Axioms of elementary probability

Here's a question about elementary (naive, finitist) probability. The proper, self-dual axioms for elementary probability are presumably P(0) = 0 P(X) = 1 P(A u B) + P(A n B) = P(A) + P(B) P's domain is a boolean algebra. P's codomain is [0,1]. What kind of algebraic structure is [0,1] in this case? What can we prove from this theory? The best I can think of is inclusion / exclusion: P(A u B u C) = P(A) + P(B) + P(C) - P(A n B) - P(A n C) - P(B n C) + P(A n B n C) P(A n B n C) = P(A) + P(B) + P(C) - P(A u B) - P(A u C) - P(B u C) + P(A u B u C) Thanks, David