popper-logic-scientific-discovery

appendix *ix 405
and that the assertion Co(x, y) will be true. That is to say, x may be
corroborated by y if y follows from x, provided only that p(y) < 1. Thus
(4) is intuitively perfectly satisfactory; but in order to operate freely
with (4), we have to have a calculus of probability in which p(y, x) is a
definite number—in our case, 1—rather than 0/0, even if p(x) = 0. In
order to achieve this, a generalization of the usual calculus has to be
provided, as explained above.
Although I had realized this by the time my note in Mind appeared
(cf. appendix *ii), the pressure of other work which I considered more
urgent prevented me from completing my researches in this field. It
was only in 1954 that I published my results concerning degree of
corroboration, in the first of the notes here reprinted; and another six
months elapsed before I published an axiom system of relative probability
3 (equivalent to, though less simple than, the one which will be
found in appendix *iv) which satisfied the demand that p(x, y) should
be a definite number even if p(y) was equal to zero. This paper provided
the technical prerequisites for a satisfactory definition of likelihood and
of degree of corroboration or confirmation.
My first note ‘Degree of Confirmation’, published in 1954 in the
B.J.P.S., contains a mathematical refutation of all those theories of induction
which identify the degree to which a statement is supported or
confirmed or corroborated by empirical tests with its degree of
probability in the sense of the calculus of probability. The refutation
consists in showing that if we identify degree of corroboration or
confirmation with probability, we should be forced to adopt a
number of highly paradoxical views, among them the following clearly
self-contradictory assertion:
(*) There are cases in which x is strongly supported by z and y is
strongly undermined by z while, at the same time, x is confirmed by z to
a lesser degree than is y.
A simple example, showing that this devastating consequence
would follow if we were to identify corroboration or confirmation
with probability, will be found under point 6 of my first note. 4 In view
3 See B.J.P.S. 6, 1955, pp. 56–57.
4 As opposed to the example here given in the text, the example given under points 5 and
6 of my first note are the simplest possible examples, that is to say, they operate with the
smallest possible number of equiprobable exclusive properties. This holds also for the