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Logics of Formal Inconsistency
 Handbook of Philosophical Logic
"... 1.1 Contradictoriness and inconsistency, consistency and noncontradictoriness In traditional logic, contradictoriness (the presence of contradictions in a theory or in a body of knowledge) and triviality (the fact that such a theory ..."
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1.1 Contradictoriness and inconsistency, consistency and noncontradictoriness In traditional logic, contradictoriness (the presence of contradictions in a theory or in a body of knowledge) and triviality (the fact that such a theory
A Taxonomy of Csystems
 PARACONSISTENCY: THE LOGICAL WAY TO THE INCONSISTENT
, 2002
"... The logics of formal inconsistency (LFIs) are paraconsistent logics which permit us to internalize the concepts of consistency or inconsistency inside our object language, introducing new operators to talk about them, and allowing us, in principle, to logically separate the notions of contradictorin ..."
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Cited by 58 (19 self)
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The logics of formal inconsistency (LFIs) are paraconsistent logics which permit us to internalize the concepts of consistency or inconsistency inside our object language, introducing new operators to talk about them, and allowing us, in principle, to logically separate the notions of contradictoriness and of inconsistency. We present the formal definitions of these logics in the context of General Abstract Logics, argue that they in fact represent the majority of all paraconsistent logics existing up to this point, if not the most exceptional ones, and we single out a subclass of them called Csystems, as the LFIs that are built over the positive basis of some given consistent logic. Given precise characterizations of some received logical principles, we point out that the gist of paraconsistent logic lies in the Principle of Explosion, rather than in the Principle of NonContradiction, and we also sharply distinguish these two from the Principle of NonTriviality, considering the next various weaker formulations of explosion, and investigating their interrelations. Subsequently, we present the syntactical formulations of some of the main Csystems based on classical logic, showing how several wellknown logics in the literature can be recast as such a kind of Csystems, and carefully study their properties and shortcomings, showing for instance how they can be used to faithfully
Formal Inconsistency and Evolutionary Databases
 LOGIC AND LOGICAL PHILOSOPHY
, 2000
"... 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 soun ..."
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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 firstorder 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 firstorder 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.
Ex contradictione non sequitur quodlibet
 Proceedings of the II Annual Conference on Reasoning and Logic, held in Bucharest, RO, July 2000
, 2001
"... We summarize here the main arguments, basic research lines, and results on the foundations of the logics of formal inconsistency. These involve, in particular, some classes of wellknown paraconsistent systems. We also present their semantical interpretations by way of possibletranslations semantic ..."
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We summarize here the main arguments, basic research lines, and results on the foundations of the logics of formal inconsistency. These involve, in particular, some classes of wellknown paraconsistent systems. We also present their semantical interpretations by way of possibletranslations semantics and their applications to human reasoning and machine reasoning. 1 1. Do we need to worry about inconsistency? Classical logic, as we all know, cannot survive contradictions. Among the principles that were gradually incorporated into the “properties of correct reasoning ” since Aristotle, the Principle of PseudoScotus (PPS), also known since medieval times as ex contradictione sequitur quodlibet (and also called the Principle of Explosion by some contemporary logicians), states that in any theory exposed to the enzymatic character of a contradiction A and ÏA one can derive any other arbitrary sentence B, so that the theory would turn out to be trivial. Another principle called the Principle of NonContradiction (PNC) states that
On the Possibility of Using Complex Values in Fuzzy Logic For Representing Inconsistencies
, 1996
"... In science and engineering, there are "paradoxical" cases when we have some arguments in favor of some statement A (so, the degree to which A is known to be true is positive (nonzero)), and we also have some arguments in favor of its negation :A, and we do not have enough information t ..."
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In science and engineering, there are "paradoxical" cases when we have some arguments in favor of some statement A (so, the degree to which A is known to be true is positive (nonzero)), and we also have some arguments in favor of its negation :A, and we do not have enough information to tell which of these two statements is correct. Traditional fuzzy logic, in which "truth values" are described by numbers from the interval [0; 1], easily describes such "paradoxical" situations: the degree a to which the statement A is true and the degree 1 \Gamma a to which its negation :A is true can be both positive. In this case, if we use traditional fuzzy &\Gammaoperations (min or product), the "truth value" a&(1 \Gamma a) of the statement A&:A is positive, indicating that there is some degree of inconsistency in the initial beliefs.
Idempotent full paraconsistent negations are not algebrizable, Notre Dame
 Journal of Formal Logic
, 1998
"... Abstract Using methods of abstract logic and the theory of valuation, we prove that there is no paraconsistent negation obeying the law of double negation and such that ¬(a ∧¬a) is a theorem which can be algebraized by a technique similar to the TarskiLindenbaum technique. 135 1What are the feature ..."
