popper-logic-scientific-discovery

APPENDIX ii
The General Calculus of Frequency
in Finite Classes (cf. sections 52
and 53)* 1
The General Multiplication Theorem: We denote the finite reference class by
‘α’, and the two property classes by ‘β’ and ‘γ’. Our first problem is to
determine the frequency of those elements which belong both to β
and to γ.
The solution is given by the formula
(1)
αF″(β.γ) = αF″(β). α.βF″(γ)
or, since β and γ may be commuted:
(1′)
αF″(β.γ) = α.γF″(β). αF″(γ)
The proof results immediately from the definition given in section
52. From (1) we obtain, by substitution in accordance with this
definition:
* 1 This appendix has since been developed by me into an axiomatic treatment of
probability. See appendices *iii to *v.