(I can’t get no) antisatisfaction

Abstract

Substructural approaches to paradoxes have attracted much attention from the philosophical community in the last decade. In this paper we focus on two substructural logics, named ST and TS, along with two structural cousins, LP and K3. It is well known that LP and K3 are duals in the sense that an inference is valid in one logic just in case the contrapositive is valid in the other logic. As a consequence of this duality, theories based on either logic are tightly connected since many of the arguments for and objections against one theory reappear in the other theory in dual form. The target of the paper is making explicit in exactly what way, if any, ST and TS are dual to one another. The connection will allow us to gain a more fine-grained understanding of these logics and of the theories based on them. In particular, we will obtain new insights on two questions concerning ST which are being intensively discussed in the current literature: whether ST preserves classical logic and whether it is LP in sheep's clothing. Explaining in what way ST and TS are duals requires comparing these logics at a metainferential level. We provide to this end a uniform proof theory to decide on valid metainferences for each of the four logics. This proof procedure allows us to show in a very simple way how different properties of inferences (unsatisfiability, supersatisfiability and antivalidity) that behave in very different ways for each logic can be captured in terms of the validity of a metainference.Substructural approaches to paradox − Non-transitive logic − Non-reflexive logic − Strong Kleene

Metainferential duality

“…It should be noted that a recently published article by Cobreros, La Rosa and Tranchini[11] independently arrived at conclusions similar to ours, also arguing in favor of the duality between ST and TS on different grounds that we do. Although in private conversation we have discussed some of the similarities and differences of the notions implemented in the two approaches, a more comprehensive study in this respect is in order.…”

“…As suggested by an anonymous reviewer, let us say (as in [5]) that a set of truth-values A is in a p-relation R to some other set B provided either some a i ∈ A does not belong to D + or some b j ∈ B belongs to D − -and similarly for a q-relation. 11 In this vein, we could define the dual relation R to R as determined by the fact that either some a i ∈ A is such that n(a i ) does not belong to the set n(D + ) made of negations of values in D + , or some b j ∈ B is such that n(b j ) belongs to the set n(D − ) of negations of values in D − . Now, applying this general idea to the case of R being the ST-consequence relation (or similarly to TS), it's easy to check that it is in fact self-dual.…”

Inferences and Metainferences in ST

“…This raises the following interesting question: 11 what is the logic corresponding to hybrid metainferences of ST + ? More precisely: we have that a meta-inference holds for ST + just in case the "lowering" renders a valid LP + inference.…”

“…Finally, the local reading states that a metainference holds when every valuation satisfying its premises, also satisfies its conclusion. See Dicher and Paoli[13], Barrio et al[5], Scambler[25], Cobreros et al[11], and Teijeiro[26]. The issue of the relation between the global, global-substitutional, and local readings -and of which should be used to capture the notion of a metainference -is of particular interest but lies beyond the scope of this paper.…”

(Meta)inferential levels of entailment beyond the Tarskian paradigm

“…Speaking more formally, a metametainference Ξ ⇒ 2 C on L is an ordered pair Ξ, C , where Ξ ⊆ SEQ 1 (L) and C ∈ SEQ 1 (L). We denote, analogously, by SEQ 2 (L) the set of all metametainferences on L. 12 In order to understand a logic related in this way to CL, we discuss the system TSST, as presented in [1]. For that purpose, it is important to 12 The same criterion of validity applies to metametainferences as to metainference.…”