Abstract
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.
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Notes
Recall that a topological space (X, OX) is Alexandroff if the arbitrary intersection of open sets is open, or, equivalently, if the arbitrary union of closed sets is closed.
Gabelaia’s terminology is somewhat different. To ensure that Gabelaia’s results are really equivalent to Theorem 2.4, the reader may consult Sect. 7 of this paper, particularly Lemmas 7.5 and 7.6. In most of the mathematical literature, Theorem 2.4 is considered “folklore” for which it is not necessary to give an exact source; see, e.g. Bezhanishvili et al. (2003).
This assumption is far from unanimously accepted. Rather, it is a quite controversial issue. For a variety of arguments in favour and against of this assumption see Bobzien (2015), Heck (1993), Williamson (1994) and Wright (2007). For a general survey see Zardini (2006). The topological perspective of this paper does not yield a direct argument in favour or against the thesis that a reasonable theory of vagueness has to be developed in the framework of S4. At least, it elucidates some of the consequences of this thesis in a particularly “intuitive” way (for those who consider the topological way of thinking as more intuitive than the set-theoretical one).
The real numbers R endowed with the familiar Euclidean topology OR and subset Q of rational numbers of R provide an elementary counterexample since bd(Q) = R and bdbd(Q) = Ø (see Sect. 7).
Another proof of this result may be found in Mormann (2013).
In some sense, the prototypical classification of colours works in a similar way as the characterization of implicitly defined mathematical concepts works. A system of geometrical concepts such as “points”, “lines”, and “planes” is grasped as a whole. Thus, “point” is an important geometrical concept defined in relation to other geometrical concepts, but “non-point” is clearly not.
The existence of the minimal basis V(x) renders (X, OX) an Alexandroff space (cf. Alexandroff 1937).
The axioms (MK)* and (B) have a certain relevance for epistemological matters: (MK)* is characteristic for modal system S4.2 corresponding to the class of extremally disconnected spaces that have proven useful for modeling the concept of knowledge (cf. Baltag et al. 2018). Brouwer’s axiom (B) is an axiom of Williamson’s logic of clarity (cf. Williamson 1994, 1999) based on modal operator C*; see Sect. 8. This emphasizes the necessity of strictly distinguishing operators C and C*, although their informal characterization in English may seem rather similar.
For the linear colour spectrum, all colour experiences can be influenced by at most two prototypical experiences, but of course, there may be polar spaces for which mixed non-paradigmatic experiences of three or more prototypical experiences exist.
This claim has to be understood with appropriate qualifications, of course. Otherwise it is obviously wrong. Clearly, colour concepts like “scarlet”, “maroon”, “turquoise”, or “olive” cannot be expressed in a language that is based on a simple pole distribution (X, m, P) possessing only the basic terms “red”, “green”, etc. Rather, what is needed to express “scarlet” is a pole distribution (X, m′, P′) such that either “scarlet” has its own pole p′∈ P′, or the extension of “scarlet” is the (connected) sum of two (or more) regular open subsets of (X, O′X), O′X defined by (X, m′, P′).
Two special cases of polar distributions may be particularly mentioned. Clearly, if P = {p} ⊆ X, every distribution (X, m, P) is trivial, i.e., m(x) = {p} for x ∈ X. More interesting is the case P = {p, p′}. In this case, p and p′ may be interpreted as prototypes of opposite concepts, e.g. “rich” and “poor”, “silly” and “intelligent” etc. In many paradigmatic cases of concepts for which a kind of Sorites makes sense, however, opposites have to be introduced more or less artificially. For instance, if X is the class of men, then the opposite of “bald” is something like “having a scalp with at least 100.000 hairs”. This, however, is not an insuperable obstacle to apply the apparatus of polar distributions also to these cases.
This definition can be found already in Tarski (1938, Definition 4.1, p. 434 with comments on p. 448)).
The expression “pedestal” is inspired by the idea that bd(A) ⊇ bd2(A) = bd3(A) = … = bdn(A) … ad infinitum.
Williamson uses the operator BD* to define two concepts of higher order vagueness: A sentence A is said to be nth-order borderline vague if BD*n(A) ≠ Ø. Somewhat more complicated is the definition of nth-order vagueness that uses first order vagueness BD*(Cn(A)) of certain “classifications” Cn(A) of A, Cn(A) being sets of sentences generated by A and the operator C*. The relation between nth-order borderline vagueness and nth-order vagueness is complicated.
The pair (h, s) is well known to be a Galois connection on PW (cf. Denecke et al. 2004). This fact, not observed by Breysse and De Glas, has many interesting consequences, but it is outside the scope of the present paper.
As Williamson points out, the logic of variable margin models does not satisfy the Brouwer axiom (B) (cf. Williamson 1994, 272). As is easily calculated, this entails that the operator hs is no longer a closure operator.
Examples of such topology-affine approaches to vagueness dealt with in this paper are that of Bobzien, Rumfitt, Williamson, Sainbury, and Wright.
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Acknowledgements
I thank Nasim Mahoozi for many intense and detailed discussions on matters of vagueness, topology, and related issues that greatly helped me improve my understanding of the topics dealt with in this paper. Further I would like to thank an anonymous referee for very insightful and constructive criticisms on a previous version of this paper. Financial support of the Ministerio de Economía y Competividad (MINECO) del Gobierno Español, project “Representación y anticipación: modelización interventiva RRI en las ciencias y técnicas emergentes”, is gratefully acknowledged.
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Mormann, T. Topological Models of Columnar Vagueness. Erkenn 87, 693–716 (2022). https://doi.org/10.1007/s10670-019-00214-2
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DOI: https://doi.org/10.1007/s10670-019-00214-2