Nous proposons une logique épistémique quadrivalente AR4.

Multialgebras have been much studied in mathematics and in computer science. In 2016 Carnielli and Coniglio introduced a class of multialgebras called swap structures, as a semantic framework for dealing with several Logics of Formal Inconsistency that cannot be semantically characterized by a single finite matrix. In particular, these LFIs are not algebraizable by the standard tools of abstract algebraic logic. In this paper, the first steps towards a theory of non-deterministic algebraization of logics by swap structures are given. Specifically, (...) a formal study of swap structures for LFIs is developed, by adapting concepts of universal algebra to multialgebras in a suitable way. A decomposition theorem similar to Birkhoff’s representation theorem is obtained for each class of swap structures. Moreover, when applied to the 3-valued algebraizable logics J3 and Ciore, their classes of algebraic models are retrieved, and the swap structures semantics become twist structures semantics. This fact, together with the existence of a functor from the category of Boolean algebras to the category of swap structures for each LFI, suggests that swap structures can be seen as non-deterministic twist structures. This opens new avenues for dealing with non-algebraizable logics by the more general methodology of multialgebraic semantics. (shrink)

It has been argued recently that dialetheist theories are unable to express the concept of naive validity. In this paper, we will show that LP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {LP}$$\end{document} can be non-trivially expanded with a naive validity predicate. The resulting theory, LPVal\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {LP}^{\mathbf {Val}}$$\end{document} reaches this goal by adopting a weak self-referential procedure. We show that LPVal\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf (...) {LP}^{\mathbf {Val}}$$\end{document} is sound and complete with respect to the three-sided sequent calculus SLPVal\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {SLP}^{\mathbf {Val}}$$\end{document}. Moreover, LPVal\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {LP}^{\mathbf {Val}}$$\end{document} can be safely expanded with a transparent truth predicate. We will also present an alternative theory LPVal∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {LP}^{\mathbf {Val}^{*}}$$\end{document}, which includes a non-deterministic validity predicate. (shrink)

In this paper, we present a non-trivial and expressively complete paraconsistent naïve theory of truth, as a step in the route towards semantic closure. We achieve this goal by expressing self-reference with a weak procedure, that uses equivalences between expressions of the language, as opposed to a strong procedure, that uses identities. Finally, we make some remarks regarding the sense in which the theory of truth discussed has a property closely related to functional completeness, and we present a sound and (...) complete three-sided sequent calculus for this expressively rich theory. (shrink)

Infectious logics are systems that have a truth-value that is assigned to a compound formula whenever it is assigned to one of its components. This paper studies four-valued infectious logics as the basis of transparent theories of truth. This take is motivated as a way to treat different pathological sentences differently, namely, by allowing some of them to be truth-value gluts and some others to be truth-value gaps and as a way to treat the semantic pathology suffered by at least (...) some of these sentences as infectious. This leads us to consider four distinct four-valued logics: one where truth-value gaps are infectious, but gluts are not; one where truth-value gluts are infectious, but gaps are not; and two logics where both gluts and gaps are infectious, in some sense. Additionally, we focus on the proof theory of these systems, by offering a discussion of two related topics. On the one hand, we prove some limitations regarding the possibility of providing standard Gentzen sequent calculi for these systems, by dualizing and extending some recent results for infectious logics. On the other hand, we provide sound and complete four-sided sequent calculi, arguing that the most important technical and philosophical features taken into account to usually prefer standard calculi are, indeed, enjoyed by the four-sided systems. (shrink)

Sumário: 1. El caso del método científico, Alberto Oliva; 2. Un capítulo de la prehistoria de las ciencias humanas: la defensa por Vico de la tópica, Jorge Alberto Molina; 3. La figura de lo cognoscible y los mundos, Pablo Vélez León; 4. Lebenswelt de Husserl y las neurociencias, Vanessa Fontana; 5. El uso estético del concepto de mundos posibles, Jairo Dias Carvalho; 6. Realismo normativo no naturalista y mundos morales imposibles, Alcino Eduardo Bonella; 7. En la lógica de pragmatismo, Hércules (...) de Araujo Feitosa, Luiz Henrique da Cruz Silvestrini; 8 La lógica, el tiempo y el lenguaje natural: un sistema formal para tiempos de portugués, Carlos Luciano Manholi. / Temas em filosofia contemporânea II / Jonas Rafael Becker Arenhart, Jaimir Conte, Cezar Augusto Mortari (orgs.) – Florianópolis : NEL/UFSC, 2016. 167 p. : tabs. - (Rumos da epistemologia ; v. 14); ISBN 978-85-87253-27-9 (papel); ISBN 978-85-87253-26-2 (e-book). (shrink)

