popper-logic-scientific-discovery

454
new appendices
is deducible from a statement function which is satisfied in all worlds that differ from our
world, if at all, only with respect to initial conditions. (*See the Addendum to the
present appendix.)
We can never know, of course, whether a supposed law is a genuine
law or whether it only looks like a law but depends, in fact, upon
certain special initial conditions prevailing in our region of the universe.
(Cf. section 79.) We cannot, therefore ever find out of any
given non-logical statement that it is in fact naturally necessary: the
conjecture that it is remains a conjecture for ever (not merely
because we cannot search our whole world in order to ensure that
no counter instance exists, but for the even stronger reason that we
cannot search all worlds that differ from ours with respect to initial
conditions.) But although our proposed definition excludes the possibility
of obtaining a positive criterion of natural necessity, we can in
practice apply our definition of natural necessity in a negative way: by
finding initial conditions under which the supposed law turns out to
be invalid, we can show that it was not necessary; that is to say, not a
law of nature. Thus the proposed definition fits our methodology
very well indeed.
The proposed definition would, of course, make all laws of nature,
together with all their logical consequences, naturally or physically
necessary. 16
It will be seen at once that the proposed definition is in perfect
agreement with the results reached in our discussion of the moa
example (cf. points 6 and 7 above): it was precisely because we thought
that moas would live longer under different conditions—under more
favourable ones—that we felt that a true universal statement about their
actual maximal age was of an accidental character.
(13) We now introduce the symbol ‘N’ as a name of the class of
statements which are necessarily true, in the sense of natural or physical
necessity; that is to say, true whatever the initial conditions may be.
With the help of ‘N’, we can define ‘a→ N b’ (or in words, ‘If a then
necessarily b’) by the following somewhat obvious definition:
16 Incidentally, logically necessary statements would (simply because they follow from
any statement) become physically necessary also; but this does not matter, of course.