Title: On Beall and Camrud’s defence of the combinatorial argument for FDE

Abstract: In their 2020 paper “FDE all the way up”, Beall and Camrud aim to defend a particular combinatorial argument for the paraconsistent and paracomplete logic FDE against a natural objection. The argument, roughly, states that, since logic must consider all possibilities, it is not enough to have the ‘True’ and the ‘False’ in the correct set of truth-values but there also must be all combinations of these two fundamental truth-values, namely ‘Both True and False’ and ‘Neither True Nor False’. Including those combinatorial truth-values yields FDE. The objection Beall and Camrud consider consists in the idea that it must be possible to iterate the processus of combining truth-values. The gist of Beall and Camrud’s defence is that iterating the operation of ‘taking combinations of truth-values’ do not change the logic after one gets to FDE.  In this talk, I argue that their defence fails because the formal definition of the operation of ‘taking combinations of truth-values’ that they use, namely Priest’s positive plurivalence, fails to deliver FDE when applied to the two-valued matrix of classical logic. Indeed, I argue, one must stay consistent in what notion of ‘taking combinations of truth-values’ one uses throughout the combinatorial argument and the defence against the objection. One can find, to my knowledge, two such notions in the literature: Priest’s positive plurivalence and Priest’s general plurivalence. Depending on which one uses, one find oneself with a combinatorial argument (immune to the natural objection) for Priest’s logic LP or Oller’s logic AL. The conclusion of the talk takes the form of a challenge: to defend the combinatorial argument for FDE against the natural objection, one must find a notion of ‘taking combinations of truth-values’ which (a) produces the four-valued matrix of FDE when applied to the two-valued matrix of classical logic and (b) always produces FDE as a logic when iterated starting from the four-valued matrix of FDE. The notion used by Beall and Camrud in their paper satisfies (b) but fails to satisfy (a). By contrast, I put forward in the talk a natural proposal which satisfies (a) but fails to satisfy (b). This leaves me quite skeptical that the challenge can be met and that any such combinatorial argument for FDE can survive the natural objection.