Standard Models of Fuzzy Quantification

Abstract

Quantifiers are at the heart of human language. A formalization of natural language (NL) quantification promises to enhance a broad range of applications including NL interfaces, linguistic data summarisation, multi-criteria decision making, database querying and others. However, the software implementations available for NL quantifiers will remain insufficient and linguistically implausible unless the inherent fuzziness of natural language is explicitly modelled. In order to remedy this situation and to provide better support for applications that need fuzzy quantifiers, the report presents an in-depth discussion of the standard models of fuzzy quantification, which best comply with our linguistic expectations. After reviewing the known classes of models that have already been identified in previous work on the axiomatic theory of fuzzy quantification (DFS theory), it introduces a novel class of models which embeds all of the previous classes. Two independent constructions are developed and thoroughly investigated which both establish the target class of models, and hence provide a justification of the resulting class from two perspectives:

• as an extension of the known class of F_xi-DFSes, it represents the full class of models definable in terms of three-valued cuts. This style of presentation lends itself to the development of algorithms which implement quantifiers in the models;
• as an abstraction of the models G, G^* and G_* proposed in [Glöckner 97], which were inspired by the fuzzification mechanism proposed by Gaines, it captures the class of models definable in terms of the extension principle, and hence links the analysis of fuzzy quantification to the fundamental principle underlying fuzzy set theory.

The report also describes some typical examples of the new models. In addition, it presents the exact conditions required to check if a model of interest obeys the adequacy properties discovered by DFS theory. The report hence reaches an important milestone in the superordinate endeavour of providing a solid theoretical foundation for the use of fuzzy quantifiers in applications.

Reference

I. Glöckner
Standard Models of Fuzzy Quantification
Technical Report TR2001-01, University of Bielefeld, Technical Faculty, P.O.-Box 100131, 33501 Bielefeld, Germany, April 2001.

Download

Portable Document Format PDF, 410 pages, 1.54MB