popper-logic-scientific-discovery

among those elements of S which are sub-elements of every element a n
belonging to the sequence A; or in symbols:
(i)
(ii)
a = πa n if, and only if, a satisfies the following two conditions
appendix *iv 351
p(an, x) � p(a, x) for all elements an of A, and for every element x
of S.
p(a, x) � p(b, x) for all elements x of S and for every element b of S
that satisfies the condition p(an, y) � p(b, y) for all elements an and
for every element y of S.
In order to show the difference between our (Boolean) product
element a of A and the set-theoretical (inner) product or meet of A, we
will now confine our discussion to examples S, satisfying our postulates
2 to 5, whose elements x, y, z, . . . are sets, such that xy is their
set-theoretic product.
Our main example S1 to which I shall refer as ‘the example of the
missing half-interval’ is the following.
S1 is a system of certain half-open sub-intervals of the universal
interval u = (0, 1]. S1 contains, precisely, (a) the decreasing sequence A
such that an = (0, 1 2 + 2 − n ], and in addition (b) the set-theoretic products
of any two of its elements and the set-theoretic complements of
any one of its elements.
Thus S1 does not contain the ‘half-interval’ h = (0, 1 2], nor any
non-empty sub-interval of h.
Since the missing half-interval h = (0, 1 2] is the set-theoretic product
of the sequence A, it is clear that S1 does not contain the set-theoretic
product of A. But S1 does contain the (Boolean) ‘product-element’ of A,
as here defined. For the empty interval trivially satisfies condition (i);
and since it is the widest interval satisfying (i), it also satisfies (ii).
It is clear, moreover, that if we add to S1, say any of the intervals
b1 = (0, 1 8], or b2 = (0, 3
16], etc., then the largest of these will be the
product element of A in the (Boolean) sense of our definition,
although none of them will be the set-theoretic product of A.
One might think, for a moment, that owing to the presence of an
empty element in every S, every S will contain, like S1, a product element
(in the sense of our definition) of any A in S; for if it does not
contain any wider element satisfying (i), the empty element will