popper-logic-scientific-discovery

A3
A4
B1
B2
B3
There are at least one x and one y such that
For every x there is a y such that
p(xx) � p(x). (Tautology)
p(x) ≠ p(y). (Existence)
p(x) � p(xy) (Monotony)
p(x) = p(xy) + p(xy¯). (Complement)
p(y) � p(x), and p(xy) = p(x)p(y). * 1
(Independence)
Here follows my old note of 1938, with a few slight stylistic
corrections.
A set of independent axioms for probability
From the formal point of view of ‘axiomatics’, probability can be
described as a two-termed functor 1 (i.e., a numerical function of two
arguments which themselves need not have numerical values), whose
arguments are variable or constant names (which can be interpreted, e.g.,
as names of predicates or as names of statements1 according to the
interpretation chosen). If we desire to accept for both of the arguments
the same rules of substitution and the same interpretation, then this
functor can be denoted by
‘p(x 1, x 2)’
which can be read as ‘the probability of x 1 with regard to x 2’.
appendix *ii 321
* 1 Without B3 the upper bound of p(x) is not fixed: p(xx¯) = k, where k may be arbitrarily
chosen. Then we get: x is independent of y if and only if p(xy) = p(x)p(y)/k; but this is
clumsy unless we choose k = 1. B3, which leads to k = 1, indicates this motive for
choosing k = 1. If in B3 we put ‘p(y) ≠ o’ instead of ‘p(y) � p(x)’, then A4 becomes
redundant.
1 For the terminology see Carnap, Logical Syntax of Language (1937); and Tarski, Erkenntnis 5,
175 (1935).