Categories and Modalities

Category theory provides various guiding principles for modal logic and its semantic modeling. In particular, Stone duality, or “syntax-semantics duality”, has been a prominent theme in semantics of modal logic since the early days of modern modal logic. This chapter focuses on duality and a few other categorical principles, and brings to light how they underlie a variety of concepts, constructions, and facts in philosophical applications as well as the model theory of modal logic. In the first half of the chapter, I review the syntax-semantics duality and illustrate some of its functions in Kripke semantics and topological semantics for propositional modal logic. In the second half, taking Kripke’s semantics for quantified modal logic and David Lewis’s counterpart theory as examples, I demonstrate how we can dissect and analyze assumptions behind different semantics for first-order modal logic from a structural and unifying perspective of category theory. (As an example, I give an analysis of the import of the converse Barcan formula that goes farther than just “increasing domains”.) It will be made clear that categorical principles play essential roles behind the interaction between logic, semantics, and ontology, and that category theory provides powerful methods that help us both mathematically and philosophically in the investigation of modal logic.