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Fuzzy Quantifiers: A Computational Theory (Studies in Fuzziness and Soft Computing)

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From a linguistic perspective, it is quantification which makes all the difference between “having no dollars” and “having a lot of dollars”. And it is the meaning of the quantifier “most” which eventually decides if “Most Americans voted Kerry” or “Most Americans voted Bush” (as it stands). Natural language (NL) quantifiers like “all”, “almost all”, “many” etc. serve an important purpose because they permit us to speak about properties of collections, as opposed to describing specific individuals only; in technical terms, quantifiers are a ‘second-order’ construct. Thus the quantifying statement “Most Americans voted Bush” asserts that the set of voters of George W. Bush comprises the majority of Americans, while “Bush sneezes” only tells us something about a specific individual. By describing collections rather than individuals, quantifiers 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 quantification available to technical applications, a number of proposals have been made how to model fuzzy quantifiers in the framework of fuzzy set theory. These approaches usually reduce fuzzy quantification to a comparison of scalar or fuzzy cardinalities [197, 132]. Thus for understanding a quantifying proposition like “About ten persons are tall”, we must know the degree to which we have j tall people in the given situation, and the degree to which the cardinal j qualifies as “about ten”. However, the results of these methods can be implausible in certain cases [3, 35, 54], i.e. they can violate some basic linguistic expectations. These problems certainly hindered the spread of the ‘traditional’ approaches into commercial applications.
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