In this article we revisit a number of disputes regarding significance logics---i.e., inferential frameworks capable of handling meaningless, although grammatical, sentences---that took place in a series of articles most of which appeared in the Australasian Journal of Philosophy between 1966 and 1978. These debates concern (i) the way in which logical consequence ought to be approached in the context of a significance logic, and (ii) the way in which the logical vocabulary has to be modified (either by restricting some notions, (...) or by adding some vocabulary) to keep as much of Classical Logic as possible. Our aim is to show that the divisions arising from these disputes can be dissolved in the context of a novel and intuitive proposal that we put forward. (shrink)

This paper discusses three relevant logics that obey Component Homogeneity - a principle that Goddard and Routley introduce in their project of a logic of significance. The paper establishes two main results. First, it establishes a general characterization result for two families of logic that obey Component Homogeneity - that is, we provide a set of necessary and sufficient conditions for their consequence relations. From this, we derive characterization results for S*fde, dS*fde, crossS*fde. Second, the paper establishes complete sequent calculi (...) for S*fde, dS*fde, crossS*fde. Among the other accomplishments of the paper, we generalize the semantics from Bochvar, Hallden, Deutsch and Daniels, we provide a general recipe to define containment logics, we explore the single-premise/single-conclusion fragment of S*fde, dS*fde, crossS*fdeand the connections between crossS*fde and the logic Eq of equality by Epstein. Also, we present S*fde as a relevant logic of meaninglessness that follows the main philosophical tenets of Goddard and Routley, and we briefly examine three further systems that are closely related to our main logics. Finally, we discuss Routley's criticism to containment logic in light of our results, and overview some open issues. (shrink)

Paraconsistent Weak Kleene Logic is the 3-valued 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 cut-free calculus for PWK that is devoid of such provisos. Moreover, we introduce a Priest-style tableaux calculus for PWK.

The aim of this article is to discuss pure variable inclusion logics, that is, logical systems where valid entailments require that the propositional variables occurring in the conclusion are included among those appearing in the premises, or vice versa. We study the subsystems of Classical Logic satisfying these requirements and assess the extent to which it is possible to characterise them by means of a single logical matrix. In addition, we semantically describe both of these companions to Classical Logic in (...) terms of appropriate matrix bundles and as semilattice-based logics, showing that the notion of consequence in these logics can be interpreted in terms of truth (or non-falsity) and meaningfulness (or meaninglessness) preservation. Finally, we use Płonka sums of matrices to investigate the pure variable inclusion companions of an arbitrary finitary logic. -/- . (shrink)

We provide a logical matrix semantics and a Gentzen-style sequent calculus for the first-degree 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.

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, (...) unlike $\textrm{PWK} $, fails to be paraconsistent. We describe its consequence relation via a variable inclusion criterion and identify its Suszko-reduced models. (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)

In this work, we propose a variant of so-called 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 (...) terms of natural deduction. This allows us to solve several problems: to develop a natural deduction calculus for |${\mathbf{S}_{\mathbf{fde}}}$| containing the standard form of disjunction elimination (in contrast to the calculus by Petrukhin), to introduce the first natural deduction calculus for |$\mathbf{dS}_{\mathbf{fde}}$| and to reflect the fundamental symmetry between |${\mathbf{S}_{\mathbf{fde}}}$| and |$\mathbf{dS}_{\mathbf{fde}}$| on proof-theoretical level forming a convenient basis for obtaining their well-known extensions |$\mathbf{K}^{\mathbf{w}}_{\mathbf{3}}$| and |$\mathbf{PWK}$|⁠. (shrink)

In this paper, we study two companions of a logic, viz., the left variable inclusion companion and the restricted rules companion, their nature and interrelations, especially in connection with paraconsistency. A sufficient condition for the two companions to coincide has also been proved. Two new logical systems—intuitionistic paraconsistent weak Kleene logic (IPWK) and paraconsistent pre-rough logic (PPRL)—are presented here as examples of logics of left variable inclusion. IPWK is the left variable inclusion companion of intuitionistic propositional logic and is also (...) the restricted rules companion of it. PPRL, on the other hand, is the left variable inclusion companion of pre-rough logic but differs from the restricted rules companion of it. We have discussed algebraic semantics for these logics in terms of Płonka sums. This amounts to introducing a contaminating truth value, intended to denote a state of indeterminacy. (shrink)

