K. Popper - Logic scientific discovery
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K. Popper - Logic scientific discovery

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makes it possible to say of a given alternative A that it is
approximately random; and it even allows us to define a degree of approximation. A
more elaborate definition may be based upon the method (of maximizing my
E-function) described under points 8 ff. of my Third Note reprinted in appendix *ix.

some structural components of a theory of experience152

begin with the segment of the first n elements of α. Next comes the
segment of the elements 2 to n + 1 of α. In general, we take as the xth
element of the new sequence the segment consisting of the elements x
to x + n − 1 of α. The new sequence so obtained may be called the
‘sequence of the overlapping n-segments of α’. This name indicates that
any two consecutive elements (i.e. segments) of the new sequence
overlap in such a way that they have n − 1 elements of the original
sequence α in common.

Now we can obtain, by selection, other n-sequences from a sequence
of overlapping segments; especially sequences of adjoining n-segments.

A sequence of adjoining n-segments contains only such n-segments
as immediately follow each other in α without overlapping. It may
begin, for example, with the n-segments of the elements numbered 1
to n, of the original sequence α, followed by that of the elements n + 1
to 2n, 2n + 1 to 3n, and so on. In general, a sequence of adjoining
segments will begin with the kth element of α and its segments will
contain the elements of α numbered k to n + k − 1, n + k to 2n + k − 1,
2n + k to 3n + k − 1, and so on.

In what follows, sequences of overlapping n-segments of α will be
denoted by ‘α(n)’, and sequences of adjoining n-segments by ‘αn’.

Let us now consider the sequences of overlapping segments α(n) a
little more closely. Every element of such a sequence is an n-segment of
α. As a primary property of an element of α(n), we might consider, for
instance, the ordered n-tuple of zeros and ones of which the segment
consists. Or we could, more simply, regard the number of its ones as the
primary property of the element (disregarding the order of the ones and
zeros). If we denote the number of ones by ‘m’ then, clearly, we have
m � n.

Now from every sequence α(n) we again get an alternative if we select
a particular m (m � n), ascribing the property ‘m’ to each element of
the sequence α(n) which has exactly m ones (and therefore n − m zeros)
and the property ‘m¯’ (non-m) to all other elements of α(n). Every
element of α(n) must then have one or the other of these two
properties.

Let us now imagine again that we are given a finite alternative α with
the primary properties ‘1’ and ‘0’. Assume that the frequency of the
ones, αF″ (1), is equal to p, and that the frequency of the zeros, αF″ (0),

probability 153

is equal to q. (We do not assume that the distribution is equal, i.e. that
p = q.)

Now let this alternative α be at least n−1-free (n being an arbitrarily
chosen natural number). We can then ask the following question: What
is the frequency with which the property m occurs in the sequence αn?
Or in other words, what will be the value of α(n)F″(m)?

Without assuming anything beyond the fact that α is at least
n−1-free, we can settle this question1 by elementary arithmetic. The
answer is contained in the following formula, the proof of which will
be found in appendix iii:

α(n)
F″ (m) = nCmp

mqn − m(1)

The right-hand side of the ‘binomial’ formula (1) was given—in
another connection—by Newton. (It is therefore sometimes called
Newton’s formula.) I shall call it the ‘first form of the binomial
formula’.*1

With the derivation of this formula, I now leave the frequency
theory as far as it deals with finite reference-classes. The formula will
provide us with a foundation for our discussion of the axiom of
randomness.

57 INFINITE SEQUENCES. HYPOTHETICAL
ESTIMATES OF FREQUENCY

It is quite easy to extend the results obtained for n-free finite sequences
to infinite n-free sequences which are defined by a generating period (cf.
section 55). An infinite sequence of elements playing the rôle of the
reference-class to which our relative frequencies are related may be

1 The corresponding problem in connection with infinite sequences of adjoining seg-
ments I call ‘Bernoulli’s problem’ (following von Mises, Wahrscheinlichkeitsrechnung, 1931,
p. 128); and in connection with infinite sequences of overlapping segments I call it ‘the
quasi-Bernoulli problem’ (cf. note 1 to section 60). Thus the problem here discussed
would be the quasi-Bernoulli problem for finite sequences.
*1 In the original text, I used the term ‘Newton’s formula’; but since this seems to be
rarely used in English, I decided to translate it by ‘binomial formula’.

some structural components of a theory of experience154

called a ‘reference-sequence’. It more or less corresponds to a
‘collective’ in von Mises’s sense.*1

The concept of n-freedom presupposes that of relative frequency; for
what its definition requires to be insensitive—insensitive to selection
according to certain predecessors—is the relative frequency with which a
property occurs. In our theorems dealing with infinite sequences I shall
employ, but only provisionally (up to section 64), the idea of a limit of
relative frequencies (denoted by F′), to take the place of relative frequency in
finite classes (F″). The use of this concept gives rise to no problem so long
as we confine ourselves to reference-sequences which are constructed
according to some mathematical rule. We can always determine for such
sequences whether the corresponding sequence of relative frequencies
is convergent or not. The idea of a limit of relative frequencies leads to
trouble only in the case of sequences for which no mathematical rule is
given, but only an empirical rule (linking, for example the sequence
with tosses of a coin); for in these cases the concept of limit is not
defined (cf. section 51).

An example of a mathematical rule for constructing a sequence is

*1 I come here to the point where I failed to carry out fully my intuitive programme—
that of analysing randomness as far as it is possible within the region of finite sequences,
and of proceeding to infinite reference sequences (in which we need limits of relative
frequencies) only afterwards, with the aim of obtaining a theory in which the existence
of frequency limits follows from the random character of the sequence. I could have
carried out this programme very easily by constructing, as my next step (finite) shortest n-
free sequences for a growing n, as I did in my old appendix iv. It can then be easily shown
that if, in these shortest sequences, n is allowed to grow without bounds, the sequences
become infinite, and the frequencies turn without further assumption into frequency
limits. (See note *2 to appendix iv, and my new appendix *vi.) All this would have
simplified the next sections which, however, retain their significance. But it would have
solved completely and without further assumption the problems of sections 63 and 64;
for since the existence of limits becomes demonstrable, points of accumulation need no
longer be mentioned.

These improvements, however, remain all within the framework of the pure frequency
theory: except in so far as they define an ideal standard of objective disorder, they
become unnecessary if we adopt a propensity interpretation of the neo-classical
(measure-theoretical) formalism, as explained in sections *53 ff of my Postscript. But even
then it remains necessary to speak of frequency hypotheses—of hypothetical estimates
and their statistical tests; and thus the present section remains relevant, as does much in
the succeeding sections, down to section 64.

probability 155

the following: ‘The nth element of the sequence α shall be 0 if, and
only if, n is divisible by four’. This defines the infinite alternative

1 1 1 0 1 1 1 0 . . .(α)

with the limits of the relative frequencies: αF′ (1) = 3/4; and αF′
(0) = 1/4. Sequences which are defined in this way by means of a
mathematical rule I shall call, for brevity, ‘mathematical sequences’.