popper-logic-scientific-discovery

288
appendices
—we obtain from (2) the special addition theorem
(2 s)
αF″(β + γ) = αF″(β) + αF″(γ).
The special addition theorem holds for all properties which are
primary properties within a class α, since primary properties are mutually
exclusive. The sum of the relative frequencies of these primary
properties is of course always equal to 1.
The division theorems state the frequency of a property γ within a class
selected from α with respect to the property β. The general formula is
obtained immediately by inversion of (1).
(3)
α.βF″(γ) = αF″(β.γ)/ αF″(β)
If we transform the general division theorem (3) with the help of the
special multiplication theorem we obtain
(3 s )
α.βF″(γ) = αF″(γ)
In this formula we recognize again the condition (1 s ); thus we see
that independence may be described as a special case of selection.
The various theorems which may be connected with the name
of Bayes are all special cases of the division theorem. Under the
assumption that (α.γ) is a sub-class of β, or in symbols
(3 bs )
α.γ ⊂ β
we obtain from (3) the first (special) form of Bayes’s rule
(3 bs)
α.βF″(γ) = αF″(γ)/ αF″(β).
We can avoid the assumption (3 bs ) by introducing, in place of ‘β’,
the sum of the classes β 1, β 2, ... β n. We shall, in analogy to our use of
the sign ‘ + ’ between class designations, use the sign ‘�’ in front of class
designations; we can then write a second (universally valid) form of Bayes’s
theorem as follows:
(3 b)
α.�β i F″(β i) = αF″(β i)/ αF″(�β i).