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

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must not be subject to any mathematical rule or
law. For the mathematical limit is nothing but a characteristic property of the
mathematical rule or law by which the sequence is determined. It is merely a prop-
erty of this rule or law if, for any chosen fraction arbitrarily close to
zero, there is an element in the sequence such that all elements follow-
ing it deviate by less than that fraction from some definite value—
which is then called their limit.

To meet such objections it has been proposed to refrain from com-
bining the axiom of convergence with that of randomness, and to
postulate only convergence, i.e. the existence of a limit. As to the axiom
of randomness, the proposal was either to abandon it altogether
(Kamke) or to replace it by a weaker requirement (Reichenbach).
These suggestions presuppose that it is the axiom of randomness
which is the cause of the trouble.

In contrast to these views, I am inclined to blame the axiom of
convergence no less than the axiom of randomness. Thus I think that
there are two tasks to be performed: the improvement of the 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″