Fuzzy Quantifiers

From a linguistic perspective, it is quanti?cation which makes all the di?- ence between having no dollars and having a lot of dollars . And it is the meaning of the quanti?er most which eventually decides if Most Ame- cans voted Kerry or Most Americans voted Bush (as it stands). Natural language(NL)quanti?erslike all , almostall , many etc. serveanimp- tant purpose because they permit us to speak about properties of collections, as opposed to describing speci?c individuals only; in technical terms, qu- ti?ers are a second-order construct. Thus the quantifying statement Most Americans voted Bush asserts that the set of voters of George W. Bush c- prisesthemajorityofAmericans,while Bushsneezes onlytellsussomething about a speci?c individual. By describing collections rather than individuals, quanti?ers extend the expressive power of natural languages far beyond that of propositional logic and make them a universal communication medium. Hence language heavily depends on quantifying constructions. These often involve fuzzy concepts like tall , and they frequently refer to fuzzy quantities in agreement like about ten , almost all , many etc. In order to exploit this expressive power and make fuzzy quanti?cation available to technical applications, a number of proposals have been made how to model fuzzy quanti?ers in the framework of fuzzy set theory. These approaches usually reduce fuzzy quanti?cation to a comparison of scalar or fuzzy cardinalities [197, 132].