The Logics of Strict-Tolerant Logic

Barrio, Eduardo Alejandro; Rosenblatt, Lucas; Tajer, Diego

doi:10.1007/s10992-014-9342-6

Cited by 76 publications

(117 citation statements)

References 24 publications

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“…The second point concerns the status of a theory based on ST as a distinctive approach to paradoxes. The connections shown by Barrio et al (2015) and Dicher and Paoli (2019) between ST and LP ground their claim that ST is nothing more than LP in sheep's clothing. Although a proper assessment of their arguments is outside the scope of this paper, the fact that we can represent xy-unsatisfiability, xy-supersatisfiability and xy-antivalidity of an inference in terms of the xy-validity of a metainference shows that all these logics are connected by different symmetries to one another.…”

Section: Final Remarksmentioning

confidence: 79%

“…Comparing trees for LP-inferences and trees for STmetainferences, we can see that there's a correspondence between the two logics. A metainference is ST-valid just in case the result of "lowering" it to an inference (that is, substituting '⇒' by '⊃' and ' ' by '⇒') is LP-valid (this connection was noticed and proved in Barrio et al (2015)). An analogous correspondence connects TS metainferences to K3 inferences (see Cobreros et al (2019) for more on this).…”

Section: Trees For Metainferencesmentioning

confidence: 94%

“…For a matter of uniformity we define metainferences with possibly multiple inferences as conclusions, although the examples we will consider below involve all a single inference in the conclusion side. The idea of validity for a metainference is also present in Barrio et al (2015) although in a slightly different ("global-substitutional") sense. In the paper Cobreros et al (2013) the validity of a metainference is used in a third different way ("simply global").…”

Section: Trees For Metainferencesmentioning

confidence: 99%

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(I can’t get no) antisatisfaction

2020

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Substructural approaches to paradoxes have attracted much attention from the philosophical community in the last decade. In this paper we focus on two substructural logics, named ST and TS, along with two structural cousins, LP and K3. It is well known that LP and K3 are duals in the sense that an inference is valid in one logic just in case the contrapositive is valid in the other logic. As a consequence of this duality, theories based on either logic are tightly connected since many of the arguments for and objections against one theory reappear in the other theory in dual form. The target of the paper is making explicit in exactly what way, if any, ST and TS are dual to one another. The connection will allow us to gain a more fine-grained understanding of these logics and of the theories based on them. In particular, we will obtain new insights on two questions concerning ST which are being intensively discussed in the current literature: whether ST preserves classical logic and whether it is LP in sheep's clothing. Explaining in what way ST and TS are duals requires comparing these logics at a metainferential level. We provide to this end a uniform proof theory to decide on valid metainferences for each of the four logics. This proof procedure allows us to show in a very simple way how different properties of inferences (unsatisfiability, supersatisfiability and antivalidity) that behave in very different ways for each logic can be captured in terms of the validity of a metainference.Substructural approaches to paradox − Non-transitive logic − Non-reflexive logic − Strong Kleene

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Section: Final Remarksmentioning

confidence: 79%

Section: Trees For Metainferencesmentioning

confidence: 94%

Section: Trees For Metainferencesmentioning

confidence: 99%

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(I can’t get no) antisatisfaction

2020

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“…About these logics we shall mention, in addition to the previous remarks, that in [4], [11] and [25] it is shown that-through some suitable translation-the set of valid inferences in LP coincides with the set of valid meta-inferences in ST, while in [14] it is conjectured that-again, through some suitable translation-the set of 8 By these rules we refer to the following, respectively.…”

Section: Definition 315 ([8] [21]) a 3-valued Ts-matrix Is A Q-matrixmentioning

confidence: 92%

What is a Paraconsistent Logic?

2018

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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.

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“…A third and final issue I will mention is that, due to the fact that in substructural logics there is a splitting of the notion of consequence into an internal and an external component (see for instance Mares & Paoli 2014;Avron 1988;Barrio, Rosenblatt, & Tajer 2015), perhaps there should be a corresponding distinction in the definition of internalization. More precisely, since there are two notions of consequence, there should be two validity predicates, one to capture each notion.…”

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confidence: 99%

Naive Validity, Internalization, and Substructural Approaches to Paradox

Rosenblatt¹

2017

Ergo, an Open Access Journal of Philosophy

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In this paper I investigate whether certain substructural theories are able to dodge paradox while at the same time containing what might be viewed as a naive validity predicate. To this end I introduce the requirement of internalization, roughly, that an adequate theory of validity should prove that its own metarules are validity-preserving. The main point of the paper is that substructural theories fail this requirement in various ways.

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The Logics of Strict-Tolerant Logic

Cited by 76 publications

References 24 publications

(I can’t get no) antisatisfaction

(I can’t get no) antisatisfaction

What is a Paraconsistent Logic?

Naive Validity, Internalization, and Substructural Approaches to Paradox

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