A Gärdenforsian Perspective on the Conceptual Geometry of LLMs

G\"ardenfors' conceptual spaces offer a geometric language for describing conceptual knowledge. We show that this cognitive theory can be approximately realized in the representation geometry of LLMs. Specifically, we propose the G\"ardenforsian Representation Hypothesis (GRH), which characterizes LLM representations across multiple levels of conceptual abstraction, from quality dimensions (graded attributes) and domains to properties (adjective), category concepts, and prototypes. GRH unifies existing representation hypotheses by showing that they can be understood as instantiations of a common conceptual geometry, each capturing a different level of conceptual abstraction. We test GRH by progressively reconstructing such a conceptual space from LLM representations. Experiments on THINGSplus and Binder show that the recovered conceptual geometry closely aligns with human conceptual organization, providing evidence that LLM representations naturally admit a G\"ardenfors'-style conceptual-space structure.