Categories and Modalities

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

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.
Original languageEnglish (US)
Title of host publicationCategories for the Working Philosopher
EditorsElaine Landry
PublisherOxford University Press
Pages163-222
Number of pages60
ISBN (Print)9780198748991
DOIs
StatePublished - Jan 18 2018
Externally publishedYes

Keywords

  • modal logic
  • Stone duality
  • syntax-semantics duality
  • Kripke semantics
  • David Lewis
  • counterpart theory
  • Barcan formula

ASJC Scopus subject areas

  • Arts and Humanities(all)

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