popper-logic-scientific-discovery

and (see page 346)
(3)
a = lim a
n →∞
n
p(a) = limp(a
n →∞
n )
It is clear that we may interpret a n as the assertion that, within the finite
sequence of elements k 1, k 2, ... k n, all elements possess the property A. This
makes it easy to apply the classical definition to the evaluation of p(a n ).
There is only one possibility that is favourable to the assertion a n : it is the
possibility that all the n individuals, k i without exception, possess the
property A rather than the property non-A. But there are in all 2 n
possibilities, since we must assume that it is possible for any individual
k i, either to possess the property A or the property non-A. Accordingly,
the classical theory gives
(4 c )
p(a n ) = 1/2 n
appendix *vii 377
But from (3) and (4 c ), we obtain immediately (1).
The ‘classical’ argument leading to (4 c ) is not entirely adequate,
although it is, I believe, essentially correct.
The inadequacy lies merely in the assumption that A and non-A are
equally probable. For it may be argued—correctly, I believe—that since
a is supposed to describe a law of nature, the various a i are instantiation
statements, and thus more probable than their negations which are
potential falsifiers. (Cf. note *1 to section 28). This objection however,
relates to an inessential part of the argument. For whatever
probability—short of unity—we attribute to A, the infinite product a
will have zero probability (assuming independence, which will be
discussed later on). Indeed, we have struck here a particularly trivial
case of the one-or-zero law of probability (which we may also call, with an
allusion to neuro-physiology, ‘the all-or-nothing principle’). In this
case it may be formulated thus: if a is the infinite product of a 1, a 2, ...,
where p(a i) = p(a j), and where every a i is independent of all others, then
the following holds: