popper-logic-scientific-discovery

APPENDIX *iv
The Formal Theory of Probability
In view of the fact that a probability statement such as ‘p(a, b) = r’ can
be interpreted in many ways, it appeared to me desirable to construct a
purely ‘formal’ or ‘abstract’ or ‘autonomous’ system, in the sense that
its ‘elements’ (represented by ‘a’, ‘b’, . . . ) can be interpreted in many
ways, so that we are not bound to any particular one of these interpretations.
I proposed first a formal axiom system of this kind in a Note in
Mind in 1938 (here re-printed in appendix *ii). Since then, I have
constructed many simplified systems. 1
1 In Brit. Journ. Phil. of Science 6, 1955, pp. 53 and 57 f., and in the first footnote to the
Appendix to my paper ‘Philosophy of Science: A Personal Report’, in British Philosophy in
Mid-Century, ed. by C. A. Mace, 1956.
It should be noted that the systems here discussed are ‘formal’ or ‘abstract’ or
‘autonomous’ in the sense explained, but that for a complete ‘formalization’, we should
have to embed our system in some mathematical formalism. (Tarski’s ‘elementary
algebra’ would suffice.)
The question may be asked whether a decision procedure might exist for a system
consisting, say, of Tarski’s elementary algebra and our system of formulae A1, B, and C + .
The answer is, no. For formulae may be added to our system which express how many
elements a, b, . . . there are in S. Thus we have in our system a theorem:
There exists an element a in S such that p(a, ā) ≠ p(ā, a).
To this we may now add the formula:
(0)
For every element a in S, p(a, ā) ≠ p(ā, a).