minimal classical logic

Minimal classical logic is an extension of minimal logic, first identified in (Zena M. Ariola and Hugo Herbelin, 2003), that validates Pierce's law but not the principle of explosion.

Minimal classical logic arises from the observation that various classical axioms, which are all equivalent relative to intuitionistic logic, are not all equivalent relative to minimal logic. In order of increasing strength:

• weak Pierce's law and excluded middle are equivalent;
• Pierce's law and generalized excluded middle are equivalent;
• double negation elimination yields full classical logic.

Additionally, adding the principle of explosion to any of the preceding axioms also yields full classical logic.

(Zena M. Ariola and Hugo Herbelin, 2003) term axioms in the first class weak classical axioms, in the second class minimal classical axioms, and in the third class full classical axioms. They also remark without elaboration that a logic that validates excluded middle but not Pierce's law “seems uninteresting”, and so do not investigate the weak classical axioms.

The system described in (Michel Parigot, 1992) is a proof system for minimal classical logic.