Kripke semantics is a type of interpretation for several non-classical logics. Most notably, modal logic

Kripke FrameEdit

A kripke frame is an ordered pair (G,R) where:

  • G is a set of points or worlds that represent possible worlds.
  • R is a relation that relates worlds to other worlds.

One could create a directed graph out of a kripke frame, where G is used as the set of nodes and R is used as the vertices.

Kripke modelEdit

A kripke frame is an ordered pair (G,R,v), where:

  • (G,R) is a kripke frame.
  • v is a mapping v:G \times \mathcal{L} \to \{t,f\}, where \mathcal{L} is the language of the formal system and \{t,f\} is a set of truth values. Alternatively, it is a relation v \subseteq G \times \mathcal{L} on the set of worlds and the language of the formal system..


Let \mathcal{M}=(G,R,v), then \mathcal{M}, w \models \phi is used to denote v(w,\phi)=t. When it is clear what the kripke model is, one sometimes uses w \models \phi.

Some authors use \Vdash instead of v. The resulting notation is w \Vdash \phi

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