On confirmation
Janina Hosiasson-Lindenbaum · Journal of Symbolic Logic · 1940
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, 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.