popper-logic-scientific-discovery

where e is evidence in favour of the theory t, may be undefined; but this
expression is very important. (It is Fisher’s ‘likelihood’ of t on the
given evidence e; see also appendix *ix.)
Thus there is a need for a probability calculus in which we may
operate with second arguments of zero absolute probability. It is, for
example, indispensable for any serious discussion of the theory of
corroboration or confirmation.
This is why I have tried for some years to construct a calculus of
relative probability in which, whenever
p(a, b) = r
is a well-formed formula i.e. true or false,
p(b, a) = r
is also a well formed formula, even if p(a) = 0. A system of this kind
may be labelled ‘symmetrical’. I published the first system of this kind
only in 1955. 4 This symmetrical system turned out to be much simpler
than I expected. But at that time, I was still pre-occupied with the
peculiarities which every system of this kind must exhibit. I am alluding
to such facts as these: in every satisfactory symmetrical system,
rules such as the following are valid:
p(a, bb¯) = 1
If p(b¯, b) ≠ 0 then p(a, b) = 1
If p(a, āb) ≠ 0 then p(a, b) = 1
These formulae are either invalid in the customary systems, or else
(the second and third) vacuously satisfied, since they involve second
arguments with zero absolute probabilities. I therefore believed, at that
time, that some of them would have to appear in my axioms. But I
found later that my axiom system could be simplified; and in simplifying
it, I found that all these unusual formulae could be derived from
formulae having a completely ‘normal’ look. I published the resulting
4 In the British Journal for the Philosophy of Science, 6, 1955, pp. 56 f.
appendix *iv 335