popper-logic-scientific-discovery

348
new appendices
S′ ={0, 1, 2, 3} and define product, complement, and absolute
probability by the matrix:
ab 0 1 2 3 ā p(a)
0 0 0 0 0 3 0
1 0 1 0 1 2 0
2 0 0 2 2 1 1
3 0 1 2 3 0 1
Relative probability is defined by
p(a, b) = 0 whenever p(a) ≠ 1 = p(b);
p(a, b) = 1 in all other cases.
This system S′ satisfies all our axioms and postulates. In order to
show the independence of the existential part of postulate 3, we now
take S to be confined to the elements 1 and 2 of S′, leaving everything
else unchanged. Obviously, postulate 3 fails, because the product of the
elements 1 and 2 is not in S; everything else remains valid. Similarly, we
can show the independence of postulate 4 by confining S to the elements
0 and 1 of S′. (We may also choose 2 and 3, or any combination
consisting of three of the four elements of S′ except the combination
consisting of 1, 2, and 3.)
The proof of the independence of postulate AP is even more trivial:
we only need to interpret S and p(a, b) in the sense of our first consistency
proof and take p(a) = constant (a constant such as 0, or 1/2, or 1, or
2) in order to obtain an interpretation in which postulate AP fails.
Thus we have shown that every single assertion made in our axiom
system is independent. (To my knowledge, no proofs of
independence for axiom systems of probability have been published
before. The reason, I suppose, is that the known systems—provided
they are otherwise satisfactory—are not independent.)