Meta-inferential classical logics, anti-validity and more

BARRIO, EDUARDO ALEJANDRO; PAILOS, FEDERICO MATÍAS

Buenos Aires

Workshop; Workshop on Metainferences and Substructural Logics; 2019

IIF-SADAF

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.