This paper introduces new logical systems which axiomatize a formal representation of inconsistency (here taken to be equivalent to contradictoriness) in classical logic. We start from an intuitive semantical account of inconsistent data, fixing some basic requirements, and provide two distinct sound and complete axiomatics for such semantics, LFI1 and LFI2, as well as their first-order extensions, LFI1* and LFI2*, depending on which additional requirements are considered. These formal systems are examples of what we dub Logics of Formal Inconsistency (LFI) (...) and form part of a much larger family of similar logics. We also show that there are translations from classical and paraconsistent first-order logics into LFI1* and LFI2*, and back. Hence, despite their status as subsystems of classical logic, LFI1* and LFI2* can codify any classical or paraconsistent reasoning. (shrink)

We present the logic K/2 which is a logic with classical implication and only the left part of classical negation.We show that it is possible to define a classical negation into K/2 and that the classical proposition logic K can be translated into this apparently weaker logic.We use concepts from model-theory in order to characterized rigorously this translation and to understand this paradox. Finally we point out that K/2 appears, following Haack's distinction, both as a deviation and an extension of (...) K. (shrink)

We present the logic K/2 which is a logic with classical implication and only the left part of classical negation.We show that it is possible to define a classical negation into K/2 and that the classical proposition logic K can be translated into this apparently weaker logic.We use concepts from model-theory in order to characterized rigorously this translation and to understand this paradox. Finally we point out that K/2 appears, following Haack's distinction, both as a deviation and an extension of (...) K. (shrink)

Maximality is a desirable property of paraconsistent logics, motivated by the aspiration to tolerate inconsistencies, but at the same time retain from classical logic as much as possible. In this paper we introduce the strongest possible notion of maximal paraconsistency, and investigate it in the context of logics that are based on deterministic or non-deterministic three-valued matrices. We show that all reasonable paraconsistent logics based on three-valued deterministic matrices are maximal in our strong sense. This applies to practically all three-valued (...) paraconsistent logics that have been considered in the literature, including a large family of logics which were developed by da Costa's school. Then we show that in contrast, paraconsistent logics based on three-valued properly nondeterministic matrices are not maximal, except for a few special cases (which are fully characterized). However, these non-deterministic matrices are useful for representing in a clear and concise way the vast variety of the (deterministic) three-valued maximally paraconsistent matrices. The corresponding weaker notion of maximality, called premaximal paraconsistency, captures the "core" of maximal paraconsistency of all possible paraconsistent determinizations of a non-deterministic matrix, thus representing what is really essential for their maximal paraconsistency. (shrink)

We define in precise terms the basic properties that an ‘ideal propositional paraconsistent logic’ is expected to have, and investigate the relations between them. This leads to a precise characterization of ideal propositional paraconsistent logics. We show that every three-valued paraconsistent logic which is contained in classical logic, and has a proper implication connective, is ideal. Then we show that for every n > 2 there exists an extensive family of ideal n -valued logics, each one of which is not (...) equivalent to any k -valued logic with k < n. (shrink)

Dialetheism is the metaphysical claim that there are true contradictions. And based on this view, Graham Priest and his collaborators have been suggesting solutions to a number of paradoxes. Those paradoxes include Russell’s paradox in naive set theory. For the purpose of dealing with this paradox, Priest is known to have argued against the presence of classical negation in the underlying logic of naive set theory. The aim of the present paper is to challenge this view by showing that there (...) is a way to handle classical negation. (shrink)

The article is devoted to the systematic study of the lattice εN4⊥ consisting of logics extending N4⊥. The logic N4⊥ is obtained from paraconsistent Nelson logic N4 by adding the new constant ⊥ and axioms ⊥ → p, p → ∼ ⊥. We study interrelations between εN4⊥ and the lattice of superintuitionistic logics. Distinguish in εN4⊥ basic subclasses of explosive logics, normal logics, logics of general form and study how they are relate.