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The Classical to the Classical
Published online by Cambridge University Press: 14 March 2022
Extract
In books on the calculus of probability, there have been many accounts as to what is the meaning of the term “probable.” We can readily divide them into three groups. The first sometimes defines probability in terms of the ratio between the number of cases favorable to an event and the number of equally possible cases. Sometimes probability is defined in some way other than this, but the above formulation, or one similar to it is used to describe the “measure of probability.” This concept is what is called “the classical concept” of probability and was held by the great workers in the field of probability from the beginning of the eighteenth and opening of the nineteenth centuries, including James Bernoulli (1713), Thomas Bayes (1763), and the Marquis de Laplace (1812). The second conceives of probability as a logical relation between hypothesis and evidence. Following this conception, one could identify “probability” with “degree of confirmation.” We will term this the “logical concept.” The third group identified probability with the relative frequency with which a property occurs in a specified class of elements. We will call this the “frequency concept.”
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References
1 Ernest Nagel, Principles of the Theory of Probability, Vol. 1 No. 6, Int. Ency. Unified Science, p. 18.
2 John Maynard Keynes, A Treatise on Probability (London: MacMillan & Co., 1921), p. 3
3 Ibid, p. 4
4 Ibid, p. 38.
5 Ibid, p. 61 passim.
6 Harold Jeffreys, Theory of Probability (Oxford: Clarendon Press, 1939), p. 20.
7 Ibid, p. 16–19.
8 Friedrich Waismann, “Logische Analyse des Wahrscheinlichkeits-begriffs” Erkenntnis, I (1930), p. 235–7.
9 Rudolf Carnap, “On Inductive Logic,” Philosophy of Science, XII (1945), pp. 74–75.
10 Richard von Mises, Probability, Statistics and Truth (New York: The MacMillan Co., 1935), p. 33.
11 Hans Reichenbach, Wahrscheinlichkeitslehre: eine Untersuchung uber die logischen und mathematischen Grundlagen der Wahrscheinlichkeitsrechnung (Leiden: A. W. Sijthoff's UitgeversMaatschappij, 1935), p. 82.
12 Von Mises, op cit., p. 20.
13 Certain problems of science concern more the redefining of concepts already existing than the construction of entirely new ones. For example, in the case of intelligence, we have a conception of what, in a rough way, we mean by intelligence before the construction of any intelligence tests. The purpose of the definition which an intelligence test provides is to replace the rather vague pre-scientific idea with one that is capable of better scientific treatment. In this sense, though, the various intelligence tests represent in a sense different concepts, each of them represents an attempt to substitute a more precise concept for the earlier one. Following Carnap and Husserl, we will call the process just described explication, the original more or less vague concept the explicandum, and the new, more exact concept the explicatum. Thus, in the example referred to above, the concepts defind by the various intelligence tests are various explicata of the same explicandum.
14 Carnap, “The Two Concepts of Probability,” Phil. and Phenom. Res. V (1945) pp. 517, 521–2.
15 Jeffreys, op. cit., p. 35.
16 Keynes, op. cit., p. 92.
17 Johannes von Kries, Principien der Wahrscheinlichkeits-Rechnung. Ein Logische Untersuchung (Freiburg: J. C. B. Mohr, 1886), p. 269.
18 Carnap, Probability, Ms., p. 105.
19 Mises, Wahrscheinlichkeitsrechnung und ihre Anwendung in Statistik und Theoretischen Physik (Leipzig: F. Deutsche, 1931), p. 3.
20 Reichenbach, Experience and Prediction, An Analysis of the Foundations and Structure of Knowledge (Chicago: University of Chicago Press, 1938), p. 301.
21 Ibid, p. 298. Reichenbach in contradistinction to Mises argues that the logical concept is also related to the frequency explicandum. From the above, it would appear that he believes the classical concept to have an even closer relation.
22 James Bernoulli, Ars Conjectandi, p. 34. German translation by R. Haussner (Leipzig: Wilhelm Engelmann, 1899). All page references will be to the original (1713) edition. Properly speaking, “hope” does not always refer to probability. Sometimes it refers to “mathematical expectation,” i.e., the probability times the size of the stake. It is used in this sense in the first three problems of the first part of the Ars Conjectandi. In the remainder of part I where this formulation is used, the values are given as fractional parts of a. If a is interpreted as the size of the stake, this is the mathematical expectation. Bernoulli, however, lists a and 1 as notations for “certainty.” If we consider a in this light, “hope” would refer to probability. In the third part, Bernoulli gives values without any reference to a factor which might refer to the size of the stake. All those examples in which the relationship between the “hope” of various players is asked for would give the same values for either interpretation.
23 Ibid, p. 164.
24 Ibid, p. 167.
25 Ibid, p. 13.
26 Ibid, p. 28.
27 Ibid, p. 182.
28 Ibid, p. 186.
29 Ibid, p. 211.
30 Keynes, op. cit., p. 10.
31 Bernoulli, op. cit., p. 211.
32 Ibid.
33 Ibid.
34 Ibid, p. 212.
35 Ibid, p. 213.
36 Ibid, p. 213.
37 Ibid, p. 214.
38 Ibid, p. 214–15.
39 Ibid, p. 215.
40 Ibid, p. 225.
41 Ibid, p. 226.
42 Like, what one “ought not doubt.” Ibid, p. 211.
43 Thomas Bayes, “An Essay Towards Solving a Problem in the Doctrine of Chances, with Richard Price's forward and Discussion.” Phil. Trans. Roy. Soc. LIII (1763) p. 376. Reprinted in, W. Edwards Deming, editor, Facsimiles of Two Papers by Bayes (Washington: Dept. of Agriculture.)
44 Ibid, p. 391.
45 Ibid, p. 384.
46 Ibid, p. 406.
47 Ibid, p. 406–410.
48 Ibid, p. 376.
49 Ibid.
50 Mises, Probability, Statistics, and Truth, p. 176.
51 Mises attributes to Laplace the fact that the recognition of the “empirical basis of probability” was so slow in developing. Mises, Wahrscheinlichkeitsrechnung. p. 3.
52 Pierre Simon, Marquis de Laplace, Theorie Analytique des Probabilities, original 1812; reprinted in: Vol. 7, Oeuvres de Laplace (Paris: Imprimerie Royale, 1847), p. V. The introduction to this edition of the Theorie Analytique is identical with the Essai Philosophique des Probabilites.
53 Ibid, p. VI.
54 Ibid, p. VIII.
55 Ibid, p. IX-X.
56 Ibid, p. X.
57 Ibid, p. XVIII.
58 John Venn, The Logic of Chance: An Essay on the Foundations and Province of the Theory of Probability with especial reference to its logical bearings and its applications to Moral and Social Science and to Statistics. (London: MacMillan and Co., 1866; 3rd. ed.) p. 191–192.
59 Laplace, op. cit., p. XIX.
60 Ibid, p. XLVIII.
61 Ibid, p. XLVIII-XLIX.
62 Ibid, p. LI.
63 Ibid, p. LXXXVII.
64 Venn, op. cit., p. 394.
65 Mises, Probability, Statistics, and Truth, p. 12.
66 Laplace, op. cit., p. 489.
67 Ibid, p. CXXIII-CXXIV.
68 Ibid, p. CLIV.
69 Ibid, p. CLVI.
70 Ibid, p. CLXIX.
71 Ibid, p. CXI.
72 Laplace, Philosophical Essay on Probabilities, trans. Frederick Wilson Truscott and Frederick Lincoln Emory (New York: John Wiley and Sons, Inc., 1917) p. 196. This sentence is omitted in the introduction to the Theorie Analytique.