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?erslikeall,almostall,manyetc. 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,whileBushsneezesonlytellsussomething 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].