A Hierarchy of Classical and Paraconsistent Logics

Abstract

In this article, we will present a number of technical results concerning Classical Logic, ST and related systems. Our main contribution consists in offering a novel identity criterion for logics in general and, therefore, for Classical Logic. In particular, we will firstly generalize the ST phenomenon, thereby obtaining a recursively defined hierarchy of strict-tolerant systems. Secondly, we will prove that the logics in this hierarchy are progressively more classical, although not entirely classical. We will claim that a logic is to be identified with an infinite sequence of consequence relations holding between increasingly complex relata: formulae, inferences, metainferences, and so on. As a result, the present proposal allows not only to differentiate Classical Logic from ST, but also from other systems sharing with it their valid metainferences. Finally, we show how these results have interesting consequences for some topics in the philosophical logic literature, among them for the debate around Logical Pluralism. The reason being that the discussion concerning this topic is usually carried out employing a rivalry criterion for logics that will need to be modified in light of the present investigation, according to which two logics can be non-identical even if they share the same valid inferences.

(I can’t get no) antisatisfaction

“…The notion of a metainference can be generalised to cover metainferences of any order, where a metainference of order n is an arrow with metainferences of order n − 1 at each side. Barrio et al (2019) use this generalisation to prove some intriguing results. See Scambler (2019) for a rejoinder.…”

“…5 Under the assumption that the notion of satisfaction applies equally to sequents at any metainferential level, see Scambler (2019) section 3.3 for a clarification of this point. Following Barrio et al (2019), Chris Scambler shows that there are uncountably many logics differing perhaps only at some metainferential level. A discussion of these results is beyond the scope of this paper.…”

“…10 These definitions come from Dicher and Paoli (2019) and are also used in Barrio et al (2019). For a matter of uniformity we define metainferences with possibly multiple inferences as conclusions, although the examples we will consider below involve all a single inference in the conclusion side.…”

“…In a companion paper in preparation we show how the trees can be "turned upside down" to obtain a sequent calculus that covers the four 3-valued logic under consideration as well the hierarchy of meta-inferential logics considered byBarrio et al (2019). As it is implicit in the construction of the trees (in particular, from the fact that to test the xy-validity of a sequent we start a tree by simply taking the formulas in the antecedent labelled by x together with the negation of those in the antecedent marked by the dual of y), the corresponding sequent calculus is two-sided and labelled, where the labels are used to reflect syntactically the key features of the semantics.…”

Metainferential duality

“…After all, we permit multiple conclusions in the context of inferences holding between (collections of) formulae. So, if metainferences (as argued in [4]) are inferences holding between different kinds of relata-in this case, inferences themselves-then there appears to be no reason to disallow multiple conclusions in this case but not in the other. Is there anything in particular about inferences as relata of inferences themselves that prevents us from having a unified picture, where both premises and conclusions can be sets?…”