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On ordered pairs

Published online by Cambridge University Press:  12 March 2014

W. V. Quine
W. V. Quine*
Affiliation:
Arlington, Virginia
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Extract

Wiener, in 1914, reduced the theory of relations to that of classes by construing relations as classes of ordered pairs and defining the ordered pair in turn on the basis of class theory alone.1 The definition, as improved by Kuratowski,2 identifies the ordered pair x;y with uxyi(ix U iy).

In terms of Russell's theory of types, x;y in the above sense is two types higher than x and y. Even when we abandon Russell's theory of fixed types of objects in favor of a theory of stratified formulae,3 there is still significance in saying that ‘x;y’ is of type 2 relative to ‘x’ and ‘y’—meaning that a test of the stratification of any context involves assigning a higher number by 2 to ‘x;y’ than to ‘x’ and ‘y’.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 10 , Issue 3 , September 1945 , pp. 95 - 96
Copyright
Copyright © Association for Symbolic Logic 1945

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References

1 Wiener, Norbert, A simplification of the logic of relations, Proceedings of the Cambridge Philosophical Society, vol. 17 (1914), pp. 387–390.Google Scholar

2 Kuratowski, Casimir, Sur la notion de l'ordre dans la theorie des ensembles, Fundamenta mathematicae, vol. 2 (1921), pp. 161–171.CrossRefGoogle Scholar

3 This is done in my New foundations for mathematical logic, The American mathematical monthly, vol. 44 (1937), pp. 70-80; also in On the theory of types, this Journal, vol. 3 (1938), esp. pp. 133 ff; also in Hailperin's, TheodoreA set of axioms for logic, this Journal, vol. 9 (1944), pp. 1–19Google Scholar; and, with ill consequences, in my Mathematical logic, pp. 157 ff (see Rosser, Barkley, The Burali-Forti paradox, this Journal, vol. 7 (1942), pp. 1–17Google Scholar, and my Element and number, this Journal, vol. 6 (1941), pp. 135-149).

4 See Goodman, Nelson, Sequences, this Journal, vol. 6 (1941), pp. 150–153Google Scholar, fifth footnote. Note that the concluding twelve words of Goodman's footnote are in error.

5 Goodman, op. cit.

6 This is done in my aforementioned New foundations, p. 71, and in Mathematical logic, pp. 122, 135.

7 See Goodman, op. cit., p. 151.