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This paper presents a semantical analysis of the Weak Kleene Logics Kw3 and PWK from the tradition of Bochvar and Halldén. These are threevalued logics in which a formula takes the third value if at least one of its components does. The paper establishes two main results: a characterisation result for the relation of logical con sequence in PWK – that is, we individuate necessary and sufficient conditions for a set. 

We provide a logical matrix semantics and a Gentzenstyle sequent calculus for the firstdegree entailments valid in W. T. Parry’s logic of Analytic Implication. We achieve the former by introducing a logical matrix closely related to that inducing paracomplete weak Kleene logic, and the latter by presenting a calculus where the initial sequents and the left and right rules for negation are subject to linguistic constraints. 

Threevalued logics are standardly used to formalize gappy languages, i.e., interpreted languages in which sentences can be true, false or neither. A threevalued logic that assigns the same truth value to all gappy sentences is, in our view, insufficient to capture important semantic differences between them. In this paper we will argue that there are two different kinds of pathologies that should be treated separately and we defend the usefulness of a fourvalued logic to represent adequately these two types of (...) 

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 Hilbertstyle axiomatization of the logics of left variable inclusion, to describe the structure of their reduced models, and to locate (...) 

In this work, we propose a variant of socalled informational semantics, a technique elaborated by Voishvillo, for two infectious logics, Deutsch’s ${\mathbf{S}_{\mathbf{fde}}}$ and Szmuc’s $\mathbf{dS}_{\mathbf{fde}}$. We show how the machinery of informational semantics can be effectively used to analyse truth and falsity conditions of disjunction and conjunction. Using this technique, it is possible to claim that disjunction and conjunction can be rightfully regarded as such, a claim which was disputed in the recent literature. Both ${\mathbf{S}_{\mathbf{fde}}}$ and $\mathbf{dS}_{\mathbf{fde}}$ are formalized in (...) 

This paper presents a sound and complete fivesided sequent calculus for firstorder weak Kleene valuations which permits not only elegant representations of four logics definable on firstorder weak Kleene valuations, but also admissibility of five cut rules by proof analysis. 