Two deductions systems for the Logic PM4N

The logic PM4N was introduced by Jean-Yves Beziau as a modal and 4-valued system. In this introductory paper, the author presented the system from a matrix logic with four values disposed in a Boolean algebra with a modal operator for the notion of necessity. From that matrix semantics, the paper shows some valid results and stresses some motivations of the matrix semantics and of the system PM4N. In this paper we develop some additional aspects of that logic and present two deductive systems for it, a simple system of tableaux and a sequent calculus

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

Knot much like tonk

Connectives such as Tonk have posed a significant challenge to the inferentialist. It has been recently argued (Button 2016; Button and Walsh 2018) that the classical semanticist faces an analogous problem due to the definability of "nasty connectives" under non-standard interpretations of the classical propositional vocabulary. In this paper, we defend the classical semanticist from this alleged problem

A strict negation operator for multivalued systems

En el siguiente trabajo abordaré algunas cuestiones sobre el significado de la constante lógica “negación”, y defenderé que posee dos rasgos fundamentales: el primero consiste en tener cierta prioridad formal y conceptual con respecto al resto de las constantes usuales conjunción y disyunción y el segundo en que su característica fundamental es la de ser un mecanismo de inversión hacia el opuesto entre valores de verdad. También discutiré algunos requisitos deseables para definir un operador de negación estricta o fuerte. Luego mostraré que es posible modelar un sistema cuatrivaluado a partir del conjunto FOUR añadiendo un operador que se comporte como una negación estricta y no clásica. In this paper I will exam some issues about the meaning of the logical constant “negation”. I will defend that it has two main features, one is being formally and conceptually prior to the other two basic logical connectives: conjunction and disjunction, and the other is having as a main character its ability to function as a switch of truth values, flip flopping one into the opposite. I will also argue for two other constraints one might ask for this type of operator to be called strict or strong. Finally, I show that is possible to design a multivalued logical system taking FOUR as a basis, and adding a strong yet non-classical negation to it.

Theories of truth based on four-valued infectious logics

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

Paraconsistency and its Philosophical Interpretations

Many authors have considered that the notions of paraconsistency and dialetheism are intrinsically connected, in many cases, to the extent of confusing both phenomena. However, paraconsistency is a formal feature of some logics that consists in invalidating the rule of explosion, whereas dialetheism is a semantical/ontological position consisting in accepting true contradictions. In this paper, we argue against this connection and show that it is perfectly possible to adopt a paraconsistent logic and reject dialetheism, and, moreover, that there are examples of non-paraconsistent logics that can be interpreted in a dialetheic way

Classical Negation and Expansions of Belnap–Dunn Logic

It is known that classical negation can be recovered in some systems of non-classical logics, such as paraconsistent logic and many-valued logic. However, the notion of classical negation needs to be examined carefully. Indeed, it is often thought that classical negation can be defined uniquely. This kind of arguments usually relies on a proof-theoretic viewpoint. But, in fact, from a semantic viewpoint, the definition of classical negation is not so straightforward as we might expect. We provide such an example by considering some expansions of FDE, and present some results which give us a new insight on the notion of classical negation in systems of non-classical logic