Substructural Logics with Minimally True Tautologies

ABSTRACT

The tautologies of propositional classical logic are always interpreted by the same (greatest) element in a Boolean algebra, but such uniformity is typically lacking in "weakening-free" substructural logics. In particular, a tautology "A implies A" may be mapped to a value in a structure for the logic that is strictly greater than the element t representing minimal truth; that is, "A implies A" is a tautology of the logic, but it is not the case that "A implies A" is equivalent to t. Several weakening-free substructural logics may be found in the literature, however, that have "minimally true" tautologies: notably, the logics of Iseki's BCI-algebras and Raftery and van Alten's sircomonoids, and the comparative logics of Casari, including Abelian logic, introduced independently by Meyer and Slaney. In this talk, based on a joint paper with José Gil-Férez and Frederik Lauridsen, I will explain how proof theory and decidability results can be obtained for these logics by adding a restricted form of weakening, justified by a Glivenko-style property, to suitable sequent calculi.

Speaker: Georg Metcalfe (University of Bern)

Research Topics:

• Proof Theory
• Non-Classical Logics
• Ordered Algebraic Structures