popper-logic-scientific-discovery

A1 There are elements c and d in S such that p(a, b) ≠ p(c, d)
(Existence)
A2 If p(a, c) = p(b, c) for every c in S, then p(d, a) = p(d, b) for every d
in S. (Substitutivity)
A3 p(a, a) = p(b, b) (Reflexivity)
Postulate 3. If a and b are in S, then ab is in S; and if, moreover, c is in S
(and therefore also bc) then the following axioms hold:
B1 p(ab, c) � p(a, c) (Monotony)
B2 p(ab, c) = p(a, bc) p(b, c) (Multiplication)
Postulate 4. If a is in S, then ā is in S; and if, moreover, b is in S, then the
following axiom holds:
C
p(a, b) + p(ā, b) = p(b, b), unless p(b, b) = p(c, b) for every c
in S. (Complementation).
This concludes the ‘elementary’ system (‘elementary’ as opposed to
its extension to Borel fields). We may, as indicated, add here the
definition of absolute probability as a fifth postulate, called ‘Postulate AP’;
alternatively, we may regard this as an explicit definition rather than as
a postulate.
Postulate AP. If a and b are in S, and if p(b, c) � p(c, b) for every c in S,
then p(a) = p(a, b) (Definition of Absolute Probability 6 )
The system of postulates and axioms given here will be shown
below to be consistent and independent. 7
6 AP is based on the theorem: If p(b, c) = 1, then p(a, bc) = p(a, c).
7 An alternative system would be the following: the postulates are the same as in the text,
and so are axioms A1 and A2, but the axioms A3 and B1 are replaced by
A3′
A4′
B1′
p(a, a) = 1
p(a, b) � 0
If p(ab, c) >p(a, c), then p(ab, c) >p(b, c)
Axiom B2 remains as in the text, and axiom C is replaced by
C′
If p(a, b) ≠ 1, then p(c, b) + p(c¯, b) = 1
appendix *iv 337