The inapplicability of (selected) paraconsistent logics

2006

A logic is paraconsistent if it allows for non-trivial inconsistent theories. Given the usual definition of inconsistency, the notion of paraconsistent logic seems to rely upon the interpretation of the sign ‘¬’. As paraconsistent logic challenges properties of negation taken to be basic in other contexts, it is disputable that an operator lacking those properties will count as real negation. The conclusion is that there cannot be truly paraconsistent logics. This objection can be met from a substructural perspective since paraconsistent sequent calculi can be built with the same operational rules as classical logic but with slightly different structural rules.

Journal of Applied Non-Classical Logics, 2008

In many real-life applications of logic it is useful to interpret a particular sentence as true together with its negation. If we are talking about classical logic, this situation would force all other sentences to be equally interpreted as true. Paraconsistent logics are exactly those logics that escape this explosive effect of the presence of inconsistencies and allow for sensible reasoning still to take effect. To provide reasonably intuitive semantics for paraconsistent logics has traditionally proven to be a challenge. Possible-translations semantics can meet that challenge by allowing for each interpretation to be composed of multiple scenarios. Using that idea, a logic with a complex semantic behavior can be understood as an appropriate combination of ingredient logics with simpler semantic behaviors into which the original logic is given a collection of translations preserving its soundness. Completeness is then achieved through the judicious choice of the admissible translating mappings. The present note provides interpretation by way of possible-translations semantics for a group of fundamental paraconsistent logics extending the positive fragment of classical propositional logic. The logics P I, Cmin, mbC, bC, mCi and Ci, among others, are all initially presented through their non-truth-functional bivaluation semantics and sequent versions and then split by way of possible-translations semantics based on 3-valued ingredients.

ACM Transactions on Computational Logic, 2014

We develop a fully algorithmic approach to "taming" logics expressed Hilbert-style, i.e., reformulating them in terms of analytic sequent calculi, and useful semantics. Our approach applies to Hilbert calculi extending the positive fragment of propositional classical logic with axioms of a certain general form that contain new unary connectives. Our work encompasses various results already obtained for specific logics. It can be applied to new logics, as well as to known logics for which an analytic calculus or a useful semantics has so far not been available. A Prolog implementation of the method is described.

Inconsistency in Data and Knowledge” workshop (KRR …, 2001

We provide a general framework for constructing natural consequence relations for paraconsistent and plausible nonmonotonic reasoning. The framework is based on preferential systems whose preferences are based on the satisfaction of formulas in models. We show that these natural preferential systems that were originally designed for paraconsistent reasoning fulfill a key condition (stopperedness or smoothness) from the theoretical research of nonmonotonic reasoning. Consequently, the nonmonotonic consequence relations that they induce fulfill the desired conditions of plausible consequence relations. Hence our frameword encompasses different types of preferential systems that were developed from different motivations of paraconsistent reasoning and non-monotonic reasoning, and reveals an important link between them.

J. Marcos, D. Batens, and WA Carnielli, organizers, …

The logics of formal inconsistency (LFIs) are logics that allow to explicitly formalize the concepts of consistency and inconsistency by means of formulas of their language. Contradictoriness, on the other hand, can always be expressed in any logic, provided its la nguage includes a symbol for negation. Besides being able to represent the distinction between contradiction and inconsistency, LFIs are non-explosive logics, in the sense that a contradiction does not entail arbitrary statements, but yet are gently explosive, in the sense that, adjoining the additional requirement of consistency, then contradictoriness does cause explosion. Several logics can be seen as LFIs, among them the great majority of paraconsistent logics developed under the Brazilian tradition, as well as the sytems developed under the Polish tradition. We present here their semantical interpretations by way of possible-translations semantics, stressing their significance and applications to human reasoning and machine reasoning. We also give tableaux systems for some important LFIs: bC, Ci and LFI1.

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.