popper-logic-scientific-discovery

probability 179
(2) Requirement of uniqueness: as above.
(2′) Axiom of convergence: for one and the same property of one and
the same chance-like alternative there exists no further middle
frequency apart from its probability p.
From the proposed system of requirements we can deduce
Bernoulli’s theorem, and with it all the theorems of the classical calculus
of probability. This solves our problem: it is now possible to deduce
the law of great numbers within the framework of the frequency theory
without using the axiom of convergence. Moreover, not only does the
formula (1) of section 61 and the verbal formulation of Bernoulli’s
theorem remain unchanged, 5 but the interpretation we have given to
it also remains unchanged: in the case of a chance-like sequence without
a frequency limit it will still be true that almost all sufficiently long
sequences show only small deviations from p. In such sequences (as in
chance-like sequences with frequency limits) segments of any length
behaving quasi-divergently will of course occur at times, i.e. segments
which deviate from p by any amount. But such segments will be comparatively
rare, since they must be compensated for by extremely long
parts of the sequence in which all (or almost all) segments behave
quasi-convergently. As calculation shows, these stretches will have to
be longer by several orders of magnitude, as it were, than the
divergently-behaving segments for which they compensate.* 3
This is also the place to solve the ‘fundamental problem of the theory of
chance’ (as it was called in section 49). The seemingly paradoxical inference
from the unpredictability and irregularity of singular events to the
applicability of the rules of the probability calculus to them is indeed
valid. It is valid provided we can express the irregularity, with a fair
degree of approximation, in terms of the hypothetical assumption that
one only of the recurring frequencies—of the ‘middle frequencies’—
so occurs in any selection according to predecessors that no after-effects
5 The quasi-Bernoulli formulae (symbol: F′) also remain unambiguous for chancelike
sequences (according to the new definition), although ‘F′’ now symbolizes only a
middle frequency.
* 3 I am in full agreement with what follows here, even though any reference to ‘middle
frequencies’ becomes redundant if we adopt the method described in section 57, note
*1, and appendix iv.