Paraconsistent Weak Kleene logic is the 3-valued logic with two designated values defined through the weak Kleene tables. This paper is a first attempt to investigate PWK within the perspective and methods of abstract algebraic logic. We give a Hilbert-style system for PWK and prove a normal form theorem. We examine some algebraic structures for PWK, called involutive bisemilattices, showing that they are distributive as bisemilattices and that they form a variety, \, generated by the 3-element algebra WK; we also (...) prove that every involutive bisemilattice is representable as the Płonka sum over a direct system of Boolean algebras. We then study PWK from the viewpoint of AAL. We show that \ is not the equivalent algebraic semantics of any algebraisable logic and that PWK is neither protoalgebraic nor selfextensional, not assertional, but it is truth-equational. We fully characterise the deductive filters of PWK on members of \ and the reduced matrix models of PWK. Finally, we investigate PWK with the methods of second-order AAL—we describe the class \ of PWK-algebras, algebra reducts of basic full generalised matrix models of PWK, showing that they coincide with the quasivariety generated by WK—which differs from \—and explicitly providing a quasiequational basis for it. (shrink)

The paper studies the containment companion of a logic \. This consists of the consequence relation \ 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 Płonka 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. (shrink)

In the logical context, ignorance is traditionally defined recurring to epistemic logic. In particular, ignorance is essentially interpreted as “lack of knowledge”. This received view has - as we point out - some problems, in particular we will highlight how it does not allow to express a type of content-theoretic ignorance, i.e. an ignorance of φ that stems from an unfamiliarity with its meaning. Contrarily to this trend, in this paper, we introduce and investigate a modal logic having a primitive (...) epistemic operator I, modeling ignorance. Our modal logic is essentially constructed on the modal logics based on weak Kleene three-valued logic introduced by Segerberg (Theoria, 33(1):53–71, 1997). Such non-classical propositional basis allows to define a Kripke-style semantics with the following, very intuitive, interpretation: a formula φ is ignored by an agent if φ is neither true nor false in every world accessible to the agent. As a consequence of this choice, we obtain a type of content-theoretic notion of ignorance, which is essentially different from the traditional approach. We dub it severe ignorance. We axiomatize, prove completeness and decidability for the logic of reflexive (three-valued) Kripke frames, which we find the most suitable candidate for our novel proposal and, finally, compare our approach with the most traditional one. (shrink)

We establish a duality between the category of involutive bisemilattices and the category of semilattice inverse systems of Stone spaces, using Stone duality from one side and the representation of involutive bisemilattices as Płonka sum of Boolean algebras, from the other. Furthermore, we show that the dual space of an involutive bisemilattice can be viewed as a GR space with involution, a generalization of the spaces introduced by Gierz and Romanowska equipped with an involution as additional operation.

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 three-valued 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 (...) scientific attitude. (shrink)

In this paper, we study logical systems which represent entailment relations of two kinds. We extend the approach of finding ‘exactly true’ and ‘non-falsity’ versions of four-valued 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). Dual-Belnap logic and anything but falsehood. Journal of Logics and their Applications, 6, 413–433; Shramko et al. (2017). First-degree entailment and its relatives. Studia Logica, 105(6), 1291–1317] to the case (...) of infectious logics, namely to the case of Deutsch's logic Sfde introduced in Deutsch [Relevant analytic entailment. The Relevance Logic Newsletter, 2(1), 26–44; The completeness of S. Studia Logica, 38(2), 137–147]. The particular systems obtained in this way are Setl and Snfl. We present them in the form of sequent calculi and prove corresponding soundness and completeness theorems. We illuminate the connection between Setl, Snfl and two well-known systems, strong Kleene three-valued logic K3 and Priest's Logic of Paradox LP. This connection allows us to investigate the characterisation of the entailment relations associated with Setl and Snfl as well as to introduce the notion of ‘infectious analogue’ of a certain logic. We also study implicative extensions of Setl and Snfl and prove soundness and completeness theorems for them as well. (shrink)

ABSTRACTThe paper deals with a problem posed by Mathieu Vidal to provide a formal representation for defective conditional in mathematics Vidal, M. [. The defective conditional in mathematics. Journal of Applied Non-Classical Logics, 24, 169–179]. The key feature of defective conditional is that its truth-value is indeterminate if its antecedent is false. In particular, we are interested in two explanations given by Vidal with the use of trivalent logics. By analysing a simple argument from plane geometry, where defective conditional is (...) in use, he gives two trivalent formal explanations for it. For both explanations, Vidal rigorously shows that trivalent logics cannot adequately represent defective conditional. Preserving Vidal's criteria of defective conditional ad max, we indicate some arguable points in his explanations and present an alternative explanation containing the original conjunction and disjunction in order to show that there are trivalent logics that might be an adequate formal explanation for defective conditional. (shrink)

Correspondence analysis is Kooi and Tamminga’s universal approach which generates in one go sound and complete natural deduction systems with independent inference rules for tabular extensions of many-valued functionally incomplete logics. Originally, this method was applied to Asenjo–Priest’s paraconsistent logic of paradox LP. As a result, one has natural deduction systems for all the logics obtainable from the basic three-valued connectives of LP -language) by the addition of unary and binary connectives. Tamminga has also applied this technique to the paracomplete (...) analogue of LP, strong Kleene logic \. In this paper, we generalize these results for the negative fragments of LP and \, respectively. Thus, the method of correspondence analysis works for the logics which have the same negations as LP or \, but either have different conjunctions or disjunctions or even don’t have them as well at all. Besides, we show that correspondence analyses for the negative fragments of \ and LP, respectively, are also suitable without any changes for the negative fragments of Heyting’s logic \ and its dual \ and LP). (shrink)

In this article we will focus our attention on the variety of distributive bisemilattices and some linguistic expansions thereof: bounded, De Morgan, and involutive bisemilattices. After extending Balbes’ representation theorem to bounded, De Morgan, and involutive bisemilattices, we make use of Hartonas–Dunn duality and introduce the categories of 2spaces and 2spaces\. The categories of 2spaces and 2spaces\ will play with respect to the categories of distributive bisemilattices and De Morgan bisemilattices, respectively, a role analogous to the category of Stone spaces (...) with respect to the category of Boolean algebras. Actually, the aim of this work is to show that these categories are, in fact, dually equivalent. (shrink)

In the present paper, various features of the class of propositional literal paralogics are considered. Literal paralogics are logics in which the paraproperties such as paraconsistence, paracompleteness and paranormality, occur only at the level of literals; that is, formulas that are propositional letters or their iterated negations. We begin by analyzing Bochvar’s three-valued nonsense logic B3, which includes two isomorphs of the propositional classical logic CPC. The combination of these two ‘strong’ isomorphs leads to the construction of two famous paralogics (...) P1 and I1, which are functionally equivalent. Moreover, each of these logics is functionally equivalent to the fragment of logic B3 consisting of external formulas only. In conclusion, we structure a four-element lattice of three-valued paralogics with respect to the possession of paraproperties. (shrink)