popper-logic-scientific-discovery

436
new appendices
statements e, f, g, h, . . . which form the elements of S 2, some kind of
frequency interpretation of probability. At the same time, we may assume the
logical interpretation of probability for the probability statements of the form
P(g, h) = r
that is to say, for the secondary probability statements which make
assertions about the degree of probability of the primary probability
statements, g and h.
Although we may not have a logical (or absolute) metric of the
primary probability statements, that is to say, although we may have no
idea of the value of p(a) or of p(b), we may have a logical or absolute
metric of the secondary probability statements: this is provided by the
Laplacean distribution, according to which P(g), the absolute probability
of g, that is to say of δ r (a) n, equals 2δ, whether g is empirically
observed, or a hypothesis; so that the typical probabilistic hypothesis, h,
gets P(h) = 0, because h has the form ‘p(a, b) = r’, with δ = 0. Since
Bernoulli’s methods allow us to calculate the value of the relative probability
P(g, h), by pure mathematical analysis, we may consider the
relative probabilities P(g, h) as likewise determined on purely logical
grounds. It therefore seems entirely adequate to adopt, on the
secondary level, the logical interpretation of the formal calculus of
probability.
To sum up, we may consider the methods of Bernoulli and Laplace as
directed towards the establishment of a purely logical metric of probabilities
on the secondary level, independently of whether or not there
exists a logical metric of probabilities on the primary level. Bernoulli’s
methods determine thereby the logical metric of relative probabilities
(secondary likelihood of primary hypotheses, in the main), and
Laplace’s the logical metric of absolute probabilities (of statistical
reports upon samples, in the main).
Their efforts were, no doubt, directed to a large extent towards
establishing a probabilistic theory of induction; they certainly tended
to identify C with p. I need not say that I believe they were mistaken in
this: statistical theories are, like all other theories, hypotheticodeductive.
And statistical hypotheses are tested, like all other hypotheses,
by attempts to falsify them—by attempts to reduce their secondary