(I can’t get no) antisatisfaction

Abstract

Substructural approaches to paradoxes have attracted much attention from the philosophical community in the last decade. In this paper we focus on two substructural logics, named ${\mathsf {ST}}$ and ${\mathsf {TS}}$, along with two structural cousins, ${\mathsf {LP}}$ and ${\mathsf {K3}}$. It is well known that ${\mathsf {LP}}$ and ${\mathsf {K3}}$ are duals in the sense that an inference is valid in one logic just in case the contrapositive is valid in the other logic. As a consequence of this duality, theories based on either logic are tightly connected since many of the arguments for and objections against one theory reappear in the other theory in dual form. The target of the paper is making explicit in exactly what way, if any, ${\mathsf {ST}}$ and ${\mathsf {TS}}$ are dual to one another. The connection will allow us to gain a more fine-grained understanding of these logics and of the theories based on them. In particular, we will obtain new insights on two questions concerning ${\mathsf {ST}}$ which are being intensively discussed in the current literature: whether ${\mathsf {ST}}$ preserves classical logic and whether it is ${\mathsf {LP}}$ in sheep’s clothing. Explaining in what way ${\mathsf {ST}}$ and ${\mathsf {TS}}$ are duals requires comparing these logics at a metainferential level. We provide to this end a uniform proof theory to decide on valid metainferences for each of the four logics. This proof procedure allows us to show in a very simple way how different properties of inferences (unsatisfiability, supersatisfiability and antivalidity) that behave in very different ways for each logic can be captured in terms of the validity of a metainference.