Algebraic approach to paraconsistent weak Kleene logic

The starting point of this work is the analysis of the logic known as Paraconsistent Weak Kleene (PWK), the 3-valued logic with two designated values defined through the weak Kleene tables. Some philosophical assumptions stand behind the introduction of non-classical logics, which is, basically allowing logics to deal with partial predicates. Despite different non-classical formalisms have found lots more success than PWK logic, this thesis highlights a very surprising connection (which can be further generalized) between such a logic on one side, and the purely algebraic theory of regular varieties, on the other. The latter has been studied in universal algebra since the '60, but had found no application in logic before. In particular, the present work is divided in two different parts, each of which makes use of different machineries and techniques: one is more logically oriented and regards the study of Paraconsistent Weak Kleene logic, under the perspective of Abstract Algebraic Logic, while the other part involves a closer study of the algebraic semantics of the mentioned logic.