popper-logic-scientific-discovery

136
some structural components of a theory of experience
aroused in us by certain assertions or conjectures. In connection with
some non-numerical statements, the word ‘probable’ may be quite
satisfactorily translated in this way; but an interpretation along these
lines does not seem to me very satisfactory for numerical probability
statements.
A newer variant of the subjective interpretation,* 1 however, deserves
more serious consideration here. This interprets probability statements
not psychologically but logically, as assertions about what may be called
the ‘logical proximity’ 2 of statements. Statements, as we all know, can
stand in various logical relations to one another, like derivability,
incompatibility, or mutual independence; and the logico-subjective
theory, of which Keynes 3 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 interpreted
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 interpretation
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: ‘Probability
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.’