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 ST and TS. In this respect, we show that this notion doesn’t coincide, in general, with a nowadays broadly used semantic approach towards metainferential validity: the 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 doesn’t 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 ST and TS.

本文的目的是在非传递性和非自反性子结构逻辑的背景下研究可推导性的概念及其语义对应物。为此，我们专注于逻辑S T和T S的研究案例. 在这方面，我们表明这个概念通常与当今广泛使用的元推断有效性的语义方法不一致：局部有效性的概念。在此之后，并以 Humberstone 之前的一些工作为基础，我们证明在这些系统中，可推导性可以根据我们称为绝对全局有效性的概念来表征。然而，得出这些结果并不会导致我们忽视局部有效性。首先，因为我们讨论了局部有效性和全局有效性与可推导性相一致的条件。其次，因为我们展示了考虑某些估值系列如何有助于描述用于表示S T和T S 的不同演算的可推导性。