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  • Correspondence and Canonicity

    in Non-Classical Logic

    Sumit Sourabh

    C o rresp

    o n d en

    ce an d C

    an o n icity

    in N

    o n -C

    lassical L o g ic

    S u m

    it S o u rab

    h

    INSTITUTE FOR LOGIC LANGUAGE AND COMPUTATION

  • Correspondence and Canonicity in Non-Classical Logic

    Sumit Sourabh

  • Correspondence and Canonicity in Non-Classical Logic

  • ILLC Dissertation Series DS-2015-04

    For further information about ILLC-publications, please contact

    Institute for Logic, Language and Computation Universiteit van Amsterdam

    Science Park 107 1098 XG Amsterdam

    phone: +31-20-525 6051 e-mail: [email protected]

    homepage: http://www.illc.uva.nl/

    The investigations were partially supported by an Erasmus Mundus scholarship from the European Commission.

    Copyright c© 2015 by Sumit Sourabh

    Cover art: White Bridge, Leonid Afremov. Source: http://www.afremov.com. Printed and bound by GVO | Ponsen & Looijen, Ede.

    ISBN: 978-90-6464-892-2

  • Correspondence and Canonicity in Non-Classical Logic

    Academisch Proefschrift

    ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam

    op gezag van de Rector Magnificus prof.dr. D.C. van den Boom

    ten overstaan van een door het college voor promoties ingestelde commissie, in het openbaar

    te verdedigen in de Aula der Universiteit op woensdag 9 september 2015, te 13.00 uur

    door

    Sumit Sourabh

    geboren te Begusarai, India.

  • Promotor: Prof. dr. Y. Venema Co-promotors: Dr. A. Palmigiano

    Dr. N. Bezhanishvili

    Overige leden: Dr. A. Baltag Prof. dr. J.F.A.K. van Benthem Dr. W.E. Conradie Prof. dr. S. Ghilardi Prof. dr. V. Goranko Prof. dr. D.H.J. de Jongh

    Faculteit der Natuurwetenschappen, Wiskunde en Informatica

  • To my Parents

    v

  • Contents

    Acknowledgments xi

    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

    vii

  • 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

    viii

  • 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 Jónsson-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 µ-inequalities . . . . . . . . . . . . . . . . . . 207