Paraconsistent Metatheory: New Proofs with Old Tools

Abstract

This paper is a step toward showing what is achievable using non-classical metatheory—particularly, a substructural paraconsistent framework. What standard results, or analogues thereof, from the classical metatheory of first order logic(s) can be obtained? We reconstruct some of the originals proofs for Completeness, Löwenheim-Skolem and Compactness theorems in the context of a substructural logic with the naive comprehension schema. The main result is that paraconsistent metatheory can ‘re-capture’ versions of standard theorems, given suitable restrictions and background assumptions; but the shift to non-classical logic may recast the meanings of these apparently ‘absolute’ theorems.

True, Untrue, Valid, Invalid, Provable, Unprovable

“…So this is one example, which seems to have some promise from field testing, but it is not presented here as final or definitive. It seems likely that there are improvements and additions (for example, restricted quantifiers (Beall et al, 2006;Badia et al, 2022), using sparingly below) that will in future make for a fuller account for framing (paraconsistent) mathematics.…”

“…2 with respect to LP, namely that definitions 5 and 7 are equivalent. This is a modification of (Weber et al, 2016;Badia et al, 2022;Omori and Weber, 2023).…”

“…Recall in Section 1.5 above that we assume some limited form of mathematical induction, a practice which seems essential and is continued in(Badia et al, 2022)  but which is not completely unproblematic. See(Weber, 2021, ch.…”

Against classical paraconsistent metatheory

“…But a metatheoretical proof of a formal system also requires valid inferences. Badia, Weber & Girard (2022) have shown that such a metalogic can be paraconsistent in the sense that a paraconsistent metalogic can 'recapture' famous classical metatheoretical theorems such as Completeness, Löwenheim-Skolem and Compactness in a substructual paraconsistent metatheory.…”

Paraconsistency in Mathematics

“…For small steps into a serious nonclassical metatheory, see Rosenblatt (2021b); cf. Weber, Badia, and Girard (2016), Badia, Weber, and Girard (2022). Meyer already attempted some work like this in Meyer (1985).…”