popper-logic-scientific-discovery

probability 171
only slightly from this average, whilst the relative frequencies of
smaller sub-segments will deviate further from this average, and the
more often, the shorter we choose them. This fact, this statistically
ascertainable behaviour of finite segments, may be referred to as their
‘quasi-convergent-behaviour’; or as the fact that random sequences are statistically
stable.* 2
Thus Bernoulli’s theorem asserts that the smaller segments of
chance-like sequences often show large fluctuations, whilst the large
segments always behave in a manner suggestive of constancy or convergence;
in short, that we find disorder and randomness in the small,
order and constancy in the great. It is this behaviour to which the
expression ‘the law of great numbers’ refers.
62 BERNOULLI’S THEOREM AND THE INTERPRETATION
OF PROBABILITY STATEMENTS
We have just seen that in the verbal formulation of Bernoulli’s theorem
the word ‘probability’ occurs twice.
The frequency theorist has no difficulty in translating this word, in
both cases, in accordance with its definition: he can give a clear interpretation
of Bernoulli’s formula and the law of great numbers. Can the
adherent of the subjective theory in its logical form do the same?
The subjective theorist who wants to define ‘probability’ as ‘degree
of rational belief’ is perfectly consistent, and within his rights, when he
interprets the words ‘The probability of . . . approaches to I as closely as
we like’ as meaning, ‘It is almost certain 1 that . . .’ But he merely
obscures his difficulties when he continues ‘. . . that the relative frequency
will deviate from its most probable value p by less than a given
amount . . .’, or in the words of Keynes, 2 ‘that the proportion of the
event’s occurrences will diverge from the most probable proportion p by less
* 2 Keynes says of the ‘Law of Great Numbers’ that ‘the “Stability of Statistical
Frequencies” would be a much better name for it’. (Cf. his Treatise, p. 336.)
1 Von Mises also uses the expression ‘almost certain’, but according to him it is of course to
be regarded as defined by ‘having a frequency close to [or equal to] I’.
2 Keynes, A Treatise on Probability, 1921, p. 338. *The preceding passage in quotation marks
had to be inserted here because it re-translates the passage I quoted from the German
edition of Keynes on which my text relied.