14 research outputs found
Prototypes, Poles, and Topological Tessellations of Conceptual Spaces
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
propositional logic.
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 GĂ€rdenfors 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, Rumfittâs and GĂ€rdenforsâs 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 âthickness problemâ (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
Topological Models of Columnar Vagueness
This paper intends to further the understanding of the formal properties of (higher-order) vagueness by connecting theories of (higher-order) vagueness with more recent work in topology. First, we provide a âtranslationâ of Bobzien's account of columnar higher-order vagueness into the logic of topological spaces. Since columnar vagueness is an essential ingredient of her solution to the Sorites paradox, a central problem of any theory of vagueness comes into contact with the modern mathematical theory of topology. Second, Rumfittâs recent topological reconstruction of Sainsburyâs theory of prototypically defined concepts is shown to lead to the same class of spaces that characterize Bobzienâs account of columnar vagueness, namely, weakly scattered spaces. Rumfitt calls these spaces polar spaces. They turn out to be closely related to GĂ€rdenforsâ conceptual spaces, which have come to play an ever more important role in cognitive science and related disciplines. Finally, Williamsonâs âlogic of clarityâ is explicated in terms of a generalized topology (âlocologyâ) that can be considered an alternative to standard topology. Arguably, locology has some conceptual advantages over topology with respect to the conceptualization of a boundary and a borderline. Moreover, in Williamsonâs logic of clarity, vague concepts with respect to a notion of a locologically inspired notion of a âslim boundaryâ are (stably) columnar. Thus, Williamsonâs logic of clarity also exhibits a certain affinity for columnar vagueness. In sum, a topological perspective is useful for a conceptual elucidation and unification of central aspects of a variety of contemporary accounts of vagueness
Modal Languages for Topology: Expressivity and Definability
In this paper we study the expressive power and definability for (extended) modal languages interpreted on topological spaces. We provide topological analogues of the van Benthem characterization theorem and the Goldblatt-Thomason definability theorem in terms of the well established first-order topological language
The modal logic of Stone spaces: diamond as derivative
Abstract. We show that if we interpret modal diamond as the derived set operator of a topological space, then the modal logic of Stone spaces is K4 and the modal logic of weakly scattered Stone spaces is K4G. As a corollary, we obtain that K4 is also the modal logic of compact Hausdorff spaces and K4G is the modal logic of weakly scattered compact Hausdorff spaces. §1. Introduction. Topological semantics of modal logic was first developed by On the other hand, if we interpret 3 as the derived set operator, then the modal logic of all topological spaces is wK4-weak K4-which is obtained from the basic normal modal logic K by adding 33 p â ( p âš 3 p) as a new axiom In this paper we are interested in the modal logic of compact Hausdorff zero-dimensional spaces, also known as Stone spaces. The interest in Stone spaces stems from the celebrated Stone duality, which establishes duality (dual equivalence) between the category of Boolean algebras and Boolean algebra homomorphisms and the category of Stone spaces and continuous maps. Under Stone duality atomless Boolean algebras correspond to densein-itself Stone spaces, atomic Boolean algebras correspond to weakly scattered Stone spaces, and superatomic Boolean algebras correspond to scattered Stone spaces. It follows fro