LLC Seminar: Federico Pailos

This work investigates the expansion of non-bivalent (in)validity from mixed logics to non-mixed connexive logics, introducing a novel framework for understanding inferential and metainferential validity. Traditionally, logical validity and invalidity have been viewed as mutually exclusive and collectively exhaustive properties. However, recent developments challenge this bivalent perspective by embracing logics that allow for inferences which are both valid and invalid, or neither — thus reflecting a non-bivalent notion of (in)validity. We focus on connexive logic C, a non-mixed, contradictory logic introduced by Herinrich Wansing. By modifying the semantic conditions for invalidity — aligning them with the falsity conditions of the connexive conditional — this work demonstrates how C can be made inferentially non-bivalent. The analysis employs a possible worlds semantics where conditionals and inferences are evaluated dynamically across worlds. Here, the conditional’s behavior under truth and falsity conditions is mirrored at the inferential and metainferential levels, achieving a precise correspondence. The resulting logics, denoted NBCs, NBCu and NBCe (for the strong, universal and existential variants of a inferentially bivalent version of C), possesses two key features: they are Mares-reflexive, capturing validities through a conditional connective while having the same behaviour of C at the level of sentential and inferential validities. Their non-bivalent inferential structure extends to the metainferential hierarchy, ensuring that the meta-principles are also non-bivalent and connexive. The philosophical implications of these variants are discussed, highlighting how this framework supports a general account of logical consequence, connexivity, and the expressive role of conditional connectives. By generalizing the non-bivalent approach beyond mixed logics, this work opens new avenues for investigating consequence relations and structural properties in non-classical logics, and suggests future research into other logical systems.