Negation and Paraconsistent Logics

Electronic Proceedings in Theoretical Computer Science

Paraconsistency is commonly defined and/or characterized as the failure of a principle of explosion. The various standard forms of explosion involve one or more logical operators or connectives, among which the negation operator is the most frequent. In this article, we ask whether a negation operator is essential for describing paraconsistency. In other words, is it possible to describe a notion of paraconsistency that is independent of connectives? We present two such notions of negation-free paraconsistency, one that is completely independent of connectives and another that uses a conjunction-like binary connective that we call fusion. We also derive a notion of quasi-negation from the former, and investigate its properties.

CLE e-prints (Section Logic), 2004

Philosophy of logic. Elsevier, 2007

In this article, we provide a survey of several paraconsistent logics (PL) and some of the philosophical issues they raise. We focus especially on the various kinds of applications that these logics have had. In particular, we consider Clogics, including their semantic properties, and the theory of descriptions associated with them. We present various kinds of paraconsistent set theories, and discuss how they can be used to develop paraconsistent mathematical theories, including theories about Russell sets, Russell relations, and paraconsistent Boolean algebras. We then examine discussive logic and its application to the foundation of physical theories and to the formal representation of partial truth. We then go on to consider different axiomatizations of annotated logics and their use in fuzzy set theory. Finally, after discussing additional developments in PL, we conclude the article by examining different applications of PL in technology, informatics, foundations of physics, morality, and law.

Synthese, 2018

Classical propositional logic can be characterized, indirectly , by means of a complementary formal system whose theorems are exactly those formulas that are not classical tautologies, i.e., contradictions and truth-functional contingencies. Since a formula is contingent if and only if its negation is also contingent, the system in question is paraconsistent. Hence classical propositional logic itself admits of a paraconsistent characterization, albeit "in the negative". More generally, any decidable logic with a syntactically incomplete proof theory allows for a paraconsistent characterization of its set of theorems. This, we note, has important bearing on the very nature of paraconsistency as standardly characterized.

2007

In traditional logic, contradictoriness (the presence of contradictions in a theory or in a body of knowledge) and triviality (the fact that such a theory entails all possible consequences) are assumed inseparable, granted that negation is available. This is an effect of an ordinary logical feature known as ‘explosiveness’: According to it, from a contradiction ‘α and ¬α’ everything is derivable. Indeed, classical logic (and many other logics) equate ‘consistency’ with ‘freedom from contradictions’. Such logics forcibly fail to distinguish, thus, between contradictoriness and other forms of inconsistency. Paraconsistent logics are precisely the logics for which this assumption is challenged, by the rejection of the classical ‘consistency presupposition’.

Paraconsistent logics are logical systems that reject the classical conception, 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 an 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.

2007

In this paper, consistency is understood as the absence of the negation of a theorem, and not, in general, as the absence of any contradiction. We define the basic constructive logic B Kc1 adequate to this sense of consistency in the ternary relational semantics without a set of designated points. Then we show how to define a series of logics extending B Kc1 within the spectrum delimited by contractionless minimal intuitionistic logic. All logics defined in the paper are paraconsistent logics.

Logic and Logical Philosophy, 2010

Two new first-order paraconsistent logics with De Morgantype negations and co-implication, called symmetric paraconsistent logic (SPL) and dual paraconsistent logic (DPL), are introduced as Gentzentype sequent calculi. The logic SPL is symmetric in the sense that the rule of contraposition is admissible in cut-free SPL. By using this symmetry property, a simpler cut-free sequent calculus for SPL is obtained. The logic DPL is not symmetric, but it has the duality principle. Simple semantics for SPL and DPL are introduced, and the completeness theorems with respect to these semantics are proved. The cut-elimination theorems for SPL and DPL are proved in two ways: One is a syntactical way which is based on the embedding theorems of SPL and DPL into Gentzen's LK, and the other is a semantical way which is based on the completeness theorems.

In some cases one is provided with inconsistent information and has to reason about various consistent scenarios contained in that information. Our goal is to argue that filtered paraconsistent logics are not the right tool to handle such cases and that the problems generalize to a large class of paraconsistent logics. A wide class of paraconsistent (inconsistency-tolerant) logics is obtained by filtration: adding conditions on the classical consequence operation (one example is weak Rescher-Manor consequence --- which bears $\Gamma$ to $\phi$ just in case $\phi$ follows classicaly from at least one maximally consistent subset of $\Gamma$). We start with surveying the most promising candidates and comparing their strength. Then we discuss the mainstream views on how non-classical logics should be chosen for an application and argue that none of these allows us to chose any of the filtered logics for action-guiding reasoning with inconsistent information, roughly because such a reasoning has to start with selecting possible scenarios and such a process does not correspond to any of the mathematical models offered by filtered paraconsistent logics. Finally, we criticize a recent attempt to defend explorative hypothetical reasoning by means of weak Rescher-Manor consequence operation by Meheus et al.

In this paper we present a philosophical motivation for the logics of formal inconsistency, a family of paraconsistent logics whose distinctive feature is that of having resources for expressing the notion of consistency within the object language in such a way that consistency may be logically independent of noncontradiction. We defend the view according to which logics of formal inconsistency may be interpreted as theories of logical consequence of an epistemological character. We also argue that in order to philosophically justify paraconsistency there is no need to endorse dialetheism, the thesis that there are true contradictions. Furthermore, we argue that an intuitive reading of the bivalued semantics for the logic mbC, a logic of formal inconsistency based on classical logic, fits in well with the basic ideas of an intuitive interpretation of contradictions. On this interpretation, the acceptance of a pair of propositions A and ¬A does not mean that A is simultaneously true and false, but rather that there is conflicting evidence about the truth value of A.