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# K. Popper - Logic scientific discovery

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```axiom of
randomness—mainly a mathematical problem; and the complete elim-
ination of the axiom of convergence—a matter of particular concern
for the epistemologist.2 (Cf. section 66.)

In what follows I propose to deal ﬁrst with the mathematical, and
afterwards with the epistemological question.

The ﬁrst of these two tasks, the reconstruction of the mathematical
theory,3 has as its main aim the derivation of Bernoulli’s theorem—
the ﬁrst ‘Law of Great Numbers’—from a modiﬁed axiom of randomness;

1 Waismann, Erkenntnis 1, 1930, p. 232.
2 This concern is expressed by Schlick, Naturwissenschaften 19, 1931. *I still believe that
these two tasks are important. Although I almost succeeded in the book in achieving
what I set out to do, the two tasks were satisfactorily completed only in the new
appendix *vi.
3 A full account of the mathematical construction will be published separately. *Cf. the
new appendix *vi.

some structural components of a theory of experience142

modiﬁed, namely, so as to demand no more than is needed to achieve
this aim. Or to be more precise, my aim is the derivation of the
Binomial Formula (sometimes called ‘Newton’s Formula’), in what I
call its ‘third form’. For from this formula, Bernoulli’s theorem and the
other limit theorems of probability theory can be obtained in the usual
way.

My plan is to work out ﬁrst a frequency theory for ﬁnite classes, and to
develop the theory, within this frame, as far as possible—that is, up to
the derivation of the (‘ﬁrst’) Binomial Formula. This frequency theory
for ﬁnite classes turns out to be a quite elementary part of the theory of
classes. It will be developed merely in order to obtain a basis for
discussing the axiom of randomness.

Next I shall proceed to inﬁnite sequences, i.e. to sequences of events
which can be continued indeﬁnitely, by the old method of introducing
an axiom of convergence, since we need something like it for our
discussion of the axiom of randomness. And after deriving and exam-
ining Bernoulli’s theorem, I shall consider how the axiom of convergence might
be eliminated, and what sort of axiomatic system we should be left with as
the result.

In the course of the mathematical derivation I shall use three diﬀerent
frequency symbols: F″ is to symbolize relative frequency in ﬁnite
classes; F′ is to symbolize the limit of the relative frequencies of an
inﬁnite frequency-sequence; and ﬁnally F, is to symbolize objective
probability, i.e. relative frequency in an ‘irregular’ or ‘random’ or
‘chance-like’ sequence.

52 RELATIVE FREQUENCY WITHIN A FINITE CLASS

Let us consider a class α of a ﬁnite number of occurrences, for example
the class of throws made yesterday with this particular die. This class α,
which is assumed to be non-empty, serves, as it were, as a frame of
reference, and will be called a (ﬁnite) reference-class. The number of
elements belonging to α, i.e. its cardinal number, is denoted by ‘N(α)’,
to be read ‘the number of α’. Now let there be another class, β, which
may be ﬁnite or not. We will call β our property-class: it may be, for
example, the class of all throws which show a ﬁve, or (as we shall say)
which have the property ﬁve.

probability 143

The class of those elements which belong to both α and β, for
example the class of throws made yesterday with this particular die and
having the property ﬁve, is called the product-class of α and β, and is
denoted by ‘α.β’, to be read ‘α and β’. Since α.β is a subclass of α, it can
at most contain a ﬁnite number of elements (it may be empty). The
number of elements in α.β is denoted by ‘N(α.β)’.

Whilst we symbolize (ﬁnite) numbers of elements by N, the relative
frequencies are symbolized by F″. For example, ‘the relative frequency
of the property β within the ﬁnite reference-class α’ is written
‘αF″(β)’, which may be read ‘the α-frequency of β’. We can now
deﬁne

αF″(β) =
N(α.β)

N(α)
(Deﬁnition 1)

In terms of our example this would mean: ‘The relative frequency of
ﬁves among yesterday’s throws with this die is, by deﬁnition, equal to
the quotient obtained by dividing the number of ﬁves, thrown yester-
day with this die, by the total number of yesterday’s throws with this
die.’*1

From this rather trivial deﬁnition, the theorems of the calculus of
frequency in ﬁnite classes can very easily be derived (more especially, the
general multiplication theorem; the theorem of addition; and the the-
orems of division, i.e. Bayes’s rules. Cf. appendix ii). Of the theorems of
this calculus of frequency, and of the calculus of probability in general,
it is characteristic that cardinal numbers (N-numbers) never appear in
them, but only relative frequencies, i.e. ratios, or F-numbers. The N-
numbers only occur in the proofs of a few fundamental theorems
which are directly deduced from the deﬁnition; but they do not occur
in the theorems themselves.*2

*1 Deﬁnition 1 is of course related to the classical deﬁnition of probability as the ratio of
the favourable cases to the equally possible cases; but it should be clearly distinguished
from the latter deﬁnition: there is no assumption involved here that the elements of α are
‘equally possible’.
*2 By selecting a set of F-formulae from which the other F-formulae can be derived, we
obtain a formal axiom system for probability; compare the appendices ii, *ii, *iv, and *v.

some structural components of a theory of experience144

How this is to be understood will be shown here with the help of
one very simple example. (Further examples will be found in appendix
ii.) Let us denote the class of all elements which do not belong to β by
‘β
-
’ (read: ‘the complement of β’ or simply: ‘non-β’). Then we may

write

αF″(β) + αF″(β
-
) = 1

While this theorem only contains F-numbers, its proof makes use of N-
numbers. For the theorem follows from the deﬁnition (1) with the
help of a simple theorem from the calculus of classes which asserts that
N(α.β) + N(α.β

-
) = N(α).

53 SELECTION, INDEPENDENCE,
INSENSITIVENESS, IRRELEVANCE

Among the operations which can be performed with relative frequen-
cies in ﬁnite classes, the operation of selection1 is of special importance
for what follows.

Let a ﬁnite reference-class α be given, for example the class of but-
tons in a box, and two property-classes, β (say, the red buttons) and γ
(say, the large buttons). We may now take the product-class α.β as a new
reference-class, and raise the question of the value of α.βF″ (γ), i.e. of the
frequency of γ within the new reference-class.2 The new reference-
class α.β may be called ‘the result of selecting β-elements from α’, or
the ‘selection from α according to the property β’; for we may think of
it as being obtained by selecting from α all those elements (buttons)
which have the property β (red).

Now it is just possible that γ may occur in the new reference-class,
α.β, with the same relative frequency as in the original reference-class
α; i.e. it may be true that

α.βF″ (γ) = αF″ (γ)

1 Von Mises’s term is ‘choice’ (‘Auswahl’).
2 The answer to this question is given by the general division theorem (cf. appendix ii).

probability 145

In this case we say (following Hausdorﬀ3) that the properties β and γ
are ‘mutually independent, within the reference-class α’. The relation of
independence is a three-termed relation and is symmetrical in the
properties β and γ.4 If two properties β and γ are (mutually)
independent within a reference-class α we can also say that the prop-
erty γ is, within α, insensitive to the selection of β-elements; or perhaps
that the reference-class α is, with respect to this property γ, insensitive
to a selection according to the property β.

The mutual independence, or insensitiveness, of β and γ within α
could also—from the point of view of the subjective theory—be inter-
preted as follows: If we are informed that a particular element of```