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# K. Popper - Logic scientific discovery

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```logico-subjective
theory, of which Keynes3 is the principal exponent, treats the probability
relation as a special kind of logical relationship between two statements.
The two extreme cases of this probability relation are derivability and
contradiction: a statement q ‘gives’,4 it is said, to another statement p
the probability 1 if p follows from q. In case p and q contradict each
other the probability given by q to p is zero. Between these extremes lie
other probability relations which, roughly speaking, may be inter-
preted in the following way: The numerical probability of a statement p
(given q) is the greater the less its content goes beyond what is already
contained in that statement q upon which the probability of p depends
(and which ‘gives’ to p a probability).

The kinship between this and the psychologistic theory may be seen
from the fact that Keynes deﬁnes probability as the ‘degree of rational
belief’. By this he means the amount of trust it is proper to accord to a
statement p in the light of the information or knowledge which we get
from that statement q which ‘gives’ probability to p.

A third interpretation, the objective interpretation, treats every numerical

*1 The reasons why I count the logical interpretation as a variant of the subjective inter-
pretation are more fully discussed in chapter *ii of the Postscript, where the subjective
interpretation is criticized in detail. Cf. also appendix *ix.
2 Waismann, Logische Analyse des Wahrscheinlichkeitsbegriﬀs, Erkenntnis 1, 1930, p. 237: ‘Prob-
ability so deﬁned is then, as it were, a measure of the logical proximity, the deductive
connection between the two statements’. Cf. also Wittgenstein, op. cit., proposition 5.15 ﬀ.
3 J. M. Keynes, A Treatise on Probability, 1921, pp. 95 ﬀ.
4 Wittgenstein, op. cit., proposition 5.152: ‘If p follows from q, the proposition q gives to
the proposition p the probability 1. The certainty of logical conclusion is a limiting case
of probability.’

some structural components of a theory of experience136

probability statement as a statement about the relative frequency with
which an event of a certain kind occurs within a sequence of occurrences.5

According to this interpretation, the statement ‘The probability of
the next throw with this die being a ﬁve equals 1/6’ is not really an
assertion about the next throw; rather, it is an assertion about a whole
class of throws of which the next throw is merely an element. The state-
ment in question says no more than that the relative frequency of ﬁves,
within this class of throws, equals 1/6.

According to this view, numerical probability statements are only
admissible if we can give a frequency interpretation of them. Those prob-
ability statements for which a frequency interpretation cannot be
given, and especially the non-numerical probability statements, are
usually shunned by the frequency theorists.

In the following pages I shall attempt to construct anew the theory
of probability as a (modiﬁed) frequency theory. Thus I declare my faith in
an objective interpretation; chieﬂy because I believe that only an objective
theory can explain the application of the probability calculus within
empirical science. Admittedly, the subjective theory is able to give a
consistent solution to the problem of how to decide probability state-
ments; and it is, in general, faced by fewer logical diﬃculties than is the
objective theory. But its solution is that probability statements are non-
empirical; that they are tautologies. And this solution turns out to be
utterly unacceptable when we remember the use which physics makes
of the theory of probability. (I reject that variant of the subjective
theory which holds that objective frequency statements should be
derived from subjective assumptions—perhaps using Bernoulli’s the-
orem as a ‘bridge’:6 I regard this programme for logical reasons as
unrealizable.)

5 For the older frequency theory cf. the critique of Keynes, op. cit., pp. 95 ﬀ., where special
reference is made to Venn’s The Logic of Chance. For Whitehead’s view cf. section 80 (note
2). Chief representatives of the new frequency theory are: R. von Mises (cf. note 1 to
section 50), Dörge, Kamke, Reichenbach and Tornier. *A new objective interpretation,
very closely related to the frequency theory, but diﬀering from it even in its mathemat-
ical formalism, is the propensity interpretation, introduced in sections *53 ﬀ. of my Postscript.
6 Keynes’s greatest error; cf. section 62, below, especially note 3. *I have not changed my
view on this point even though I now believe that Bernoulli’s theorem may serve as a
‘bridge’ within an objective theory—as a bridge from propensities to statistics. See also
appendix *ix and sections *55 to *57 of my Postscript.

probability 137

49 THE FUNDAMENTAL PROBLEM OF
THE THEORY OF CHANCE

The most important application of the theory of probability is to what
we may call ‘chance-like’ or ‘random’ events, or occurrences. These
seem to be characterized by a peculiar kind of incalculability which
makes one disposed to believe—after many unsuccessful attempts—
that all known rational methods of prediction must fail in their case.
We have, as it were, the feeling that not a scientist but only a prophet
could predict them. And yet, it is just this incalculability that makes us
conclude that the calculus of probability can be applied to these
events.

This somewhat paradoxical conclusion from incalculability to cal-
culability (i.e. to the applicability of a certain calculus) ceases, it is true,
to be paradoxical if we accept the subjective theory. But this way of
avoiding the paradox is extremely unsatisfactory. For it entails the view
that the probability calculus is not a method of calculating predictions,
in contradistinction to all the other methods of empirical science. It is,
according to the subjective theory, merely a method for carrying out
logical transformations of what we already know; or rather what we do
not know; for it is just when we lack knowledge that we carry out these
transformations.1 This conception dissolves the paradox indeed, but it
does not explain how a statement of ignorance, interpreted as a frequency statement, can be
empirically tested and corroborated. Yet this is precisely our problem. How can
we explain the fact that from incalculability—that is, from ignorance—
we may draw conclusions which we can interpret as statements about
empirical frequencies, and which we then ﬁnd brilliantly corroborated
in practice?

Even the frequency theory has not up to now been able to give a
satisfactory solution of this problem—the fundamental problem of the theory
of chance, as I shall call it. It will be shown in section 67 that this problem
is connected with the ‘axiom of convergence’ which is an integral part

1 Waismann, Erkenntnis 1, 1930, p. 238, says: ‘There is no other reason for introducing the
concept of probability than the incompleteness of our knowledge.’ A similar view is held
by C. Stumpf (Sitzungsberichte der Bayerischen Akademie der Wissenschaften, phil.-hist. Klasse, 1892,
p. 41). *I believe that this widely held view is responsible for the worst confusions. This
will be shown in detail in my Postscript, chapters *ii and *v.

some structural components of a theory of experience138

of the theory in its present form. But it is possible to ﬁnd a satisfactory
solution within the framework of the frequency theory, after this
axiom has been eliminated. It will be found by analysing the assump-
tions which allow us to argue from the irregular succession of single
occurrences to the regularity or stability of their frequencies.

50 THE FREQUENCY THEORY OF VON MISES

A frequency theory which provides a foundation for all the principal
theorems of the calculus of probability was ﬁrst proposed by Richard
von Mises.1 His fundamental ideas are as follows.

The calculus```