Prototypes, Poles, and Topological Tessellations of Conceptual Spaces

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The aim of this paper is to present a general method for constructing natural tessellations of conceptual spaces that is based on their topological structure. This method works for a class of spaces that was defined some 80 years ago by the Russian mathematician Pavel Alexandroff. Alexandroff spaces, as they are called today, are distinguished from other topological spaces by the fact that they exhibit a 1-1 correspondence between their specialization orders and their topological structures. Recently, Ian Rumfitt (apparently not being aware of Alexandroff’s work) used a very special case of Alexandroff’s method to elucidate the logic of vague concepts in a new way. Elaborating his approach, the color circle’s conceptual space can be shown to define an atomistic Boolean algebra of regular open concepts. In a similar way Gärdenfors’ geometrical discretization of conceptual spaces by Voronoi tessellations also can be shown to be a kind of geometrical version of Alexandroff’s topological construction. More precisely, a discretization à la Gärdenfors is extensionally equivalent to a topological discretization constructed by Alexandroff’s method. Rumfitt’s and Gärdenfors’s constructions turn out to be special cases of an approach that works much more generally, namely, for Alexandroff spaces. For these spaces (X, OX) the Boolean algebras O*X of regular open sets are still atomistic and yield natural tessellations of X.
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First archival date: 2018-10-31
Latest version: 6 (2020-04-25)
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Vagueness: A Conceptual Spaces Approach.Douven, Igor; Decock, Lieven; Dietz, Richard & Égré, Paul
Vagueness: A Reader.Keefe, Rosanna & Smith, Peter (eds.)

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