A Categorical Semantics of Fuzzy Concepts in Conceptual Spaces
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We define a symmetric monoidal category modelling fuzzy concepts and fuzzy conceptual reasoning within Gärdenfors' framework of conceptual (convex) spaces. We propose log-concave functions as models of fuzzy concepts, showing that these are the most general choice satisfying a criterion due to Gärdenfors and which are well-behaved compositionally. We then generalise these to define the category of log-concave probabilistic channels between convex spaces, which allows one to model fuzzy reasoning with noisy inputs, and provides a novel example of a Markov category.
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