Fuzzy Quantifiers

From a linguistic perspective, it is quanti?cation which makes all the di?- ence between "e;having no dollars"e; and "e;having a lot of dollars"e;. And it is the meaning of the quanti?er "e;most"e; which eventually decides if "e;Most Ame- cans voted Kerry"e; or "e;Most Americans voted Bush"e; (as it stands). Natural language(NL)quanti?erslike"e;all"e;,"e;almostall"e;,"e;many"e;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 "e;Most Americans voted Bush"e; asserts that the set of voters of George W. Bush c- prisesthemajorityofAmericans,while"e;Bushsneezes"e;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 "e;tall"e;, and they frequently refer to fuzzy quantities in agreement like "e;about ten"e;, "e;almost all"e;, "e;many"e; 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].