Metainferential Duality

Abstract

The aim of this article is to discuss the extent to which certain substructural logics are related through the phenomenon of duality. Roughly speaking, metainferences are inferences between collections of inferences, and thus substructural logics can be regarded as logics that have fewer valid metainferences that Classical Logic. In order to investigate duality in substructural logics, we will focus on the case study of the logics (Formula presented.) and (Formula presented.), the former lacking Cut, the latter Reflexivity. The sense in which these logics, and metainferences, are dual has yet to be explained in the context of a thorough exposition of duality for frameworks of this sort. Thus, we try to elucidate whether this way of talking holds some ground–specially generalizing one notion of duality available in the specialized literature, the so-called notion of negation duality. In doing so, we hope to shed light on the phenomenon of duality in substructural logics.