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.

Inconsistency, paraconsistency and ω-inconsistency

In this paper I'll explore the relation between ω-inconsistency and plain inconsistency, in the context of theories that intend to capture semantic concepts. In particular, I'll focus on two very well known inconsistent but non-trivial theories of truth: LP and STTT. Both have the interesting feature of being able to handle semantic and arithmetic concepts, maintaining the standard model. However, it can be easily shown that both theories are ω-inconsistent. Although usually a theory of truth is generally expected to be ω-consistent, all conceptual concerns don't apply to inconsistent theories. Finally, I'll explore if it's possible to have an inconsistent, but ω-consistent theory of truth, restricting my analysis to substructural theories.Fil: Da Re, Bruno. Universidad de Buenos Aires; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Instituto de Investigaciones Filosóficas - Sadaf; Argentin

What is a Paraconsistent Logic?

Paraconsistent logics are logical systems that reject the classical principle, usually dubbed Explosion, that a contradiction implies everything. However, the received view about paraconsistency focuses only the inferential version of Explosion, which is concerned with formulae, thereby overlooking other possible accounts. In this paper, we propose to focus, additionally, on a meta-inferential version of Explosion, i.e. which is concerned with inferences or sequents. In doing so, we will offer a new characterization of paraconsistency by means of which a logic is paraconsistent if it invalidates either the inferential or the meta-inferential notion of Explosion. We show the non-triviality of this criterion by discussing a number of logics. On the one hand, logics which validate and invalidate both versions of Explosion, such as classical logic and Asenjo–Priest’s 3-valued logic LP. On the other hand, logics which validate one version of Explosion but not the other, such as the substructural logics TS and ST, introduced by Malinowski and Cobreros, Egré, Ripley and van Rooij, which are obtained via Malinowski’s and Frankowski’s q- and p-matrices, respectively

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

Relevant Arithmetic and Mathematical Pluralism

In The Consistency of Arithmetic and elsewhere, Meyer claims to “repeal” Goedel’s second incompleteness theorem. In this paper, I review his argument, and then consider two ways of understanding it: from the perspective of mathematical pluralism and monism, respectively. Is relevant arithmetic just another legitimate practice among many, or is it a rival of its classical counterpart—a corrective to Goedel, setting us back on the path to the (One) True Arithmetic? To help answer, I sketch a few worked examples from relevant mathematics, to see what a non-classical (re)formulation of mathematics might look like in practice. I conclude that, while it is unlikely that relevant arithmetic describes past and present mathematical practice, and so might be most acceptable as a pluralist enterprise, it may yet prescribe a more monistic future venture

Relevant Arithmetic and Mathematical Pluralism

In The Consistency of Arithmetic and elsewhere, Meyer claims to “repeal” Goedel’s second incompleteness theorem. In this paper, I review his argument, and then consider two ways of understanding it: from the perspective of mathematical pluralism and monism, respectively. Is relevant arithmetic just another legitimate practice among many, or is it a rival of its classical counterpart—a corrective to Goedel, setting us back on the path to the (One) True Arithmetic? To help answer, I sketch a few worked examples from relevant mathematics, to see what a non-classical (re)formulation of mathematics might look like in practice. I conclude that, while it is unlikely that relevant arithmetic describes past and present mathematical practice, and so might be most acceptable as a pluralist enterprise, it may yet prescribe a more monistic future venture

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

Non-Boolean classical relevant logics I

Under embargo until: 2020-12-13Relevant logics have traditionally been viewed as paraconsistent. This paper shows that this view of relevant logics is wrong. It does so by showing forth a logic which extends classical logic, yet satisfies the Entailment Theorem as well as the variable sharing property. In addition it has the same S4-type modal feature as the original relevant logic E as well as the same enthymematical deduction theorem. The variable sharing property was only ever regarded as a necessary property for a logic to have in order for it to not validate the so-called paradoxes of implication. The Entailment Theorem on the other hand was regarded as both necessary and sufficient. This paper shows that the latter theorem also holds for classical logic, and so cannot be regarded as a sufficient property for blocking the paradoxes. The concept of suppression is taken up, but shown to be properly weaker than that of variable sharing.acceptedVersio

Relevant First-Order Logic LP#LP^\# and Curry's Paradox resolution

In 1942 Haskell B.Curry presented what is now called Curry paradox which can be found in a logic independently of its stand on negation.In recent years there has been a revitalised interest in non-classical solutions to the semantic paradoxes. In this article the non-classical resolution of Curry's Paradox and Shaw-Kwei paradox without rejection any contraction postulate is proposed.Comment: 7page