popper-logic-scientific-discovery

320
new appendices
logical interpretation of an ‘absolute probability’; that is to say, of a
probability p(x, y), with tautological y. Since a tautology may be written
not-(x and not-x), or xx¯, in the symbols used in my note, we can define
the absolute probability of x (for which we may write ‘p(x)’ or ‘pa(x)’)
in terms of relative probability as follows:
p(x) = p(x, xx¯), or pa(x) = p(x, xx¯) = p(x, yy¯)
A similar definition is given in my note.
When I wrote this note I did not know Kolmogorov’s book Foundations
of Probability, although it had been first published in German in
1933. Kolmogorov had very similar aims; but his system is less ‘formal’
than mine, and therefore susceptible to fewer interpretations. The
main point of difference is this. He interprets the arguments of the
probability functor as sets; accordingly, he assumes that they have
members (or ‘elements’). No corresponding assumption was made in
my system: in my theory, nothing whatever is assumed concerning these arguments
(which I call ‘elements’) except that their probabilities behave in the manner required by
the axioms. Kolmogorov’s system can be taken, however, as one of
the interpretations of mine. (See also my remarks on this topic in
appendix *iv.)
The actual axiom system at the end of my note is somewhat clumsy,
and very shortly after its publication I replaced it by a simpler and more
elegant one. Both systems, the old and the new, were formulated in
terms of product (or conjunction) and complement (negation), as were also
my later systems. At that time, I had not succeeded in deriving the
distributive law from simpler ones (such as A1 to A3 and B2 below),
and I therefore stated it as an axiom. But, written in terms of product
and complement, the distributive law is very clumsy. I have therefore
here omitted the end of the note, with the old axiom system; instead I
will restate here my simpler system (cf. Brit. Journal Phil. Sc., loc. cit.), based,
like the old system, on absolute probability. It is, of course, derivable
from the system based on relative probability given in appendix *iv. I
am stating the system here in an order corresponding to that of my old
note.
A1
A2
p(xy) � p(yx) (Commutation)
p((xy)z) � p(x(yz)) (Association)