popper-logic-scientific-discovery

corroboration, or how a theory stands up to tests 261
problem is a consequence of the conception of science for which I have
been arguing.* 6
81 INDUCTIVE LOGIC AND PROBABILITY LOGIC
The probability of hypotheses cannot be reduced to the probability of
events. This is the conclusion which emerges from the examination
carried out in the previous section. But might not a different approach
lead to a satisfactory definition of the idea of a probability of hypotheses?
I do not believe that it is possible to construct a concept of the
probability of hypotheses which may be interpreted as expressing a
‘degree of validity’ of the hypothesis, in analogy to the concepts ‘true’
and ‘false’ (and which, in addition, is sufficiently closely related to the
concept ‘objective probability’, i.e. to relative frequency, to justify the
use of the word ‘probability’). 1 Nevertheless, I will now, for the sake of
argument, adopt the supposition that such a concept has in fact been
successfully constructed, in order to raise the question: how would this
affect the problem of induction?
Let us suppose that a certain hypothesis—say Schrödinger’s theory
—is recognized as ‘probable’ in some definite sense; either as ‘probable
to this or that numerical degree’, or merely as ‘probable’, without
specification of a degree. The statement that describes Schrödinger’s
theory as ‘probable’ we may call its appraisal.
* 6 The last two paragraphs were provoked by the ‘naturalistic’ approach sometimes
adopted by Reichenbach, Neurath, and others; cf. section 10, above.
1 (Added while the book was in proof.) It is conceivable that for estimating degrees of
corroboration, one might find a formal system showing some limited formal analogies
with the calculus of probability (e.g. with Bayes’s theorem), without however having
anything in common with the frequency theory. I am indebted to Dr. J. Hosiasson for
suggesting this possibility to me. I am satisfied, however, that it is quite impossible to
tackle the problem of induction by such methods with any hope of success. *See also note 3 to
section *57 of my Postscript.
* Since 1938, I have upheld the view that ‘to justify the use of the word probability’, as
my text puts it, we should have to show that the axioms of the formal calculus are
satisfied. (Cf. appendices *ii to *v, and especially section *28 of my Postscript.) This would
of course include the satisfaction of Bayes’s theorem. As to the formal analogies between
Bayes’s theorem on probability and certain theorems on degree of corroboration, see appendix
*ix, point 9 (vii) of the first note, and points (12) and (13) of section *32 of my Postscript.