Proof-theoretic pluralism

Abstract

Starting from a proof-theoretic perspective, where meaning is determined by the inference rules governing logical operators, in this paper we primarily aim at developing a proof-theoretic alternative to the modeltheoretic meaning-invariant logical pluralism discussed in [1]. We will also outline how this framework can be easily extended to include a form of meaningvariant logical pluralism. In this respect, the framework developed in this paper -which we label two-level proof-theoretic pluralism -is much broader in scope than the one discussed in [1] L¬

Change of logic, without change of meaning

“…While it is clear that the framework advanced by Paoli is not appropriate to capture a wide array of interesting systems, some other tricks may do so. This is basically the claim by Hjortland (2013) (and Ferrari and Orlandeli (2021), follow the suggestion closely). In a nutshell, the plan consists in advancing an ‐sided sequent calculus.…”

“…This paper is a contribution precisely to the third kind of strategy. Differently from the already available attempts (by, e.g., Paoli, 2003; Hjortland 2013; Ferrari and Orlandeli, 2021), we offer a general semantic framework which, we shall argue, brings a common semantic basis to important systems differing on their laws and valid inferences. This will have some interesting consequences for the main threat advanced by the meaning‐variance thesis.…”

“…3. Claims attempting to resist the argument by arguing that meaning variance is restricted or perhaps non-existent, and, as a result, genuine rivalry is made possible (as defended, among others, by Paoli (2003) and, in different form, by Hjortland (2013) and Ferrari and Orlandeli (2021)). Notice that this requires rejection of premise (P1).…”

“…Basically, we could classify them as follows: Claims that although there is meaning variance, that does not prevent legitimate rivalry about logic; this is advanced by Priest (2006) and Hjortland (2022) (and perhaps even by Quine himself, according to the readings offered by these authors). Notice that this involves rejecting premise (P2) of the meaning‐variance argument. Claims that the argument goes through, so that there is meaning variance and troubles for some kinds of rivalry about logic; this is defended by Warren (2018) (and Quine, according to some standard readings). Claims attempting to resist the argument by arguing that meaning variance is restricted or perhaps non‐existent, and, as a result, genuine rivalry is made possible (as defended, among others, by Paoli (2003) and, in different form, by Hjortland (2013) and Ferrari and Orlandeli (2021)). Notice that this requires rejection of premise (P1). …”

Limits of Abductivism About Logic

“…Here I am ignoring the Shapiro-Cook "logic as modeling" view(Shapiro, 2014;Cook, 2010). On that conception, there may be several equally good models of the meanings of the natural language analogues of logical constants.20 Advocates of a semantic conception could endorse a meaning-invariant pluralism if they locate the source of plurality in the structural rules; see(Dicher, 2016;Ferrari and Orlandelli, 2020).21 Notice that it won't do to claim that different logics capture different subsets of the most general truths and no logic captures all the most general truths. That would suggest that we should accept the union of the consequence relations of the "correct" logics as our logical theory.…”