popper-logic-scientific-discovery

330
new appendices
There are three main characteristics which distinguish a theory of
this kind from others. (i) It is formal; that is to say, it does not assume
any particular interpretation, although allowing for at least all known
interpretations. (ii) It is autonomous; that is to say, it adheres to the
principle that probability conclusions can be derived only from probability
premises; in other words, to the principle that the calculus of
probabilities is a method of transforming probabilities into other probabilities.
(iii) It is symmetrical; that is to say, it is so constructed that
whenever there is a probability p(b, a)—i.e. a probability of b given a—
then there is always a probability p(a, b) also—even when the absolute
probability of b, p(b), equals zero; that is, even when p(b) = p(b, aā) = 0.
Apart from my own attempts in this field, a theory of this kind,
strange to say, does not seem to have existed hitherto. Some other
authors have intended to construct an ‘abstract’ or ‘formal’ theory—
for example Kolmogorov—but in their constructions they have always
assumed a more or less specific interpretation. For example, they assumed
that in an equation like
p(a, b) = r
the ‘elements’ a and b are statements, or systems of statements; or they
assumed that a and b are sets, or systems of sets; or perhaps properties;
or perhaps finite classes (ensembles) of things.
Kolmogorov writes 2 ‘The theory of probability, as a mathematical
discipline, can and should be developed from axioms in exactly the
same way as geometry and algebra’; and he refers to ‘the introduction
of basic geometric concepts in the Foundations of Geometry by Hilbert’, and
to similar abstract systems.
And yet, he assumes that, in ‘p(a, b)’—I am using my own symbols,
not his—a and b are sets; thereby excluding, among others, the logical
But if this formula is added to our system, then it can be proved that there are exactly two
elements in S. The examples by which we prove, below, the consistency of our axioms
show however that there may be any number of elements in S. This shows that (0), and
all similar formulae determining the number of elements in S, cannot be derived; nor can
the negations of these formulae be derived. Thus our system is incomplete.
2 The quotations here are all from p. 1 of A. Kolmogorov, Foundation of the Theory of Probability,
1950. (First German edition 1933.)