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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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Cited by 45 (19 self)
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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
Modulated Fibring and the Collapsing Problem
, 2001
"... Fibring is recognized as one of the main mechanisms in combining logics, with great signicance in the theory and applications of mathematical logic. However, an open challenge to bring is posed by the collapsing problem: even when no symbols are shared, certain combinations of logics simply collapse ..."
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Cited by 20 (12 self)
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Fibring is recognized as one of the main mechanisms in combining logics, with great signicance in the theory and applications of mathematical logic. However, an open challenge to bring is posed by the collapsing problem: even when no symbols are shared, certain combinations of logics simply collapse to one of them, indicating that bring imposes unwanted interconnections between the given logics. Modulated bring allows a ner control of the combination, solving the collapsing problem both at the semantic and deductive levels. Main properties like soundness and completeness are shown to be preserved, comparison with bring is discussed, and some important classes of examples are analyzed with respect to the collapsing problem. 1
Nondeterministic Semantics for Logics with a Consistency Operator
 IN THE INTERNATIONAL JOURNAL OF APPROXIMATE REASONING
, 2006
"... In order to handle inconsistent knowledge bases in a reasonable way, one needs a logic which allows nontrivial inconsistent theories. Logics of this sort are called paraconsistent. One of the oldest and best known approaches to the problem of designing useful paraconsistent logics is da Costa’s appr ..."
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Cited by 19 (11 self)
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In order to handle inconsistent knowledge bases in a reasonable way, one needs a logic which allows nontrivial inconsistent theories. Logics of this sort are called paraconsistent. One of the oldest and best known approaches to the problem of designing useful paraconsistent logics is da Costa’s approach, which seeks to allow the use of classical logic whenever it is safe to do so, but behaves completely differently when contradictions are involved. Da Costa’s approach has led to the family of logics of formal (in)consistency (LFIs). In this paper we provide in a modular way simple nondeterministic semantics for 64 of the most important logics from this family. Our semantics is 3valued for some of the systems, and infinitevalued for the others. We prove that these results cannot be improved: neither of the systems with a threevalued nondeterministic semantics has either a finite characteristic ordinary matrix or a twovalued characteristic nondeterministic matrix, and neither of the other systems we investigate has a finite characteristic nondeterministic matrix. Still, our semantics provides decision procedures for all the systems investigated, as well as easy proofs of important prooftheoretical properties of them.
Logical Nondeterminism as a Tool for Logical Modularity: An Introduction
 in We Will Show Them: Essays in Honor of Dov Gabbay, Vol
, 2005
"... It is well known that every propositional logic which satisfies certain very ..."
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Cited by 13 (11 self)
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It is well known that every propositional logic which satisfies certain very
Coherent Integration of Databases by Abductive Logic Programming
 Journal of Artificial Intelligence Research
, 2004
"... We introduce an abductive method for a coherent integration of independent datasources. ..."
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Cited by 11 (4 self)
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We introduce an abductive method for a coherent integration of independent datasources.
An Encompassing Framework for Paraconsistent Logic Programs
 J. Applied Logic
, 2003
"... We propose a framework which extends Antitonic Logic Programs [13] to an arbitrary complete bilattice of truthvalues, where belief and doubt are explicitly represented. Inspired by Ginsberg and Fitting 's bilattice approaches, this framework allows a precise de nition of important operato ..."
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Cited by 9 (5 self)
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We propose a framework which extends Antitonic Logic Programs [13] to an arbitrary complete bilattice of truthvalues, where belief and doubt are explicitly represented. Inspired by Ginsberg and Fitting 's bilattice approaches, this framework allows a precise de nition of important operators found in logic programming, such as explicit and default negation. In particular, it leads to a natural semantical integration of explicit and default negation through the Coherence Principle [38], according to which explicit negation entails default negation. We then de ne Coherent Answer Sets, and the Paraconsistent Wellfounded Model semantics, generalising many paraconsistent semantics for logic programs. In particular, Paraconsistent WellFounded Semantics with eXplicit negation (WFSXp ) [3, 11]. The framework is an extension of Antitonic Logic Programs for most cases, and is general enough to capture Probabilistic Deductive Databases, Possibilistic Logic Programming, Hybrid Probabilistic Logic Programs, and Fuzzy Logic Programming. Thus, we have a powerful mathematical formalism for dealing simultaneously with default, paraconsistency, and uncertainty reasoning. Results are provided about how our semantical framework deals with inconsistent information and with its propagation by the rules of the program.
Preservation of interpolation features by fibring
 Journal of Logic and Computation
"... Fibring is a metalogical constructor that permits to combine different logics by operating on their deductive systems under certain natural restrictions, as for example that the two given logics are presented by deductive systems of the same type. Under such circumstances, fibring will produce a new ..."
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Cited by 9 (9 self)
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Fibring is a metalogical constructor that permits to combine different logics by operating on their deductive systems under certain natural restrictions, as for example that the two given logics are presented by deductive systems of the same type. Under such circumstances, fibring will produce a new deductive system by means of the free use of inference rules from both deductive systems, provided the rules are schematic, in the sense of using variables that are open for application to formulas with new linguistic symbols (from the point of view of each logic component). Fibring is a generalization of fusion, a less general but wider developed mechanism which permits results of the following kind: if each logic component is decidable (or sound, or complete with respect to a certain semantics) then the resulting logic heirs such a property. The interest for such preservation results for combining logics is evident, and they have been achieved in the more general setting of fibring in several cases. The Craig interpolation property and the Maehara interpolation have a special significance when combining logics, being related to certain problems of complexity theory, some properties of model theory and to the usual (global) metatheorem of deduction. When the peculiarities of the distinction between local and global deduction interfere, justifying what we call careful reasoning, the question of preservation of interpolation becomes more subtle and other forms of interpolation can be distinguished. These questions are investigated and several (global and local) preservation results for interpolation are obtained for fibring logics that fulfill mild requirements. AMS Classification: 03C40, 03B22, 03B45 1
Possibletranslations semantics for some weak classicallybased paraconsistent logics
, 2004
"... ..."
On negation: Pure local rules
, 2003
"... This is an initial systematic study of the properties of negation from the point of view of abstract deductive systems. A unifying framework of multipleconclusion consequence relations is adopted so as to allow us to explore symmetry in exposing and matching a great number of positive contextua ..."
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Cited by 8 (6 self)
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This is an initial systematic study of the properties of negation from the point of view of abstract deductive systems. A unifying framework of multipleconclusion consequence relations is adopted so as to allow us to explore symmetry in exposing and matching a great number of positive contextual subclassical rules involving this logical constant among others, wellknown forms of proof by cases, consequentia mirabilis and reductio ad absurdum. Finer definitions of paraconsistency and the dual paracompleteness can thus be formulated, allowing for pseudoscotus and ex contradictione to be di#erentiated and for a comprehensive version of the Principle of NonTriviality to be presented. A final proposal is made to the e#ect that pure positive rules involving negation being often fallible a characterization of what most negations in the literature have in common should rather involve, in fact, a reduced set of negative rules.