Modal logic: a semantic perspective

This chapter discusses the semantic ideas underlying modern modal logic, and in particular, Kripke semantics—or relational semantics. It introduces the basic model theoretic constructions, explores links between modal logic and classical logic, both on models and on frames, and examines the extent to which the key semantic ideas transfer to richer modal logics and languages while maintaining a relatively low computational complexity. The basic modal languages and the graphs over which they are interpreted are discussed. The chapter also introduces the notion of bisimulation, based on which, modal logic as a fragment of first-order logic is characterized. The computability and computational complexity of modal logic is examined. The level of frames and the link between modal and classical logic are explored. Three alternatives to relational semantics––namely, algebraic semantics, neighborhood semantics, and topological semantics are also discussed.