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Is ‘Congruence’ a Peculiar Predicate?

Published online by Cambridge University Press:  28 February 2022

Gerald J. Massey
Gerald J. Massey*
Affiliation:
University of Pittsburgh
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Extract

The logical status of spatial and temporal congruence has been much debated. Recently the topic of debate has shifted from the alleged conventionality of congruence to the alleged eccentricity of congruence predicates. For example, espousing a point of view suggested to him by Nuel Belnap, Adolf Grünbaum has recently remarked that the predicate ‘x is (spatially) congruent to y’ deviates from the ‘classical account’ of the interrelations between the intensional and extensional components of the meaning of a predicate. By the classical account (hereafter CA), Grünbaum means the rather common view that the extension of a predicate is completely determined by its intension. Contrary to CA, Grünbaum maintains that “the fact that ‘being spatially congruent’ means sustaining the relation of spatial equality does not suffice at all to determine its extension uniquely in the class of spatial intervals.”

Type
Contributed Papers
Information
PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association , Volume 1970 , 1970 , pp. 606 - 615
Copyright
Copyright © Philosophy of Science Association 1970

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References

Notes

1 Grünbaum, A., ‘Reply to Hilary Putnam’s “An Examination of Grünbaum's Philosophy of Geometry’”, Boston Studies in the Philosophy of Science Vol. V (1968) p. 45Google Scholar.

2 Ibid., p. 82.

3 Massey, G. J., ‘Toward a Clarification of Grünbaum’s Conception of an Intrinsic Metric’, Philosophy of Science 36 (1969) 341CrossRefGoogle Scholar.

4 Concerning the thesis of geochronometric conventionalism see Grünbaum, op. cit.; Grünbaum, , Philosophical Problems of Space and Time, Knopf, New York, 1963Google Scholar, Chapters 1-6; Grünbaum, , Geometry and Chronometry in Philosophical Perspective, University of Minnesota Press, Minneapolis, 1968Google Scholar; Grünbaum, , ‘Space, Time, and Falsifiability’, Philosophy of Science 37 (1970)Google Scholar, Part A.

5 See Putnam, H., ‘An Examination of Grünbaum’s Philosophy of Geometry’, in Philosophy of Science, The Delaware Seminar 2 (1963) 205-55Google Scholar. and Grünbaum's Reply (pp. cit.).

6 R. G. Swinburne, Review of Grünbaum's Geometry and Chronometry in Philosophical Perspective, forthcoming in British Journal of the Philosophy of Science.

7 Butrick, R., ‘Quine on the “is“ in “is analytic’”, Mind 74 (1970) 262Google Scholar.

8 Ibid.

9 Concerning tensed predicates, see Massey, G. J., Understanding Symbolic Logic, Harper and Row, New York, 1970, pp. 265-6 and pp. 404ffGoogle Scholar.

10 Butrick, op. cit., p. 263.

11 Two intervals are quasi-disjoint if and only if neither is included in the other. (This statement is not entirely accurate; for an unexceptionable definition of quasi-disjointedness, see Grünbaum, ‘Space, Time, and Falsifiability’, op. cit., p. 548.

12 In his recent ‘Space, Time, and Falsifiability’, op. cit., pp. 538ff., Grünbaum points out that in the primary sense of ‘nontrivial’ that he prefers, there are in discrete space (without extreme elements) two nontrivial congruence classes generated by dyadic properties intrinsic to the intervals of the space. One of these is the cardinality-based congruence class; the other is the partition effected by the identity relation among intervals. Any metric such that equality of measure relative to that metric generates either of these congruence classes is nontrivial and intrinsic to the intervals. My ‘progression metric’ (discussed in Grünbaum, ibid., pp. 539ff) generates the identity-based congruence class, so it is both nontrivial, intrinsic to the intervals, and discrepant with the cardinality-based metrics. But Grünbaum observes that in a stricter sense of ‘nontrivial’, my progression metric must be judged trivial because it renders no two distinct intervals congruent. Because the rudimentary or unadorned intension of ‘x is congruent to y’ seems to have nontriviality, in the stricter sense, built into it, I will use ‘nontrivial’ throughout in the stricter of Grünbaum’s two senses.

13 See especially Grünbaum, ibid., pp. 538ff.

14 See especially Grünbaum, , ‘Simultaneity by Slow Clock Transport in the Special Theory of Relativity’, Philosophy of Science, 36 (1969) 5–43CrossRefGoogle Scholar.

15 For details, see ibid.

16 For details, see ibid.

17 In preparing this paper I have profited greatly from discussions with my colleagues Nuel Belnap, Joseph Camp, and Richard Gale. I am especially indebted to Adolf Grünbaum for detailed criticisms and suggestions concerning a number of specific points.