Applications of Convexity in Semantics for Natural Language

The purpose of the article is to present an overview of how convexity serves as a constraint on the semantics of natural language. I begin by presenting four question that a semantic theory should be able to answer. The semantic theory I propose is based on conceptual spaces, which are geometrical or topological structures provided with a betweenness relation. Previously, I have proposed the criterion that a natural concept is a convex region in a conceptual space. I show how this criterion can be expanded to an analysis of how convexity plays a role in the semantics of different word classes. I will show that the convexity criterion also improves the learnability of word meanings. Voronoi tessellations based on prototypes of categories can function as an efficient mechanism behind such learning processes. For many applications, a Euclidean or a city-block metric provides this relation, but for some word classes, for example color words and prepositions, polar convexity is required. Another topic concerns how the mappings between words and regions in conceptual spaces can be aligned between different individuals – in other words, how we know that we mean the same things when we use a word. If continuity and convexity of the mappings are assumed, Brouwer’s fixpoint theorem assures that there exist “meetings of minds” as regards word meanings.