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Relative Identity and Cardinality1

Published online by Cambridge University Press:  01 January 2020

Patricia Blanchette
Patricia Blanchette*
Affiliation:
University of Notre Dame Notre Dame, IN46556USA
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Extract

Peter Geach famously holds that there is no such thing as absolute identity. There are rather, as Geach sees it, a variety of relative identity relations, each essentially connected with a particular monadic predicate. Though we can strictly and meaningfully say that an individual a is the same man as the individual b, or that a is the same statue as b, we cannot, on this view, strictly and meaningfully say that the individual a simply is b.

It is difficult to find anything like a persuasive argument for this doctrine in Geach’s work. But one claim made by Geach is that his account of identity is the account most naturally aligned with Frege's widely admired treatment of cardinality. And though this claim of an affinity between Frege's and Geach's doctrines has been challenged, the challenge has been resisted. William Alston and Jonathan Bennett, indeed, go further than Geach to argue that Frege's doctrine implies Geach's.

Type
Research Article
Information
Canadian Journal of Philosophy , Volume 29 , Issue 2 , June 1999 , pp. 205 - 223
Copyright
Copyright © The Authors 1999

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Footnotes

1

An earlier version of this paper was read at the 1997 MMM conference at the University of Notre Dame; thanks to members of that audience, and particularly to John O'Leary-Hawthorne, for helpful comments. Thanks also to Leora Weitzman, Palle Yourgrau, and an anonymous referee for helpful comments.

References

2 See Perry, John ‘Relative Identity and Number,’ Canadian Journal of Philosophy 8 (1978) 1–14CrossRefGoogle Scholar.

3 Alston, William and Bennett, Jonathan ‘Identity and Cardinality: Geach and Frege,’ The Philosophical Review 93 (1984) 553–67CrossRefGoogle Scholar

4 Frege, Gottlob Die Grundlagen der Arithmetik (Breslau: Koebner 1884)Google Scholar; trans. Austin, J.L. as The Foundations of Arithmetic (Evanston: Northwestern University Press 1980), §22Google Scholar

5 Ibid., §46

6 Ibid., §54

7 Though in 1884 in The Foundations of Arithetic Frege takes the notion of ‘set’ to be problematically vague, and hence resists the identification of his extensions with sets (see e.g. §45), he is by the 1890s ready to treat extensions as equivalent with sets. See e.g. the letter to Russell of 28 July 1902; English version in Frege, Gottlob Philosophical and Mathematical Correspondence, McGuinness, B. ed. (Chicago: University of Chicago Press 1980) 139–42Google Scholar.

Regarding Husserl's claim that number belongs primarily to an extension and only secondarily to a concept, Frege replies: ‘This actually concedes all that I am maintaining: a statement of number states something about a concept. I will not quibble over whether the statement is directly about the concept and indirectly about its extension, or indirectly about the concept and directly about its extension, for one goes with the other’ (Frege, Gottlob Collected Papers on Mathematics, Logic, and Philosophy McGuinness, B. ed. (Oxford: Blackwell 1984), 322)Google Scholar. See also Frege's letter to Russell. To say that a set S has n members, for Frege, is to say that n things fall under a concept (e.g., member of S) of which S is the extension.

Palle Yourgrau has argued that since Frege’s ‘relativity argument’ applies to sets as well as to other objects, Frege cannot unproblematically take number to apply to sets (Yourgrau, Palle ‘Sets, Aggregates, and Numbers,’ Canadian Journal of Philosophy 15 (1985) 581–92)CrossRefGoogle Scholar. Yourgrau is right that sets can be ‘numbered’ in various ways; for each set, we can ask how many members it has, how many subsets it has, perhaps how many fingers its members have, and so on. Sets do not have, intrinsically, a unique number. But this is not an issue for Frege, since Frege's assignment of number to sets simply requires that each set have a unique number of members.

Yourgrau has argued further (Yourgrau 1997) that the relativity argument applies to concepts as well, with a similarly problematic result for Frege's attribution of number to concepts (Yourgrau, Palle ‘What is Frege's Relativity Argument?’ Canadian Journal of Philosophy 27 (1997) 137–71)CrossRefGoogle Scholar. This is not the place for a detailed response to this claim; the important point here is that on Frege's account, an attribution of number to a concept is always a claim about how many things fall under that concept. Frege’ s position is not that each concept (or set) somehow comes along with a unique number intrinsically associated with it. It is rather that all assignments of number can (and should) be treated as claims about the number of entities falling under a given concept.

8 Geach, Peter ‘Identity,’ Review of Metaphysics 21 (1967)Google Scholar; reprinted in Geach, Peter Logic Matters (Berkeley: University of California Press 1972) 238–47, at 238Google Scholar

9 Ibid., 241

10 This is Geach's procedure. See, e.g., Geach, Peter Reference and Generality (Ithaca, NY: Cornell University Press 1962)Google Scholar, §109; and Geach, Peter ‘Ontological Relativity and Relative Identity,’ Logic and Ontology Munitz, M. ed. (New York: New York University Press 1973), 291Google Scholar.

11 Geach, ‘Ontological Relativity,’ 292Google Scholar

12 Geach, ‘Ontological Relativity,’ 294–5Google Scholar

13 Geach, ‘Identity,’ 246Google Scholar

14 Geach, ‘Identity,’ 238Google Scholar

15 Geach, Reference and Generality, 39Google Scholar

16 Frege, Foundations of Arithmetic, §29Google Scholar

17 See §VI for a defense of this claim.

18 Perry, ‘Relative Identity and Number’

19 Perry, ‘Relative Identity and Number,’ 7Google Scholar

20 The doctrine of relative numbers is what Alston and Bennett call the ‘Relative Cardinality Thesis’; see Alston & Bennett, 555.

21 Alston & Bennett, 557

22 Alston & Bennett, 558

23 Ibid., 560

24 That there are such disjunctive concepts follows immediately from Frege’ s view that every (1-place, first-level) predicative phrase refers to a concept. Such a phrase is, as Frege sees it, obtained from a sentence by removing one or more occurrences of a singular term; thus e.g. ‘_ = 2 or_ = 7’ refers to a concept under which exactly 2 and 7 fall. See e.g. Frege, Gottlob Begriffsschrift (Halle: Louis Nebert 1879)Google Scholar, trans. Bauer-Mengelberg, S. in From Frege to Godel, Heijenoort, J. van ed. (Cambridge, MA: Harvard University Press 1967), §9;Google Scholar Frege, The Foundations of Arithmetic §65n; Frege, ‘Uber Begriff und Gegenstand,’ Vierteljahrsscchrift fur wissenschaftliche Philosophie 16 (1892) 192–205Google Scholar, trans. Geach, P. as ‘On Concept and Object’ in Collected Papers on Mathematics, Logic, and Philosophy 182–94Google Scholar; Frege, Grundgesetze der Arithmetik Gena: Hermann Pohle 1893)Google Scholar, partially trans. by Furth, M. as The Basic Laws of Arithmetic (Berkeley and Los Angeles: University of California Press 1964), §26Google Scholar. The use of phrases of the form ‘_ = a’ (for ‘a’ a referring singular term) to refer to concepts is essential to Frege's construction of the natural numbers; see e.g. Frege, The Foundations of Arithmetic, §77; Frege, The Basic Laws of Arithmetic §42.

25 Alston & Bennett, 561