On a Suggested Logic for Paraconsistent Mathematics

The logic subDL and its quantified extension subDLQ were proposed by Badia and Weber (Dialethism and its Applications, 2019: 155-176) as a basis for developing a version of mathematics in which paradoxes are harmless. In the present paper, subDL as defined in the literature is shown to be too strong to support the theories which motivate it. The crucial point is that contraction is derivable in subDL. It follows that the semantic structure used by Badia and Weber to invalidate contraction is not, in fact, a model of subDL. Here we identify the axioms responsible for contraction in subDL and prove that the logic, weakened by removal of these axioms, is contraction-free and paraconsistent.