La contrastación de teorías inconsistentes no triviales

This dissertation offers a proof of the logical possibility of testing empirical/factual theories that are inconsistent, but non-trivial. In particular, I discuss whether or not such theories can satisfy Popper's principle of falsifiablility. An inconsistent theory Ƭ closed under a classical consequence relation implies every statement of its language because in classical logic the inconsistency and triviality are coextensive. A theory Ƭ is consistent iff there is not a α such that Ƭ ⊢ α ∧ ¬α, otherwise it is inconsistent. We say, instead, that Ƭ is non-trivial iff there is at least one α such that Ƭ ⊢ α, otherwise we say that it trivial. This happens because classical logic satisfies the principle of explosion, according ex contradictione sequitur quodlibet (from a contradiction anything follows). Under these conditions inconsistent classical theories would be compatible with any well-formed formula, which makes them useless for science. There are, however, so-called paraconsistent logics in which the principle of explosion does not generally hold and in which a theory can be (simply) inconsistent, but also absolutely consistent. It is in this logical framework that we can prove that some inconsistent theories can be falsifiable.