Conjunction and Disjunction in Infectious Logics

Omori, Hitoshi; Szmuc, Damián Enrique

doi:10.1007/978-3-662-55665-8_19

Cited by 23 publications

(36 citation statements)

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“…Thus, we might 11 As mentioned by an anonymous reviewer, the interpretation sketched (but not endorsed) in [19] seems less motivated than Priest's, which belongs to a well-developed tradition in paraconsistency. Moreover, we agree with the reviewer that, in fact, there is still a need for a better story regarding PWK-although a recent discussion concerning the reasonability of conjunction and disjunction as featured in infectious logics can be found in [51]. However, important as it is, in this paper we take the discussion as presented in the relevant literature, and leave deeper considerations about the philosophical motivations, e.g.…”

Section: Why Infectious Logics?supporting

confidence: 65%

“…1 More recently, however, it has also been argued that they are useful to formalize aboutness-preserving entailments and inferences [52], [35], [20], [72], pooling procedures for expert opinions and database reasoning [38], Buddhist philosophy [61], [63], faulty computers and programming languages [34], among many other deeply interesting philosophical issues. Additionally, from a more algebraic point of view, these systems were studied in relation with various structures such as bilattices, trilattices and operations on them [38], [36], power algebras and power matrices [44], bisemilattices [15], and plurivalent logics [62], [44], [50], [51]. 2 All these topics involve philosophical or conceptual reflections that we carry out in natural language.…”

Section: Introductionmentioning

confidence: 99%

“…This definition is extended to cover non-deterministic matrices in[72], where also alternative notions of infectiousness related to non-deterministic semantics are discussed.17 By this we mean that systems referred to as infectious logics in the recent literature (e.g. in[35],[72],[51]) do not need to be considered undesirable systems, for they can be proven to be interesting, useful and illuminating in many respects-which we do not have space to review here, for it will take us too far afield. We do, nevertheless, share the opinion of an anonymous referee, who called to attention that referring to something as "infectious" clearly does carry a disease metaphor and sounds negative.…”

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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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Section: Why Infectious Logics?supporting

confidence: 65%

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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Theories of truth based on four-valued infectious logics

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Pailós

Szmuc

2018

Logic Journal of the IGPL

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“…Thanks to an anonymous referee for this comment. 3 For a detailed overview and discussion, see Ciuni 2017, Ferguson 2017, Omori & Szmuc 2017 and Szmuc 2017. 4 Lewis 1986.…”

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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“…in [29], Fitting refers to them (and by extension to their eventual four-valued generalizations, on which more below) as strong Kleene logics. Among the weak Kleene logics, an increasing amount of recent work has been focused on Paraconsistent Weak Kleene [see, e.g., 9,11,12,13,39,40,44,51]. However, what is salient among these works is the absence of a cogent philosophical interpretation for it.…”

Section: Introductionmentioning

confidence: 99%

An Epistemic Interpretation of Paraconsistent Weak Kleene Logic

Szmuc

2019

LLP

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This paper extends Fitting's epistemic interpretation of some Kleene logics to also account for Paraconsistent Weak Kleene logic. To achieve this goal, a dualization of Fitting's "cut-down" operator is discussed, leading to the definition of a "track-down" operator later used to represent the idea that no consistent opinion can arise from a set including an inconsistent opinion. It is shown that, if some reasonable assumptions are made, the truth-functions of Paraconsistent Weak Kleene coincide with certain operations defined in this track-down fashion. Finally, further reflections on conjunction and disjunction in the weak Kleene logics accompany this paper, particularly concerning their relation with containment logics. These considerations motivate a special approach to defining sound and complete Gentzen-style sequent calculi for some of their four-valued generalizations.

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Conjunction and Disjunction in Infectious Logics

Cited by 23 publications

References 16 publications

Theories of truth based on four-valued infectious logics

Theories of truth based on four-valued infectious logics

Models & Proofs: LFIs Without a Canonical Interpretations

An Epistemic Interpretation of Paraconsistent Weak Kleene Logic

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