popper-logic-scientific-discovery

456
new appendices
A little reflection will show that the class N of naturally necessary
statements comprises not only the class of all those statements
which, like true universal laws of nature, can be intuitively described
as being unaffected by changes of initial conditions, but also all
those statements which follow from true universal laws of nature, or
from the true structural theories about the world. There will be
statements among these that describe a definite set of initial conditions;
for example, statements of the form ‘if in this phial under
ordinary room temperature and a pressure of 1000 g per cm 2 ,
hydrogen and oxygen are mixed . . . then . . .’. If conditional statements
of this kind are deducible from true laws of nature, then their
truth will be also invariant with respect to all changes of initial
conditions: either the initial conditions described in the antecedent
will be satisfied, in which case the consequent will be true (and
therefore the whole conditional); or the initial conditions described
in the antecedent will not be satisfied and therefore factually untrue
(‘counter-factual’). In this case the conditional will be true as vacuously
satisfied. Thus the much discussed vacuous satisfaction plays its
proper part to ensure that the statements deducible from naturally
necessary laws are also ‘naturally necessary’ in the sense of our
definition.
Indeed, we could have defined N simply as the class of natural
laws and their logical consequences. But there is perhaps a slight
advantage in defining N with the help of the idea of initial conditions
(of a simultaneity class of singular statements). If we define N
as, for example, the class of statements which are true in all worlds
that differ from our world (if at all) only with respect to initial
conditions, then we avoid the use of subjunctive (or counter-factual)
wording, such as ‘which would remain true even if different
initial conditions held (in our world) than those which actually do
hold’.
Nevertheless, the phrase in (N°) ‘all worlds which differ (if at all)
from our world only with respect to the initial conditions’ undoubt-
‘∼�(a). (�(a) ⊃ ψ(a))’. I wonder whether Kneale realized that this expression of his was
only a complicated way of saying ‘∼� (a)’; for who would ever think of asserting that
‘∼� (a)’ was deducible from the law ‘(x) (�(x) ⊃ ψ (x))’?