popper-logic-scientific-discovery

To the numerator of this formula we can apply the special addition
theorem (2 s ) if we assume that the β i have no members in common in
α. This assumption can be written:
(3/2 s )
N(α.β i.β j) = 0. (i≠ j)
Under this assumption we obtain the third (special) form of Bayes’s
theorem, which is always applicable to primary properties β i:
(3/2 s)
α.�β i F″(β i) = αF″(β i)/(� αF″(β i)).
The fourth and most important special form of Bayes’s theorem may
be obtained from the last two formulae together with its constituent
assumptions (3/2 s ) and 4 bs ):
(4 bs )
α.γ ⊂ �β i
which is always satisfied if γ ⊂ �β i is satisfied.
Substituting in (3/2 s)‘β iγ’ for ‘β i’, we apply to the left side of the
result the formula
(4 α.�βi.γ = α.γ.
bs .1) (4bs )
To the right side of the result we apply (1′), both to the denumerator
and denominator. Thus we obtain:
(4 s)
α,γF″(β i) = α.βiF″(γ). αF″(β i)/�( α.βi F″ (γ). αF″(β i))
appendix ii 289
Thus if β i is an exclusive system of property classes, and any property
class which is (within α) part of β i, then (4 s) gives the frequency of
every one of the properties β i within a selection with respect to γ.