Logics of left variable inclusion and Płonka sums of matrices

The paper aims at studying, in full generality, logics defined by imposing a variable inclusion condition on a given logic ⊢. We prove that the description of the algebraic counterpart of the left variable inclusion companion of a given logic ⊢ is related to the construction of Płonka sums of the matrix models of ⊢. This observation allows to obtain a Hilbert-style axiomatization of the logics of left variable inclusion, to describe the structure of their reduced models, and to locate them in the Leibniz hierarchy

Probability over Płonka sums of Boolean algebras: States, metrics and topology

The paper introduces the notion of state for involutive bisemilattices, a variety which plays the role of algebraic counterpart of weak Kleene logics and whose elements are represented as Płonka sums of Boolean algebras. We investigate the relations between states over an involutive bisemilattice and probability measures over the (Boolean) algebras in the Płonka sum representation and, the direct limit of these algebras. Moreover, we study the metric completion of involutive bisemilattices, as pseudometric spaces, and the topology induced by the pseudometric

An algebraic study of logics of variable inclusion and analytic containment

This thesis focuses on a wide family of logics whose common feature is to admit a syntactic definition based on specific variable inclusion principles. This family has been divided into three main components: logics of left variable inclusion, containment logics, and the logic of demodalised analytic implication. We offer a general investigation of such logics within the framework of modern abstract algebraic logic