Validities, antivalidities and contingencies: A multi-standard approach

Abstract

It is widely accepted that classical logic is trivialized in the presence of a transparent truth-predicate. In this paper, we will explain why this point of view must be given up. The hierarchy of metainferential logics defined in Barrio et al. (Journal of Philosophical Logic, 1–28, 2019) and Pailos (The Review of Symbolic Logic, Forthcoming) 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 Barrio et al. (Journal of Philosophical Logic, 1–28, 2019)), or because a logic of a transfinite level defined in terms of the hierarchy shares its validities with classical logic. Each of these logics is consistent with transparent truth—as is shown in Pailos (The Review of Symbolic Logic, Forthcoming)—, and this suggests that, contrary to standard opinions, transparent truth is after all consistent with classical logic. However, Scambler (Journal of Philosophical Logic, 49, 351–370, 2020) presents a major challenge to this approach. He argues that this hierarchy cannot be identified with classical logic in any way, because it recovers no classical antivalidities. We embrace Scambler’s challenge and develop a new logic based on these hierarchies. This logic recovers both every classical validity and every classical antivalidity. Moreover, we will follow the same strategy and show that contingencies need also be taken into account, and that none of the logics so far presented is enough to capture classical contingencies. Then, we will develop a multi-standard approach to elaborate a new logic that captures not only every classical validity, but also every classical antivalidity and contingency. As a€truth-predicate can be added to this logic, this result can be interpreted as showing that, despite the claims that are extremely widely accepted, classical logic does not trivialize in the context of transparent truth.