Algebraic semantics for substructural logics

BUSANICHE, MANUELA

Buenos Aires

Workshop; VIII Workshop on Philosophical logic; 2019

BA logic group, SADAF

Substructural logics encompass many of the interesting nonclassical logics: intuitionistic logic, fuzzy logic, relevance logic, linear logic, besides including classical logic as a limit case. They are logics that, when formulated as Gentzen-Style systems, lack some of the three basic structural rules: contraction, weakening and exchange. Residuated lattices are the algebraic semantics of substructural logics, that is why their investigation is one of the main tools to understand and study those logical systems uniformly. But the multitude of different structures makes the study fairly complicated, thus the investigation of interesting subvarieties of residuated lattices is an appealing problem to address. Examples of subvarieties of residuated lattices include Boolean algebras, Heyting algebras, MV-algebras, BL-algebras and lattice ordered groups.The study of substructural logics from the semantical point of view, as systems whose algebraic models are residuated structures settles a new perspective, where mathematics becomes the main tool of investigation. In this talk we will present some mathematical constructions of residuated lattices from simpler or better known structures. We will define subvarieties of residuated lattices whose members are built using these constructions, and we will explain the logical applications of our study.