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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.
Two’s company: “The humbug of many logical values
 In Logica Universalis
, 2005
"... How was it possible that the humbug of many logical values persisted over the last fifty years? —Roman Suszko, 1976. Abstract. The Polish logician Roman Suszko has extensively pleaded in the 1970s for a restatement of the notion of manyvaluedness. According to him, as he would often repeat, “there ..."
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Cited by 16 (12 self)
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How was it possible that the humbug of many logical values persisted over the last fifty years? —Roman Suszko, 1976. Abstract. The Polish logician Roman Suszko has extensively pleaded in the 1970s for a restatement of the notion of manyvaluedness. According to him, as he would often repeat, “there are but two logical values, true and false. ” As a matter of fact, a result by WójcickiLindenbaum shows that any tarskian logic has a manyvalued semantics, and results by Suszkoda CostaScott show that any manyvalued semantics can be reduced to a twovalued one. So, why should one even consider using logics with more than two values? Because, we argue, one has to decide how to deal with bivalence and settle down the tradeoff between logical 2valuedness and truthfunctionality, from a pragmatical standpoint. This paper will illustrate the ups and downs of a twovalued reduction of logic. Suszko’s reductive result is quite nonconstructive. We will exhibit here a way of effectively constructing the twovalued semantics of any logic that has a truthfunctional finitevalued semantics and a sufficiently expressive language. From there, as we will indicate, one can easily go on to provide those logics with adequate canonical systems of sequents or tableaux. The algorithmic methods developed here can be generalized so as to apply to many nonfinitely valued logics as well —or at least to those that admit of computable quasi tabular twovalued semantics, the socalled dyadic semantics.
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 (10 self)
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It is well known that every propositional logic which satisfies certain very
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
"... ..."
Dyadic Semantics for ManyValued Logics
, 2003
"... This paper obtains an effective method which assigns twovalued semantics to every finitevalued truthfunctional logic (in the direction of the socalled "Suszko's Thesis"), provided that its truthvalues can be individualized by means of its linguistic resources. Such twovalued semantics permit u ..."
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Cited by 7 (5 self)
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This paper obtains an effective method which assigns twovalued semantics to every finitevalued truthfunctional logic (in the direction of the socalled "Suszko's Thesis"), provided that its truthvalues can be individualized by means of its linguistic resources. Such twovalued semantics permit us to obtain new tableau proof systems for a wide class of finitevalued logics, including the main manyvalued paraconsistent logics.
On a problem of da Costa
 CLE ePrints
, 2001
"... Abstract. The two main founders of paraconsistent logic, Stanis̷law Ja´skowski and Newton da Costa, built their systems on distinct grounds. Starting from different projects, they used different tools and ultimately designed quite different calculi to attend their needs. How successful were their en ..."
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Cited by 7 (2 self)
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Abstract. The two main founders of paraconsistent logic, Stanis̷law Ja´skowski and Newton da Costa, built their systems on distinct grounds. Starting from different projects, they used different tools and ultimately designed quite different calculi to attend their needs. How successful were their enterprises? Here we discuss the problem of defining paraconsistent logics following the original instructions laid down by da Costa. We present a new approach to P 1, the first full solution —proposed by Antônio Mário Sette — to the problem of da Costa, and argue in favor of yet another solution we shall study here: the logic P 2. Both P 1 and P 2 constitute maximal 3valued paraconsistent fragments of classical logic. Constructive completeness proofs are here presented for both logics. 1 Requisites to paraconsistent calculi When proposing the first paraconsistent propositional system, in 1948, Ja´skowski expected it to enjoy the following properties (see [17]): Jas1 when applied to inconsistent systems it should not always entail their trivialization; Jas2 it should be rich enough to enable practical inferences; Jas3 it should have an intuitive justification. A few years later, in 1963, da Costa would independently tackle a similar problem, this time proposing a whole hierarchy of paraconsistent propositional calculi, known as Cn, for 0 < n < ω. His requisites to these calculi were the following (see [12]): NdC1 in these calculi the principle of noncontradiction, in the form ¬(A ∧ ¬A), should not be a valid schema; NdC2 from two contradictory formulae, A and ¬A, it would not in general be possible to deduce an arbitrary formula B; NdC3 it should be simple to extend these calculi to corresponding predicate calculi (with or without equality); NdC4 they should contain the most part of the schemata and rules of the classical propositional calculus which do not interfere with the first conditions.
5valued Nondeterministic Semantics for The Basic Paraconsistent Logic mCi
, 2008
"... One of the most important paraconsistent logics is the logic mCi, which is one of the two basic logics of formal inconsistency. In this paper we present a 5valued characteristic nondeterministic matrix for mCi. This provides a quite nontrivial example for the utility and effectiveness of the use o ..."
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Cited by 3 (3 self)
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One of the most important paraconsistent logics is the logic mCi, which is one of the two basic logics of formal inconsistency. In this paper we present a 5valued characteristic nondeterministic matrix for mCi. This provides a quite nontrivial example for the utility and effectiveness of the use of nondeterministic manyvalued semantics. 1
Ineffable Inconsistencies
 Paraconsistency with no Frontiers
"... For any given consistent tarskian logic it is possible to find another nontrivial logic that allows for an inconsistent model yet completely coincides with the initial given logic from the point of view of their associated singleconclusion consequence relations. A paradox? This short note... ..."
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Cited by 3 (3 self)
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For any given consistent tarskian logic it is possible to find another nontrivial logic that allows for an inconsistent model yet completely coincides with the initial given logic from the point of view of their associated singleconclusion consequence relations. A paradox? This short note...