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Free logics are a family of firstorder logics which came about as a result of examining the existence assumptions of classical logic. What those assumptions are varies, but the central ones are that the domain of interpretation is not empty, every name denotes exactly one object in the domain and the quantifiers have existential import. Free logics reject the claim that names need to denote in. Positive free logic concedes that some atomic formulas containing nondenoting names are true, negative free (...) 

Paraconsistent Weak Kleene Logic is the 3valued propositional logic defined on the weak Kleene tables and with two designated values. Most of the existing proof systems for PWK are characterised by the presence of linguistic restrictions on some of their rules. This feature can be seen as a shortcoming. We provide a cutfree calculus for PWK that is devoid of such provisos. Moreover, we introduce a Prieststyle tableaux calculus for PWK. 

Paraconsistent weak Kleene 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} $. 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, (...) 

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. 

Individuating the logic of scientific discovery appears a hopeless enterprise. Less hopeless is trying to figure out a logical way to model the epistemic attitude distinguishing the practice of scientists. In this paper, we claim that classical logic cannot play such a descriptive role. We propose, instead, one of the threevalued logics in the Kleene family that is often classified as the less attractive one, namely Hallden’s logic. By providing it with an appropriate epistemic interpretation, we can informally model the (...) 

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 (...) 

In this paper, we study logical systems which represent entailment relations of two kinds. We extend the approach of finding ‘exactly true’ and ‘nonfalsity’ versions of fourvalued logics that emerged in series of recent works [Pietz & Rivieccio (2013). Nothing but the truth. Journal of Philosophical Logic, 42(1), 125–135; Shramko (2019). DualBelnap logic and anything but falsehood. Journal of Logics and their Applications, 6, 413–433; Shramko et al. (2017). Firstdegree entailment and its relatives. Studia Logica, 105(6), 1291–1317] to the case (...) 

In this paper we discuss the extent to which conjunction and disjunction can be rightfully regarded as such, in the context of infectious logics. Infectious logics are peculiar manyvalued logics whose underlying algebra has an absorbing or infectious element, which is assigned to a compound formula whenever it is assigned to one of its components. To discuss these matters, we review the philosophical motivations for infectious logics due to Bochvar, Halldén, Fitting, Ferguson and Beall, noticing that none of them discusses (...) 