In the framework of conceptual spaces, objects, concepts and similarity relations are represented geometrically as points, regions and distances in a mathematical space. This approach fits well the prototype theory of concepts. Conceptual spaces have been used to study vagueness, natural concepts, inductive and non-monotonic inferences and lexical semantics.
My approach makes use of some ideas of Rudolf Carnap that were developed by Thomas Mormann (see the Section on Carnap’s early work). Some of my publications in this area are:
- Belastegui, J. (2025): A Conceptual Space for Classical Concepts. Synthese, 206, 71, T.C. Conceptual Spaces: A Mathematical Framework for Concept Engineering. DOI: 10.1007/s11229-025-05148-7
Usually conceptual spaces are combined with the prototype theory of concepts. In this paper, I extend the approach by studying classical concepts, i.e. concepts defined by necessary and sufficient conditions, geometrically. For this, I select a non-Euclidean space, namely the space of infinite sequences of 0-s and 1-s (the Cantor space).
I show that one can naturally represent classical concepts as regions in this space, that these regions satisfy the conditions in the literature of being closed and convex, that they can be combined by combining their definitions and that one can use them to explicate a Kantian notion of analyticity.
The space is actually quite simple. One starts with an infinite set of basic qualities. The points of the space are all the sets of these qualities and represent objects (or more accurately, the qualitative states of these objects). Degrees of similarity (which give the topology of the space) are represented in terms of (not) sharing these qualities. A classical concept is represented as a region whose points are exactly those that contain exactly the qualities in the definition of the concept.
- Belastegui, J. (2022): A Qualitative Approach to Conceptual Spaces: Prototypes as Qualitative Atoms. Erkenntnis, 89, 319–354. DOI: 10.1007/s10670-022-00535-9
Since similarity relations are represented as distances in a conceptual space, similarity comes in degrees. In this paper I present an alternative approach based on categorical (yes-no) similarity relations. Roughly speaking, categorical similarities are to degree similarities what the categorical notion of belief is to the Bayesian one.
I introduce several conditions that I think that such an approach should satisfy concerning the structure of similarity, concepts and categorization. Then I introduce three different models that I show to be mathematically equivalent. One of them is based on Carnap’s preAufbau work and another one was introduced as a model of vagueness by Ian Rumfitt. I also show how Carnap’s model avoids Goodman’s objections to similarity.