Logic and Substructurality
Traditional approaches to paradoxes reject particular inferences involving logical vocabulary in order to avoid the paradoxical argument. In a Kripke-style theory of truth, for example, the law of excluded middle A or not-A is rejected. With this, other related principles like the axiom of identity If A then A, are likewise rejected. Many recent approaches to paradoxes, however, focus their attention on structural properties: principles that govern the behaviour of the consequence relation itself, without reference to specific items in the logical vocabulary. Among these we can find non-reflexive approaches, non-monotonic approaches, non-contractive approaches and non-transitive approaches. So-called substructural logics are promising solutions to the paradoxes since they provide a way to avoid triviality with a minimal disruption of inferences involving logical vocabulary and, therefore, without modifying the meaning of logical constants (or so the defenders of these approaches claim). The study of substructural logics is nowadays one of hot topics in Philosophical Logic.
The research question that motivates this project is the following: do we have reasonable evidence for the claim that our reasoning is substructural? Our initial hypothesis is that the answer to this question is 'yes'. In order to try to address the question we propose, in the first place, studying the possibilities of a semantic characterization of substructural logics and, in the second place, the philosophical analysis of the arguments against substructurality.