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.

The paper studies the containment companion (or, right variable inclusion companion) of a logic . This consists of the consequence relation r which satisfies all the inferences of , where the variables of the conclusion are contained into those of the set of premises, in case this is not inconsistent. In accordance with the work started in [10], we show that a different generalization of the P lonka sum construction, adapted from algebras to logical matrices, allows to provide a matrix-based semantics for containment logics. In particular, we provide an appropriate completeness theorem for a wide family of containment logics, and we show how to produce a complete Hilbert style axiomatization.

Bochvar algebras consist of the quasivariety BCA playing the role of equivalent algebraic semantics for Bochvar (external) logic, a logical formalism introduced by Bochvar [3] in the realm of (weak) Kleene logics. In this paper, we provide an algebraic investigation of the structure of Bochvar algebras. In particular, we prove a representation theorem based on Płonka sums and investigate the lattice of subquasivarieties, showing that Bochvar (external) logic has only one proper extension (apart from classical logic), algebraized by the subquasivariety N BCA of BCA. Finally, we prove that both BCA and N BCA enjoy the Amalgamation Property (AP).

Paraconsistent weak Kleene ($\textrm{PWK}$) logic is the $3$-valued logic based on the weak Kleene matrices and with two designated values. In this paper, we investigate the poset of prevarieties of generalized involutive bisemilattices, focussing in particular on the order ideal generated by Α$\textrm{lg} (\textrm{PWK})$. Applying to this poset a general result by Alexej Pynko, we prove that, exactly like Priest’s logic of paradox, $\textrm{PWK}$ has only one proper nontrivial extension apart from classical logic: $\textrm{PWK}_{\textrm{E}}\textrm{,}$ PWK logic plus explosion. This $6$-valued logic, unlike $\textrm{PWK} $, fails to be paraconsistent. We describe its consequence relation via a variable inclusion criterion and identify its Suszko-reduced models.

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