Logics based on weak Kleene algebra (WKA) and related structures have been recently proposed as a tool for reasoning about flaws in computer programs. The key element of this proposal is the presence, in WKA and related structures, of a non-classical truth-value that is “contaminating” in the sense that whenever the value is assigned to a formula ϕ, any complex formula in which ϕ appears is assigned that value as well. Under such interpretations, the contaminating states represent occurrences of a (...) flaw. However, since different programs and machines can interact with (or be nested into) one another, we need to account for different kind of errors, and this calls for an evaluation of systems with multiple contaminating values. In this paper, we make steps toward these evaluation systems by considering two logics, HYB1 and HYB2, whose semantic interpretations account for two contaminating values beside classical values 0 and 1. In particular, we provide two main formal contributions. First, we give a characterization of their relations of (multiple-conclusion) logical consequence—that is, necessary and sufficient conditions for a set Δ of formulas to logically follow from a set Γ of formulas in HYB1 or HYB2 . Second, we provide sound and complete sequent calculi for the two logics. (shrink)

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

Our aim is to provide a sequent calculus whose external consequence relation coincides with the three-valued paracomplete logic `of nonsense' introduced by Dmitry Bochvar and, independently, presented as the weak Kleene logic K3W by Stephen C. Kleene. The main features of this calculus are (i) that it is non-reflexive, i.e., Identity is not included as an explicit rule (although a restricted form of it with premises is derivable); (ii) that it includes rules where no variable-inclusion conditions are attached; and (iii) (...) that it is hybrid, insofar as it includes both left and right operational introduction as well as elimination rules. (shrink)

In one of his papers, JC Beall advanced a new and interesting interpretation of Weak Kleene logic, in terms of on-topic/off-topic. In brief, Beall suggests to read the third value as _off-topic_, whereas the two classical values are read as _true and on-topic_ and _false and on-topic_. Building on Beall’s new interpretation, the aim of this paper is threefold. First, we discuss two motivations to enrich Beall’s interpretation, and offer an alternative semantic framework that reflects our motivations. Second, by making (...) use of our new framework, we will offer a new interpretation of the logic of Catuskoti which combines Beall’s proposal of having _FDE_ as the correct logic with the on-topic/off-topic divide. Finally, we will offer a general result that will allow us to make sense of a family of infectious logics in terms of Beall’s on-topic/off-topic reading. (shrink)

This is the Editorial Introduction to “S.I.: Strong and Weak Kleene Logics”.

Hintikka’s game theoretical approach to semantics has been successfully applied also to some non-classical logics. A recent example is Başkent (A game theoretical semantics for logics of nonsense, 2020. arXiv:2009.10878), where a game theoretical semantics based on three players and the notion of dominant winning strategy is devised to fit both Bochvar and Halldén’s logics of nonsense, which represent two basic systems of the family of weak Kleene logics. In this paper, we present and discuss a new game theoretic semantics (...) for Bochvar and Halldén’s logics, GTS-2, and show how it generalizes to a broader family of logics of variable inclusions. (shrink)

Our aim is to provide a sequent calculus whose external consequence relation coincides with the three-valued paracomplete logic ‘of nonsense’ introduced by Dmitry Bochvar and, independently, presented as the weak Kleene logic $$\textbf{K}_{\textbf{3}}^{\textbf{w}}$$ by Stephen C. Kleene. The main features of this calculus are (i) that it is non-reflexive, i.e., Identity is not included as an explicit rule (although a restricted form of it with premises is derivable); (ii) that it includes rules where no variable-inclusion conditions are attached; and (iii) (...) that it is hybrid, insofar as it includes both left and right operational introduction as well as elimination rules. (shrink)

Logical pluralism is a general idea that there is more than one correct logic. Carnielli and Rodrigues [2019a] defend an epistemic interpretation of the paraconsistent logic N4, according to which an argument is valid in this logic just in case it necessarily preserves evidence. The authors appeal to this epistemic interpretation to briefly motivate a kind of logical pluralism: “different accounts of logical consequence may preserve different properties of propositions”. The aim of this paper is to study the prospect of (...) a logical pluralism based on different interpretations of logical systems. First, we give our analysis of what it means to interpret a logic – and make some hopefully useful distinctions along the way. Second, we present what we call an interpretational logical pluralism: there is more than one correct logic and a logic is correct only if it has some adequate interpretation. We consider four variants of this idea, bring up some possible objections, and try to find plausible solutions on behalf of the pluralist. We will argue that interpretations of logical systems provide a promising – albeit not unproblematic – route to logical pluralism. (shrink)