adjunction rules . . . . 107
5.4.3 Normal forms and normalization . . . . . 108
5.4.4 Computing the adjoints of normal inner formulas . 111
5.5 Adjunction rules for normal inner formulas . . . . 115
5.6 Examples . . . . . . . . . 119
5.7 Conclusions . . . . . . . . . 124
6 Pseudocorrespondence and relativized canonicity 125
6.1 Preliminaries . . . . . . . . . 127
6.1.1 Language, basic axiomatization and algebraic semantics of DLE and DLE . . . . . . . 127
6.1.2 Inductive DLE and DLE inequalities . . . 128
6.2 Pseudo-correspondence and relativized canonicity and correspondence . . . . . . 130
6.3 An alternative proof of the canonicity of additivity . . 135
6.3.1 A purely order-theoretic perspective . . . . 135
6.3.2 Canonicity of the additivity of DLE-term functions . 140
6.4 Towards extended canonicity results: enhancing ALBA . . 141
6.5 Meta-inductive inequalities and success of ALBAe . . . 145
6.6 Relativized canonicity via ALBAe . . . . . 146
6.7 Examples . . . . . . . . . 147
6.8 Conclusions . . . . . . . . . 149
7 Subordinations, closed relations, and compact Hausdorff spaces151
7.1 Preliminaries . . . . . . . . . 152
7.2 Subordinations on Boolean algebras . . . . . 154
7.3 Subordinations and closed relations . . . . . 156
7.4 Subordinations, strict implications, and Jonsson-Tarski duality . 160
7.5 Modally definable subordinations and Esakia relations . . 163
7.6 Further duality results . . . . . . . 166
7.7 Lattice subordinations and the Priestley separation axiom . 169
7.8 Irreducible equivalence relations, compact Hausdorff spaces, and
de Vries duality . . . . . . . . . 171
7.9 Conclusion . . . . . . . . . 175
8 Sahlqvist preservation for topological fixed-point logic 177
8.1 Preliminaries . . . . . . . . . 179
8.2 Topological fixed-point semantics . . . . . 181
8.2.1 Open fixed-point semantics . . . . . 184
8.3 Algebraic semantics . . . . . . . 186
8.4 Sahlqvist preservation . . . . . . . 189
8.4.1 An alternative fixed-point semantics . . . 189
8.4.2 Esakia's lemma . . . . . . . 194
8.4.3 Sahlqvist formulas . . . . . . . 196
8.5 Sahlqvist correspondence . . . . . . . 197
8.6 Conclusion . . . . . . . . . 201
A Success of ALBA on inductive and recursive inequalities 203
A.1 ALBAr succeeds on inductive inequalities . . . . 203
A.2 Success on recursive -inequalities . . . . . 207
A.3 Preprocess, first approximation and approximation . . 207
A.4 Application of adjunction rules . . . . . . 210
A.5 The "-Ackermann shape . . . . . . . 211
B Topological Ackermann Lemmas 213
B.1 Intersection lemmas . . . . . . . 213
B.2 Topological Ackermann Lemma . . . . . . 216
Bibliography 219
Index 231
Samenvatting 235
Abstract 237
