popper-logic-scientific-discovery

The first problem I have in mind is, again, that of the metric of logical
probability (cf. the second note, point 3), and its relation to the distinction
between what I am going to call primary and secondary probability
statements. My thesis is that on the secondary level, Laplace’s
and Bernoulli’s distribution provide us with a metric.
We may operate with a system S 1 = {a, b, c, a 1, b 1, c 1, . . . } of elements
(in the sense of our system of postulates in appendix *iv). These elements
will give rise to probability statements of the form ‘p(a, b) = r’. We
may call them ‘primary probability statements’. These primary probability
statements may now be considered as the elements of a secondary
system of elements, S 2 = {e, f, g, h, . . . }; so that ‘e’, ‘f’, etc., are now
names of statements of the form ‘p(a, b) = r’.
Now Bernoulli’s theorem tells us, roughly, the following: let h be
‘p(a, b) = r’; then if h is true, it is extremely probable that in a long
sequence of repetitions of the conditions b, the frequency of the occurrence
of a will be equal to r, or very nearly so. Let ‘δ r(a) n’ denote the
statement that a will occur in a long sequence of n repetitions with a
frequency r ± δ. Then Bernoulli’s theorem says that the probability of
δ r(a) n will approach 1, with growing n, given h, i.e. given that p(a, b) = r.
(It also says that this probability will approach 0, given that p(a, b) = s,
wheever s falls outsider r ± δ; which is important for the refutation of
probabilistic hypotheses.)
Now this means that we may write Bernoulli’s theorem in the form
of a (secondary) statement of relative probability about elements g and h
of S 2; that is to say, we can write it in the form
lim p(g, h) = 1
n →∞
appendix *ix 435
where g = δ r(a) n and where h is the information that p(a, b) = r; that is to
say, h is a primary probability statement and g is a primary statement of
relative frequency.
These considerations show that we have to admit, in S 2, frequency
statements such as g, that is to say, δ r(a) n, and probabilistic assumptions,
or hypothetical probabilistic estimates, such as h. It seems for this reason
proper, in the interest of a homogeneous S 2, to identify all the
probability statements which form the elements of S 2, with frequency
statements, or in other words, to assume, for the primary probability