popper-logic-scientific-discovery

350
new appendices
B1 is violated because 2 = p(1.2, 1) > p(1, 1) = 1
C is violated because p(2¯ , 1) + p(2, 1) = 2, though p(0, 1) ≠ p(1, 1)
The fact that our system remains independent even if we postulate
Boolean algebra and (*) may be expressed by saying that our system is
‘autonomously independent’. If we replace our axiom B1 by A4′ and
B1′ (see note 7 above), then our system ceases, of course, to be
autonomously independent. Autonomous independence seems to me
an interesting (and desirable) property of axiom systems for the
calculus of probability.
In conclusion I wish to give a definition, in the ‘autonomous’ i.e.
probabilistic terms of our theory, of an ‘admissible system’ S, and of a
‘Borel field of probabilities’ S. The latter term is Kolmogorov’s; but I am
using it in a sense slightly wider than his. I will discuss the difference
between Kolmogorov’s treatment of the subject and mine in some
detail because it seems to me illuminating.
I first define, in probabilistic terms, what I mean by saying that a is a
super-element of b (and wider than, or equal to, b) or that b is a subelement
of a (and logically stronger than, or equal to, a). The definition
is as follows. (See also the end of appendix *v.)
a is a super-element of b, or b is a sub-element of a—in symbols,
a � b—if, and only if, p(a, x) � p(b, x) for every x in S.
Next I define what I mean by the product-element a of an infinite
sequence, A = a 1, a 2, ..., all of whose members a n are elements of S.
Let some or perhaps all elements of S be ordered in an infinite sequence
A = a 1, a 2, ..., such that any element of S is permitted to recur in the
sequence. For example, let S consist only of the two elements, 0 and 1;
then A = 0, 1, 0, 1, ..., and B = 0, 0, 0, ..., will both be infinite
sequences of elements of S, in the sense here intended. But the more
important case is of course that of an infinite sequence A such that
all, or almost all, of its members are different elements of S which,
accordingly, will contain infinitely many elements.
A case of special interest is a decreasing (or rather non-increasing)
infinite sequence, that is to say, a sequence A = a 1, a 2, ..., such that
a n � a n + 1 for every consecutive pair of members of A.
We can now define the (Boolean, as opposed to set-theoretical)
product element a of the infinite sequence A = a 1, a 2, ..., as the widest