An Algebraic Approach to Incompleteness in Modal Logic [PhD Thesis]

1 Introduction 2
2 Basic notions 7
2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4 Wolter’s Splitting Theorem . . . . . . . . . . . . . . . . . . . . 18
3 Duality theory and minimal extensions 21
3.1 Basics of duality theory, general case . . . . . . . . . . . . . . 21
3.2 Duality theory for AV-baos . . . . . . . . . . . . . . . . . . . 27
3.3 Conservativity of minimal extensions . . . . . . . . . . . . . . 31
3.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4 Non-equivalence of completeness notions 40
4.1 A-inconsistency . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2 wC-inconsistency . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3 C-inconsistency . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.4 AV U T -inconsistency . . . . . . . . . . . . . . . . . . . . . . . 50
4.5 T -inconsistency . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5 The Generalized Blok Alternative 56
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.2 Degrees of wC-incompleteness . . . . . . . . . . . . . . . . . . 61
5.3 Degrees of AV U T -incompleteness . . . . . . . . . . . . . . . 69
5.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6 Subdirectly irreducible algebras 72
6.1 Subdirect indeterminacy of C . . . . . . . . . . . . . . . . . . 72
6.2 Positive results . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.3 S and master modality based on well-order . . . . . . . . . . . 76
6.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
7 Strong completeness 80
7.1 Local and global consequence . . . . . . . . . . . . . . . . . . 80
7.2 Application: modal definability for discrete frames . . . . . . . 82
7.3 Second-order character of C-consequence . . . . . . . . . . . . 84
8 Tense logics of linear time flows 86
8.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
8.2 AV-completeness result . . . . . . . . . . . . . . . . . . . . . . 87
8.3 !C-inconsistency once again . . . . . . . . . . . . . . . . . . . 92
8.4 Aside on computational complexity . . . . . . . . . . . . . . . 94
8.5 The gap between !C-completeness and C-completeness . . . . 100
8.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
9 Open problems 104
Appendix A: Algebraic and model-theoretical preliminaries 106
Appendix B: set-theoretical preliminaries 113
Appendix C: Second-order logic and strong consequence 121
References 131
Publications 137