Workshop On Metainferences and Substructural Logics

November 29, 2019

Argentinean Society of Philosophical Analysis (SADAF)

Buenos Aires, Argentina

15:00 to 16:10 – Bogdan Dicher (University of Lisbon) Substructurality on metainferential conceptions on logic (with Francesco Paoli)

16:20 to 17:30 – Eduardo Barrio and Federico Pailos (UBA and IIF-SADAF-CONICET) Meta-inferential classical logics, anti-validity and more

18:00 to 19:10 – Bruno Da Ré, Federico Pailos, Damián Szmuc, and Paula Teijeiro (UBA and IIF-SADAF-CONICET) Duality, metainferences and hierarchies of metainferential logics

19:20 to 20:30 – Andreas Fjellstad (University of Bergen) Metainferential reasoning with nested sequents 

Bogdan Dicher and Francesco Paoli: “Substructurality on metainferential conceptions on logic”

Recent debates on substructural solutions to the paradoxes have brought to the fore the importance of metainferences in establishing the identity of a logic. Among those that agree that logics should be identified metainferentially, there is, nonetheless, disagreement as to which metainferences matter. Some believe that the matter is settled at the first metainferential level: the identity of a logic is determined by the set of locally valid inferences between inferences. Others believe that the matter can be settled only in the transfinite: a logic is fully determined only as the transfinite union of all its metainferential levels.

Upholding the first option is not without costs, as the defenders of the second option have astutely pointed out. Among these costs is the need to account in some fashion for substructurality: On this view, a logical consequence relation is always Tarskian and so it appears as though on this account there is no room for substructural logics. The idea to be fleshed out is that while indeed there are no logics that are substructural, plenty of the logics usually called substructural are, in fact, logics—i.e., are structural consequence relations.

This paper takes a few steps in that direction, building on some programmatic remarks in Dicher & Paoli, The original sin of proof-theoretic semantics (Synthese, 2018). Generalising by overcoming a model suggested by, among others, Restall’s bilateralism, the argument will be that logic is concerned with sentences in contexts or, with a better choice of words, inferential networks. The familiar and less familiar structural rules of a logic stipulate principles for manipulating these networks. They are, in terms of conceptual priority, on a par with the operational rules of the calculus and also interconnected with them.

This may not provide much of a positive incentive to favour finitist metainferentialism over its more radical brethren. Yet one hopes that it will subvert one reason for going transfinite: The putative loss of substructurality is a mere appearance, being no more than the redistribution of the roles usually played by these rules.

Eduardo Barrio & Federico Pailos: “Meta-inferential classical logics, anti-validity and more”

The hierarchy of metainferential logics defined in Barrio et al ([3]) and Pailos ([22]) recovers classical logic, either in the sense that every classical(meta)inferential validity is valid at some point in the hierarchy (as is stressed in [3]), or because a logic of a transfinite level defined in terms of the hierarchy shares its validities with classical logic (as in [22]). Scambler ([31]) presents a major criticism to this approach. He argues that this hierarchy cannot be identified with classical logic in any way, because it recovers no classical antivalidities. And if it can do that, so is the case with a parallel hierarchy based on TS, that recovers every classical antivalidity, but none of its validities. We will do two things: (1) argue that there are good reasons to reject Scambler’s criticism, because the importance of antivalidities has not been well established yet; (2) we will take Scambler’s criticism for granted and develop a new hierarchy based on the previous two. This new hierarchy recovers both every classical validity and every classical antivalidity. Moreover, we will show that contingencies need to be taken into account, and that none of the logics so far presented are enough to capture classical contingencies. But we will redefine our new hierarchy in such a way that it captures not only every classical validity, but also every classical antivalidity and contingency.

Bruno Da Ré, Federico Pailos, Damián Szmuc, and Paula Teijeiro: “Duality, metainferences and hierarchies of metainferential logics”

According to the usual notions of duality, at the level of the inferences, ST and TS are self-dual. However, there are some intuitions regarding the duality of ST and TS, as metainferential logics, and of cut and reflexivity, as metainferences, and, so far, there has not been provided any appropriate notion of duality that fullfils these intuitions. In this talk, we offer such a notion and show that ST and TS are dual, we discuss the case of cut and reflexivity, and build a hierarchy of dual logics that capture classical logic (not only its validities, but also its antivalidities).

Andreas Fjellstad: “Metainferential reasoning with nested sequents”

The aim of this talk is to illustrate how we can utilise sequent calculi for nested sequents to reproduce and expand on recent work within philosophical logic involving metainferences. With the help of nested sequents we will (1) obtain cut-elimination theorems for a calculi capturing Barrio, Pailos and Szmuc’s metainferential hierarchies on strong Kleene valuations and Pailos’ “fully classical” metainferential logic, and (2) obtain a completeness theorem with regard to certain Kripke frames for a calculus capturing Hlobil’s approach to a validity-predicate for non-transitive logics.

Bruno Ré (University of Buenos Aires and IIF-SADAF-CONICET

Mariela Rubin (University of Buenos Aires and IIF-SADAF-CONICET)

Isis Urgell (University of Buenos Aires)

We are thankful for the support provided by CONICET