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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 53 (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
Hypersequents and the proof theory of intuitionistic fuzzy logic
 Computer Science Logic CSL’2000. Proceedings, LNCS 1862
, 2000
"... Abstract. Takeuti and Titani have introduced and investigated a logic they called intuitionistic fuzzy logic. This logic is characterized as the firstorder Gödel logic based on the truth value set [0,1]. The logic is known to be axiomatizable, but no deduction system amenable to prooftheoretic, and ..."
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Cited by 20 (10 self)
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Abstract. Takeuti and Titani have introduced and investigated a logic they called intuitionistic fuzzy logic. This logic is characterized as the firstorder Gödel logic based on the truth value set [0,1]. The logic is known to be axiomatizable, but no deduction system amenable to prooftheoretic, and hence, computational treatment, has been known. Such a system is presented here, based on previous work on hypersequent calculi for propositional Gödel logics by Avron. It is shown that the system is sound and complete, and allows cutelimination. A question by Takano regarding the eliminability of the TakeutiTitani density rule is answered affirmatively. 1
Herbrand’s theorem for prenex Gödel logic and its consequences for theorem proving
 IN LOGIC FOR PROGRAMMING AND AUTOMATED REASONING LPAR’2001, 201–216. LNAI 2250
, 2001
"... Herbrand’s Theorem £¥ ¤ ¦ for, i.e., Gödel logic enriched by the projection § operator is proved. As a consequence we obtain a “chain normal form” and a translation of £ ¤ ¦ prenex into (order) clause logic, referring to the classical theory of dense total orders with endpoints. A chaining calc ..."
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Cited by 15 (12 self)
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Herbrand’s Theorem £¥ ¤ ¦ for, i.e., Gödel logic enriched by the projection § operator is proved. As a consequence we obtain a “chain normal form” and a translation of £ ¤ ¦ prenex into (order) clause logic, referring to the classical theory of dense total orders with endpoints. A chaining calculus provides a basis for efficient theorem proving.
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
Quantified propositional Gödel logics
 In Proceedings of LPAR’2000, LNAI 1955
, 2000
"... Entailment in propositional Gödel logics can be defined in a natural way. While all infinite sets of truth values yield the same sets of tautologies, the entailment relations differ. It is shown that there is a rich structure of infinitevalued Gödel logics, only one of which is compact. It is also ..."
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Cited by 14 (7 self)
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Entailment in propositional Gödel logics can be defined in a natural way. While all infinite sets of truth values yield the same sets of tautologies, the entailment relations differ. It is shown that there is a rich structure of infinitevalued Gödel logics, only one of which is compact. It is also shown that the compact infinitevalued Gödel logic is the only one which interpolates, and the only one with an r.e. entailment relation. 1
Hypersequent calculi for Gödel logics: a survey
 Journal of Logic and Computation
, 2003
"... Hypersequent calculi arise by generalizing standard sequent calculi to refer to whole contexts of sequents instead of single sequents. We present a number of results using hypersequents to obtain a Gentzenstyle characterization for the family of Gödel logics. We first describe analytic calculi for ..."
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Cited by 13 (4 self)
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Hypersequent calculi arise by generalizing standard sequent calculi to refer to whole contexts of sequents instead of single sequents. We present a number of results using hypersequents to obtain a Gentzenstyle characterization for the family of Gödel logics. We first describe analytic calculi for propositional finite and infinitevalued Gödel logics. We then show that the framework of hypersequents allows one to move straightforwardly from the propositional level to firstorder as well as propositional quantification. A certain type of modalities, enhancing the expressive power of Gödel logic, is also considered. 1
Mathematical fuzzy logic as a tool for the treatment of vague information
 Information Sciences
, 2005
"... The paper considers some of the main trends of the recent development of mathematical fuzzy logic as an important tool in the toolbox of approximate reasoning techniques. Particularly the focus is on fuzzy logics as systems of formal logic constituted by a formalized language, by a semantics, and by ..."
