16 Modal logic in mathematics

Abstract

Publisher Summary Formal modal logic is mostly mathematical in its methods, regardless of area of application. This chapter presents a wide variety of mathematical techniques developed over decades of studying the intricate details of modal logic. Mathematics normally finds a proper language and level of abstraction for the study of its objects. Propositional modal logic offers a new paradigm of applying logical methods: instead of using the traditional languages with quantification to describe a structure, an appropriate quantifier-free language with additional logic operators that represent the phenomenon at hand, is used. Mathematics is one of modal logic's oldest application areas. There are two major ideas that dominate the landscape of modal logic application in mathematics: Godel's provability semantics and Tarski's topological semantics. Godel's use of modal logic to describe provability, gave the first exact semantics of modality. This approach led to comprehensive provability semantics for a broad class of modal logics. It also proved vital for applications such as the Brouwer–Heyting–Kolmogorov provability semantics for intuitionistic logic, for introducing justification into formal epistemology and tackling its logical omniscience problem, and for introducing self reference into combinatory logic and lambda-calculi.