This paper investigates the question of characterizing first-order LFIs (logics of formal inconsistency) by means of two-valued semantics. LFIs are powerful paraconsistent logics that encode classical logic and permit a finer distinction between contradictions and inconsistencies, with a deep involvement in philosophical and foundational questions. Although focused on just one particular case, namely, the quantified logic QmbC, the method proposed here is completely general for this kind of logics, and can be easily extended to a large family of quantified paraconsistent (...) logics, supplying a sound and complete semantical interpretation for such logics. However, certain subtleties involving term substitution and replacement, that are hidden in classical structures, have to be taken into account when one ventures into the realm of nonclassical reasoning. This paper shows how such difficulties can be overcome, and offers detailed proofs showing that a smooth treatment of semantical characterization can be given to all such logics. Although the paper is well-endowed in technical details and results, it has a significant philosophical aside: it shows how slight extensions of classical methods can be used to construct the basic model theory of logics that are weaker than traditional logic due to the absence of certain rules present in classical logic. Several such logics, however, as in the case of the LFIs treated here, are notorious for their wealth of models precisely because they do not make indiscriminate use of certain rules; these models thus require new methods. In the case of this paper, by just appealing to a refined version of the Principle of Explosion, or Pseudo-Scotus, some new constructions and crafty solutions to certain non-obvious subtleties are proposed. The result is that a richer extension of model theory can be inaugurated, with interest not only for paraconsistency, but hopefully to other enlargements of traditional logic. (shrink)

Strong negation is a well-known alternative to the standard negation in intuitionistic logic. It is defined virtually by giving falsity conditions to each of the connectives. Among these, the falsity condition for implication appears to unnecessarily deviate from the standard negation. In this paper, we introduce a slight modification to strong negation, and observe its comparative advantages over the original notion. In addition, we consider the paraconsistent variants of our modification, and study their relationship with non-constructive principles and connexivity.

In this paper we consider the logics L(i,n) obtained from the (n+1)-valued Lukasiewicz logics L(n+1) by taking the order filter generated by i/n as the set of designated elements. In particular, the conditions of maximality and strong maximality among them are analyzed. We present a very general theorem that provides sufficient conditions for maximality between logics. As a consequence of this theorem, it is shown that L(i,n) is maximal w.r.t. CPL whenever n is prime. Concerning strong maximality (i.e. maximality w.r.t. (...) rules instead of only axioms), we provide algebraic arguments in order to show that the logics L(i,n) are not strongly maximal w.r.t. CPL, even for n prime. Indeed, in such case, we show that there is just one extension between L(i,n) and CPL obtained by adding to L(i,n) a kind of graded explosion rule. Finally, using these results, we show that the logics L(i,n) with n prime and i/n < 1/2 are ideal paraconsistent logics. (shrink)

The logics of formal inconsistency are logics tolerant to some amount of contradiction, but in which some versions of explosion still hold. The main result of this paper is a reconstruction of two such logics in the dialogical framework. By doing so, we achieve two things. On the one hand, we provide a formal approach to argumentative situations where some contradictions may occur while keeping the idea that there may still be situations in which some propositions are ‘safe’ in the (...) sense of immunity to the contradictions. On the other hand, we open a new line of study on these logics, in the context of the game-theoretical approach to semantics born in the 1960s, with various interesting perspectives, some of which are discussed at the end of this article. (shrink)

We generalize the construction of lattice-valued models of set theory due to Takeuti, Titani, Kozawa and Ozawa to a wider class of algebras and show that this yields a model of a paraconsistent logic that validates all axioms of the negation-free fragment of Zermelo-Fraenkel set theory.

We present a sequent calculus of a paraconsistent logic QMPT0, which has the paraconsistent-type excluded middle law (PEML) as an initial sequent. Our system shows that the presence of PEML is essentially important for QMPT0. It also has special rules when the set of constant symbols is finite. We also discuss the cut-elimination property of our system.