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On the Confirmation of Laws

Published online by Cambridge University Press:  14 March 2022

Jared Darlington
Jared Darlington*
Affiliation:
Connecticut College
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Abstract

The author discusses some difficulties involved in the application of “degree of confirmation” to the confirmation of lawlike-statements. An alternative analysis is proposed, which is based on interval estimation. It is argued that this analysis is superior to the criticized method, in that it is better able to show how instantial confirmations are inductively relevant to a law, and in that it requires fewer undesirable extra-logical assumptions.

Type
Research Article
Information
Philosophy of Science , Volume 26 , Issue 1 , January 1959 , pp. 14 - 24
Copyright
Copyright © 1959 by Philosophy of Science Association

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Footnotes

1

This paper developed out of the author's doctoral thesis, A Priori Probability and Inductive Estimation, Yale University, 1957. He is indebted to Professor Arthur Pap for his careful reading of both the article and the dissertation.

References

2 C. G. Hempel: “The function of general laws in history.” Journal of Philosophy, 39, 1942.

C. G. Hempel and P. Oppenheim: “Studies in the logic of explanation.” Philosophy of Science, 15, 1948, pp. 135-175.

3 See R. Carnap: Logical Foundations of Probability. Chicago, 1950, University of Chicago Press, Sect. 60.

4 R. Carnap: op. cit., p. 571.

5 J. M. Keynes: A Treatise on Probability. London, 1921, Macmillan and Co., Ltd., pp. 235-237.

6 J. M. Keynes: op. cit., p. 258.

7 R. Carnap: op. cit.

8 E. Nagel: Principles of the Theory of Probability. Chicago, 1939, University of Chicago Press.

9 J. O. Wisdom: Foundations of Inference in Natural Science. London, 1952, Methuen and Co., Ltd.

10 R. Carnap: op. cit. Sect. 100.

11 A. Wald: Statistical Decision Functions. New York, 1950, John Wiley and Sons, Inc. Sect. 1.4.

12 R. A. Fisher: “On the mathematical foundations of theoretical statistics.” Phil. Trans. Royal Soc. Series A, 222, pp. 309-368, 1922.

“Theory of statistical estimation.” Proc. Cambridge Phil. Soc, 22, 1923-25, pp. 700-725. 13 A. Wald: op. cit.

14 J. Neyman: Lectures and Conferences on Mathematical Statistics and Probability. Washington, 1952, Graduate School, U. S. Dept. of Agriculture.

15 R. B. Braithwaite: Scientific Explanation. Cambridge, 1953, University Press.15

16 W. Kneale: Probability and Induction. Oxford, 1949, The Clarendon Press.

17 S. S. Wilks: Elementary Statistical Analysis. Princeton, 1948, Princeton University Press, esp. Chapter 10.

18 D. Williams: The Ground of Induction. Cambridge, Mass., 1947, Harvard University Press.

“On the direct probability of inductions.” Mind, LXII, N. S., No. 248, October, 1953.

19 W. Kneale: op, cit. Sect. 16.

20 R. B. Braithwaite: op. cit. pp. 9-10.

21 R. B. Braithwaite: op. cit. p. 116.

22 See R. Carnap: op. cit. p. 499.

23 See R. Carnap: op. cit. Sect. 96.

24 See R. Carnap: op. cit. Sect. 90.

25 R. Carnap: op. cit. pp. 500-501, 505-506.

26 See R. Carnap: op. cit. p. 505.

In this theorem, and in the remainder of the paper, we are following Carnap (op. cit., pp. 493-494) in not requiring that the sample be known to be “random,” i.e. such that each member of the population has an equal probability of being selected. Carnap, following Keynes (op. cit.), points out that the usual definition of “randomness” in terms of “all individuals being selected with equal frequencies in the long run” renders it impossible in most cases ever to know whether a given sample actually is random.

Another point that should be made in connection with this theorem is that it is based on the normal approximation to the binomial distribution, while the situation with which we shall be dealing is one in which the Poisson approximation would also be applicable. Our reasons for preferring the normal distribution in this paper are that interval theorems based on the normal distribution are more familiar than those based on the Poisson, and that the procedure we adopt enables us to obtain the simple formulas in VI.

27 S. S. Wilks: op. cit. Chapter 10.

28 D. Williams: references cited.

29 S. S. Wilks: op. cit. Chapter 10.

30 S. S. Wilks: op. cit. p. 200.

31 See reference number 5.

32 See R. Carnap: op. cit. p. 505.

33 This matter is discussed in a clear way by M. R. Cohen and E. Nagel: Introduction to Logic and Scientific Method. New York, 1934, Harcourt, Brace, and Co., Inc., Ch. 14, Sect. 2. They point out that repetition of a large number of confirming instances of a universal proposition is important only in the case of “... universal propositions that in our knowledge are relatively isolated from one another.”

34 R. Carnap: op. cit. p. 572.

35 Ibid.

36 K. Popper: “The demarcation between science and metaphysics.” p. 42. Library of Living Philosophers, P. A. Schilpp, ed. Forthcoming volume on Carnap.

H. Putnam: “Degree of confirmation and inductive logic.” Forthcoming Schilpp volume on Carnap.

37 R. Carnap: op. cit. p. 497.

38 R. Carnap: The Continuum of Inductive Methods. Chicago, 1952, University of Chicago Press, pp. 75-76.