Based On Fuzzy Measure And Fuzzy Integral Language Quantifiers,

It is well-known that quantification is a very important topic in knowledge representation and reasoning, since quantifiers have the ability of summarizing the properties of a class of objects without enumerating them. Linguistic quantification models are employed in coping with problems such as data summarization, database querying, information fusion, decision making, expert systems, etc. Accordingly, linguistic quantification has attracted a good deal of attention from researchers.Zadeh first introduced fuzzy quantifiers and classified them into two classes: absolute quantifiers and relative ones. According to his description, fuzzy quantifiers were considered as fuzzy numbers. Moreover, the truth value of a linguistically quantified statement is obtained by calculating the cardinality of the fuzzy set determined by the fuzzy predicate and then finding the compatibility between this cardinality and the involved quantifier. Since then, a large amount of investigations relating to linguistic quantifiers have been made based on fuzzy set theory. For example, Yager used ordered weighted averaging (OWA) operators for the evaluation of quantified propositions. However, most of these methods are limited to the finite discourse universes.M. S. Ying proposed a novel approach to model linguistic quantifiers, which is suitable for the infinite discourse universes and need not distinguish between absolute quantifiers and relative one. More precisely, a fuzzy quantifier Q is seen as a family of fuzzy measures, and the truth value of a quantified proposition is evaluated by using Sugeno's integral. Prom a formal point of view, the Choquet integral and the Sugeno integral differ only with the operators used in their definition; respectively＋,×and∨,∧. Nevertheless, they are very different in essence, since the Sugeno integral is more adapted to qualitative problems and the Choquet integral is better suited for quantitative problems. It is natural to consider whether the Choquet integral, which is widely used as an aggregation operator in multi-criteria decision making problems, could establish the corresponding framework.Hence in this paper, linguistic quantifiers are still represented by fuzzy measures, but the truth value of a quantified proposition is evaluated by using the Choquet integral. This model enables us to have some nice logical properties of linguistic quantifiers. More precisely, according to the first order language with linguistic quantifiers, we present its semantics based on the Choquet integral, then we carefully investigate its logical properties, including prenex normal form theorem and duality. Moreover, our model fulfills most of the desired properties in fuzzy quantification, thus illustrating the reasonableness of our Choquet integral semantics of linguistic quantifiers.In addition, this paper generalizes M. S. Ying's model of linguistic quantifiers to in-tuitionistic linguistic quantifiers. An intuitionistic linguistic quantifier is represented by a family of intuitionistic fuzzy-valued fuzzy measures and the intuitionistic truth value (the degree of satisfaction and non-satisfaction) of a quantified proposition is calculated by using intuitionsitic fuzzy-valued fuzzy integral. Some elegant logical properties of intuitionistic linguistic quantifiers are obtained, including prenex normal form theorem and duality. Moreover, M. S. Ying's model of linguistic quantifiers is a special case of this paper. Description of a quantifier by intuitionistic fuzzy-valued fuzzy measures allows us to consider differences in understanding the meaning of the quantifier by different persons.