popper-logic-scientific-discovery

the problem of the empirical basis 83
and a basic statement can contradict each other. Condition (b) can only
be satisfied if it is possible to derive the negation of a basic statement
from the theory which it contradicts. From this and condition (a) it
follows that a basic statement must have a logical form such that its
negation cannot be a basic statement in its turn.
We have already encountered statements whose logical form is
different from that of their negations. These were universal statements
and existential statements: they are negations of one another,
and they differ in their logical form. Singular statements can be constructed
in an analogous way. The statement: ‘There is a raven in the
space-time region k’ may be said to be different in its logical form—
and not only in its linguistic form—from the statement ‘There is no
raven in the space-time region k’. A statement of the form ‘There is a
so-and-so in the region k’ or ‘Such-and-such an event is occurring in
the region k’ (cf. section 23) may be called a ‘singular existential
statement’ or a ‘singular there-is statement’. And the statement which
results from negating it, i.e. ‘There is no so-and-so in the region k’ or
‘No event of such-and-such a kind is occurring in the region k’, may
the same logical form as ‘There are no swans’, for it is equivalent to ‘There are no
non-white swans’.)
Now if this is admitted, it will be seen at once that the singular statements which can be
deduced from purely universal statements cannot be basic statements. I have in mind
statements of the form: ‘If there is a swan at the place k, then there is a white swan at the
place k.’ (Or, ‘At k, there is either no swan or a white swan.’) We see now at once why
these ‘instantial statements’ (as they may be called) are not basic statements. The reason
is that these instantial statements cannot play the role of test statements (or of potential falsifiers)
which is precisely the role which basic statements are supposed to play. If we were to
accept instantial statements as test statements, we should obtain for any theory (and
thus both for ‘All swans are white’ and for ‘All swans are black’) an overwhelming
number of verifications—indeed, an infinite number, once we accept as a fact that the
overwhelming part of the world is empty of swans.
Since ‘instantial statements’ are derivable from universal ones, their negations must be
potential falsifiers, and may therefore be basic statements (if the conditions stated below
in the text are satisfied). Instantial statements, vice versa, will then be of the form of
negated basic statements (see also note *4 to section 80). It is interesting to note that
basic statements (which are too strong to be derivable from universal laws alone) will
have a greater informative content than their instantial negations; which means that the
content of basic statements exceeds their logical probability (since it must exceed 1/2).
These were some of the considerations underlying my theory of the logical form of
basic statements. (See my Conjectures and Refutations, 1963, pp. 386 f.)