Multi-Dimensional Modal Logic

Modal Logic is a branch of logic with applications in many related disciplines such as computer science, philosophy, linguistics and artificial intelligence. Over the last twenty years, in all of these neighbouring fields, modal systems have been developed that we call multi-dimensional. (Our definition of multi-dimensionality in modal logic is a technical one: we call a modal formalism multi-dimensional if, in its intended semantics, the universe of a model consists of states that are tuples over some more basic set.)
This book treats such multi-dimensional modal logics in a uniform way, linking their mathematical theory to the research tradition in algebraic logic. We will define and discuss a number of systems in detail, focusing on such aspects as expressiveness, definability, axiomatics, decidability and interpolation. Although the book will be mathematical in spirit, we take care to give motivations from the disciplines mentioned earlier on.

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.