Derivability and Metainferential Validity

DAMIAN SZMUC; BRUNO DA RE; PAULA TEIJEIRO

Seminario; WIP; 2020

The aim of this article is to study the notion of Derivability and its semantic counterpart in the context of non-transitive and non-reflexive substructural logics. For this purpose we focus on the study cases of the logics ${\bf ST}$ and ${\bf TS}$. In this respect, we show that this notion does not coincide, in general, with a nowadays broadly used semantic approach towards metainferential validity: the so-called notion of Local validity. Following this, and building on some previous work by Humberstone, we prove that in these systems Derivability can be characterized in terms of a notion we call Absolute Global validity. However, arriving at these results does not lead us to disregard Local validity. First, because we discuss the conditions under which Local, and also Global validity, can be expected to coincide with Derivability. Secondly, because we show how taking into account certain families of valuations can be useful to describe Derivability for different calculi used to present ${\bf ST}$ and ${\bf TS}$.