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A Treatise on ManyValued Logics
 Studies in Logic and Computation
, 2001
"... The paper considers the fundamental notions of many valued logic together with some of the main trends of the recent development of infinite valued systems, often called mathematical fuzzy logics. Besides this logical approach also a more algebraic approach is discussed. And the paper ends with som ..."
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Cited by 84 (5 self)
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The paper considers the fundamental notions of many valued logic together with some of the main trends of the recent development of infinite valued systems, often called mathematical fuzzy logics. Besides this logical approach also a more algebraic approach is discussed. And the paper ends with some hints toward applications which are based upon actual theoretical considerations about infinite valued logics. Key words: mathematical fuzzy logic, algebraic semantics, continuous tnorms, leftcontinuous tnorms, Pavelkastyle fuzzy logic, fuzzy set theory, nonmonotonic fuzzy reasoning 1 Basic ideas 1.1 From classical to manyvalued logic Logical systems in general are based on some formalized language which includes a notion of well formed formula, and then are determined either semantically or syntactically. That a logical system is semantically determined means that one has a notion of interpretation or model 1 in the sense that w.r.t. each such interpretation every well formed formula has some (truth) value or represents a function into
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 69 (24 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
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
Limits for Paraconsistent Calculi
 Notre Dame Journal of Formal Logic
, 2001
"... This paper discusses how to define logics as deductive limits of sequences of other logics. The case of da Costa's hierarchy of increasingly weaker paraconsistent calculi, known as $C_n$, $\leq n\leq\omega$, is carefully studied. The calculus $C_\omega$, in particular, constitutes n ..."
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This paper discusses how to define logics as deductive limits of sequences of other logics. The case of da Costa's hierarchy of increasingly weaker paraconsistent calculi, known as $C_n$, $&#61489;\leq n\leq\omega$, is carefully studied. The calculus $C_\omega$, in particular, constitutes no more than a lower deductive bound to this hierarchy, and differs considerably from its companions. A long standing problem in the literature (open for more than 35 years) is to define the deductive limit to this hierarchy, that is its greatest lower deductive bound. The calculus $C_{min}$, stronger than $C_\omega$, is first presented as a step towards this limit. As an alternative to the bivaluation semantics of $C_{min}$ presented thereupon, possibletranslations semantics are then introduced and suggested as the standard technique both to give this calculus a more reasonable semantics and to derive some interesting properties about it. Possibletranslations semantics are then used to provide both a semantics and a decision procedure for $C_{Lim}$, the real deductive limit of da Costas hierarchy. Possibletranslations semantics also make it possible to characterize a precise sense of duality: as an example, $D_{min}$ is proposed as the dual to $C_{min}$.
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 9 (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.
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 sem ..."
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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.
Fibring in the Leibniz Hierarchy
"... This article studies preservation of certain algebraic properties of propositional logics when combined by fibring. The logics analyzed here are classified in protoalgebraic, equivalential and algebraizable. By introducing new categories of algebrizable logics and of deductivizable quasivarieties, ..."
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This article studies preservation of certain algebraic properties of propositional logics when combined by fibring. The logics analyzed here are classified in protoalgebraic, equivalential and algebraizable. By introducing new categories of algebrizable logics and of deductivizable quasivarieties, it is stated an isomorphism between these categories. This constitutes an alternative to a similar result found in the literature. 1
How to Build Your Own Paraconsistent Logic: An Introduction To . . .
"... 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 negat ..."
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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 nonexplosive 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 possibletranslations 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.
1 Splitting Logics
"... abstract. This paper addresses the question of factoring a logic into families of (generally simpler) components, estimating the top– down perspective, splitting, versus the bottom–up, splicing. Three methods are carefully analyzed and compared: possible–translations semantics, nondeterministic sema ..."
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abstract. This paper addresses the question of factoring a logic into families of (generally simpler) components, estimating the top– down perspective, splitting, versus the bottom–up, splicing. Three methods are carefully analyzed and compared: possible–translations semantics, nondeterministic semantics and plain fibring (joint with its particularization, direct union of matrices). The possibilities of inter–definability between these methods are also examined. Finally, applications to some well–known logic systems are given and their significance evaluated. 1 Splitting logics, splicing logics and their use One of fundamental questions in the philosophy of logic, “Why there are so many logics instead of just one? ” (or even, instead of none), is naturally counterposed by another: If there are indeed many logics, are they excluding alternatives, or are they compatible? Is it possible to combine them into coherent systems, with the purpose of using them in applications and of taking profit of this composionality capacity to better understand logics? And if we can compose, why not decompose logics? One of the first, and one of the most general, approaches for the question of combining logics is the concept of fibring introduced by D. Gabbay in [Gabbay, 1996]. Fibring is able to combine logics creating new and expressive systems, in the direction of what we call splicing logics. The other direction is called splitting logics. Though, as we shall argue, there is no essential distinction between splicing and splitting, there are important differences with respect to the aims one may have in mind. Splitting as a process for investigating logics has been under–appreciated, and we intend to stress here some results and some views that we believe to be of interest for the sake of splitting in the trade of combining logics. 1 1 The process tags “splicing ” and “splitting ” logics were introduced in [Carnielli and Coniglio, 1999]. As a noun, “splitting ” is also used in the literature in a completely different sense, viz., to designate a “logic that splits a class”, as e.g. in W.J. Blok, “On