Paraconsistency: Logic and Applications - Springer (original) (raw)
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Paraconsistent Logic: A Proof-Theoretical Approach*
2006
A logic is paraconsistent if it allows for non-trivial inconsistent theories. Given the usual definition of inconsistency, the notion of paraconsistent logic seems to rely upon the interpretation of the sign ‘¬’. As paraconsistent logic challenges properties of negation taken to be basic in other contexts, it is disputable that an operator lacking those properties will count as real negation. The conclusion is that there cannot be truly paraconsistent logics. This objection can be met from a substructural perspective since paraconsistent sequent calculi can be built with the same operational rules as classical logic but with slightly different structural rules.
What is a Paraconsistent Logic
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.
Paraconsistency is the study of logical systems with a non-explosive negation such that a pair of contradictory formulas (with respect to such negation) does not necessarily imply triviality, discordant to what would be expected by contemporary logical orthodoxy. From a purely logical point of view, the significance of paraconsistency relies on the meticulous distinction between the general notions of contradictoriness and triviality of a theory—respectively, the fact that a given theory proves a proposition and its negation, and the fact that a given theory proves any proposition (in the language of its underlying logic). Aside from this simple rationale, the formal techniques and approaches that meet the latter definitional requirement are manifold. Furthermore, it is not solely the logical-mathematical properties of such systems that are open to debate. Rather, there are several foundational and philosophical questions worth studying, including the very question about the nature of the contradictions allowed by paraconsistentists. This entry aims to advance a brief account of some distinct approaches to paraconsistency, providing a panorama on the development of paraconsistent logic.
Paraconsistent logics and paraconsistency
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.
Remarks on Paraconsistency and Contradiction
In this paper we propose to take seriously the claim that at least some kinds of paraconsistent negations are subcontrariety forming operators. We shall argue that from an intuitive point of view, by considering paraconsistent negations that way, one needs not worry with true contradictions and the like, given that "true contradictions" are not involved in these paraconsistent logics. Our strategy consists in showing that the natural translation for subcontrariety in formal languages is not a contradiction in natural language, and vice versa. This move shall provide for an intuitive interpretation for paraconsistent negation, which we also discuss here. By putting all those pieces together, we hope a clearer sense of paraconsistency can be made, one which may free us from the need to tame contradictions.
Remarks on the Epistemic Interpretation of Paraconsistent Logic
Principia: an international journal of epistemology
In a recent work, Walter Carnielli and Abilio Rodrigues present an epistemically motivated interpretation of paraconsistent logic. In their view, when there is conflicting evidence with regard to a proposition A (i.e. when there is both evidence in favor of A and evidence in favor of ¬A) both A and ¬A should be accepted without thereby accepting any proposition B whatsoever. Hence, reasoning within their system intends to mirror, and thus, should be constrained by, the way in which we reason about evidence. In this article we will thoroughly discuss their position and suggest some ways in which this project can be further developed. The aim of the paper is twofold. On the one hand, we will present some philosophical critiques to the specific epistemic interpretation of paraconsistent logic proposed by Carnielli & Rodrigues. First, we will contend that Carnielli & Rodrigues's interpretation implies a thesis about what evidence rationally justifies to accept or believe, called Extreme Permissivism, which is controversial among epistemologists. Second, we will argue that what agents should do, from an epistemic point of view, when faced with conflicting evidence, is to suspend judgment. On the other hand, despite these criticisms we do not believe that the epistemological motivation put forward by Carnielli & Rodrigues is entirely wrong. In the last section, we offer an alternative way in which one might account for the epistemic rationality of accepting contradictions and, thus, for an epistemic understanding of paraconsistency, which leads us to discuss the notion of diachronic epistemic rationality.