Intermediate Logic - Definition

Definition

A superintuitionistic logic is a set L of propositional formulas in a countable set of variables pi satisfying the following properties:

  1. all axioms of intuitionistic logic belong to L;
  2. if F and G are formulas such that F and F → G both belong to L, then G also belongs to L (closure under modus ponens);
  3. if F(p1, p2, ..., pn) is a formula of L, and G1, G2, ..., Gn are any formulas, then F(G1, G2, ..., Gn) belongs to L (closure under substitution).

Such a logic is intermediate if furthermore

  1. L is not the set of all formulas.

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