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 those logics which 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 ST and TS, the former lacking Cut, the latter Reflexivity. The sense in which these logics, and these metainferences, are dual has yet to be explained in the context of a thorough and detailed exposition of duality for frameworks of this sort. Thus, our intent here is to try to elucidate whether or not 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 hint at broader points that might need to be addressed when studying duality in relation to substructural logics.

Metainferential Paraconsistency

“…The interesting thing about them is that even if inferentially there is no difference between ST and CL, as we will see, the metainferences they determine are quite different. In order to state it in more formal terms, we introduce the following definition that we take from [Da Ré et al, 2020]:…”

“…The main idea is that L 1 represents the standard for the premises of the metainference and L 2 stands for the canon for the conclusion. This can be generalized to metainferential logics of any level (we adapt this definition from [Da Ré et al, 2020]):…”

“…Before doing so, we will review the BPS proposal and we will argue against it. Finally, to deal with the problems the BPS proposal has, we will introduce some changes in the framework and, following [Da Ré et al, 2020], we will introduce a metainferential negation, which will allow us to define metainferential paraconsistency.…”

“…In the next section, we will introduce in detail such definition. Afterward, we will extend the work made by Barrio et al [2018], and using some machinery recently developed in [Da Ré et al, 2020] we will present our own definition.…”

“…In a nutshell, BPS define meta Explosion using the internal operator ¬, which seems inadequate to define a concept of contradiction useful for metainferential logics. Instead, we will extend the framework introducing the kind of negation for (meta)inferences Da Ré et al [2020] use to characterise duality.…”