Novas classes de lógicas na hierarquia de Leibniz.

Abstract

The Leibniz hierarchy constitutes a classification system for propositional logics based on the behavior of the Leibniz operator associated to each logic, which is one of the main subjects of interest pertaining to the field of Abstract Algebraic Logic (AAL). In a series of recent papers [15, 16, 17], there was an attempted formalization of the concepts of Leibniz conditions, classes and hierarchy, which was inspired by the Maltsev hierarchy of Universal Algebra. The goal of this master’s thesis is answering the following problem included in those articles: “is there a precise relation between Leibniz classes and the behavior of the Leibniz operator?” With this in mind, we start off with the basic concepts and results from AAL and present a few of the main classes of logics in the Leibniz hierarchy, along with their respective characterizations, properties and examples of logics belonging to each of them. Moreover, we analyze the proposed formalization of the Leibniz classes by way of reframing it from the standpoint of Category Theory. In conclusion, we introduce and discuss a series of new classes of logics which are related to, yet distinct from those present in the Leibniz hierarchy; we also test these new classes against the proposed formalization of Leibniz classes, thus obtaining a negative answer to the aforementioned open problem, as our main result.