A case against convexity in conceptual spaces

Abstract

The notion of conceptual space, proposed by Gardenfors as a framework for the representation of concepts and knowledge, has been highly influential over the last decade or so. One of the main theses involved in this approach is that the conceptual regions associated with properties, concepts, verbs, etc. are convex. The aim of this paper is to show that such a constraint—that of the convexity of the geometry of conceptual regions—is problematic; both from a theoretical perspective and with regard to the inner workings of the theory itself. On the one hand, all the arguments provided in favor of the convexity of conceptual regions rest on controversial assumptions. Additionally, his argument in support of a Euclidean metric, based on the integral character of conceptual dimensions, is weak, and under non-Euclidean metrics the structure of regions may be non-convex. Furthermore, even if the metric were Euclidean, the convexity constraint could be not satisfied if concepts were differentially weighted. On the other hand, Gardenfors’ convexity constraint is brought into question by the own inner workings of conceptual spaces because: (i) some of the allegedly convex properties of concepts are not convex; (ii) the conceptual regions resulting from the combination of convex properties can be non-convex; (iii) convex regions may co-vary in non-convex ways; and (iv) his definition of verbs is incompatible with a definition of properties in terms of convex regions. Therefore, the mandatory character of the convexity requirement for regions in a conceptual space theory should be reconsidered.