PRE2023-The Shape of Nonclassical Logics (PID2022-141529NB-C22)

This research project aims at investigating nonclassical logics through the lenses of algebraic and geometrical constructions. We ground our investigation in the realm of substructural logics, a family of deductive systems that includes most of the interesting non-classical logics: intuitionistic logic, relevance logics, many-valued logics, linear logic, besides including classical logic as a limit case. Of paramount importance is the fact that each substructural logic admits an algebraic semantics in terms of structures known as residuated lattices, i.e., lattice ordered monoids endowed with two division operations that are related to the monoidal multiplication by the so-called residuation law. This makes substructural logics amenable to techniques typical of algebraic logic which allow one to study metalogical properties by means of their algebraic counterparts, which in turn are amenable to the powerful methods of algebra and topology. While the algebraic interpretation makes it amenable to the methods of universal algebra, the topological interpretation has the advantage of carrying a visual intuition on shapes and deformations which makes it easier to construct new objects from old ones. The aim of the project is to develop a systematic study of new and existing construction for residuated structures in order to study metalogical properties of the corresponding systems, and complement the universal algebraic approach with powerful methods.