Eduardo Barrio

El objetivo de este artículo es investigar diversos resultados limitativos acerca del concepto de validez. En particular, argumento que ninguna teoría lógica de orden superior con semántica estándar puede tener recursos expresivos suficientes como para capturar su propio concepto de validez. Además, muestro que la lógica de la verdad transparente que Hartry Field desarrolló recientemente conduce a resultados limitativos similares.

Adding a transparent truth predicate to a language completely governed by classical logic is not possible. The trouble, as is well-known, comes from paradoxes such as the Liar and Curry. Recently, Cobreros, Egré, Ripley and van Rooij have put forward an approach based on a non-transitive notion of consequence which is suitable to deal with semantic paradoxes while having a transparent truth predicate together with classical logic. Nevertheless, there are some interesting issues concerning the set of metainferences validated by this logic. In this paper, we show that this logic, once it is adequately understood, is weaker than classical logic. Moreover, the logic is in a way similar to the paraconsistent logic LP.

En un célebre artículo de 1936 Alfred Tarski presenta sus ideas sobre la noción de consecuencia lógica, discute distintas dificultades conectadas con ella, y señala que una definición formal aceptable de este concepto debe traer luz sobre la noción intuitiva de consecuencia. Con ...

It is widely accepted that classical logic is trivialized in the presence of a transparent truth-predicate. In this paper, we will explain why this point of view must be given up. The hierarchy of metainferential logics defined in Barrio et al. (Journal of Philosophical Logic, 1–28, 2019) and Pailos (The Review of Symbolic Logic, Forthcoming) recovers classical logic, either in the sense that every classical (meta)inferential validity is valid at some point in the hierarchy (as is stressed in Barrio et al. (Journal of Philosophical Logic, 1–28, 2019)), or because a logic of a transfinite level defined in terms of the hierarchy shares its validities with classical logic. Each of these logics is consistent with transparent truth—as is shown in Pailos (The Review of Symbolic Logic, Forthcoming)—, and this suggests that, contrary to standard opinions, transparent truth is after all consistent with classical logic. However, Scambler (Journal of Philosophical Logic, 49, 351–370, 2020) presents a major challenge to this approach. He argues that this hierarchy cannot be identified with classical logic in any way, because it recovers no classical antivalidities. We embrace Scambler’s challenge and develop a new logic based on these hierarchies. This logic recovers both every classical validity and every classical antivalidity. Moreover, we will follow the same strategy and show that contingencies need also be taken into account, and that none of the logics so far presented is enough to capture classical contingencies. Then, we will develop a multi-standard approach to elaborate a new logic that captures not only every classical validity, but also every classical antivalidity and contingency. As a€truth-predicate can be added to this logic, this result can be interpreted as showing that, despite the claims that are extremely widely accepted, classical logic does not trivialize in the context of transparent truth.

The main idea that we want to defend in this paper is that the question of what a logic is should be addressed differently when structural properties enter the game. In particular, we want to support the idea according to which it is not enough to identify the set of valid inferences to characterize a logic. In other words, we will argue that two logical theories could identify the same set of validities (e.g. its logical truths and valid inferences), but not be the same logic.

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 ...

In a forthcoming paper, Rodrigues & Carnielli (2016) present two logics motivated by the idea of capturing contradictions as conflicting evidence. The first logic is called BLE (the Basic Logic of Evidence) and the second-that is a conservative extension of BLE-is named LETJ (the Logic of Evidence and Truth). Roughly, BLE and LETJ are two non-classical (paraconsistent and paracomplete) logics in which the Laws of Explosion (EXP) and Excluded Middle (PEM) are not admissible. LETJ is built on top of BLE. Moreover, LETJ is a Logic of Formal Inconsistency (an LFI). This means that there is an operator that, roughly speaking, identifies a formula as having classical behavior. Both systems are motivated by the idea that there are different conditions for accepting or rejecting a sentence of our natural language. So, there are some special introduction and elimination rules in the theory that are capturing different conditions of use. Rodrigues & Carnielli's paper has an interesting and challenging idea. According to them, BLE and LETJ are incompatible with dialetheia. It seems to show that these paraconsistent logics cannot be interpreted using truth-conditions that allow true contradictions. In short, BLE and LETJ talk about conflicting evidence avoiding to talk about gluts. I am going to argue against this point of view. Basically, I will firstly offer a new interpretation of BLE and LETJ that is compatible with dialetheia. The background of my position is to reject the one canonical interpretation thesis: the idea according to which a logical system has one standard interpretation. Then, I will secondly show that there is no logical basis to fix that Rodrigues & Carnielli's interpretation is the canonical way to establish the content of logical notions of BLE and LETJ. Furthermore, the system LETJ captures inside classical logic. Then, I am also going to use this technical result to offer some further doubts about the one canonical interpretation thesis. §0 Introduction In this paper, I am interested in analyzing the relationship between pure logics and their interpretations. For example, it could be stimulating to discuss if there are some intrinsic characteristic in pure modal logic S5 (presented by axioms or by Kripke-models) and the well-known David Lewis' interpretation using actual possible worlds and accessibility relations. Is there a single canonical interpretation for S5? Are there any central features S5 sufficient to determine a single canonical interpretation? Questions like those raised for all kinds of logics. In particular, non-classical logics have been motivated considering specific Interpretations that serve as reasons for challenging classical logic. In the context of paraconsistent logics, logicians have provided different ways of interpreting how we should reason in the presence of contradictions. In this way, in a recently work, Rodrigues & Carnielli present 2 two logics motivated by the idea of capturing contradictions as conflicting evidence. The first logic is called BLE (the Basic Logic of Evidence) and the second-that is a conservative extension of BLE-is named LETJ (the Logic of Evidence and Truth). Roughly, BLE and LETJ are two non-classical (paraconsistent and paracomplete) logics in which the Laws of Explosion (EXP) and Excluded Middle (PEM) are not admissible. LETJ is built on top of BLE. Moreover, LETJ is a Logic of Formal Inconsistency (an LFI). This means that there is an operator that, roughly speaking, identifies a formula as having classical behavior. In particular, there are instances of EXP that can be captured inside LETJ using a circle operator. Both formal theories are presented by a natural deduction system (a set of introduction and elimination rules for the conjunction, disjunction and the material conditional), But, they focus on the negation and introduce special rules for refutability. The main idea is that there are different conditions for accepting or rejecting a sentence of our natural language. So, there are some special introduction and elimination rules in the theory that are capturing different conditions of use. As I said before, the underlying motivation provided for BLE and LETJ is that rules should preserve evidence for an assertion rather than its truth. Rodrigues & Carnielli's paper has an interesting and challenging idea. BLE and LETJ are incompatible with dialetheia. It seems to show that these paraconsistent logics cannot be interpreted using truth-conditions that allow true Acknowledgments: I wish to express my gratitude to

