Fuzzy Quantifiers: A Computational Theory

by Ingo Glöckner

Overview

From a linguistic perspective, it is quantification which makes all the diffence 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)quantifierslike“all”,“almostall”,“many”etc. serveanimp- tant 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].

Product Details

ISBN-13:	9783642067433
Publisher:	Springer Berlin Heidelberg
Publication date:	11/23/2010
Series:	Studies in Fuzziness and Soft Computing , #193
Edition description:	Softcover reprint of hardcover 1st ed. 2006
Pages:	460
Product dimensions:	6.00(w) x 9.00(h) x 1.10(d)

About the Author

Ingo Glöckner received his M.A. in Computational Linguistics and Artificial Intelligence from University of Osnabrück in 1996. He then became a research assistant at the University of Bielefeld, where he pursued research on fuzzy set theory and its application to information retrieval. In 2003, I. Glöckner received his PhD for his thesis on the semantical interpretation and implementation of fuzzy quantifiers. He then joined the Intelligent Information and Communication Systems Group (Prakt. Informatik VII) of Prof. H. Helbig at the FernUniversität in Hagen. His current research activities are centered on the representation and processing of knowledge expressed in natural language.

Table of Contents

An Introduction to Fuzzy Quantification: Origins and Basic Concepts.- A Framework for Fuzzy Quantification.- The Axiomatic Class of Plausible Models.- Semantic Properties of the Models.- Special Subclasses of Models.- Special Semantical Properties and Theoretical Limits.- Models Defined in Terms of Three-Valued Cuts and Fuzzy-Median Aggregation.- Models Defined in Terms of Upper and Lower Bounds on Three-Valued Cuts.- The Full Class of Models Defined in Terms of Three-Valued Cuts.- The Class of Models Based on the Extension Principle.- Implementation of Quantifiers in the Models.- Multiple Variable Binding and Branching Quantification.- Discussion.