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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 52 (3 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
Monotonic and Residuated Logic Programs
, 2001
"... In this paper we define the rather general framework of Monotonic Logic Programs, where the main results of (definite) logic programming are validly extrapolated. Whenever defining new logic programming extensions, we can thus turn our attention to the stipulation and study of its intuitive algebrai ..."
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Cited by 44 (9 self)
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In this paper we define the rather general framework of Monotonic Logic Programs, where the main results of (definite) logic programming are validly extrapolated. Whenever defining new logic programming extensions, we can thus turn our attention to the stipulation and study of its intuitive algebraic properties within the very general setting. Then, the existence of a minimum model and of a monotonic immediate consequences operator is guaranteed, and they are related as in classical logic programming. Afterwards we study the more restricted class of residuated logic programs which is able to capture several quite distinct logic programming semantics. Namely: Generalized Annotated Logic Programs, Fuzzy Logic Programming, Hybrid Probabilistic Logic Programs, and Possibilistic Logic Programming. We provide the embedding of possibilistic logic programming.
MultiAdjoint Logic Programming with Continuous Semantics
 Lect. Notes in Artificial Intelligence 2173
, 2001
"... Considering different implication operators, such as Lukasiewicz, Gödel or product implication in the same logic program, naturally leads to the allowance of several adjoint pairs in the lattice of truthvalues. In this paper we apply this idea to introduce multiadjoint logic programs as an extensi ..."
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Cited by 39 (18 self)
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Considering different implication operators, such as Lukasiewicz, Gödel or product implication in the same logic program, naturally leads to the allowance of several adjoint pairs in the lattice of truthvalues. In this paper we apply this idea to introduce multiadjoint logic programs as an extension of monotonic logic programs. The continuity of the immediate consequences operators is proved and the assumptions required to get continuity are further analysed.
A Complete ManyValued Logic With ProductConjunction
, 1996
"... this paper we investigate some logics whose set of truth values is the real interval [0; 1] and we concentrate our attention to logics having a conjunction whose truth function t(x; y) is a tnorm, and having a corresponding residuated implication (or, as Pavelka [14] observes, the conjunction and t ..."
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Cited by 28 (3 self)
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this paper we investigate some logics whose set of truth values is the real interval [0; 1] and we concentrate our attention to logics having a conjunction whose truth function t(x; y) is a tnorm, and having a corresponding residuated implication (or, as Pavelka [14] observes, the conjunction and the implication form an adjoint couple); i.e., if i(x; y) is the truth function of the implication then
Local Possibilistic Logic
 Journal of Applied NonClassical Logic
, 1997
"... Possibilistic states of information are fuzzy sets of possible worlds. They constitute a complete lattice, which can be endowed with a monoidal operation (a tnorm) to produce a quantal. An algebraic semantics is presented which links possibilistic formulae with information states, and gives a natur ..."
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Cited by 21 (14 self)
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Possibilistic states of information are fuzzy sets of possible worlds. They constitute a complete lattice, which can be endowed with a monoidal operation (a tnorm) to produce a quantal. An algebraic semantics is presented which links possibilistic formulae with information states, and gives a natural interpretation of logical connectives as operations on fuzzy sets. Due to the quantal structure of information states, we obtain a system which shares several features with (exponentialfree) intuitionistic linear logic. Soundness and completeness are proved, parametrically on the choice of the tnorm operation.
Hybrid Probabilistic Logic Programs as Residuated Logic Programs
, 2002
"... In this paper we show the embedding of Hybrid Probabilistic Logic Programs into the rather general framework of Residuated Logic Programs, where the main results of (definite) logic programming are validly extrapolated, namely the extension of the immediate consequences operator of van Emden and Kow ..."
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Cited by 19 (4 self)
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In this paper we show the embedding of Hybrid Probabilistic Logic Programs into the rather general framework of Residuated Logic Programs, where the main results of (definite) logic programming are validly extrapolated, namely the extension of the immediate consequences operator of van Emden and Kowalski. The importance of this result is that for the first time a framework encompassing several quite distinct logic programming semantics is described, namely Generalized Annotated Logic Programs, Fuzzy Logic Programming, Hybrid Probabilistic Logic Programs, and Possibilistic Logic Programming. Moreover, the embedding provides a more general semantical structure paving the way for defining paraconsistent probabilistic reasoning with a logic programming semantics.
