In mathematical logic, a superintuitionistic logic is a propositional logic extending intuitionistic logic. Classical logic is the strongest consistent superintuitionistic logic. Thus consistent superintuitionistic logics are called intermediate logics (the logics are intermediate between intuitionistic logic and classical logic). Specifically, a superintuitionistic logic is a set L of propositional formulas in a countable set of variables pi satisfying the following properties: all axioms of intuitionistic logic belong to L; 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); 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 L is not the set of all formulas.

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