popper-logic-scientific-discovery

appendix *ix 431
9. One may see from all this that the testing of a statistical
hypothesis is deductive—as is that of all other hypotheses: first a
test-statement is constructed in such a way that it follows (or almost
follows) from the hypothesis, although its content or testability is high;
and afterwards it is confronted with experience.
It is interesting to note that if e were chosen so as to be a full report
of our observations—say, a full report about a long sequence of tosses,
head, head, tail, . . . , etc., a sequence of one thousand elements—then
e would be useless as evidence for a statistical hypothesis; for any actual
sequence of the length n has the same probability as any other
sequence (given h). Thus we should arrive at the same value for P(e, h),
and thus for E and C—viz. E = C = 0—whether e contains, say only
heads, or whether it contains exactly half heads and half tails. This
shows that we cannot use, as evidence for or against h, our total observational
knowledge, but that we must extract, from our observational
knowledge, such statistical statements as can be compared with statements
which either follow from h, or which have at least a high
probability, given h. Thus if e consists of the complete results of a long
sequence of tosses, then e is, in this form, completely useless as a teststatement
of a statistical hypothesis. But a logically weaker statement of
the average frequency of heads, extracted from the same e, could be used.
For a probabilistic hypothesis can explain only statistically interpreted findings,
and it can therefore be tested and corroborated only by statistical
abstracts—and not, for example, by the ‘total available evidence’, if
this consists of a full observation report; not even if its various
‘First Note’ above. It should be stressed that by formulating a theory in the form ‘(x)Ax’,
we are in general forced to make ‘A’ a highly complex and non-observational predicate.
(See also appendix *vii, especially footnote 1.)
I believe that it is of some interest to mention here that the method developed in the
text allows us to obtain numerical results—that is, numerical degrees of corroboration—in
all cases envisaged either by Laplace or by those modern logicians who introduce artificial
language systems, in the vain hope of obtaining in this way an a priori metric for the
probability of their predicates, believing as they do that this is needed in order to get
numerical results. Yet I get numerical degrees of corroboration in many cases far beyond
those language systems, since measurable predicates do not create any new problem for
our method. (And it is a great advantage that we do not have to introduce a metric for the
logical probability of any of the ‘predicates’ dealt with; see my criticism in point 3 of the
‘Second Note’, above. See also my second Preface, 1958.)