Many-Dimensional Modal Logic [PhD Thesis]

Cover ......Page 1
Table of contents ......Page 5
Chapter 1. Dimensions in Modal Logic ......Page 7
Chapter 2. Rules for the Undefinable ......Page 11
2.1. Introduction ......Page 12
2.2. Sahlqvist theorems ......Page 15
2.3. Sahlqvist tense formulas ......Page 17
2.4. The D-operator ......Page 22
2.5. The main proof ......Page 25
2.6. Uni-directional complications ......Page 32
2.7. The SD-theorem ......Page 37
2.8. The SNS-theorem ......Page 42
2.9. Conclusions, remarks and questions ......Page 44
Chapter 3. The Square Universe ......Page 49
3.1. Introduction ......Page 50
3.2.1. Two-dimensional cylindric logic ......Page 54
3.2.2. Two-dimensional cylindric completeness ......Page 57
3.3.1. (Representable) relation algebras ......Page 60
3.3.2. A modal logic of binary relations ......Page 62
3.3.3. CCd and classical logic ......Page 63
3.3.4. CCd and relation algebras ......Page 65
3.3.5. Characterizing squares ......Page 69
3.3.6. Axiomatizing squares ......Page 71
3.3.7. Some arrow-arithmetic ......Page 73
3.4. A two-dimensional temporal logic ......Page 76
3.5.1. Two-dimensional cylindric algebras ......Page 80
3.5.2. Simplifying Henkin's equation ......Page 81
3.6. Conclusions, remarks and questions ......Page 84
Chapter 4. Quantifiers and Cubes ......Page 87
4.1. Algebraizing restricted first order logic ......Page 88
4.2. Cylindric modal logic ......Page 94
4.3. Characterizing n-cubes ......Page 99
4.4. Axiomatizing n-cubes ......Page 103
4.5. Harvest ......Page 105
4.6. Using the Dn-irreflexivity rule ......Page 108
4.7.1. General conclusions ......Page 114
4.7.2. Unrestricted first order logic ......Page 115
4.7.2. Interpolation and amalgamation ......Page 116
4.7.3. Questions ......Page 117
Chapter 5. Periods in Planes ......Page 119
5.1.1. Time in periods ......Page 120
5.1.2. Intervals as two-dimensional points ......Page 123
5.2.1. Introduction ......Page 125
5.2.2. Expressiveness ......Page 127
5.2.3. (In)Completeness ......Page 130
5.3. Intervals as computation paths ......Page 135
5.4. A modal operator for chopping intervals ......Page 139
5.5. Conclusions, remarks and questions ......Page 143
Chapter 6. Conclusions ......Page 145
Appendix A. Modal Similarity Types ......Page 149
A2. Similarity types ......Page 150
A3. Frames, models and correspondence ......Page 151
A4. Boolean S-algebras ......Page 153
A5. Frames and algebras ......Page 155
A6. Modal logics ......Page 156
A7. Algebraic derivations ......Page 158
A8. Canonical structures ......Page 160
A9. Versatility ......Page 161
Appendix B. Consequences of Derivation Systems ......Page 163
Index ......Page 167
Samenvatting ......Page 172
Bibliography ......Page 175
Acknowledgements ......Page 184