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.
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—Ralph Waldo Emerson (1803–1882)