The logics of a universal language

Semantic paradoxes pose a real threat to logics that attempt to be capable of expressing their own semantic concepts. Particularly, Curry paradoxes seem to show that many solutions must change our intuitive concepts of truth or validity or impose limits on certain inferences that are intuitively valid. In this way, the logic of a universal language would have serious problems. In this paper, we explore a different solution that tries to avoid both limitations as much as possible. Thus, we argue that it is possible to capture the naive concepts of truth and validity without losing any of the valid inferences of classical logic. This approach is called the Buenos Aires plan. We present the logic of truth and validity, \(\mathsf {STTV}_{\omega }\) based on the hierarchy of logics \(\textsf{ST}_{\omega }\), whose validity predicate has the same semantic conditions as the material conditional. We argue that \(\mathsf {STTV}_{\omega }\) is capable of blocking the problematic results while keeping the deductive power of classical logic as much as possible and offering an adequate semantic theory. On the other hand, one could object that it is not possible to reason with \(\mathsf {STTV}_{\omega }\) because it is not closed under its logical principles. We respond to this objection and argue that the local characterization of validity shows how to make inferences using the logic \(\textsf{ST}_{\omega }\).