Restall’s Proof-Theoretic Pluralism and Relevance Logic

Abstract

Restall (2014) proposes a new, proof-theoretic, logical pluralism. This is in contrast to the model-theoretic pluralism he and Beall proposed in Beall and Restall (2000) and in Beall and Restall (2006). What I will show is that Restall has not described the conditions on being admissible to the proof-theoretic logical pluralism in such a way that relevance logic is one of the admissible logics. Though relevance logic is not hard to add formally, one critical component of Restall's pluralism is that the relevance logic that gets added must have connectives which mean the same thing as the connectives in the already admitted logic. This is what I will show is not possible.

Proof-theoretic pluralism

“…This gives us a form of MILP because the calculi for these different logics, while differing at the structural level, are based on the same set of operational rules. In developing the framework we will also show how our proposal fully addresses the criticisms moved by Teresa Kouri [19] to a version of proof-theoretic MILP presented by Greg Restall [30]. According to Kouri, Restall's proposal cannot allow for any relevance logics as admissible while at the same time avoiding the charge of meaning-variance.…”

“…Section 2 introduces the basic framework of proof-theoretic MILP and Section 3 shows how classical and intuitionistic logics can coexist within the basic framework. Section 4 presents a problem for the basic framework that has been raised in [19], namely that it is not clear whether the basic framework can be supplemented with an admissiblity criterion that includes not only classical and intuitionistic logics but also some relevance logic. Then, in Section 5 we will argue that Belnap's proof-theoretic notion of harmony provides such a criterion and in Section 6 we will show how relevance logics behave in an harmony-based proof-theoretic pluralism.…”

“…Section 7 develops our proposal in more detail by arguing that it gives us a two-level proof-theoretic pluralism that admits of both a form of MILP and a In this section we are going to introduce what we call the basic framework of proof-theoretic MILP. We call it the basic framework because it presents only some core philosophical ideas and the technical framework of sequent calculi that we share with other proponents of proof-theoretic MILP -e.g., with minor variations the basic framework is adopted, among others, by Dicher [6], Dosen [9], Hjortland [18], Kouri [19], Paoli [25], and Restall [30]. 1 Let us start by introducing sequent calculi.…”

Logical Pluralism from a Pragmatic Perspective

“…We find that any time we settle on one meaning more specific than 'somehow closely related to the accepted clauses' or 'sharing a family resemblance', we rule out some things we would want to count as negation, and allow some things we would not. For further work on this ruling in and ruling out problem see Kouri [2016], Priest [1987], and Hjortland [2013]. This makes sense of our intuitions that we can only find robust meanings of connectives within a specified context, and that the connectives are fluid, in the sense of Lynch [1998].…”

Pluralistic perspectives on logic: an introduction

“…Ferrari and Orlandelli argue that it is perfectly coherent for the pluralist to endorse this framework and, as a result, to treat single-and multiple-conclusion calculi as equally legitimate. They then turn to a problem raised by Kouri (2016). Every logical pluralist needs to have criteria for admissible logics, but for a pluralist with a proof-theoretic perspective, some criteria have unappealing results (e.g.…”