A theory of truth is usually demanded to be consistent, but ω- consistency is less frequently requested. Recently, Yatabe [21] has argued in favor of ω-inconsistent first-order theories, minimizing their odd consequences. In view of this fact, in this paper we present five arguments against ω-inconsistent theories of truth. In order to bring out this point, we will focus on two very well-known ω-inconsistent theories of truth: the classical theory of symmetric truth FS and the non-classical theory of naïve truth based on L  ukasiewicz infinitely-valued logic: PALT.

"Adding a transparent truth predicate to a language completely governed by classical logic is not possible. The trouble, as is well-known, comes from paradoxes such as the Liar and Curry. Recently, Cobreros, Egre, Ripley and van Rooij have put forward an approach based on a non-transitive notion of logical consequence. In a way, their framework is suitable to deal with semantic paradoxes while having a transparent truth predicate together with classical logic. Nevertheless, there are some disturbing facts concerning, on the one hand, the set of metainferences validated by this logic and, on the other, what is sometimes identified as its `external notion of consequence'. In this paper, we show that the strength of this logic, once it is adequately understood, is far weaker than classical logic. Moreover, the logic is in a way similar to the paraconsistent logic LP.
"

Adding a transparent truth predicate to a language completely governed by classical logic is not possible. The trouble, as is well-known, comes from paradoxes such as the liar and curry. Because of these paradoxes, theories of truth typically use a non- transparent truth predicate preserving classical logic. Revisionary approaches to logic take a different path. Semantic paradoxes are a central motivation for standard non-classical-logic-based truth theories: usually a logic weaker than classical logic. But curry paradox makes this task really complicated. It is well known that any transitive logic with transparent truth is in trouble because the Curry paradox, if it features a conditional connective validating identity, modus ponens, and contraction. Recently, Cobreros, Egre, Ripley and van Rooij have proposed an alternative approach based on a non-transitive notion of logical consequence. Adopting 3- valued models, they develop a permissive consequence in which the validity of an argument goes from strictly true premises to a tolerant true conclusion. In the internal level, this framework allows to deal with semantic paradoxes adopting transparent truth to mesh with classical logic. Nevertheless, in the external level where we are interested on the closure properties on the set of valid arguments, some results are disturbing. In this paper, I show that transitivity is not the only metainference that is missed. In particular, I prove that the strict-tolerant external notion of validity is just LP validity. This result shows that permissive consequence is too weak to express a suitable notion of external entailment.

It is widely accepted that a theory of truth for arithmetic should be consistent, but ω-consistency is less frequently requested. This paper argues that ω-consistency is a highly desirable feature for such theories. The point has already been made for first-order languages, though it is not entirely conclusive. We show that in the second-order case the consequence of adopting ω-inconsistent truth theories for arithmetic is unsatisfiability. In order to bring out this point, very well known ω- inconsistent theories of truth are considered: the revision theory of nearly stable truth T# and the classical theory of symmetric truth FS. Briefly, some conceptual problems are shown as well as new technical results that support the position taken here.

The aim of this paper is to show that it’s not a good idea to have a theory of truth that is consistent but ω-inconsistent. In order to bring out this point, it is useful to consider a particular case: Yablo’s Paradox. In theories of truth without standard models, the introduction of the truth-predicate to a first order theory does not maintain the standard ontology. Firstly, I exhibit some conceptual problems that follow from so introducing it. Secondly, I show that in second order theories with standard semantic the same procedure yields a theory that doesn’t have models. So, while having an ω-inconsistent theory is a bad thing, having an unsatisfiable theory is actually worse. This casts doubts on whether the predicate in question is, after all, a truth-predicate of that language. Finally, I present some alternative to prove an inconsistency adding plausible principles to certain theories of truth.

En Paradojas, Paradojas y más Paradojas se analiza de modo claro y preciso un problema fundamental: ¿Qué es una paradoja y cuál es su importancia en la filosofía contemporánea? El libro pone al alcance del lector no especializado una selección de las paradojas más conocidas e inquietantes, y realiza un análisis y discusión que permite que el lector pueda tener una comprensión profunda de ellas y de las alternativas que surgen frente a ellas. Las paradojas son un fenómeno multidisciplinario: la verdad, la racionalidad, el conocimiento, la justificación y el infinito son algunos de los conceptos que en diferentes campos del saber han sido afectados por ellas.