A bipolar model of vague concepts based on random set and prototype theory Original Research Article

We argue that vagueness is a multi-faceted phenomenon requiring a framework for concept representation incorporating aspects of typicality, semantic uncertainty and indeterminism. In this paper we propose a bipolar model for vague concepts within the framework of prototype theory where concepts are represented by prototypical regions of an underlying conceptual space, and in which the appropriateness of a concept label to describe a given instance is determined on the basis of both a lower and an upper threshold on the distance from the defining prototype. Essentially, the label is absolutely appropriate as a description, providing that the distance to the prototype is less than the lower threshold, and not absolutely inappropriate if it is less than the upper threshold. Hence, in effect a concept is defined by lower and upper neighbourhoods of the prototype within the conceptual space, and the borderline region between the neighbourhoods identifies those elements of the space for which the concept label is neither absolutely appropriate nor absolutely inappropriate to describe. Semantic uncertainty is then represented by a joint probability density function on the lower and upper thresholds so that the lower and upper neighbourhoods correspond to nested random sets. This naturally results in lower and upper appropriateness measures quantifying the belief that a concept label is absolutely appropriate and not absolutely inappropriate to describe a given element of the space. These measures can then be related to the random set interpretation of fuzzy sets and in particular to lower and upper membership functions in interval fuzzy set theory.