Abstract. The aim of this paper is to present a topological method for constructing
discretizations (tessellations) of conceptual spaces. The method works for a class of
topological spaces that the Russian mathematician Pavel Alexandroff defined more than
80 years ago. Alexandroff spaces, as they are called today, have many interesting
properties that distinguish them from other topological spaces. In particular, they exhibit
a 1-1 correspondence between their specialization orders and their topological structures.
Recently, a special type of Alexandroff spaces was used by Ian Rumfitt to elucidate the
logic of vague concepts in a new way. According to his approach, conceptual spaces such
as the color spectrum give rise to classical systems of concepts that have the structure
of atomic Boolean algebras. More precisely, concepts are represented as regular open
regions of an underlying conceptual space endowed with a topological structure.
Something is subsumed under a concept iff it is represented by an element of the
conceptual space that is maximally close to the prototypical element p that defines that
concept. This topological representation of concepts comes along with a representation
of the familiar logical connectives of Aristotelian syllogistics in terms of natural settheoretical
operations that characterize regular open interpretations of classical Boolean
In the last 20 years, conceptual spaces have become a popular tool of dealing with a
variety of problems in the fields of cognitive psychology, artificial intelligence, linguistics
and philosophy, mainly due to the work of Peter Gärdenfors and his collaborators. By using
prototypes and metrics of similarity spaces, one obtains geometrical discretizations of
conceptual spaces by so-called Voronoi tessellations. These tessellations are extensionally
equivalent to topological tessellations that can be constructed for Alexandroff spaces.
Thereby, Rumfitt’s and Gärdenfors’s constructions turn out to be special cases of an
approach that works for a more general class of spaces, namely, for weakly scattered
Alexandroff spaces. This class of spaces provides a convenient framework for conceptual
spaces as used in epistemology and related disciplines in general. Alexandroff spaces are
useful for elucidating problems related to the logic of vague concepts, in particular they
offer a solution of the Sorites paradox (Rumfitt). Further, they provide a semantics for the
logic of clearness (Bobzien) that overcomes certain problems of the concept of higher2
order vagueness. Moreover, these spaces help find a natural place for classical syllogistics
in the framework of conceptual spaces. The crucial role of order theory for Alexandroff
spaces can be used to refine the all-or-nothing distinction between prototypical and nonprototypical
stimuli in favor of a more fine-grained gradual distinction between more-orless
prototypical elements of conceptual spaces. The greater conceptual flexibility of the
topological approach helps avoid some inherent inadequacies of the geometrical approach,
for instance, the so-called “thickness problem” (Douven et al.) and problems of selecting
a unique metric for similarity spaces. Finally, it is shown that only the Alexandroff account can deal with an issue that is gaining more and more importance for the theory of conceptual spaces, namely, the role that digital conceptual spaces play in the area of artificial intelligence, computer science and related disciplines.
Keywords: Conceptual Spaces, Polar Spaces, Alexandroff Spaces, Prototypes, Topological Tessellations, Voronoi Tessellations, Digital Topology.