Kripke Semantics - Semantics of Intuitionistic Logic

Semantics of Intuitionistic Logic

Kripke semantics for the intuitionistic logic follows the same principles as the semantics of modal logic, but it uses a different definition of satisfaction.

An intuitionistic Kripke model is a triple, where is a preordered Kripke frame, and satisfies the following conditions:

  • if p is a propositional variable, and, then (persistency condition),
  • if and only if and ,
  • if and only if or ,
  • if and only if for all, implies ,
  • not .

The negation of A, ¬A, could be defined as an abbreviation for A → ⊥. If for all u such that wu, not u A, then w A → ⊥ is vacuously true, so w ¬A.

Intuitionistic logic is sound and complete with respect to its Kripke semantics, and it has FMP.

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