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.