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A Mathematical Setting for Fuzzy Logics
 International Journal of Uncertainty, Fuzziness and KnowledgeBased Systems
"... The setup of a mathematical propositional logic is given in algebraic terms, describing exactly when two choices of truth value algebras give the same logic. The propositional logic obtained when the algebra of truth values is the real numbers in the unit interval equipped with minimum, maximum and: ..."
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Cited by 12 (12 self)
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The setup of a mathematical propositional logic is given in algebraic terms, describing exactly when two choices of truth value algebras give the same logic. The propositional logic obtained when the algebra of truth values is the real numbers in the unit interval equipped with minimum, maximum and:x = 1 x for conjunction, disjunction and negation, respectively, is the standard propositional fuzzy logic. This is shown to be the same as threevalued logic. The propositional logic obtained when the algebra of truth values is the set f(a; b) j a b and a; b 2 [0; 1]g of subintervals of the unit interval with componentwise operations, is propositional intervalvalued fuzzy logic. This is shown to be the same as the logic given by a certain four element lattice of truth values. Since both of these logics are equivalent to ones given by …nite algebras, it follows that there are …nite algorithms for determining when two statements are logically equivalent within either of these logics. On this topic, normal forms are discussed for both of these logics.
Algebraic aspects of fuzzy sets and fuzzy logics
 Proc. Work. on Current Trends and Development in Fuzzy Logic, ThessalonikiGreece
, 1998
"... This paper is expository. It is mainly a survey of some of our work on the algebraic systems that arise in fuzzy set theory and logic. We include some of the proofs here ..."
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Cited by 10 (5 self)
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This paper is expository. It is mainly a survey of some of our work on the algebraic systems that arise in fuzzy set theory and logic. We include some of the proofs here
On the Possibility of Using Complex Values in Fuzzy Logic For Representing Inconsistencies
, 1996
"... In science and engineering, there are "paradoxical" cases when we have some arguments in favor of some statement A (so, the degree to which A is known to be true is positive (nonzero)), and we also have some arguments in favor of its negation :A, and we do not have enough information to tell wh ..."
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Cited by 6 (5 self)
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In science and engineering, there are "paradoxical" cases when we have some arguments in favor of some statement A (so, the degree to which A is known to be true is positive (nonzero)), and we also have some arguments in favor of its negation :A, and we do not have enough information to tell which of these two statements is correct. Traditional fuzzy logic, in which "truth values" are described by numbers from the interval [0; 1], easily describes such "paradoxical" situations: the degree a to which the statement A is true and the degree 1 \Gamma a to which its negation :A is true can be both positive. In this case, if we use traditional fuzzy &\Gammaoperations (min or product), the "truth value" a&(1 \Gamma a) of the statement A&:A is positive, indicating that there is some degree of inconsistency in the initial beliefs.
Fuzzy Logics Arising from Strict De Morgan Systems
 IN PROCEEDINGS OF LINZ ’99: TOPOLOGICAL AND ALGEBRAIC STRUCTURES
, 1999
"... ..."
Towards Combining Fuzzy and Logic Programming Techniques
, 1998
"... Many problems from AI have been successfully solved using fuzzy techniques. On the other hand, there are many other AI problems, in which logic programming (LP) techniques have been very useful. Since we have two successful techniques, why not combine them? ..."
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Cited by 3 (3 self)
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Many problems from AI have been successfully solved using fuzzy techniques. On the other hand, there are many other AI problems, in which logic programming (LP) techniques have been very useful. Since we have two successful techniques, why not combine them?
Towards More Adequate Representation of Uncertainty: From Intervals to Set Intervals, with the Possible Addition of Probabilities and Certainty Degrees
"... In the ideal case of complete knowledge, for each property Pi (such as “high fever”, “headache”, etc.), we know the exact set Si of all the objects that satisfy this property. In practice, we usually only have partial knowledge. In this case, we only know the set Si of all the objects about which w ..."
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Cited by 2 (1 self)
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In the ideal case of complete knowledge, for each property Pi (such as “high fever”, “headache”, etc.), we know the exact set Si of all the objects that satisfy this property. In practice, we usually only have partial knowledge. In this case, we only know the set Si of all the objects about which we know that Pi holds and the set Si about which we know that Pi may hold (i.e., equivalently, that we have not yet excluded the possibility of Pi). This pair of sets is called a set interval. Based on the knowledge of the original properties, we would like to describe the set S of all the values that satisfy some combination of the original properties: e.g., high fever and headache and not rash. In the ideal case when we know the exact set Si of all the objects satisfying each property, it is suf cient to apply the corresponding set operation (composition of union, intersection, and complement) to the known sets Si. In this paper, we describe how to compute the class S of all possible sets S.
How to Interpret Neural Networks In Terms of Fuzzy Logic?
, 2001
"... Neural networks are a very efficient learning tool, e.g., for transforming an experience of an expert human controller into the design of an automatic controller. It is desirable to reformulate the neural network expression for the inputoutput function in terms most understandable to an expert cont ..."
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Cited by 1 (1 self)
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Neural networks are a very efficient learning tool, e.g., for transforming an experience of an expert human controller into the design of an automatic controller. It is desirable to reformulate the neural network expression for the inputoutput function in terms most understandable to an expert controller, i.e., by using words from natural language. There are several methodologies for transforming such naturallanguage knowledge into a precise form; since these methodologies have to take into consideration the uncertainty (fuzziness) of natural language, they are usually called fuzzy logics. 1
Some comments on fuzzy normal forms
"... Abstract — In this paper, we examine and compare de Morgan, Kleene, and Booleandisjunctive and conjunctive normal forms in fuzzy settings. This generalizes papers of Turksen on the subject of Booleannormal forms. I. ..."
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Abstract — In this paper, we examine and compare de Morgan, Kleene, and Booleandisjunctive and conjunctive normal forms in fuzzy settings. This generalizes papers of Turksen on the subject of Booleannormal forms. I.
Fuzzy Normal Forms and Truth Tables
"... Abstract — In this paper, we examine and compare De Morgan, Kleene, and Booleandisjunctive and conjunctive normal forms and consider their role in fuzzy settings. In particular, we show that there are normal forms and truth tables for classical fuzzy propositional logic and intervalvalued fuzzy p ..."
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Abstract — In this paper, we examine and compare De Morgan, Kleene, and Booleandisjunctive and conjunctive normal forms and consider their role in fuzzy settings. In particular, we show that there are normal forms and truth tables for classical fuzzy propositional logic and intervalvalued fuzzy propositional logic that are completely analogous to those for Boolean propositional logic. Thus, determining logical equivalence of two expressions in classical fuzzy propositional logic is a nite problem, and similarly for the intervalvalued case. Turksen's work on intervalvalued fuzzy sets is examined in light of these results. I.