This chapter is devoted to the evaluation of quantified statements which can be found in many applications as decision making, expert systems, or flexible querying of relational databases using fuzzy set theory. Its contribution is to introduce the main techniques to evaluate such statements and to propose a new theoretical background for the evaluation of quantified statements of type “Q X are A” and “Q B X are A.” In this context, quantified statements are interpreted using an arithmetic on gradual numbers from Nf, Zf, and Qf. It is shown that the context of fuzzy numbers provides a framework to unify previous approaches and can be the base for the definition of new approaches.
Key Terms in this Chapter
Fuzzy Predicate: Predicate defined by a fuzzy set. A fuzzy predicate delivers a degree of satisfaction.
Linguistic Quantifiers: Quantifiers defined by linguistic expressions like “around 5” or “most of.” Such quantifiers allow an intermediate attitude between the conjunction (expressed by the universal quantifier ?) and the disjunction (expressed by the existential quantifier ?).
Gradual Integer: Integer which takes the form of a fuzzy subset of the set of naturals (interpreted as a conjunction). Such integers differ from fuzzy numbers which are interpreted as disjunctions of candidates.
Gradual Relative Integer: Gradual number represented by a fuzzy subset of the set of relatives (interpreted as a conjunction). It is defined as the substraction of two gradual integers.
Sugeno Fuzzy Integral: Aggregate operator which can be viewed as a compromise between two aspects: (1) a certain quantity (a fuzzy measure) and (2) a quality of information (a fuzzy set).
OWA Operator: Ordered Weighted Average Operator. The inputs are assumed to be sorted and the weights of this average are associated to input data depending on their rank (weight w1 is associated to the largest input, weight w2 is associated to the second largest input, and so forth).
Gradual Relational Number: Gradual number interpreted as a conjunction and defined as the ratio of two relative integers.
Complete Chapter List
Maria Amparo Vila, Miguel Delgado
Slawomir Zadrozny, Guy de Tré, Rita de Caluwe, Janusz Kacprzyk
Balazs Feil, Janos Abonyi
Didier Dubois, Henri Prade
Noureddine Mouaddib, Guillaume Raschia, W. Amenel Voglozin, Laurent Ughetto
P Bosc, A Hadjali, O Pivert
Guy De Tré, Marysa Demoor, Bert Callens, Lise Gosseye
Bordogna Bordogna, Guiseppe Psaila
Ludovic Liétard, Daniel Rocacher
Angélica Urrutia, Leonid Tineo, Claudia Gonzalez
Rallou Thomopoulos, Patrice Buche, Ollivier Haemmerlé
Troels Andreasen, Henrik Bulskov
Mohamed Ali Ben Hassine, Amel Grissa Touzi, José Galindo, Habib Ounelli
Geraldo Xexéo, André Braga
Aleksandar Takaci, Srdan Škrbic
Carlos D. Barranco, Jesús R. Campaña, Juan M. Medina
Yauheni Veryha, Jean-Yves Blot, Joao Coelho
Yan Chen, Graham H. Rong, Jianhua Chen
R. A. Carrasco, F. Araque, A. Salguero, M. A. Vila
Andreas Meier, Günter Schindler, Nicolas Werro
Shyue-Liang Wang, Ju-Wen Shen, Tuzng-Pei Hong
Radim Belohlavek, Vilem Vychodil
Awadhesh Kumar Sharma, A. Goswami, D. K. Gupta
Hamid Haidarian Shahri
J. I. Peláez, J. M. Doña, D. La Red