The inapplicability of (selected) paraconsistent logics

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.

Synthese, 2018

Classical propositional logic can be characterized, indirectly , by means of a complementary formal system whose theorems are exactly those formulas that are not classical tautologies, i.e., contradictions and truth-functional contingencies. Since a formula is contingent if and only if its negation is also contingent, the system in question is paraconsistent. Hence classical propositional logic itself admits of a paraconsistent characterization, albeit "in the negative". More generally, any decidable logic with a syntactically incomplete proof theory allows for a paraconsistent characterization of its set of theorems. This, we note, has important bearing on the very nature of paraconsistency as standardly characterized.

International Journal of Intelligent Systems, 1994

Paraconsistent logics are examined as an approach to knowledge representation devoted to the formalization of reasoning in the presence of contradictions. The adequacy of paraconsistent logics in such a perspective is described both on a general level and on a more specific level: discussion involves representative examples as well as special features (in the form of logical principles) of some significant paraconsistent logics. There is also a comparison of the paraconsistent logics approach with two alternative approaches, namely belief revision and non-monotonic logics.

1998

We present a novel approach to paraconsistent reasoning, that is, to reasoning from inconsistent information. The basic idea is the following. We transform an inconsistent theory into a consistent one by renaming all literals occurring in the theory. Then, we restore some of the original contents of the theory by introducing progressively formal equivalences linking the original literals to their renamings. This is done as long as consistency is preserved. The restoration of the original contents of the theory is done by appeal to default logic. The overall approach provides us with a family of paraconsistent consequence relations.

Data & Knowledge Engineering, 1995

Classical logic has many appealing features for knowledge representation and reasoning. But unfortunately it is awed when reasoning about inconsistent information, since anything follows from a classical inconsistency. This problem is addressed by introducing the notions of \argument" and of \acceptability" of an argument. These notions are used to introduce the concept of \argumentative structures". Each de nition of acceptability selects a subset of the set of arguments, and an argumentative structure is a subset of the power set of arguments. In this paper, we consider, in detail, a particular argumentative structure, where each argument is de ned as a classical inference together with the applied premisses. For such arguments, a variety of de nitions of acceptability are provided, the properties of these de nitions are explored, and their inter-relationship described. The de nitions of acceptability induce a family of logics called argumentative logics which we explore. The relevance of this work is considered and put in a wider perspective.

Logic and Logical Philosophy, 2018

This article describes a refutation method of proving the maximality of three-valued paraconsistent logics. After outlining the philosophical background related to paraconsistent logics and the refutation approach to modern logic, we briefly describe how these two areas meet in the case of maximal paraconsistent logics. We focus on a method of proving maximality introduced in [34] and [37] that has the benefit of being simple and effective. We show how the method works on a number of examples, thus emphasising the fact that it provides a unifying approach to the search for maximal paraconsistent logics. Finally, we show how the method can be generalised to cover a wide range of paraconsistent logics. We also conduct a small experiment that confirms the theoretical results.

Logic Journal of the IGPL

In 2016, Béziau introduces a restricted notion of paraconsistency, the so-called genuine paraconsistency. A logic is genuine paraconsistent if it rejects the laws $\varphi ,\neg \varphi \vdash \psi $ and $\vdash \neg (\varphi \land \neg \varphi )$. In that paper, the author analyzes, among the three-valued logics, which of them satisfy this property. If we consider multiple-conclusion consequence relations, the dual properties of those above-mentioned are $ \vdash \varphi , \neg \varphi $ and $\neg (\psi \vee \neg \psi ) \vdash $. We call genuine paracomplete logics those rejecting the mentioned properties. We present here an analysis of the three-valued genuine paracomplete logics. A very natural twist structures semantics for these logics is also found in a systematic way. This semantics produces automatically a simple and elegant Hilbert-style characterization for all these logics. Finally, we introduce the logic LGP which is genuine paracomplete is not genuine paraconsistent, no...

CLE e-prints (Section Logic), 2004

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.

New Essays on Belnap-­Dunn Logic, 2019

Epistemic logics based on normal modal logic are notoriously bad at handling inconsistent and yet non-trivial information. This fact motivates epistemic logics based on paraconsistent logic, examples of which can be traced back at least to the 1980s. These logics handle inconsistent and non-trivial information, but they usually do not articulate sources of the inconsistency. Yet, making the origin of an inconsistency present in a body of information explicit is important to assess the body—can we trace the mutually conflicting pieces of information to sources of information relevant to the body or is the inconsistency a result of an error unrelated to any outside sources? Is the inconsistency derived from various equally trustworthy sources or from a single inconsistent source? In this article we show that a paraconsistent modal logic, namely, the logic BK introduced by Odintsov and Wansing, is a first step toward a formalism capable of making these distinctions explicit. We interpr...