Paraconsistent logic

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

Univalent Foundations of Mathematics and Paraconsistency

Abstract: Vladimir Voevodsky in his Univalent Foundations Project writes that univalent foundations can be used both for constructive and for non-constructive mathematics. The last is of extreme interest since this project would be understood in a sense that this means an opportunity to extend univalent approach on non-classical mathematics. In general, Univalent Foundations Project allows the exploitation of the structures on homotopy types instead of structures on sets or structures on categories as in case of set-level mathematics or category-level mathematics. Non-classical mathematics should be respectively considered either as non-classical set-level mathematics or as non-classical category-level (toposes-level) mathematics. Since it is possible to directly formalize the world of homotopy types using in particular Martin-Lof type systems then the task is to pass to non-classical type systems e.g. da Costa paraconsistent type systems in order to formalize the world of non-classical homotopy types. Taking into account that the univalent model takes values in the homotopy category associated with a given set theory and to construct this model one usually first chooses a locally cartesian closed model category (in the sense of homotopy theory) then trying to extend this scheme for a case of non-classical set theories (e.g. paraconsistent ones) we need to evaluate respective non-classical homotopy types not in cartesian closed categories but in more suitable ones. In any case it seems that such Non-Classical Univalent Foundations Project should be directly developed according to Logical Pluralism paradigma and and it seems that it is difficult to find counter-argument of logical or mathematical character against such an opportunity except the globality and complexity of a such enterprise.

The Dissolution of Bar-Hillel-Carnap Paradox by Semantic Information Theory Based on a Paraconsistent Logic

Abstract: Several logical puzzles, riddles and problems are defined based on the notion of games in informative contexts. Hintikka argues that epistemology or the theory of knowledge must be considered from the notion of information. So, knowledge cannot just be based on the notions of belief and justification. The present proposal will focus on the logical structure of information, and not only on the quantification of information as suggested by Claude A. Shannon (1916-2001) (Shannon 1948). In many cases, the information bits, although seemingly or factually contradictory, are quite relevant. The paraconsistent systems of logic offer a formalization of reasoning that can support certain contradictions. The well-known “Bar-Hillel–Carnap Paradox” (Bar-Hillel, 1964) causes embarrassment when it concludes that the informational content of a contradiction would be maximum, opposing the traditional notion that the semantic information must be true, and that contradictions are necessarily false.

Extended Stepping Theories of Active Logic: Declarative Semantics

Abstract: Active Logic is a conceptual system with reasoning formalism that allows for correlation of their results with specific points in time and that has tolerance to inconsistencies. Currently, tolerance to inconsistencies (paraconsistency) in Active Logic systems is theoretically justified in the works of the authors of this paper and is attributed to the so-called formalisms of stepping theories, which integrate the principles of Active Logic and Logic Programming. More specifically, the argumentation semantics of so-called formalisms of stepping theories with two kinds of negation has been proved to be paraconsistent. This formalism has more expressive power than the other formalisms of stepping theories and to a greater extent satisfies the principles of Logic Programming. This case study proposes the declarative semantics for formalisms of stepping theories and represents its equivalency with respect to the argumentation semantics of this type of formalism. This, in turn, means that the proposed declarative semantics is also paraconsistent, and logical inconsistencies existing in these theories do not result in their destruction.