A paraconsistent route to semantic closure

Barrio, Eduardo Alejandro; Pailós, Federico Matías; Szmuc, Damián Enrique

doi:10.1093/jigpal/jzx009

Cited by 19 publications

(34 citation statements)

References 20 publications

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“…For some further discussion of these issues, see[23] 2. This thesis, as plausible as it sounds, we think is not true.…”

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confidence: 89%

Paraconsistency and its Philosophical Interpretations

Barrio¹,

Ré²

2018

AJL

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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.

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“…For some further discussion of these issues, see[23] 2. This thesis, as plausible as it sounds, we think is not true.…”

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confidence: 89%

Paraconsistency and its Philosophical Interpretations

Barrio¹,

Ré²

2018

AJL

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“…In a nutshell, Field embraces a view on disjunctions that complies with the classical understanding of its truth and falsity conditions, according to which a disjunction is true if and only if it has at least one true disjunct, and is false if and only if it has two false disjuncts. Given this 9 Let us clarify that we are not claiming that if a given sentence can be true and also can be false, then this is a good reason for treating it as a truth-value glut. (For, primarily, propositional letters share this feature with hypodoxical sentences, but we do not by any means support the idea that propositional letters should be regarded as truth-value gluts) What we claim, instead, is that if a given pathological sentence can be true and also can be false, then this is a plausible reason for treating such pathological sentences as a truth-value glut.…”

Section: Why Infectious Logics?mentioning

confidence: 99%

“…a glut. 9 Furthermore, this alternative path seems to suggest that not only will it be reasonable to work with an underlying four-valued logic with gaps and gluts for transparent theories of truth, but something else. Namely, that all paradoxical sentences should be taken as truth-value gaps, and that all hypodoxical sentences should be treated as truth-value gluts.…”

Section: Introductionmentioning

confidence: 99%

“…Hitoshi Omori's paper [25] shows how this technique can be applied to render a non-trivial and highly expressive naive set-theory based on the three-valued logic LP. Barrio, Pailos and Szmuc's paper [9] show how to apply this technique to build a functionally complete transparent theory of truth based on a paraconsistent logic closely related to Coniglio and Silvestrini's MPT from [22]. In this vein, Graham Priest's paper [64] also presents a discussion of Goodship's suggestions in connection with his favored position toward semantic paradoxes.…”

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confidence: 99%

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Theories of truth based on four-valued infectious logics

Ré

Pailós

Szmuc

2018

Logic Journal of the IGPL

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Infectious logics are systems which 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 (i) as a way to treat different pathological sentences (like the Liar and the Truth-Teller) differently, namely, by allowing some of them to be truth-value gluts and some others to be truthvalue gaps, and (ii) 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.

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“…5 For example, one could use a paraconsistent logic to capture the notion of transparent truth. See Barrio et al (2017). 6 For example, Priest 2014 and 2016.…”

Section: Notesmentioning

confidence: 99%

Models & Proofs: LFIs Without a Canonical Interpretations

Barrio¹

2018

Principia

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In different papers, Carnielli, W. & Rodrigues, A. (2012), Carnielli, W. Coniglio, M. & Rodrigues, A. (2017) and 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.

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A paraconsistent route to semantic closure

Cited by 19 publications

References 20 publications

Paraconsistency and its Philosophical Interpretations

Paraconsistency and its Philosophical Interpretations

Theories of truth based on four-valued infectious logics

Models & Proofs: LFIs Without a Canonical Interpretations

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