Many-Dimensional Modal Logic

This is a doctoral dissertation of Yde Venema under the supervision of prof. Johan van Benthem. Chapter 1. Dimensions in Modal Logic Chapter 2. Rules for the Undefinable 2.1. Introduction 2.2. Sahlqvist theorems 2.3. Sahlqvist tense formulas 2.4. The D-operator 2.5. The Main Proof 2.6. Uni-directional complications 2.7. The SD-theorem 2.8. The SNS-theorem 2.9. Conclusions, remarks and questions Chapter 3. The Square Universe 3.1. Introduction 3.2. Two-dimensional cylindric logic 3.3. A modal logic of binary relations 3.4. A two-dimensional temporal logic 3.5. Two-dimensional algebras 3.6. Conclusions, remarks and questions Chapter 4. Quantifiers and Cubes 4.1. Algebraizing restricted first order logic 4.2. Cylindric modal logic 4.3. Characterizing n-cubes 4.4. Axiomatizing n-cubes 4.5. Harvest 4.6. Using the Dn-irreflexivity rule 4.7. Conclusions, remarks and questions Chapter 5. Periods in Planes 5.1. Introduction 5.2. The system HS 5.3. Intervals as computation paths 5.4. A modal operator for chopping intervals 5.5. Conclusions, remarks and questions Chapter 6. Conclusions Appendix A. Modal Similarity Types Appendix B. Consequences of Derivation Systems