popper-logic-scientific-discovery

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some structural components of a theory of experience
The class of those elements which belong to both α and β, for
example the class of throws made yesterday with this particular die and
having the property five, is called the product-class of α and β, and is
denoted by ‘α.β’, to be read ‘α and β’. Since α.β is a subclass of α, it can
at most contain a finite number of elements (it may be empty). The
number of elements in α.β is denoted by ‘N(α.β)’.
Whilst we symbolize (finite) numbers of elements by N, the relative
frequencies are symbolized by F″. For example, ‘the relative frequency
of the property β within the finite reference-class α’ is written
‘ αF″(β)’, which may be read ‘the α-frequency of β’. We can now
define
(Definition 1)
αF″(β) = N(α.β)
N(α)
In terms of our example this would mean: ‘The relative frequency of
fives among yesterday’s throws with this die is, by definition, equal to
the quotient obtained by dividing the number of fives, thrown yesterday
with this die, by the total number of yesterday’s throws with this
die.’* 1
From this rather trivial definition, the theorems of the calculus of
frequency in finite classes can very easily be derived (more especially, the
general multiplication theorem; the theorem of addition; and the theorems
of division, i.e. Bayes’s rules. Cf. appendix ii). Of the theorems of
this calculus of frequency, and of the calculus of probability in general,
it is characteristic that cardinal numbers (N-numbers) never appear in
them, but only relative frequencies, i.e. ratios, or F-numbers. The Nnumbers
only occur in the proofs of a few fundamental theorems
which are directly deduced from the definition; but they do not occur
in the theorems themselves.* 2
* 1 Definition 1 is of course related to the classical definition of probability as the ratio of
the favourable cases to the equally possible cases; but it should be clearly distinguished
from the latter definition: there is no assumption involved here that the elements of α are
‘equally possible’.
* 2 By selecting a set of F-formulae from which the other F-formulae can be derived, we
obtain a formal axiom system for probability; compare the appendices ii, *ii, *iv, and *v.