popper-logic-scientific-discovery

378
new appendices
(4)
p(a) = lim p(a
n →∞
n ) = 0, unless p(a) = p(an ) = 1
But it is clear that p(a) = 1 is unacceptable (not only from my point
of view but also from that of my inductivist opponents who clearly
cannot accept the consequence that the probability of a universal law
can never be increased by experience). For ‘all swans are black’ would
have the probability 1 as well as ‘all swans are white’—and similarly for
all colours; so that ‘there exists a black swan’ and ‘there exists a white
swan’, etc., would all have zero probability, in spite of their intuitive
logical weakness. In other words, p(a) = 1 would amount to asserting
on purely logical grounds with probability 1 the emptiness of the
universe.
Thus (4) establishes (1).
Although I believe that this argument (including the assumption of
independence to be discussed below) is incontestable, there are a
number of much weaker arguments which do not assume independence
and which still lead to (1). For example we might argue as
follows.
It was assumed in our derivation that for every k i, it is logically
possible that it has the property A, and alternatively, that it has the
property non-A: this leads essentially to (4). But one might also
assume, perhaps, that what we have to consider as our fundamental
possibilities are not the possible properties of every individual in the
universe of n individuals, but rather the possible proportions with
which the properties A and non-A may occur within a sample of individuals.
In a sample of n individuals, the possible proportions with
which A may occur are: 0, 1/n, ..., n/n. If we consider the occurrences
of any of these proportions as our fundamental possibilities, and thus
treat them as equi-probable (‘Laplace’s distribution’ 2 ), then (4) would
have to be replaced by
(5)
p(a n ) = 1/(n + 1); so that lim p(a n ) = 0.
2 It is the assumption underlying Laplace’s derivation of his famous ‘rule of succession’;
this is why I call it ‘Laplace’s distribution’. It is an adequate assumption if our problem is
one of mere sampling; it seems inadequate if we are concerned (as was Laplace) with a
succession of individual events. See also appendix *ix, points 7 ff. of my ‘Third Note’; and
note 10 to appendix *viii.