Vagueness: A Conceptual Spaces Approach

Abstract

The conceptual spaces approach has recently emerged as a novel account of concepts. Its guiding idea is that concepts can be represented geometrically, by means of metrical spaces. While it is generally recognized that many of our concepts are vague, the question of how to model vagueness in the conceptual spaces approach has not been addressed so far, even though the answer is far from straightforward. The present paper aims to fill this lacuna.

Tolerant, Classical, Strict

“…5 See also [7] where a very similar definition of borderlineness is proposed, but in a metric setting, in terms of Voronoi diagrams. More generally, our present definition of borderline cases bears a direct analogy to the definition of the boundary of a set in topology.…”

“…Both relations are always interpreted classically; there is no difference between a model's strictly satisfying, classically satisfying, or tolerantly satisfying any sentence built entirely from identity or similarity relations. 7 …”

“…One is that more or less equidistant cases between focal points (conceived as prototypes) are part and parcel of what it means to be a borderline case (see [7]). Similarly here, borderline cases between P and not P can be viewed as cases equisimilar to P and not P cases.…”

Applications of Conceptual Spaces

“…This seems clear on the basis of a priori reflection, but it is also supported by empirical data: in their famous color naming experiment, Berlin and Kay (1969) found that their participants often designated more than one Munsell color chip if they were asked to point at the chip or chips they regarded as typical for a given color. In Douven et al (2013), we concluded from this that, rather than with prototypical points, conceptual spaces Douven et al (2013) is at bottom very simple. This modification starts by considering the set of all possible selections of one single point from each prototypical region in a space, and noting that each element of that set can be used to generate a Voronoi diagram of the space.…”

“…The modification then constructs a new type of Voronoi diagram -called 'collated Voronoi diagram' -by projecting all the Voronoi diagrams in V S onto each other, so to speak. It is shown in Douven et al (2013) that the collated Voronoi diagram of a space still carves up the space in such a way that the regions representing the concepts are convex. The great advantage of the collated construction over simple Voronoi diagrams is that the former have 'thick' boundary regions.…”