popper-logic-scientific-discovery

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some structural components of a theory of experience
and which may be called the ‘event-sequence’ or the ‘propertysequence’.
As a simple example of a collective I choose what we may call an
‘alternative’. By this term we denote a sequence of events supposed to
have two properties only—such as a sequence of tosses of a coin. The one
property (heads) will be denoted by ‘1’, and the other (tails) by ‘0’.
A sequence of events (or sequence of properties) may then be
represented as follows:
(A)
0 1 1 0 0 0 1 1 1 0 1 0 1 0 ....
Corresponding to this ‘alternative’—or, more precisely, correlated
with the property ‘1’ of this alternative—is the following sequence of
relative frequencies, or ‘frequency-sequence’: 2
(A′)
O 1
2
2
3
2
4
2
5
2
6
3
7
4
8
5
9
5
10
6
11
6
12
7
13
7
14 ....
Now the axiom of convergence (or ‘limit-axiom’) postulates that, as the
event-sequence becomes longer and longer, the frequency-sequence
shall tend towards a definite limit. This axiom is used by von Mises
because we have to make sure of one fixed frequency value with which we
can work (even though the actual frequencies have fluctuating values).
In any collective there are at least two properties; and if we are given
the limits of the frequencies corresponding to all the properties of a
collective, then we are given what is called its ‘distribution’.
The axiom of randomness or, as it is sometimes called, ‘the principle of the
excluded gambling system’, is designed to give mathematical expression
to the chance-like character of the sequence. Clearly, a gambler
would be able to improve his chances by the use of a gambling system
2 We can correlate with every sequence of properties as many distinct sequences of
relative frequencies as there are properties defined in the sequence. Thus in the case of an
alternative there will be two distinct sequences. Yet these two sequences are derivable
from one another, since they are complementary (corresponding terms add up to 1). For
this reason I shall, for brevity, refer to ‘the (one) sequence of relative frequencies correlated
with the alternative (α)’, by which I shall always mean the sequence of frequencies
correlated with the property ‘1’ of this alternative (α).