Generalized Quantifiers and Measure Theory

Dissertation, Syracuse University (1996)

Charles Kurtz
Richland College
In natural language there are determiners which combine with nouns to form noun phrases. The mathematical model of these natural language determiners is the generalized quantifier, given first by Mostowski, and further generalized by Lindstrom. This work aims to improve our understanding of four features of the generalized quantifier. The first involves a fundamental change to the definition of a generalized quantifier, to expand it from its current position of working on count nouns, to working on mass nouns as well. To accomplish this, instead of a model, the generalized quantifier will be defined on a measure space. The second aim is to improve the characterization of the generalized quantifier. Intriguingly, the generalized quantifier can be characterized pictorially, and the reworked definition of a generalized quantifier leads naturally to an improved pictographic rendering. The third aim is to consider unstudied aspects of the algebra of quantifiers. Though results have been found for the various boolean connectives, little has been done on approximation, or operators on generalized quantifiers such as 'at least'. The final aim is to create a framework for the generalized aristotelian syllogism. Current systems focus on certain classes of generalized quantifiers, while in this dissertation a general approach is discovered. It is the reworked definition of the generalized quantifier that provides the background and the tools for the work in the subsequent portions of the paper
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