popper-logic-scientific-discovery

344
new appendices
The second example is a generalization of the first, showing that the idea
underlying the first example can be extended to a number of elements
exceeding any chosen number, provided these elements form a
Boolean Algebra, which means that the number of elements has to be
equal to 2 n . Here n may be taken to be the number of the smallest
exclusive areas or classes into which some universe of discourse is
divided. We can freely correlate with each of these classes some positive
fraction, 0 � r � 1, as its absolute probability, taking care that
their sum equals 1. With any of the Boolean sums, we correlate the
arithmetical sum of their probabilities, and with any Boolean complement,
the arithmetical complement with respect to 1. We may assign to
one or several of the smallest (non-zero) exclusive areas or classes the
probability 0. If b is such an area or class, we put p(a, b) = 0 in case
ab = 0; otherwise p(a, b) = 1. We also put p(a, 0) = 1; and in all other
cases, we put p(a, b) = p(ab)/p(b).
In order to show that our system is consistent even under the
assumption that S is denumerably infinite, the following interpretation
may be chosen. (It is of interest because of its connection with the
frequency interpretation.) Let S be the class of rational fractions in
diadic representation, so that, if a is an element of S, we may write a as a
sequence, a = a 1, a 2 ..., where a i is either 0 or 1. We interpret ab as the
sequence ab = a 1b 1, a 2b 2, . . . so that (ab) i = a ib i and ā as the sequence
ā = 1−a 1, 1 − a 2, ..., so that ā i = 1 − a i. In order to define p(a, b), we
introduce an auxiliary expression, A n, defined as follows:
so that we have
A n = � n
(AB) n = � n
a i
a ib i;
moreover, we define an auxiliary function, q:
q(an, bn) = 1 whenever Bn = 0
q(an, bn) = (AB) n/Bn, whenever Bn ≠ 0.