Multi-Dimensional Modal Logic

1 Multi-dimensional modal logic.- 1.1 What is multi-dimensional modal logic?.- 1.2 Manifestations of multi-dimensional modal logics.- 1.3 Themes and questions.- 1.4 Overview of the book.- 1.5 How to read this book.- 2 Two-dimensional modal logics.- 2.1 Operations on the square universe.- 2.2 Axiomatizing S5-square.- 2.3 Cylindric modal logic of squares.- 2.4 The modal logic of composition.- 2.5 A two-dimensional temporal logic.- 2.6 Historical notes.- 3 Arrow logic.- 3.1 Introduction.- 3.2 Motivation.- 3.3 Arrow logic and relation algebras.- 3.4 Connection with first order logic.- 3.5 Characterizing (local) squares.- 3.6 Axiomatizing (local) squares.- 3.7 Decidability and interpolation.- 3.8 Temporal arrow logic.- 3.9 Other directions in arrow logic.- 4 Modal logics of intervals.- 4.1 Introduction.- 4.2 The System HS: Introduction.- 4.3 The system HS: expressiveness.- 4.4 The System HS: Axiomatics.- 5 Modal logics of relations.- 5.1 Introduction.- 5.2 Modalizing first-order logic.- 5.3 Abstract and generalized assignment frames.- 5.4 Characterizing cubes and local cubes.- 5.5 Meta-properties.- 5.6 Infinite dimensions.- 5.7 Connections.- 6 Multi-dimensional semantics for every modal language.- 6.1 Logics with one modality.- 6.2 Logics with arbitrary many modalities.- 6.3 Versatile similarity types.- 6.4 The modal logic of composition and its conjugates.- Open problems.- Appendices.- A Modal Similarity Types.- A.1 Introduction.- A.2 Modal similarity types.- A.3 Frames, models and correspondence.- A.4 Structural frame operations.- A.5 Boolean S-algebras.- A.6 Frames and algebras.- A.7 Modal logics and derivation systems.- A.8 Algebraic derivations.- A.9 Canonical structures.- B A Modal Toolkit.- B.1 Sahlqvist theory.- B.1.1 Definitions.- B.1.2 Sahlqvist correspondence.- B.1.3 Canonicity & completeness.- B.1.4 Algebraic aspects of Sahlqvist theory.- B.2 Logical operators.- B.2.1 The universal modality.- B.2.2 Versatile similarity types.- B.2.3 The D-operator.- B.3 Negative definability and unorthodox axiomatics.- B.4 Interpolation.- B.5 Filtrations.- B.6 A local and a global paradigm.- List of symbols.