popper-logic-scientific-discovery

386
new appendices
‘universe’—that of possible positions—may here be chosen to be a
finite one, there will be nevertheless an infinity of mathematically
incompatible explanatory theories. I am aware, of course, that instrumentalists
or operationalists might say that the differences between any
two theories determining the same squares would be ‘meaningless’.
But apart from the fact that this example does not form part of my argument—so
that I need really not reply to this objection—the following should be
noted. It will be possible, in many cases, to give ‘meaning’ to these
‘meaningless’ differences by making our mesh sufficiently fine, i.e.
subdividing our squares.)
The detailed discussion of the fact that my consistency condition
cannot be satisfied will be found in appendix *viii. I will now leave the
problem of the validity of formulae (1) and (2), in order to proceed to
the discussion of a formal problem arising from the fact that these
formulae are valid, so that all universal theories, whatever their
content, have zero probability.
There can be no doubt that the content or the logical strength of two
universal theories can differ greatly. Take the two laws a 1 = ‘All planets
move in circles’ and a 2 = ‘All planets move in ellipses’. Owing to the
fact that all circles are ellipses (with eccentricity zero), a 1 entails a 2, but
not vice versa. The content of a 1 is greater by far than the content of a 2.
(There are, of course, other theories, and logically stronger ones, than
a 1; for example, ‘All planets move in concentric circles round the sun’.)
The fact that the content of a 1 exceeds that of a 2 is of the greatest
significance for all our problems. For example, there are tests of a 1—that
is to say, attempts to refute a 1 by discovering some deviation from
circularity—which are not tests of a 2; but there could be no genuine
test of a 2 which would not, at the same time, be an attempt to refute a 1.
Thus a 1 can be more severely tested than a 2, it has the greater degree of
testability; and if it stands up to its more severe tests, it will attain a
higher degree of corroboration than a 2 can attain.
Similar relationships may hold between two theories, a 1 and a 2, even
if a 1 does not logically entail a 2, but entails instead a theory to which a 2
is a very good approximation. (Thus a 1 may be Newton’s dynamics and
a 2 may be Kepler’s laws which do not follow from Newton’s theory,
but merely ‘follow with good approximation’; see also section *15 of