β Scribed by Ulrich Endriss
No coin nor oath required. For personal study only.
1 Zooming in 1
1.1 Adding a Zoom to Linear Temporal Logic . . . . . . . . . . . . . . . . . . 1
1.2 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Modal and Temporal Logics 13
2.1 Introducing Modalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Possible World Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Axioms and Correspondences . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4 Temporal Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.5 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3 Semantic Characterisation 39
3.1 Syntax and Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Some Defined Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3 Correspondences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.4 Ontological Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.5 General Models and P-morphisms . . . . . . . . . . . . . . . . . . . . . . 60
3.6 Loosely Ordered Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4 Axiomatic Characterisation 97
4.1 Axioms and Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.2 Some Derived Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.3 Completeness without Descendant Modalities . . . . . . . . . . . . . . . . 113
4.4 Alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
5 Decidability 147
5.1 Monadic Second-order Logic . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.2 Finite Binary Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
5.3 Discretely Ordered Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
5.4 General Ordered Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
6 Zooming out 185
6.1 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
6.2 Summary of Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
6.3 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
Appendices 197
A Relations and Orders 199
A.1 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
A.2 Linear Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
B Derivations 203
B.1 Proof of Lemma 4.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
Bibliography 205
List of Figures 211
List of Tables 213
Index 215
π SIMILAR VOLUMES
This is a doctoral dissertation of Edith Spaan under the supervision of prof. Johan van Benthem.
In this thesis we investigate modal logics from a lattice theoretic point of view. There are essentially two well-known methods of research on lattices of modal logics. The investigation of the lattice of modal logics as a whole, and the local investigation of the lattice of extensions of some stron
In this thesis we investigate modal logics from a lattice theoretic point of view. There are essentially two well-known methods of research on lattices of modal logics. The investigation of the lattice of modal logics as a whole, and the local investigation of the lattice of extensions of some stron
This is a doctoral dissertation of Edith Spaan under the supervision of prof. Johan van Benthem.
This dissertation is about extending modal logic. It tells you what a system of extended modal logic is, it gives you three case studies of systems of modal logic, and it gives you very general approaches to two important themes in modal logic.
This dissertation is about extending modal logic. It tells you what a system of extended modal logic is, it gives you three case studies of systems of modal logic, and it gives you very general approaches to two important themes in modal logic.