Kripke semantics

Kripke semantics (also known as relational semantics or frame semantics, and often confused with possible world semantics) is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke and André Joyal. It was first conceived for modal logics, and later adapted to intuitionistic logic and other non-classical systems. The development of Kripke semantics was a breakthrough in the theory of non-classical logics, because the model theory of such logics was almost non-existent before Kripke (algebraic semantics existed, but were considered 'syntax in disguise'). Modal logic The language of propositional modal logic consists of a countably infinite set of propositional variables, a set of truth-functional connectives (in this article and ), and the modal operator ("necessarily"). The modal operator ("possibly") is (classically) the dual of and may be defined in terms of necessity like so: ("possibly A" is defined as equivalent to "not necessarily not A"). A Kripke frame or modal frame is a pair , where W is a (possibly empty) set, and R is a binary relation on W. Elements of W are called nodes or worlds, and R is known as the accessibility relation. A Kripke model is a triple , where is a Kripke frame, and is a relation between nodes of W and modal formulas, such that for all w ∈ W and modal formulas A and B: if and only if , if and only if or , if and only if for all such that . We read as “w satisfies A”, “A is satisfied in w”, or “w forces A”. The relation is called the satisfaction relation, evaluation, or forcing relation. The satisfaction relation is uniquely determined by its value on propositional variables. A formula A is valid in: a model , if for all w ∈ W, a frame , if it is valid in for all possible choices of , a class C of frames or models, if it is valid in every member of C. We define Thm(C) to be the set of all formulas that are valid in C. Conversely, if X is a set of formulas, let Mod(X) be the class of all frames which validate every formula from X.