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Abstract Using methods of abstract logic and the theory of valuation, we prove that there is no paraconsistent negation obeying the law of double negation and such that ¬(a ∧¬a) is a theorem which can be algebraized by a technique similar to the TarskiLindenbaum technique. 135 1What are the features of a paraconsistent negation? Since paraconsistent logic was launched by da Costa in his seminal paper [4], one of the fundamental problems has been to determine what exactly are the theoretical or metatheoretical properties of classical negation that can have a unary operator not obeying the principle of noncontradiction, that is, a paraconsistent operator. What the result presented here shows is that some of these properties are not compatible with each other, so that in constructing a paraconsistent negation as close as possible to classical negation, we have to make a choice among classical properties compatible with the idea of paraconsistency. In particular, there is no paraconsistent negation more classical than all the others. The incompatibility appearing here is between theoretical properties (double negation and ¬(a ∧¬a) as a theorem) and a metatheoretical property (replacement theorem). One who chooses the theoretical properties will not be able to algebraize his system with the usual TarskiLindenbaum method and should use some alternative treatments such as that in da Costa [5]. On the other hand, one who chooses the metatheoretical property will have to sacrifice at least one fundamental theoretical property of negation, risking the possibility of dealing with an operator that is a modality rather than a negation. The result presented here is of the same kind as some previous results concerning the incompatibility between the replacement theorem and the paraconsistent logic C1 of [4]. It was soon realized that the replacement theorem is not valid in C1. Mortensen [14] proved that, in fact, it was impossible to define a nontrivial congruence in C1. Urbas [19] proved that the addition of the replacement theorem to
Cutfree Sequent Calculi for Csystems with Generalized Finitevalued Semantics
"... In [5], a general method was developed for generating cutfree ordinary sequent calculi for logics that can be characterized by finitevalued semantics based on nondeterministic matrices (Nmatrices). In this paper, a substantial step towards automation of paraconsistent reasoning is made by applyin ..."
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In [5], a general method was developed for generating cutfree ordinary sequent calculi for logics that can be characterized by finitevalued semantics based on nondeterministic matrices (Nmatrices). In this paper, a substantial step towards automation of paraconsistent reasoning is made by applying that method to a certain crucial family of thousands of paraconsistent logics, all belonging to the class of Csystems. For that family, the method produces in a modular way uniform Gentzentype rules corresponding to a variety of axioms considered in the literature. 1
Modelling Rational Inquiry in NonIdeal Agents
, 1997
"... The construction of rational agents is one of the goals that has been pursued in Artificial Intelligence (AI). In most of the architectures that have been proposed for this kind of agents, its behaviour is guided by its set of beliefs. In our work, rational agents are those systems that are permanen ..."
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The construction of rational agents is one of the goals that has been pursued in Artificial Intelligence (AI). In most of the architectures that have been proposed for this kind of agents, its behaviour is guided by its set of beliefs. In our work, rational agents are those systems that are permanently engaged in the process of rational inquiry; thus, their beliefs keep evolving in time, as a consequence of their internal inference procedures and their interaction with the environment. Both AI researchers and philosophers are interested in having a formal model of this process, and this is the main topic in our work. Beliefs have been formally modelled in the last decades using doxastic logics. The possible worlds model and its associated Kripke semantics provide an intuitive semantics for these logics, but they seem to commit us to model agents that are logically omniscient and perfect reasoners. We avoid these problems by replacing possible worlds by conceivable situations, which ar...
Literalparaconsistent and literalparacomplete matrices
 Math. Logic Quart
, 2006
"... We introduce a family of matrices that define logics in which paraconsistency and/or paracompleteness occurs only at the level of literals, that is, formulas that are propositional letters or their iterated negations. We give a sound and complete axiomatization for the logic defined by the class of ..."
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We introduce a family of matrices that define logics in which paraconsistency and/or paracompleteness occurs only at the level of literals, that is, formulas that are propositional letters or their iterated negations. We give a sound and complete axiomatization for the logic defined by the class of all these matrices, we give conditions for the maximality of these logics and we study in detail several relevant examples.
Paraconsistent Computation Tree Logic
, 2010
"... It is known that paraconsistent logical systems are more appropriate for inconsistencytolerant and uncertainty reasoning than other types of logical systems. In this paper, a paraconsistent computation tree logic, PCTL, is obtained by adding paraconsistent negation to the standard computation tre ..."
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It is known that paraconsistent logical systems are more appropriate for inconsistencytolerant and uncertainty reasoning than other types of logical systems. In this paper, a paraconsistent computation tree logic, PCTL, is obtained by adding paraconsistent negation to the standard computation tree logic CTL. PCTL can be used to appropriately formalize inconsistencytolerant temporal reasoning. A theorem for embedding PCTL into CTL is proved. The validity, satisfiability, and modelchecking problems of PCTL are shown to be decidable. The embedding and decidability results indicate that we can reuse the existing CTLbased algorithms for validity, satisfiability, and modelchecking. An illustrative example of medical reasoning involving the use of PCTL is presented.