This book provides a unified mathematical treatment of multi-dimensional modal logics, linking their theory to algebraic logic. The author, Maarten Marx, a researcher in modal logic, defines and discusses various multi-dimensional modal systems, focusing on their expressiveness, definability, axiomatics, decidability, and interpolation properties. The book is mathematical in spirit but includes motivations from computer science, philosophy, linguistics, and artificial intelligence, where these multi-dimensional systems have found applications. The core technical definition of multi-dimensionality used is that the universe of a model consists of states that are tuples over a more basic set.

The book is presented as a comprehensive and unified treatment of multi-dimensional modal logics, aimed at a mathematically inclined audience. Its focus on technical aspects like expressiveness and decidability suggests it is a resource for researchers and advanced students in logic, computer science, and philosophy. The integration of motivations from various applied fields indicates an effort to bridge theoretical advancements with practical relevance.