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Intermediate logic
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).
==Definition==
A superintuitionistic logic is a set ''L'' of propositional formulas in a countable set of
variables ''p''''i'' 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''(''p''1, ''p''2, ..., ''p''''n'') is a formula of ''L'', and ''G''1, ''G''2, ..., ''G''''n'' are any formulas, then ''F''(''G''1, ''G''2, ..., ''G''''n'') belongs to ''L'' (closure under substitution).
Such a logic is intermediate if furthermore
:4. ''L'' is not the set of all formulas.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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