popper-logic-scientific-discovery

y mixing water and alcohol’, illustrates one kind of statement which,
suitably interpreted, might perhaps be transformed into a numerical
probability statement. (For example, ‘The probability of obtaining . . .
is very near to 1’.) A very different kind of non-numerical probability
statement would be, for instance, ‘The discovery of a physical effect
which contradicts the quantum theory is highly improbable’; a statement
which, I believe, cannot be transformed into a numerical probability
statement, or put on a par with one, without distorting its
meaning. I shall deal first with numerical probability statements; nonnumerical
ones, which I think less important, will be considered
afterwards.
In connection with every numerical probability statement, the question
arises: ‘How are we to interpret a statement of this kind and, in
particular, the numerical assertion it makes?’
48 SUBJECTIVE AND OBJECTIVE INTERPRETATIONS
probability 135
The classical (Laplacean) theory of probability defines the numerical
value of a probability as the quotient obtained by dividing the number
of favourable cases by the number of equally possible cases. We might
disregard the logical objections which have been raised against this
definition, 1 such as that ‘equally possible’ is only another expression
for ‘equally probable’. But even then we could hardly accept this definition
as providing an unambiguously applicable interpretation. For
there are latent in it several different interpretations which I will
classify as subjective and objective.
A subjective interpretation of probability theory is suggested by the frequent
use of expressions with a psychological flavour, like ‘mathematical
expectation’ or, say, ‘normal law of error’, etc.; in its original form it is
psychologistic. It treats the degree of probability as a measure of the feelings
of certainty or uncertainty, of belief or doubt, which may be
1 Cf. for example von Mises, Wahrscheinlichkeit, Statistik und Wahrheit, 1928, pp. 62 ff.; 2nd
edition, 1936, pp. 84 ff.; English translation by J. Neyman, D. Sholl, and E. Rabinowitsch,
Probability, Statistics and Truth, 1939, pp. 98 ff. *Although the classical definition is often
called ‘Laplacean’ (also in this book), it is at least as old as De Moivre’s Doctrine of Chances,
1718. For an early objection against the phrase ‘equally possible’, see C. S. Peirce, Collected
Papers 2, 1932 (first published 1878), p. 417, para. 2, 673.