Defining LFIs and LFUs in extensions of infectious logics

The aim of this paper is to explore the peculiar case of infectious logics, a group of systems obtained generalizing the semantic behavior characteristic of the (¬, ∧, ∨) -fragment of the logics of nonsense, such as the ones due to Bochvar and Halldén, among others. Here, we extend these logics with classical negations, and we furthermore show that some of these extended systems can be properly regarded as logics of formal inconsistency (LFIs) and logics of formal undeterminedness (LFUs).Fil: Szmuc, Damián Enrique. Universidad de Buenos Aires. Facultad de Filosofía y Letras; Argentina. Instituto de Investigaciones Filosóficas - Sadaf; Argentin

Track-Down Operations on Bilattices

This paper discusses a dualization of Fitting's notion of a "cut-down" operation on a bilattice, rendering a "track-down" operation, later used to represent the idea that a consistent opinion cannot arise from a set including an inconsistent opinion. The logic of track-down operations on bilattices is proved equivalent to the logic d_Sfde, dual to Deutsch's system S_fde. Furthermore, track-down operations are employed to provide an epistemic interpretation for paraconsistent weak Kleene logic. Finally, two logics of sequential combinations of cut-and track-down operations allow settling positively the question of whether bilattice-based semantics are available for subsystems of S_fde

Logics of variable inclusion and the lattice of consequence relations

In this paper, firstly, we determine the number of sublogics of variable inclusion of an arbitrary finitary logic L with partition function. Then, we investigate their position into the lattice of consequence relations over the language of L.Comment: arXiv admin note: text overlap with arXiv:1804.08897, arXiv:1809.0676

Dualities for Plonka sums

Plonka sums consist of an algebraic construction similar, in some sense to direct limits, which allows to represent classes of algebras defined by means of regular identities (namely those equations where the same set of variables appears on both sides). Recently, Plonka sums have been connected to logic, as they provide algebraic semantics to logics obtained by imposing a syntactic filter to given logics. In this paper, I present a very general topological duality for classes of algebras admitting a Plonka sum representation in terms of dualisable algebras.Comment: 12 pages; the paper was awarded with the "SILFS Logic Prize" and is appearing on "Logica Universalis

ЛЮБОПЫТНЫЙ ФАКТ ОБ ИНФЕКЦИОННЫХ ЛОГИКАХ

This work is devoted to the study of some properties of logical consequence in infectious logics. Two four-valued systems—Deutsch’s Sfde and Szmuc’s dSfde—are of particular interest. In this short note, I prove the following two facts: (1) if we define a consequence relation via preservation of truth and non-falsity in the context of dSfde, then the resulting consequence relation is equivalent to the one from recently discovered logic Setl, (2) if we define a consequence relation via preservation of non-falsity (in the strong sense) in the context of dSfde, then the resulting consequence relation is equivalent to the one from recently discovered logic Snfl. Since both Setl and Snfl were originally obtained by analogous modification of Sfde, the main thesis of my work can be formulated as follows: the exactly true and non-falsity versions of Sfde and dSfde are equivalent. From the philosophical point of view, their equivalence allows to broaden the domain of possible applications, because it suggests that they can be equally interpreted in the context of infectious gaps as well as in the context of infectious gluts.Работа посвящена исследованию свойств логического следования в контек- сте так называемых инфекционных логик. Среди последних особый интерес представляют две четырехзначные логики, которые также можно отнести к классу релевантных: логи- ка Дойча Sfde и логика Шмуца dSfde. В этой короткой заметке я докажу, что если в dSfde отношение следования определить через сохранность истинности и неложности, то результат эквивалентен отношению следования в другой недавно открытой логике Setl, а если определить его через сохранность неложности (в сильном смысле), то мы получим следование, эквивалентное следованию в логике Snfl, также предложенной совсем недав- но. Поскольку обе логики Setl и Snfl получаются в результате аналогичной модификации логики Sfde, главный тезис моей работы сводится к следующему: «в точности истинност- ная» и «неложностная» версии логик Sfde и dSfde эквивалентны. С содержательной точки зрения эквивалентность исследуемых систем позволяет расширить область их возможных приложений, поскольку она свидетельствует о том, что эти системы могут быть интер- претированы как в контексте семантики с инфекционными истинностно-значными прова- лами, так и в контексте семантики с инфекционными пресыщенными оценками

Conjunction and Disjunction in Infectious Logics

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 many-valued 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 our main question. This is why we finally turn to the analysis of the truth-conditions for conjunction and disjunction in infectious logics, employing the framework of plurivalent logics, as discussed by Priest. In doing so, we arrive at the interesting conclusion that —in the context of infectious logics— conjunction is conjunction, whereas disjunction is not disjunction

Pure Refined Variable Inclusion Logics

In this article, we explore the semantic characterization of the (right) pure refined variable inclusion companion of all logics, which is a further refinement of the nowadays well-studied pure right variable inclusion logics. In particular, we will focus on giving a characterization of these fragments via a single logical matrix, when possible, and via a class of finite matrices, otherwise. In order to achieve this, we will rely on extending the semantics of the logics whose companions we will be discussing with infectious values in direct and in more subtle ways. This further establishes the connection between infectious logics and variable inclusion logics

A game theoretical semantics for logics of nonsense

Logics of non-sense allow a third truth value to express propositions that are nonsense. These logics are ideal formalisms to understand how errors are handled in programs and how they propagate throughout the programs once they appear. In this paper, we give a Hintikkan game semantics for logics of non-sense and prove its correctness. We also discuss how a known solution method in game theory, the iterated elimination of strictly dominated strategies, relates to semantic games for logics of nonsense. Finally, we extend the logics of nonsense only by means of semantic games, developing a new logic of nonsense, and propose a new game semantics for Priest’s Logic of Paradox

Peirce’s Triadic Logic and Its (Overlooked) Connexive Expansion

In this paper, we present two variants of Peirce’s Triadic Logic within a language containing only conjunction, disjunction, and negation. The peculiarity of our systems is that conjunction and disjunction are interpreted by means of Peirce’s mysterious binary operations Ψ and Φ from his ‘Logical Notebook’. We show that semantic conditions that can be extracted from the definitions of Ψ and Φ agree (in some sense) with the traditional view on the semantic conditions of conjunction and disjunction. Thus, we support the conjecture that Peirce’s special interest in these operations is due to the fact that he interpreted them as conjunction and disjunction, respectively. We also show that one of our systems may serve as a suitable base for an interesting implicative expansion, namely the connexive three-valued logic by Cooper. Sound and complete natural deduction calculi are presented for all systems examined in this paper

Containment Logics: Algebraic Completeness and Axiomatization

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 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