We present an account of semantics that is not construed as a mapping ofudlanguage to theworld but rather as amapping between individual meaning spaces. Theudmeanings of linguistic entities are established via a “meeting of minds.” The conceptsudin the minds of communicating individuals are modeled as convex regions in conceptualudspaces. We outline a mathematical framework, based on fixpoints in continuousudmappings between conceptual spaces, that can be used to model such a semantics.udIf concepts are convex, it will in general be possible for interactors to agree on jointudmeaning even if they start out from different representational spaces. Language isuddiscrete, while mental representations tend to be continuous—posing a seeming paradox.udWe show that the convexity assumption allows us to address this problem. Usingudexamples, we further show that our approach helps explain the semantic processesudinvolved in the composition of expressions.
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