popper-logic-scientific-discovery

178
some structural components of a theory of experience
probabilities; to assume, that is to say, that for one and the same property
there can be one and only one probability p, within one and the
same reference sequence.
However, the difficulty of defining a unique probability concept
without the limit axiom can easily be overcome. We may introduce the
requirement of uniqueness (as is, after all, the most natural procedure)
as the last step, after having postulated that the sequence shall be ‘absolutely
free’. This leads us to propose, as a solution of our problem, the
following modification of our definition of chance-like sequences, and
of objective probability.
Let α be an alternative (with one or several middle frequencies). Let
the ones of α have one and only one middle frequency p that is ‘absolutely
free’; then we say that α is chance-like or random, and that p is
the objective probability of the ones, within α.
It will be helpful to divide this definition into two axiomatic
requirements.* 2
(1) Requirement of randomness: for an alternative to be chancelike,
there must be at least one ‘absolutely free’ middle frequency, i.e.
its objective probability p.
(2) Requirement of uniqueness: for one and the same property of
one and the same chance-like alternative, there must be one and only one
probability p.
The consistency of the new axiomatic system is ensured by the
example previously constructed. It is possible to construct sequences
which, whilst they have one and only one probability, yet possess no
frequency limit (cf. section b of appendix iv). This shows that the new
axiomatic demands are actually wider, or less exacting, than the old
ones. This fact will become even more evident if we state (as we may)
our old axioms in the following form:
(1) Requirement of randomness: as above.
* 2 It is possible to combine the approach described in note *1 to section 57, and in
appendices iv and *vi, with these two requirements by retaining requirement (1) and
replacing requirement (2) by the following:
(+ 2) Requirement of finitude: the sequence must become, from its commencement,
as quickly n-free as possible, and for the largest possible n; or in other words, it must be
(approximately) a shortest random-like sequence.