What is a Paraconsistent Logic?

Abstract

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 Nontransitive Approaches

“…It is usually presented as a first order logic, but for the sake of simplicity, we will deal with a propositional version of these systems and their expansions. 2 To move forward, it will be helpful to fix some definitions first. If L is a propositional language, then with FOR(L) we will denote the absolutely free algebra of L, whose universe is F O R(L).…”

Metainferential duality

“…Furthermore, a simple inference schema of a language L is the set of all and only the inference tokens that can be obtained from one of its members-its "basic instance"-by uniformly substituting some propositional variable p in it by some formula ϕ. Lastly, an inference schema with contexts is the union of a simple inference schema ρ with a subset of { Γ ∪ Σ, ∆ ∪ Π | Γ, ∆ ∈ ρ}. 3 When it generates no ambiguity, we will indistinctly refer as "schemata" to both simple schemata and to those with contexts.…”

“…Such a valuation satisfies both Γ ⇒ + ∆ and Γ ⇒ − ∆ in ST, but does not satisfy neither Γ ⇒ + ∆ nor Γ ⇒ − ∆ in TS. Some could, following the path initiated by [3], see this as a sort of revelation of the paraconsistent nature of ST and the paracomplete nature of TS, but we will not discuss this issue at length here. Now, with the help of these modifications to the metainferential framework, we are in a position to describe what a metainference token and a metainference schema are, in this new sense and-most importantly-to precisely state when two metainferences are dual, which will lead us to understand when two logics are metainferentially dual.…”

“…Given that our main concern are the logics ST and TS, for the most part of the article, these collections will simply be taken to be sets of formulae. The need to consider multisets will only briefly come into play when considering the failure of the structural rule of Contraction 3. We thank an anonymous referee for pointing out the need to clarify this generalization.…”

Paraconsistency and its Philosophical Interpretations

“…Finally, let's mention that recently, Barrio, Pailos and Szmuc ( [3]) have proposed a generalization of the notion of paraconsistency. They analyze the meta-inferential formulation of Explosion (Meta-Explosion):…”