Maarten McKubre-Jordens: Material implication over minimal logic

Abstract: The “atrocities” of material implication have motivated the development of many non-classical logics over the years. In this talk, we outline some of these paradoxes and classify them over minimal logic. We provide some proofs of equivalence and semantic models separating the paradoxes where appropriate. A number of equivalent groups arise, all of which collapse with unrestricted use of double negation elimination. Principles weaker than ex falso quodlibet turn out to be distinguishable, even with a logic so little “mutilated” as minimal logic. This can be interpreted in at least two ways: even minor changes to the logic underpinning mathematical reasoning allow many more distinctions than the classical understanding; and supporting motivation for adopting minimal logic as the ambient logic for mathematical reasoning in the possible presence of inconsistency.