Kripke frame

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 discovery 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').

The language of propositional modal logic consists of a countably infinite set of propositional variables, a set of truth-functional connectives (in this article ${\displaystyle \to }$ and ${\displaystyle \neg }$), and the modal operator ${\displaystyle \Box }$ ("necessarily"). The modal operator ${\displaystyle \Diamond }$ ("possibly") is (classically) the dual of ${\displaystyle \Box }$ and may be defined in terms of necessity like so: ${\displaystyle \Diamond A:=\neg \Box \neg A}$ ("possibly A" is defined as equivalent to "not necessarily not A").

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