  The highly influential framework of conceptual spaces provides a geometric
way of representing knowledge. Instances are represented by points in a
high-dimensional space and concepts are represented by regions in this space.
In this article, we extend our recent mathematical formalization of this
framework by providing quantitative mathematical definitions for measuring
relations between concepts: We develop formal ways for computing concept size,
subsethood, implication, similarity, and betweenness. This considerably
increases the representational capabilities of our formalization and makes it
the most thorough and comprehensive formalization of conceptual spaces
developed so far.
