Towards an Operational Interpretation of Membership Grades - On H-Valued Fuzzy Sets and Their Use for Fuzzy Quantification

Advances in fuzzy quantification have rendered possible a consistent interpretation of quantifying expressions involving vague quantifiers and fuzzy arguments (Glockner 2006, Diaz-Hermida et al 2005). However, the assumption of these approaches that the modeller is able to specify precise [0,1]-valued membership functions for the involved fuzzy sets and fuzzy quantifiers can be too strong in certain cases. To alleviate this problem, we extend the existing theory of fuzzy quantification to lattice-valued fuzzy sets which no longer require a specification of precise numerical membership grades. The paper focuses on a special type of so called H-lattices whose Hasse diagram has an hourglass shape. In this setting, we can achieve an operational interpretation of membership values, which can be calculated automatically provided that the modeller (a) decides on the basic tendency of the membership assessments and (b) specifies the salient ordering relationships between the confidence levels. The generalization of the existing theory of fuzzy quantification to H-valued fuzzy sets is a straightforward task and few properties of the models will be lost when turning from [0,1] to the generalized valuations. It is even possible to devise a generic construction which assigns a plausible model of fuzzy quantification to any given H-lattice