LP, K3, and FDE as Substructural Logics

Building on recent work, I present sequent systems for the non-classical logics LP, K3, and FDE with two main virtues. First, derivations closely resemble those in standard Gentzen-style systems. Second, the systems can be obtained by reformulating a classical system using nonstandard sequent structure and simply removing certain structural rules (relatives of exchange and contraction). I clarify two senses in which these logics count as “substructural.

Normality Operators and Classical Collapse

In this paper, we extend the expressive power of the logics K3, LP and FDE with anormality operator, which is able to express whether a for-mula is assigned a classical truth value or not. We then establish classical recapture theorems for the resulting logics. Finally, we compare the approach via normality operator with the classical collapse approach devisedby Jc Beall

Ordered pair semantics and negation in LP

In this note, I present a modified semantic framework for the multi-valued paraconsistent logic LP, which allows for a straightforward preservation of a significant classical intuition about negation, namely that the negation operator reverses truth-value

Normality operators and Classical Recapture in Extensions of Kleene Logics

In this paper, we approach the problem of classical recapture for LP and K3 by using normality operators. These generalize the consistency and determinedness operators from Logics of Formal Inconsistency and Underterminedness, by expressing, in any many-valued logic, that a given formula has a classical truth value (0 or 1). In particular, in the rst part of the paper we introduce the logics LPe and Ke3 , which extends LP and K3 with normality operators, and we establish a classical recapture result based on the two logics. In the second part of the paper, we compare the approach in terms of normality operators with an established approach to classical recapture, namely minimal inconsistency. Finally, we discuss technical issues connecting LPe and Ke3 to the tradition of Logics of Formal Inconsistency and Underterminedness

Validity, dialetheism and self-reference

It has been argued recently (Beall in Spandrels of truth, Oxford University Press, Oxford, 2009; Beall and Murzi J Philos 110:143–165, 2013) that dialetheist theories are unable to express the concept of naive validity. In this paper, we will show that (Formula presented.) can be non-trivially expanded with a naive validity predicate. The resulting theory, (Formula presented.) reaches this goal by adopting a weak self-referential procedure. We show that (Formula presented.) is sound and complete with respect to the three-sided sequent calculus (Formula presented.). Moreover, (Formula presented.) can be safely expanded with a transparent truth predicate. We will also present an alternative theory (Formula presented.), which includes a non-deterministic validity predicate.Fil: Pailos, Federico Matias. Instituto de Investigaciones Filosóficas - Sadaf; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

A recovery operator for non-transitive approaches

In some recent articles, Cobreros, Egré, Ripley, & van Rooij have defended the idea that abandoning transitivity may lead to a solution to the trouble caused by semantic paradoxes. For that purpose, they develop the Strict-Tolerant approach, which leads them to entertain a nontransitive theory of truth, where the structural rule of Cut is not generally valid. However, that Cut fails in general in the target theory of truth does not mean that there are not certain safe instances of Cut involving semantic notions. In this article we intend to meet the challenge of answering how to regain all the safe instances of Cut, in the language of the theory, making essential use of a unary recovery operator. To fulfill this goal, we will work within the so-called Goodship Project, which suggests that in order to have nontrivial naïve theories it is sufficient to formulate the corresponding self-referential sentences with suitable biconditionals. Nevertheless, a secondary aim of this article is to propose a novel way to carry this project out, showing that the biconditionals in question can be totally classical. In the context of this article, these biconditionals will be essentially used in expressing the self-referential sentences and, thus, as a collateral result of our work we will prove that none of the recoveries expected of the target theory can be nontrivially achieved if self-reference is expressed through identities.Fil: Barrio, Eduardo Alejandro. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Instituto de Investigaciones Filosóficas - Sadaf; ArgentinaFil: Pailos, Federico Matias. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Instituto de Investigaciones Filosóficas - Sadaf; ArgentinaFil: Szmuc, Damián Enrique. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Instituto de Investigaciones Filosóficas - Sadaf; Argentin

A recovery operator for nontransitive approaches

In some recent articles, Cobreros, Egré, Ripley, & van Rooij have defended the idea that abandoning transitivity may lead to a solution to the trouble caused by semantic paradoxes. For that purpose, they develop the Strict-Tolerant approach, which leads them to entertain a nontransitive theory of truth, where the structural rule of Cut is not generally valid. However, that Cut fails in general in the target theory of truth does not mean that there are not certain safe instances of Cut involving semantic notions. In this article we intend to meet the challenge of answering how to regain all the safe instances of Cut, in the language of the theory, making essential use of a unary recovery operator. To fulfill this goal, we will work within the so-called Goodship Project, which suggests that in order to have nontrivial naïve theories it is sufficient to formulate the corresponding self-referential sentences with suitable biconditionals. Nevertheless, a secondary aim of this article is to propose a novel way to carry this project out, showing that the biconditionals in question can be totally classical. In the context of this article, these biconditionals will be essentially used in expressing the self-referential sentences and, thus, as a collateral result of our work we will prove that none of the recoveries expected of the target theory can be nontrivially achieved if self-reference is expressed through identities

A SIMPLE SEQUENT SYSTEM FOR MINIMALLY INCONSISTENT LP

Minimally inconsistent LP (MiLP) is a nonmonotonic paraconsistent logic based on Graham Priest’s logic of paradox (LP). Unlike LP, MiLP purports to recover, in consistent situations, all of classical reasoning. The present paper conducts a proof-theoretic analysis of MiLP. I highlight certain properties of this logic, introduce a simple sequent system for it, and establish soundness and completeness results. In addition, I show how to use my proof system in response to a criticism of this logic put forward by J. C. Beall

Two-valued Logics for Transparent Truth Theory

It is part of the current wisdom that the Liar and similar semantic paradoxes can be taken care of by the use of certain non-classical multivalued logics. It is also well-known that some of these logics can be characterized by means of two-valued semantics. An immediate consequence of this is that there are two-valued logics that support a transparent truth predicate. In this paper I want to suggest that these logics are not just interesting from a formal point of view but also from a philosophical perspective. In particular, I will argue that the two-valued presentation of these logics has a number of advantages over the more usual presentations.Fil: Rosenblatt, Lucas Daniel. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires; Argentina. Instituto de Investigaciones Filosóficas - Sadaf; Argentin