popper-logic-scientific-discovery

(An example of an interpretation of ‘pa(x 1)’ in the sense of (3), i.e. of
the logical interpretation, is the concept ‘logical probability’ as used by
me in a previous publication. 1 )
Now it is possible to proceed with the whole construction from the
other end: instead of introducing ‘p(x 1, x 2)’ as primitive concept
(primitive functor) of an axiom system s 1 and defining ‘pa(x 1)’
explicitly, we can construct another axiom system s 2 in which ‘pa(x 1)’
appears as (undefined) primitive variable, and we can then proceed to
define ‘pa(x 1,x 2)’ explicitly, with the help of ‘pa(x 1)’; as follows.
Df 2
p(x 1, x 2) = pa(x 1x 2)
pa(x 2)
appendix *ii 323
The formulae adopted in s 1 as axioms (and Df 1) now become theorems
within s 2, i.e. they can be deduced with the help of the new system of
axioms s 2.
It can be shown that the two methods described—the choice of s 1
and Df 1, or s 2 and Df 2 respectively—are not equally convenient from
the viewpoint of formal axiomatics. The second method is superior to
the first in certain respects, the most important of which is that it is
possible to formulate in s 2 an axiom of uniqueness which is much
stronger than the corresponding axiom of s 1 (if the generality of s 1 is
not restricted). This is due to the fact that if pa(x 2) = 0, the value of
p(x 1, x 2) becomes indeterminate.* 2
A system of independent axioms, s 2, as described above, is here
subjoined. (It is easy to construct a system s 1 with the help of it.)
Combined with definition Df 2 it is sufficient for the deduction of the
mathematical theory of probability. The axioms can be divided into
two groups. Group A is formed by the axioms for the junctional
6 See Popper, l.c., 71, 151.
* 2 The absolute system (s 2) has an advantage over the relative system (s 1) only as long as
the relative probability p(x, y) is considered as indeterminate if pa(y) = 0. I have since
developed a system (see appendix *iv) in which relative probabilitiés are determinate
even in case pa(y) = 0. This is why I now consider the relative system superior to the
absolute system. (I may also say that I now consider the term ‘axiom of uniqueness’ as
badly chosen. What I intended to allude to was, I suppose, something like postulate 2 or
axiom A2 of the system of appendix *iv.)