Semantic Closure and Classicality

Abstract

The semantic paradoxes show that semantic theories that internalize their semantic concepts, such as truth and validity, cannot validate all classical logic. That is, it is necessary to weaken some connective of the object language, taken as the guilty of the paradoxes, or give up some property of the consequence relation of the logical theory. Both strategies may distance us from classical logic, the logic commonly used in our current mathematical theories. So, a desirable solution to semantic paradoxes cannot distance us from classical logic. This paper analyzes two interesting proposals that aim to maintain classical logic as most as possible. The first strategy, the Barrio & Pailos & Szmuc-approach (2017) (BPS-approach), proposes the paraconsistent logic MSC that contains in its object language a connective capable of recovering classical inference whenever the sentences at issue are consistent. So, they show that it is possible to build a semantic theory over this logic that is immune to the semantic paradoxes. The second approach is based on the hierarchy STω of non-transitive systems STn, proposed by Pailos (2020a). This hierarchy recovers classical metainferences as many as possible in higher levels of the hierarchy. We argue in favor of the second approach by arguing that the first strategy must adopt weak self-referential procedures to avoid the paradoxes.