popper-logic-scientific-discovery

404
new appendices
I will now briefly explain how the problem of p(y, x)—of the likelihood
of x—arises.
If we are asked to give a criterion of the fact that the evidence y
supports or corroborates or confirms a statement x, the most obvious
reply is: ‘that y increases the probability of x’. We can put this in symbols
if we write ‘Co(x, y)’ for ‘x is supported or corroborated or confirmed
by y’. We then can formulate our criterion as follows.
(1)
Co(x, y) if, and only if, p(x, y) >p(x).
This formulation, however, has a defect. For if x is a universal theory
and y some empirical evidence, then, as we have seen in the two
preceding appendices, 2
(2)
p(x) = 0 = p(x, y).
But from this it would follow that, for a theory x and evidence y,
Co(x, y) is always false; or in other words, that a universal law can never
be supported, or corroborated, or confirmed by empirical evidence.
(This holds not only for an infinite universe but also for any
extremely large universe, such as ours; for in this case, p(x, y) and p(x)
will be both immeasurably small, and thus practically equal.)
This difficulty however can be got over, as follows. Whenever p(x) ≠
0 ≠ p(y), we have
(3)
p(x, y) >p(x) if, and only if, p(y, x) >p(y),
so that we can transform (1) into
(4)
Co(x, y) if, and only if, p(x, y) >p(x) or p(y, x) >p(y).
Now let x be again a universal law, and let y be empirical evidence
which, say, follows from x. In this case, that is whenever y follows from
x, we shall say, intuitively, that p(y, x) = 1. And since y is empirical, so
that p(y) will certainly be less than 1, we find that (4) can be applied,
2 See, especially, appendix *vii, formulae (1) and (2), and appendix *viii, formula (2).