Modeling vague quantifiers in mathematical fuzzy logic

Vague quantifiers like `many`, `few`, or `about a half` present a problem in computerized natural language processing. Designing a satisfactory theory of vague quantifiers calls for constructing formal models and to evaluate them with regard to linguistic adequateness, embeddability in logical frameworks and to amenability to automated deduction. This constitutes a serious research challenge involving computer science and logic, but also linguistics, and analytic philosophy. The fuzzy logic paradigm, based on the notion of degrees of truth, provides a mathematical apparatus for dealing with several aspects of vagueness. The applications of fuzzy methods to vague quantifiers have so far largely neglected the potential of deductive systems studied by mathematical fuzzy logic. The aim of the project is to deepen and extend the logical and computational foundations for adequate modeling of vague quantifiers by employing formalisms and results of mathematical fuzzy logic. The research objectives of the project can be grouped into three clusters: 1. Development of variations and extensions of the existing frameworks for vague quantifiers, exploiting the potentialities of mathematical fuzzy logic in four levels of increasing complexity: semi-fuzzy quantifiers, modal fuzzy logics with two-tiered syntax, general first-order formalisms, and higher- order formalisms. 2. Development of formal dialogue and betting game frameworks to justify concrete choices of particular truth functions in modeling logical reasoning with vague quantifiers. 3. Extension of analytic proof systems and automated deduction formalisms for fuzzy logics enriched by fuzzy quantifiers.

This project deepened and extended the logical and computational foundations for adequate models of vague quantifiers like many, few, about half. The main tools, partly crafted specifically for the project, were formalisms from mathematical fuzzy logic. At the core of the Austrian part of the bilateral project were formal games, which model the meaning of fuzzy quantifiers by rules that govern the decomposition of complex statements into simpler statements. For example, if attacked by an opponent, the proponent of a statement many doors are locked will have to defend more concrete statements involving one or more randomly chosen doors. The indicated principle of random witness selection lead to a host of new computer oriented models for a wide family of vague quantifiers. Moreover, we explored the application of our models to flexible information retrieval, where the users have the option to use vague quantifier expressions in formulating their queries.