Abstract

Weber argues in Paradoxes and Inconsistent Mathematics (2021) for a substructural foundation of mathematics. The mathematical theories investigated by Weber are all axiomatized using the paraconsistent logic. This paper shows how to modify the logic so as to ensure that the deduction theorem Weber claims to hold in fact does. What is a more significant contribution is the suggestion for how to analyze Weber’s notion of “bad” assumptions. A new form of a Hilbert calculus is presented in which premises are taken as a pair consisting of a premise set—the formulas of which can be drawn upon unrestrictedly—and a premise multiset—the formulas of which can be used at most as many times as they occur in the multiset. The idea is that “bad” assumptions go into the multiset, whereas non-bad assumptions go into the set. It is shown that the consequence relation restricted to only “non-bad” axiom sets is fully structural. Weber’s mathematical theories are non-classical; some are even contra-classical in that they are provably inconsistent. Since Weber regards the axioms of his mathematical theories as non-bad, they are, however, non-classical in a more standard sense than what Weber seems to claim: they are fully structural, Tarskian, and closed under both modus ponens and adjunction.