popper-logic-scientific-discovery

probability > 1/2. (They even failed to establish that a universal law, or
a theory, could ever have a probability other than zero.) What was
needed was a perfectly general treatment. I therefore aimed at constructing
a formal probability calculus which could be interpreted in
various senses. I had in mind (i) the logical sense, outlined in my book
as (absolute) logical probability of statements; (ii) the sense of relative
logical probability of statements or propositions, as envisaged by Keynes;
(iii) the sense of a calculus of relative frequencies in sequences;
(iv) the sense of a calculus of a measure of ranges, or of predicates,
classes, or sets.
The ultimate aim was, of course, to show that degree of corroboration was
not a probability; that is to say that it was not one of the possible interpretations of the
probability calculus. Yet I realized that the task of constructing a formal
calculus was not only needed for this purpose, but was interesting in
itself.
This led to my paper in Mind, reprinted here as appendix *ii, and to
other work extending over many years and aimed both at simplifying
my axiom systems and at producing a probability calculus in which
p(a, b)—the probability of a given b—could have definite values, rather
than 0/0, even if p(b) was equal to zero. The problem arises, of course,
because the definition.
p(a, b) = p(ab)/p(b)
appendix *ix 403
breaks down if p(b) = 0.
A solution of this last problem was needed because I soon found
that, in order to define C(x, y)—the degree of corroboration of the
theory x by the evidence y—I had to operate with some converse p(y,
x), called by Fisher the ‘likelihood of x’ (in the light of the evidence y, or
given y; note that both, my ‘corroboration’ and Fisher’s likelihood, are
intended to measure the acceptability of the hypothesis x; it is thus x
which is important, while y represents merely the changing empirical
evidence or, as I prefer to say, the reports of the results of tests). Now I was
convinced that, if x is a theory, p(x) = 0. I saw therefore that I had to
construct a new probability calculus in which the likelihood, p(y, x),
could be a definite number, other than 0/0, even if x was a universal
theory with p(x) = 0.