popper-logic-scientific-discovery

Thus I prefer to look upon (2) and (3) as highly adequate definitions
of explanatory power—of E(x, y) and E(x, y, z)—rather than of degree of
confirmation. The latter may be defined, on the basis of explanatory
power, in many different ways so as to satisfy viii(c). One way is as
follows (I think that better ways may be found):
(4)
(5)
C(x, y) = E(x, y)/((1 + nP(x))P(x¯, y))
C(x, y, z) = E(x, y, z)/((1 + nP(x, z)) P(x¯, yz))
Here we may choose n � 1. And if we wish viii(c) to have a marked
effect, we can make n a large number.
In case x is a universal theory with P(x) = 0 and y is empirical evidence,
the difference between E and C disappears, as in my original
definitions, and as demanded by desideratum (vi). It also disappears if
x follows from y. Thus at least some of the advantages of operating with
a logarithmic measure remain: as explained by Hamblin, the concept
defined by (1) becomes closely related to the fundamental idea
of information theory. Good also comments on this point (see
footnote 4).
The transition from the old to the new definitions is orderpreserving.
(This holds also for explanatory power, as Hamblin’s
observations imply.) Thus the difference is metrical only.
2. The definitions of explanatory power, and even more of degree of
confirmation (or corroboration or acceptability or attestation, or whatever
name may be chosen for it) give of course full weight to the ‘weight
of evidence’ (or the ‘weight of an argument’ as Keynes called it in his
chapter vi).* 1 This becomes obvious with the new definitions, based
upon Hamblin’s suggestions, which seem to have considerable
advantages if we are at all interested in metrical questions.
3. However, we must realize that the metric of our C will depend
entirely upon the metric of P. But there cannot be a satisfactory metric of P; that is
to say, there cannot be a metric of logical probability which is based upon purely logical
considerations. To show this we consider the logical probability of any
measurable physical property (non-discrete random variable) such as
* 1 See the ‘Third Note’, below.
appendix *ix 421