Formal Ways for Measuring Relations between Concepts in Conceptual Spaces

TL;DR

This paper enhances the conceptual spaces framework by introducing formal mathematical measures for relations between concepts, such as size, subsethood, implication, similarity, and betweenness, improving its expressiveness.

Contribution

It provides the most comprehensive formalization of conceptual spaces by defining quantitative measures for various concept relations.

Findings

Formal definitions for concept size and subsethood

Quantitative measures for implication and similarity

Enhanced representational capabilities of the framework

Abstract

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.