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 “translation” 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, Rumfitt’s recent topological reconstruction (...) of Sainsbury’s theory of prototypically defined concepts is shown to lead to the same class of spaces that characterize Bobzien’s account of columnar vagueness, namely, weakly scattered spaces. Rumfitt calls these spaces polar spaces. They turn out to be closely related to Gärdenfors’ conceptual spaces, which have come to play an ever more important role in cognitive science and related disciplines. Finally, Williamson’s “logic of clarity” is explicated in terms of a generalized topology (“locology”) 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 Williamson’s logic of clarity, vague concepts with respect to a notion of a locologically inspired notion of a “slim boundary” are (stably) columnar. Thus, Williamson’s 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. (shrink)

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 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. (shrink)

The aim of this paper is to present a topological method for constructing discretizations of topological conceptual spaces. The method works for a class of topological spaces that the Russian mathematician Pavel Alexandroff defined more than 80 years ago. The aim of this paper is to show that Alexandroff spaces, as they are called today, have many interesting properties that can be used to explicate and clarify a variety of problems in philosophy, cognitive science, and related disciplines. For instance, recently, (...) Ian Rumfitt used a special type of Alexandroff spaces to elucidate the logic of vague concepts in a new way. Moreover, Rumfitt’s class of Alexandroff spaces can be shown to provide a natural topological semantics for Susanne Bobzien’s “logic of clearness”. Mainly due to the work of Peter Gärdenfors and his collaborators, conceptual spaces have become an increasingly popular tool of dealing with a variety of problems in the fields of cognitive psychology, artificial intelligence, linguistics and philosophy. For Gärdenfors’s conceptual spaces, geometrically defined discretizations play an essential role. These tessellations can be shown to be extensionally equivalent to topological tessellations that can be constructed for Alexandroff spaces in general. 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. The main aim of this paper is to show that this class of spaces provides a convenient framework for conceptual spaces as used in epistemology and related disciplines in general. Weakly scattered Alexandroff spaces are useful for elucidating problems related to the logic of vague concepts, in particular they offer a solution of the Sorites paradox. Further, they provide a semantics for the logic of clearness that overcomes certain problems of the concept of higher-order vagueness. Moreover, these spaces help find a natural place for classical syllogistics in the framework of conceptual spaces. The specialization order of Alexandroff spaces can be used to refine the all-or-nothing distinction between prototypical and non-prototypical stimuli in favor of a gradual distinction between more or less 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”. Finally, it is shown that the Alexandroff spaces offer an appropriate framework to deal with digital conceptual spaces that are gaining more and more importance in the areas of artificial intelligence, computer science and related disciplines. (shrink)

It is notoriously difficult to model the range of application of vague predicates relative to a suitable sorites series. In this paper I offer some critical remarks against an interesting view that has received little attention in the literature. According to it, the sharp cut-offs we find in our semantic models are just artifacts of the theory, and, as such, they are harmless. At the end I discuss a contextualist view that, at a cost, may be able to get around (...) the problems related to sharp cut-offs incurred in by other theories of vagueness. Es sumamente difícil modelar el rango de aplicación de predicados vagos relativo a una serie sorites apropiada. En este artículo ofrezco algunos comentarios críticos en contra de una postura interesante que ha recibido poca atención en la literatura. De acuerdo con ella, los cortes precisos que encontramos en modelos semánticos son simplemente artefactos de la teoría, y, como tales, son inofensivos. Al final del artículo discuto una postura contextualista que, no sin sacrificios, podría ser capaz de evadir los problemas relacionados con cortes precisos que otras teorías de la vaguedad poseen. (shrink)

According to columnar higher-order vagueness, all orders of vagueness coincide: any borderline case is a borderline borderline case, and a third-order borderline case, etc. Bobzien has worked out many details of such a theory and models it with a modal logic closely related to S4. I take up a range of questions about the framework and argue that it is not suitable for modelling the structure of vagueness and higher-order vagueness.

This paper is an expanded written version of my reply to Rosanna Keefe’s paper ‘Modelling higher-order vagueness: columns, borderlines and boundaries’ (Keefe 2015), which in turn is a reply to my paper ‘Columnar higher-order vagueness, or Vagueness is higher-order vagueness’ (Bobzien 2015). Both papers were presented at the Joint Session of the the Aristotelian Society and the Mind Association in July, 2015. At the Joint Session meeting, there was insufficient time to present all of my points in response to Keefe’s (...) paper. In addition, the audio of the session, which is available online, becomes inaudible at the beginning of my reply to Keefe’s comments due to a technical defect. The following is a full version of my remarks. (shrink)

ABSTRACT: This chapter offers a revenge-free solution to the liar paradox (at the centre of which is the notion of Gestalt shift) and presents a formal representation of truth in, or for, a natural language like English, which proposes to show both why -- and how -- truth is coherent and how it appears to be incoherent, while preserving classical logic and most principles that some philosophers have taken to be central to the concept of truth and our use of (...) that notion. The chapter argues that, by using a truth operator rather than truth predicate, it is possible to provide a coherent, model-theoretic representation of truth with various desirable features. After investigating what features of liar sentences are responsible for their paradoxicality, the chapter identifies the logic as the normal modal logic KT4M (= S4M). Drawing on the structure of KT4M (=S4M), the author proposes that, pace deflationism, truth has content, that the content of truth is bivalence, and that the notions of both truth and bivalence are semideterminable. (shrink)

This paper deals with higher-order vagueness in Williamson's 'logic of clarity'. Its aim is to prove that for 'fixed margin models' (W,d,α ,[ ]) the notion of higher-order vagueness collapses to second-order vagueness. First, it is shown that fixed margin models can be reformulated in terms of similarity structures (W,~). The relation ~ is assumed to be reflexive and symmetric, but not necessarily transitive. Then, it is shown that the structures (W,~) come along with naturally defined maps h and s (...) that define a Galois connection on the power set PW of W. These maps can be used to define two distinct boundary operators bd and BD on W. The main theorem of the paper states that higher-order vagueness with respect to bd collapses to second-order vagueness. This does not hold for BD, the iterations of which behave in quite an erratic way. In contrast, the operator bd defines a variety of tolerance principles that do not fall prey to the sorites paradox and, moreover, do not always satisfy the principles of positive and negative introspection. (shrink)