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 w ≤ u, 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.