Saul Kripke - Intuitionistic Logic

Intuitionistic Logic

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

An intuitionistic Kripke model is a triple, where is a partially ordered 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 .

Intuitionistic logic is sound and complete with respect to its Kripke semantics, and it has the Finite Model Property.

Intuitionistic first-order logic

Let L be a first-order language. A Kripke model of L is a triple, where is an intuitionistic Kripke frame, Mw is a (classical) L-structure for each node wW, and the following compatibility conditions hold whenever uv:

  • the domain of Mu is included in the domain of Mv,
  • realizations of function symbols in Mu and Mv agree on elements of Mu,
  • for each n-ary predicate P and elements a1,…,anMu: if P(a1,…,an) holds in Mu, then it holds in Mv.

Given an evaluation e of variables by elements of Mw, we define the satisfaction relation :

  • if and only if holds in Mw,
  • if and only if and ,
  • if and only if or ,
  • if and only if for all, implies ,
  • not ,
  • if and only if there exists an such that ,
  • if and only if for every and every, .

Here e(xa) is the evaluation which gives x the value a, and otherwise agrees with e.

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