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Relative Identity and Number

Published online by Cambridge University Press:  01 January 2020

John Perry*
Stanford University


Geach has claimed that Frege had an insight about number which should have led him to the doctrine of relative identity:

I maintain it makes no sense to judge whether x and y are “the same or whether x remains “the same” unless we add or understand some general term — the same F. That in accordance with which we thus judge as to the identity, I call a criterion of identity; … Frege sees clearly that “one” cannot significantly stand as a predicate of objects unless it is (at least understood as) attached to a general term; I am surprised he did not see that this holds for the closely allied expression, “the same”.

Frege has clearly explained that the predication of “one endowed with wisdom” … does not split up into predications of “one” and “endowed with wisdom” … It is surprising that Frege should on the contrary have constantly assumed that “x is the same A as Y” Does split up into “x is an A (and y is an A)” and “x is the same as … y.” We have already by implication rejected this analysis.

Research Article
Copyright © The Authors 1978

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1 Geach, Peter Reference and Generality (Ithaca, 1962), p. 39Google Scholar.

2 Ibid., pp. 151–152.

3 Frege, Gottlob Grundlagen der Arithmetik, (Breslau, 1884Google Scholar); Translated by Austin, J. L. as Foundations of Arithmetic, 2nd edition (New York, 1960) p. 40Google Scholar. The Grundlagen was not Frege's major work, and not his last word on the issues here discussed. When I speak of “Frege's views,” I mean his views in the Grundlagen.

4 Ibid, p. 29.

5 “The Same F,” Philosophical Review (April, 1970). See also Wiggins, David Identity and Spatio-Temporal Continuity (Oxford, 1967Google Scholar); Nelson, Jack “Relative Identity,Nous (September, 1970CrossRefGoogle Scholar); Feldman, Fred“Geach and Relative Identity,” Review of Metaphysics (March, 1969Google Scholar). Geach, replies to Feldman, 's criticisms in “A Reply,” Review of Metaphysics (March, 1969Google Scholar) and seems to include other critics in the scope of this reply in its somewhat altered version in Logic Matters (Berkeley, 1972). In correspondence, Geach has informed me that my criticisms are so completely based on misunderstandings as not to be worth replying to in print. I am unconvinced, however, of any misunderstandings relevant to my criticisms of relative identity. I did, as McIntosh, Jack has observed, take Geach to say “polluted” at one point where he said “pullulated.” An excellent discussion of Geach's views on identity and related matters appears in Dummett, M. Frege (London, 1973), chapter 16Google Scholar. On the whole, Dummett, does an excellent job separating the insightful from the implausible in Geach's writings on these issues. However, Dummett and also Quine, W. V. in Roots of Reference (La Salle, Ill., 1973Google Scholar) maintain that something like Geach's doctrine of relative identity is true “as long as the sides of the identity sentence are demonstrative pronouns.“ (Quine, p. 59; see Dummett, pp. 570ff.) It is true that some sentences, and many others not involving demonstratives, are in some sense incomplete or deficient, and can be completed by inserting an appropriate general term after the word “same.” (See “The Same F,” p. 184, and below, section A.) But the problem is not, as Dummett supposes, that in such sentences “it is correct to say, with Geach, that ‘the same’ is a fragmentary expression.“ (p. 570) For surely the indeterminateness of “This is the same as that” (said, e.g., by Heraclitus’ wife, rather slowly, with two paintings towards the Cayster) has the same source as the indeterminateness of “This was here yesterday“ (said in similar circumstances), in which the allegedly fragmentary expression does not occur. The information required, in both cases, to make good the indeterminancy, is which objects are referred to, not what kind of identity is predicated. (If we suppose, with Quine, that we are at a stage of language at which experience has not yet been clumped into objects, it is surely not the kind of identity that is in question. No identity, we might say, without entity.) In both cases, inserting the appropriate general term after the demonstrative will suffice. In spite of his admirable analysis of a wide variety of examples, I believe Wiggins is also not completely clear about this; his treatment seems to involve retaining different kinds of identity, while not allowing the possibility of “same F, different Gs” emphasized by Geach. See Shoemaker, Sydney “Wiggins on Identity,” Philosophical Review (October, 1970), pp. 533ffGoogle Scholar, and my review of Wiggins's, book, journal of Symbolic Logic, 39, (1970), pp. 447-48.Google Scholar

6 Reference and Generality, pp. 151–52.

7 Ibid., p. 157; Logic Matters, p. 249.

8 Geach, Mental Acts (New York, 1957), p. 69Google Scholar; “Identity,” Review of Metaphysics (1967-68), passim. In his reply Geach does not disavow doctrines (i), (ii), and (iii).

9 Frege, Foundations of Arithmetic, p. 79Google Scholar. Frege's word for the “possibility of correlating one to one the objects which fall under the one concept with those which fall under the other” is Gleichzahlig, which Austin translated “equal”; I prefer “equinumerous.”

10 Ibid.

11 Ibid., p. 75. I say “almost,” for Frege finds a difficulty: we haven't yet explained why England, for example, is not a direction. He solves this problem by defining the direction of line a as the extension of the concept parallel to line a.

12 See “Can the Self Divide?” Journal of Philosophy (September 7, 1972).

13 Frege, Foundations of Arithmetic, p. 28.

14 Ibid.

15 Ibid., p. 29.

16 Ibid.

17 Ibid., p. 59.

18 Geach, Reference and Generality, p. 39.Google Scholar

19 Frege, Foundations of Arithmetic, pp. 67ft.

20 Ibid., p. 61.

21 This paper was completed while I was a Guggenheim Fellow and on sabbatical leave from Stanford University. I would like to thank both institutions for their support. Tyler Burge gave me extensive and helpful comments in an earlier version.