popper-logic-scientific-discovery

How this is to be understood will be shown here with the help of
one very simple example. (Further examples will be found in appendix
ii.) Let us denote the class of all elements which do not belong to β by
‘β - ’ (read: ‘the complement of β’ or simply: ‘non-β’). Then we may
write
αF″(β) + αF″(β - ) = 1
While this theorem only contains F-numbers, its proof makes use of Nnumbers.
For the theorem follows from the definition (1) with the
help of a simple theorem from the calculus of classes which asserts that
N(α.β) + N(α.β - ) = N(α).
53 SELECTION, INDEPENDENCE,
INSENSITIVENESS, IRRELEVANCE
Among the operations which can be performed with relative frequencies
in finite classes, the operation of selection 1 is of special importance
for what follows.
Let a finite reference-class α be given, for example the class of buttons
in a box, and two property-classes, β (say, the red buttons) and γ
(say, the large buttons). We may now take the product-class α.β as a new
reference-class, and raise the question of the value of α.βF″ (γ), i.e. of the
frequency of γ within the new reference-class. 2 The new referenceclass
α.β may be called ‘the result of selecting β-elements from α’, or
the ‘selection from α according to the property β’; for we may think of
it as being obtained by selecting from α all those elements (buttons)
which have the property β (red).
Now it is just possible that γ may occur in the new reference-class,
α.β, with the same relative frequency as in the original reference-class
α; i.e. it may be true that
α.βF″ (γ) = αF″ (γ)
probability 145
1 Von Mises’s term is ‘choice’ (‘Auswahl’).
2 The answer to this question is given by the general division theorem (cf. appendix ii).