Paraconsistent logic

About: Paraconsistent logic is a research topic. Over the lifetime, 1610 publications have been published within this topic receiving 28842 citations.

Subprevarieties Versus Extensions. Application to the Logic of Paradox

Abstract: Abstract In the present paper we prove that the poset of all extensions of the logic defined by a class of matrices whose sets of distinguished values are equationally definable by their algebra reducts is the retract, under a Galois connection, of the poset of all subprevarieties of the prevariety generated by the class of the algebra reducts of the matrices involved. We apply this general result to the problem of finding and studying all extensions of the logic of paradox (viz., the implication-free fragment of any non-classical normal extension of the relevance-mingle logic). In order to solve this problem, we first study the structure of prevarieties of Kleene lattices. Then, we show that the poset of extensions of the logic of paradox forms a four-element chain, all the extensions being finitely many-valued and finitely-axiomatizable logics. There are just two proper consistent extensions of the logic of paradox. The first is the classical logic that is relatively axiomatized by the Modus ponens rule for the material implication. The second extension, being intermediate between the logic of paradox and the classical logic, is the one relatively axiomatized by the Ex Contradictione Quodlibet rule.

Paraconsistent Double Negation That Can Simulate Classical Negation

Abstract: A new classical paraconsistent logic (CP), which is a variant of Nelson's paraconsistent four-valued logic, is introduced as a Gentzen-type sequent calculus. The logic CP can simulate the classical negation in classical logic by paraconsistent double negation in CP. Some theorems for syntactically and semantically embedding CP into a Gentzen-type sequent calculus LK for classical logic and vice versa are proved. The cut-elimination and completeness theorems for CP are also shown using these embedding theorems.

Applied Logic without Psychologism

Abstract: Logic is a celebrated representation language because of its formal generality. But there are two senses in which a logic may be considered general, one that concerns a technical ability to discriminate between different types of individuals, and another that concerns constitutive norms for reasoning as such. This essay embraces the former, permutation-invariance conception of logic and rejects the latter, Fregean conception of logic. The question of how to apply logic under this pure invariantist view is addressed, and a methodology is given. The pure invariantist view is contrasted with logical pluralism, and a methodology for applied logic is demonstrated in remarks on a variety of issues concerning non-monotonic logic and non-monotonic inference, including Charles Morgan’s impossibility results for non-monotonic logic, David Makinson’s normative constraints for non-monotonic inference, and Igor Douven and Timothy Williamson’s proposed formal constraints on rational acceptance.

From paraconsistent three-valued logics to multiple-source epistemic logic

Abstract: Several interpretations can be given to the third truth value in three-valued logics. Here, we consider the case when it refers to the epistemic notion of contradictory, or both true and false at the same time. We study several paraconsistent three-valued logics that carry this concern and show that they can be translated into a fragment of a simple epistemic logic where modalities can only appear in front of literals. This logic is unusual in the sense that necessity modalities distribute over disjunctions instead of conjunctions. An equivalent translation into a fragment of KD modal logic can be obtained by exchanging the role of possibility and necessity modalities, highlighting the perfect symmetry between three-valued logics of contradiction and three-valued logics of incomplete information.

Logical pluralism meets logical dynamics

Abstract: Introduction Where is logic heading today? There is a general feeling that the discipline is broadening its scope and agenda beyond classical foundational issues, and maybe even a concern that, like Stephen Leacock’s famous horseman, it is ‘riding off madly in all directions’. So, what is the resultant vector? There seem to be two broad answers in circulation today. One is logical pluralism, locating the new scope of logic in charting a wide variety of reasoning styles, often marked by non-classical structural rules of inference. This is the new program that I subscribed to in my work on sub-structural logics around 1990, and it is a powerful movement today. 1 But gradually, I have changed my mind about the crux of what logic should become. I would now say that the main issue is not variety of reasoning styles and notions of consequence, but the variety of informational tasks performed by intelligent interacting agents, of which inference is only one among many, involving observation, memory, questions and answers, dialogue, or general communication. And logical systems should deal with a wide variety of these, making information-carrying events first-class citizens in their set-up. This program of logical dynamics was proposed in van Benthem 1996. The purpose of this brief paper is to contrast and compare the two approaches, drawing freely on some insights from earlier published papers. In particular, I will argue that logical dynamics sets itself the more ambitious diagnostic goal of explaining why substructural phenomena occur, by ‘deconstructing’ them into classical logic plus an explicit account of the relevant informational events. I see this as a still more challenging departure from traditional logic. Diehard mathematicians still feel at ease with logical pluralism since it is all still a ‘science of formal systems’ describing ‘inference’, while to me, inference is just one way of producing information, at best on a par, even for logic itself, with others.