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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
A Survey on Different Triangular NormBased Fuzzy Logics
, 1999
"... Among various approaches to fuzzy logics, we have chosen two of them, which are built up in a similar way. Although starting from different basic logical connectives, they both use interpretations based on Frank tnorms. Different interpretations of the implication lead to different axiomatizati ..."
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Cited by 13 (1 self)
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Among various approaches to fuzzy logics, we have chosen two of them, which are built up in a similar way. Although starting from different basic logical connectives, they both use interpretations based on Frank tnorms. Different interpretations of the implication lead to different axiomatizations, but most logics studied here are complete. We compare the properties, advantages and disadvantages of the two approaches. Key words: Fuzzy logic, manyvalued logic, Frank tnorm 1 Introduction A manyvalued propositional logic with a continuum of truth values modelled by the unit interval [0; 1] is quite often called a fuzzy logic. In such a logic, the conjunction is usually interpreted by a triangular norm. In this context, a (propositional) fuzzy logic is considered as an ordered pair P = (L; Q) of a language (syntax ) L and a structure (semantics) Q described as follows: (i) The language of P is a pair L = (A; C), where A is an at most countable set of atomic symbols and C is ...
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 10 (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
Intervalvalued preference structures
 European Journal of Operational Research
, 1998
"... Different languages that are offered to model vague preferences are reviewed and an intervalvalued language is proposed to resolve a particular difficulty encountered with other languages. It is shown that intervalvalued languagesare well definedfor De Morgan triples constructedby continuoustriang ..."
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Cited by 5 (2 self)
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Different languages that are offered to model vague preferences are reviewed and an intervalvalued language is proposed to resolve a particular difficulty encountered with other languages. It is shown that intervalvalued languagesare well definedfor De Morgan triples constructedby continuoustriangular norms, conorms and a strong negation function. A new transitivity condition for vague preferences is suggestedand its relationships to known transitivity conditions are established. A complete characterization of intervalvalued preference structures is also provided.
General logical formalism for fuzzy mathematics: Methodology and apparatus
 Fuzzy Logic, Soft Computing and Computational Intelligence: Eleventh International Fuzzy Systems Association World Congress. Tsinghua
, 2005
"... ABSTRACT: There is a programme in the formal foundations of fuzzy mathematics proposed by the authors, the goal of which is to encompass a large part of existing fuzzy mathematics within a general logical formalism. This paper presents the methodology behind this programme and reviews the technical ..."
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Cited by 4 (1 self)
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ABSTRACT: There is a programme in the formal foundations of fuzzy mathematics proposed by the authors, the goal of which is to encompass a large part of existing fuzzy mathematics within a general logical formalism. This paper presents the methodology behind this programme and reviews the technical aspects of a particular apparatus for this enterprise.
Conditionally Firing Rules Extend the Possibilities of Fuzzy Controllers
 In: Proc. Int. Conf. Computational Intelligence for Modelling, Control and Automation
, 1999
"... s with the corresponding consequent, i.e., 8i 2 f1; : : : ; ng : Int \Theta (X i ) = Y i ; [Int2] for each normal observation X 2 F(X), the conclusion, Int \Theta (X ), is not contained in all consequents, i.e., there is an index i 2 f1; : : : ; ng with Int \Theta (X ) 6 Y i . [Int3] the conclusi ..."
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Cited by 2 (1 self)
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s with the corresponding consequent, i.e., 8i 2 f1; : : : ; ng : Int \Theta (X i ) = Y i ; [Int2] for each normal observation X 2 F(X), the conclusion, Int \Theta (X ), is not contained in all consequents, i.e., there is an index i 2 f1; : : : ; ng with Int \Theta (X ) 6 Y i . [Int3] the conclusion Int \Theta (X ) belongs to the convex hull of (Y i ) i2F , where F = fi j Supp X i " Supp X 6= ;g. Axiom [Int1] states that the "typical inputs" used in the rules produce the corresponding outputs
On approximate reasoning with graded rules
 Fuzzy Sets and Systems
"... This contribution presents a comprehensive view on problems of approximate reasoning with imprecise knowledge in the form of a collection of fuzzy IFTHEN rules formalized by approximating formulas of a special type. Two alternatives that follow from the dual character of approximating formulas are ..."
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Cited by 2 (1 self)
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This contribution presents a comprehensive view on problems of approximate reasoning with imprecise knowledge in the form of a collection of fuzzy IFTHEN rules formalized by approximating formulas of a special type. Two alternatives that follow from the dual character of approximating formulas are developed in parallel. The link to the theory of fuzzy control systems is also explained.
Neutrosophic logics on NonArchimedean Structures
 Critical Review, Creighton University, USA
"... We present a general way that allows to construct systematically analytic calculi for a large family of nonArchimedean manyvalued logics: hyperrationalvalued, hyperrealvalued, and padic valued logics characterized by a special format of semantics with an appropriate rejection of Archimedes ’ ax ..."
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Cited by 2 (0 self)
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We present a general way that allows to construct systematically analytic calculi for a large family of nonArchimedean manyvalued logics: hyperrationalvalued, hyperrealvalued, and padic valued logics characterized by a special format of semantics with an appropriate rejection of Archimedes ’ axiom. These logics are built as different extensions of standard manyvalued logics (namely, Lukasiewicz’s, Gödel’s, Product, and Post’s logics). The informal sense of Archimedes ’ axiom is that anything can be measured by a ruler. Also logical multiplevalidity without Archimedes ’ axiom consists in that the set of truth values is infinite and it is not wellfounded and wellordered. We consider two cases of nonArchimedean multivalued logics: the first with manyvalidity in the interval [0, 1] of hypernumbers and the second with manyvalidity in the ring Zp of padic integers. On the base of nonArchimedean valued logics, we construct nonArchimedean valued interval neutrosophic logics by which we can describe neutrality phenomena.
Mathematical fuzzy control. A survey of some recent results (submitted
"... The core point of fuzzy control approaches are finite lists of linguistic control rules. For computerbased automatic control these lists have to be transformed into control algorithms which can be realized on a computer. The main general idea of this fuzzy control approach is that such an algorithm ..."
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Cited by 1 (1 self)
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The core point of fuzzy control approaches are finite lists of linguistic control rules. For computerbased automatic control these lists have to be transformed into control algorithms which can be realized on a computer. The main general idea of this fuzzy control approach is that such an algorithm should yield a fuzzy subset of the output space of the control problem if confronted with a fuzzy subset of the input space. This paper surveys mathematical problems which are connected with, and arose out of these basic ideas. The main formal tools used in these mathematical considerations are fuzzy sets and fuzzy relations together with some generalized, viz. manyvalued logic which underlies these considerations. And the essential way of understanding the mathematical context of fuzzy control is to look at it as an interpolation problem: one has to determine a fuzzy control function out of a finite list of interpolation nodes.
Toward problems for mathematical fuzzy logic, in
 Proc. of IEEE International Conference on Fuzzy Systems
, 2006
"... The paper discusses some open problems in the field of mathematical fuzzy logic which may have a decisive influence for the future development of fuzzy logic within the next decade. 1 ..."
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The paper discusses some open problems in the field of mathematical fuzzy logic which may have a decisive influence for the future development of fuzzy logic within the next decade. 1