Abstract
In the context of classical (crisp, precise) sets, there is a familiar connection between the notions of counting, ordering and cardinality. When it comes to vague collections, the connection has not been kept in central focus: there have been numerous proposals regarding the cardinality of vague collections, but these proposals have tended to be discussed in isolation from issues of counting and ordering. My main concern in this paper is to draw focus back onto the connection between these notions. I propose a natural generalisation to the vague case of the familiar process of counting precise collections. I then discuss the relationships between this process of counting and various notions of ordering and cardinality for vague sets. Some existing views concerning the cardinality of vague collections fit better than others with my proposal about how to count the members of such a collection. In particular, the idea that we should approach cardinality via certain formulas of a logical language—which has been prominent in the recent literature—is less attractive than other existing proposals.
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Notes
- 1.
For example, it is reviewed on the first page of a recent handbook article on set theory (Bagaria 2008, 616).
- 2.
A word of explanation concerning my title: it is a reference to the Count, a character from the television show Sesame Street. He loved to count things—and when he had finished doing so, would laugh maniacally (Aahh Aahh Aahh Aahh Aaaahhhh!) to the accompaniment of thunder and lightning.
- 3.
A function f: S → T is said to be total if it satisfies the condition that every member of S gets sent to some member of T; onto (aka surjective, a surjection) if it satisfies the condition that every member of T gets hit at least once; and one-one (aka one-to-one, into, injective, an injection) if no member of T gets hit more than once. A bijection (aka correspondence) is a function that is total, onto and one-one. If there is a bijection f from S to T, then there is a bijection (the inverse of f) from T to S; hence it is common to talk nonspecifically of a bijection between S and T.
- 4.
An ordinal, as Cantor (1915) put it, results from a single act of abstraction: we ignore the particular identity of each object in the set and simply look at the order in which these objects appear; a cardinal results from a double act of abstraction, in which we ignore both the particular identity of each object in the set and the order in which these objects appear, paying attention only to the number of objects in the set.
- 5.
- 6.
The teletransporter is playing the role of a disrupter. Readers who do not like the example should substitute their favourite case from the personal identity literature of a disruptive process where it is unclear whether the person who enters the process is the same as the person who exits the process.
- 7.
I said that much in this paper could be applied, mutatis mutandis, both to approaches to vagueness that do not employ fuzzy sets and to counting issues arising from vague identity. The story that I have just told about counting vague collections extends in an obvious way to any treatment of vagueness wherein the extension of a vague predicate can be modelled as a function from the domain of discourse to a set of membership values—for example, supervaluationist (or subvaluationist) treatments and treatments employing a many-valued or gappy (or glutty) logic. For the case of vague identity, the extent to which the next counting number is attached to the next object in the set should reflect both the extent to which that object is a member of the set and the extent to which it is distinct from all other objects in the set.
- 8.
The notation is Cantor’s. Each bar represents an act of abstraction: one for an ordinal, two for a cardinal (see Footnote 4 above).
- 9.
The symbol × represents the Cartesian product. S × T is the set of all ordered pairs whose first element is a member of the set S and whose second element is a member of the set T.
- 10.
Recall that \(\bar{\bar{S_{{\ast}}}}\) is the number of elements in the support of S—and we may think of this number as a set.
- 11.
Of course there is also a reverse version of this ordering, where we begin with the lowest degree members and work up.
- 12.
Recall Cantor’s second act of abstraction.
- 13.
- 14.
Compare the way that universities count students for certain purposes: a full-time student adds 1 to the count; a half-time student adds 0.5 to the count; and so on. (Thanks to David Braddon-Mitchell for this example.)
- 15.
For more details on the foregoing material see, e.g. Smith (2012, Sect. 13.5).
- 16.
Parsons does not work with fuzzy sets or degrees of truth. Here and below I adapt his ideas to the present context, in which we use fuzzy sets to model vagueness.
- 17.
See, for example, Parsons (2000, p. 135): “It follows that the question of how many persons there are all told has no correct answer. … in this case it seems clear that these are the right things to say: any answer less than two or more than three is wrong, and either “two” or “three” is such that it is indeterminate whether it is correct” and p. 136: “It appears that in this case any answer less than one or more than three is definitely wrong, but the answers “one”, “two”, or “three” should all have indeterminate truth-value.”
- 18.
Thanks to Siegfried Gottwald for helpful discussion and an anonymous referee for useful comments. Thanks also to audiences at a seminar at the Department of Philosophy at the University of Sydney on 22 May 2013, at a workshop on Metaphysical Indeterminacy at the University of Leeds on 12 June 2013 and at the LENLS 10 workshop (Logic and Engineering of Natural Language Semantics) at Keio University in Kanagawa on 27 October 2013.
References
Bagaria, J. (2008). Set theory. In T. Gowers (Ed.), The Princeton companion to mathematics (pp. 615–634). Princeton: Princeton University Press.
Cantor, G. (1915). Contributions to the founding of the theory of transfinite numbers. New York: Dover. ( Translated by Philip E.B. Jourdain)
Hyde, D. (2008). Vagueness, logic and ontology. Aldershot: Ashgate.
Parsons, T. (2000). Indeterminate identity: Metaphysics and semantics. Oxford: Clarendon Press.
Smith, N. J. J. (2008a). Why sense cannot be made of vague identity. Noûs, 42, 1–16.
Smith, N. J. J. (2008b). Vagueness and degrees of truth. Oxford: Oxford University Press. Paperback 2013.
Smith, N. J. J. (2012). Logic: The laws of truth. Princeton: Princeton University Press.
Wygralak, M. (2003). Cardinalities of fuzzy sets. Berlin: Springer.
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Smith, N.J.J. (2014). One Bald Man … Two Bald Men … Three Bald Men—Aahh Aahh Aahh Aahh Aaaahhhh!. In: Akiba, K., Abasnezhad, A. (eds) Vague Objects and Vague Identity. Logic, Epistemology, and the Unity of Science, vol 33. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7978-5_9
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