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Cited by 11 (1 self)
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The paper considers some of the main trends of the recent development of mathematical fuzzy logic as an important tool in the toolbox of approximate reasoning techniques. Particularly the focus is on fuzzy logics as systems of formal logic constituted by a formalized language, by a semantics, and by a calculus for the derivation of formulas. Besides this logical approach also a more algebraic approach is discussed. And the paper ends with some hints toward applications which are based upon these theoretical considerations. Key words: mathematical fuzzy logic, algebraic semantics, continuous tnorms, leftcontinuous tnorms, Pavelkastyle fuzzy logic, fuzzy set theory, nonmonotonic fuzzy reasoning 1
Commodious Axiomatization of Quantifiers in MultipleValued Logic
, 1997
"... . We provide tools for a concise axiomatization of a broad class of quantifiers in manyvalued logic, socalled distribution quantifiers. Although sound and complete axiomatizations for such quantifiers exist, their size renders them virtually useless for practical purposes. We show that for quantif ..."
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Cited by 11 (2 self)
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. We provide tools for a concise axiomatization of a broad class of quantifiers in manyvalued logic, socalled distribution quantifiers. Although sound and complete axiomatizations for such quantifiers exist, their size renders them virtually useless for practical purposes. We show that for quantifiers based on finite distributive lattices compact axiomatizations can be obtained schematically. This is achieved by providing a link between skolemized signed formulas and filters/ideals in Boolean set lattices. Then lattice theoretic tools such as Birkhoff's representation theorem for finite distributive lattices are used to derive tableaustyle axiomatizations of distribution quantifiers. Introduction The aim of this paper 1 is to provide concise axiomatizations of certain quantifiers in manyvalued logic which were introduced by Mostowski (1961) and baptized distribution quantifiers by Carnielli (1987). The task of axiomatizing such quantifiers has been solved satisfactorily in theor...
Terminological difficulties in fuzzy set theory  the case of “intuitionistic fuzzy sets”, Fuzzy Sets and Systems 156 (3
, 2005
"... Abstract: This note points out a terminological clash between Atanassov's “intuitionistic fuzzy sets ” and what is currently understood as intuitionistic logic. They differ both by their motivations and their underlying mathematical structure. Furthermore, Atanassov's construct is isomorph ..."
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Cited by 10 (1 self)
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Abstract: This note points out a terminological clash between Atanassov's “intuitionistic fuzzy sets ” and what is currently understood as intuitionistic logic. They differ both by their motivations and their underlying mathematical structure. Furthermore, Atanassov's construct is isomorphic to intervalvalued fuzzy sets and other similar notions, even if their interpretive settings and motivation are quite different, the latter capturing the idea of illknown membership grade, while the former starts from the idea of evaluating degrees of membership and nonmembership independently. This paper is a plea for a clarification of terminology, based on mathematical resemblances and the comparison of motivations between “intuitionistic fuzzy sets ” and other theories. 1.
Firstorder Gödel logics
, 2006
"... Firstorder Gödel logics are a family of infinitevalued logics where the sets of truth values V are closed subsets of [0,1] containing both 0 and 1. Different such sets V in general determine different Gödel logics GV (sets of those formulas which evaluate to 1 in every interpretation into V). It i ..."
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Cited by 9 (3 self)
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Firstorder Gödel logics are a family of infinitevalued logics where the sets of truth values V are closed subsets of [0,1] containing both 0 and 1. Different such sets V in general determine different Gödel logics GV (sets of those formulas which evaluate to 1 in every interpretation into V). It is shown that GV is axiomatizable iff V is finite, V is uncountable with 0 isolated in V, or every neighborhood of 0 in V is uncountable. Complete axiomatizations for each of these cases are given. The r.e. prenex, negationfree, and existential fragments of all firstorder Gödel logics are also characterized.