¹ Kemble, E. C., Phys. Rev. 56, 1013-23 (1939). Cf. especially pp. 1018-19.Google Scholar

² Cf., e.g., Bridgman, P. W., Phil. Sci. 5, 114 (1938).CrossRefGoogle Scholar

³ Cf. Kemble, E. C., Fundamental Principles of Quantum Mechanics, New York, 1937, pp. 53-55.Google Scholar

⁴ The term “likelihood” is used in its nontechnical sense.Google Scholar

⁵ Jeffreys, H., Theory of Probability, Oxford, 1939, p. 8.Google Scholar

⁶ In defining probabilities as numbers we go a short distance with the advocates of the frequency definition!Google Scholar

⁷ Keynes has amply set forth the related absurdity of supposing that a definite numerical probability can be assigned to the logical relation between any set of premises and any possible conclusion. Cf. his Treatise on Probability, London, 1921, Chapter III.Google Scholar

⁸ Cf. Born, Max, Proc. Roy. Soc. Edinburgh, 57, 1, (1936).CrossRefGoogle Scholar

⁹ Cf. Mach, E., Principien der Wärmelehre, 4th (posthumous) ed., Leipsic, 1922, p. 46. The doctrine that science is an analysis of experience is central in the positivistic philosophy of Mach. This is to my mind the quintessence of operationalism. I hold no brief here for positivism as a complete philosophy, but because I believe that natural science must avoid hidden assumptions and obscurantism as it would the pestilence, I am convinced that this kind of activity should be carried out on a positivistic basis.Google Scholar

¹⁰ Beliefs do not always relate to the future and are to a large extent staged in the external world of common sense. Nevertheless they can be reduced in large part to operational terms and are then equivalent to expectancies regarding what would happen if certain experiments were performed. I shall not concern myself with probabilities originating in beliefs not reducible to operational terms.Google Scholar

¹¹ The view of Keynes that probability should be defined as an objective logical relation between propositions is closely related to my conclusion that probability is best regarded as a number relating given data with a suggested event and computed by rules designed to secure that reliability which we associate with the adjective “objective”. The difference is primarily one of philosophic bias and reflects in part my skepticism of the traditional view of logic and mathematics as repositories of external objective “truth”.Google Scholar

¹² As an example consider the case of a gas at a very low pressure through which an electrical discharge is passing. In such a case the free electrons can have one effective temperature, the translational motion of the molecules a second, and the rotational motion of the molecules a third. For the system as a whole there is no definite temperature.Google Scholar

¹³ This fact was first impressed upon me by my colleague, Prof. Bridgman. 14 Cf. D. L. Miller, Philosophy of Science, 7, 26 (1940).Google Scholar

¹⁵ Here I use the word “implies” as quivalent to “creates a practical certainty that the event in question will occur”. It does not signify a logical connection.Google Scholar

¹⁶ Cf. Introduction to Mathematical Probability, J. V. Uspensky, New York, 1937, Chapter 2.Google Scholar

¹⁷ Peirce, C. S., Theory of Probable Inference, pp. 172-3.Google Scholar

¹⁸ Keynes, J. M., A Treatise on Probability, 1921, pp. 56-7.Google Scholar

¹⁹ Jeffreys, H., l.c., p. 100.Google Scholar

²⁰ See, for example, Tolman, R. C., The Principles of Statistical Mechanics, Oxford, 1938, pp. 59-63.Google Scholar

²¹ Cf. von Neumann, J.Zeits. f. Physik 57 30-70 (1929); Kemble, E. C., Phys. Rev. 56, 1146-64 (1939).Google Scholar

²² I take the nomenclature from Jeffreys, l.c., p. 41.Google Scholar

²³ I mean tests to locate the center of gravity and fix the ellipsoid of inertia.Google Scholar

²⁴ At this point we touch on the great problem of statistical control. Cf. e.g., W. A. Shewhart, Statistical Method from the Viewpoint of Quality Control, Washington, 1939.Google Scholar

²⁵ Broad, C. D., Mind, 27, 389 404 (1918); Jeffreys, l.c., p. 105. Jeffreys also gives a formula for the probability that w‘ of the next n‘ draws will be white, a result again independent of N.CrossRefGoogle Scholar

²⁶ Jeffreys, H., l.c., p. 113. The notation of Eq. (3) comes from Jeffreys.Google Scholar