A multiplicative ingredient for omega-inconsistency

Abstract

This paper presents a distinctively multiplicative quantificational principle that arguably captures the problematic aspects of Zardini's infinitary rules for a multiplicative quantifier within the context of the semantic paradoxes and the theoretical goal to obtain a (omega)-consistent theory of transparent truth. After showing that the principle is derivable with Zardini's rules and that one obtains through vacuous quantification an inconsistent theory of truth if truth is transparent, the paper presents two results regarding the principle and omega-inconsistency. First, the principle is used to obtain a non-classical variant of McGee's omega-inconsistency result for certain classical theories of truth. Second, it is demonstrated that the conditions for a truth-theoretic variant of Bacon's omega-inconsistency result for certain non-classical theories of transparent truth implies that the principle holds for the paradoxical formula. Finally, the paper argues that the paradoxical reasoning that the principle enables is structurally similar to the kind of infinitary reasoning popularised by Hilbert's Grand Hotel.