Taking for the relation of confirmation the following obvious axioms, we obtain several more or less well-known theorems and are able to solve in a definite and strict manner several problems concerning confirmation.
Let a, b, and c be variable names of sentences belonging to a certain class, the operations a·b, a + b, and ā the (syntactical) product, the sum, and the negation of them. Let us further assume the existence of a real non-negative function c(a, b) of a and b, when b is not self-contradictory. Let us read ‘c(a, b)’ ‘degree of confirmation of a with respect to b' and take the following axioms:
Axiom I. If a is a consequence of b, c(a, b) = 1.
Axiom II. If
is a consequence of c,
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Axiom III. c(a·b, c) = c(a, c)·c(b, a·c).
Axiom IV. If b is equivalent to c, c(a, b) = c(a, c).
As may be easily seen, the interval of variation for c is (0, 1); this is quite conventional.