popper-logic-scientific-discovery

‘p(2) + p(4)’ we can write, more generally, ‘p(ab)’, that is to say, the
probability of an even throw other than a six. And instead of writing
‘p(1) + p(2) + ...+ p(5)’ or, what amounts to the same, ‘1 − p(6)’, we
can write ‘p(b)’, that is to say, the probability of throwing a number
other than six. It is clear that these calculations are quite general, and
assuming p(b) ≠ 0, we are led to the formula,
(1)
p(a, b) = p(ab)/p(b)
or to the formula (more general because it remains meaningful even if
p(b) = 0),
(2)
p(ab) = p(a, b) p(b).
This is the general multiplication theorem for the absolute probability
of a product ab.
By substituting ‘bc’ for ‘b’, we obtain from (2): 1
p(abc) = p(a, bc) p(bc)
and therefore, by applying (2) to p(bc):
or, assuming p(c) ≠ 0,
This, in view of (1), is the same as
(3)
p(abc) = p(a, bc) p(b, c) p(c)
p(abc) /p(c) = p(a, bc) p(b, c).
p(ab, c) = p(a, bc) p(b, c).
appendix *iii 327
This is the general multiplication theorem for the relative probability
of a product ab.
1 I omit brackets round ‘bc’ because my interest is here heuristic rather than formal, and
because the problem of the law of association is dealt with at length in the next two
appendices.