Fuzzy Logic and Probability
 In Uncertainty in Artificial Intelligence. Proc. of 11th conference
, 1995
"... In this paper we deal with a new approach to probabilistic reasoning in a logical framework. Nearly almost all logics of probability that have been proposed in the literature are based on classical twovalued logic. After making clear the differences between fuzzy logic and probability theory, here ..."
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Cited by 17 (4 self)
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In this paper we deal with a new approach to probabilistic reasoning in a logical framework. Nearly almost all logics of probability that have been proposed in the literature are based on classical twovalued logic. After making clear the differences between fuzzy logic and probability theory, here we propose a fuzzy logic of probability for which completeness results (in a probabilistic sense) are provided. The main idea behind this approach is that probability values of crisp propositions can be understood as truthvalues of some suitable fuzzy propositions associated to the crisp ones. Moreover, suggestions and examples of how to extend the formalism to cope with conditional probabilities and with other uncertainty formalisms are also provided. 1 Introduction Discussions about the relation between fuzzy logic and probability are still numerous and sometimes rather controversial. In particular, using fuzzy logic to reason in a probabilistic way may be a priori considered as a "danger...
The Relation between Inference and Interpolation in the Framework of Fuzzy Systems
, 1996
"... This papers aims at clarifying the meaning of different interpretations of the MaxMin or, more generally, the Maxtnorm rule in fuzzy systems. It turns out that basically two distinct approaches play an important role in fuzzy logic and its applications: fuzzy interpolation on the basis of an impr ..."
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Cited by 16 (1 self)
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This papers aims at clarifying the meaning of different interpretations of the MaxMin or, more generally, the Maxtnorm rule in fuzzy systems. It turns out that basically two distinct approaches play an important role in fuzzy logic and its applications: fuzzy interpolation on the basis of an imprecisely known function and logical inference in the presence of fuzzy information. Keywords: Fuzzy logic; fuzzy control; MaxMin rule, fuzzy interpolation. 1 Introduction This is a synthesizing paper which returns to the question, what is the role of the MaxMin (Maxtnorm) rule in fuzzy logic from the viewpoint of logical inference. We aim at demonstrating that two basic, more or less complementary approaches in fuzzy logic and its applications can be distinguished, namely: fuzzy interpolation of a fuzzily specified precise function and logical inference in the presence of fuzzy information. The first task is solved using the Maxtnorm rule which essentially leads to search of a fuzzy...
On Triangular NormBased Propositional Fuzzy Logics
 Fuzzy Sets and Systems
, 1995
"... Fuzzy logics based on triangular norms and their corresponding conorms are investigated. An affirmative answer to the question whether in such logics a specific level of satisfiability of a set of formulas can be characterized by the same level of satisfiability of its finite subsets is given. T ..."
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Cited by 16 (3 self)
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Fuzzy logics based on triangular norms and their corresponding conorms are investigated. An affirmative answer to the question whether in such logics a specific level of satisfiability of a set of formulas can be characterized by the same level of satisfiability of its finite subsets is given. Tautologies, contradictions and contingencies with respect to such fuzzy logics are studied, in particular for the important cases of minmax and Lukasiewicz logics. Finally, fundamental tnormbased fuzzy logics are shown to provide a gradual transition between minmax and Lukasiewicz logics. Key words: Fuzzy Logics, Minmax logic, Lukasiewicz Logic, Triangular Norms, Satisfiability. AMSClassification: 03B52, 03B50, 03B05 0
An algebraic semantics for possibilistic logic
 Uncertainty in Artificial Intelligence (UAI 95
, 1995
"... The first contribution of this paper is the presentation of a Pavelka–like formulation of possibilistic logic in which the language is naturally enriched by two connectives which represent negation (¬) and a new type of conjunction (⊗). The space of truth values for this logic is the lattice of poss ..."
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Cited by 15 (9 self)
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The first contribution of this paper is the presentation of a Pavelka–like formulation of possibilistic logic in which the language is naturally enriched by two connectives which represent negation (¬) and a new type of conjunction (⊗). The space of truth values for this logic is the lattice of possibility functions, that, from an algebraic point of view, forms a quantal. A second contribution comes from the understanding of the new conjunction as the combination of tokens of information coming from different sources, which makes our language ”dynamic”. A Gentzen calculus is presented, which is proved sound and complete with respect to the given semantics. The problem of truth functionality is discussed in this context. 1