Paraconsistent logic

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

Inconsistency in Science: A Partial Perspective

Abstract: The prevalence of inconsistency in both our scientific and ‘everyday’ belief structures is something which is being increasingly recognised.1 In the world of scientific representations, Bohr’s theory, of course, is one of the more well known examples, described by Lakatos as ‘... sat like a baroque tower upon the Gothic base of classical electrodynamics’ (Lakatos 1970, 142; see also Brown 1992); others that have been put forward include the old quantum theory of black-body radiation, the Everett-Wheeler interpretation of quantum mechanics, Newtonian cosmology, the (early) theory of infinitesimals in calculus, the Dirac δ-function, Stokes’ analysis of pendulum motion and Michelson’s ‘single-ray’ analysis of the Michelson-Morley interferometer arrangement. The problem, of course, is how to accommodate this aspect of scientific practice given that within the framework of classical logic an inconsistent set of premises yields any well-formed statement as a consequence. The result is disastrous: the set of consequences of an inconsistent theory will explode into triviality and the theory is rendered useless. Another way of expressing this descent into logical anarchy which will be useful for our discussion to follow is to say that under classical logic the closure of any inconsistent set of sentences includes every sentence. It is this which lies behind Popper’s famous declaration that the acceptance of inconsistency ‘... would mean the complete breakdown of science’ since an inconsistent system is ultimately uninformative (Popper 1940, 408; 1972, 91–92).

Large-scale organizational computing requires unstratified reflection and strong paraconsistency

Abstract: Organizational Computing is a computational model for using the principles, practices, and methods of human organizations. Organizations of Restricted Generality (ORGs) have been proposed as a foundation for Organizational Computing. ORGs are the natural extension of Web Services, which are rapidly becoming the overwhelming standard for distributed computing and application interoperability in Organizational Computing. The thesis of this paper is that large-scale Organizational Computing requires reflection and strong paraconsistency for organizational practices, policies, and norms. Strong paraconsistency is required because the practices, policies, and norms of large-scale Organizational Computing are pervasively inconsistent. By the standard rules of logic, anything and everything can be inferred from an inconsistency, e.g., "The moon is made of green cheese." The purpose of strongly paraconsistent logic is to develop principles of reasoning so that irrelevances cannot be inferred from the fact of inconsistency while preserving all natural inferences that do not explode in the face of inconsistency. Reflection is required in order that the practices, policies, and norms can mutually refer to each other and make inferences. Reflection and strong paraconsistency are important properties of Direct Logic [Hewitt 2007] for large software systems. Godel first formalized and proved that it is not possible to decide all mathematical questions by inference in his 1st incompleteness theorem. However, the incompleteness theorem (as generalized by Rosser) relies on the assumption of consistency! This paper proves a generalization of the Godel/Rosser incompleteness theorem: theories of Direct Logic are incomplete. However, there is a further consequence. Although the semi-classical mathematical fragment of Direct Logic is evidently consistent, since the Godelian paradoxical proposition is selfprovable, every theory in Direct Logic has an inconsistency!

A Model-Theoretic Approach for Recovering Consistent Data from Inconsistent Knowledge Bases

Abstract: One of the most significant drawbacks of classical logic is its being useless in the presence of an inconsistency. Nevertheless, the classical calculus is a very convenient framework to work with. In this work we propose means for drawing conclusions from systems that are based on classical logic, although the information might be inconsistent. The idea is to detect those parts of the knowledge base that ‘cause’ the inconsistency, and isolate the parts that are ‘recoverable’. We do this by temporarily switching into Ginsberg/Fitting multivalued framework of bilattices (which is a common framework for logic programming and nonmonotonic reasoning). Our method is conservative in the sense that it considers the contradictory data as useless and regards all the remaining information unaffected. The resulting logic is nonmonotonic, paraconsistent, and a plausibility logic in the sense of Lehmann.

da Costa Meets Belnap and Nelson

Abstract: There are various approaches to develop a system of paraconsistent logic, and those we focus on in this paper are approaches of da Costa, Belnap, and Nelson. Our main focus is da Costa, and we deal with a system that reflects the idea of da Costa. We understand that the main idea of da Costa is to make explicit, within the system, the area in which you can infer classically. The aim of the paper is threefold. First, we introduce and present some results on a classicality operator which generalizes the consistency operator of Logics of Formal Inconsistency. Second, we show that we can introduce the classicality operator to the systems of Belnap. Third, we demonstrate that we can generalize the classicality operator above to the system of Nelson. The paper presents both the proof theory and semantics for the systems to be introduced, and also establishes some completeness theorems.