On the Triviality of High-Order Probabilistic Beliefs

Several axioms concerning probabilistic beliefs are examined here, and the relations between them are established, using belief spaces that generalize Harsanyi type spaces. Two axioms concerning high-order probabilistic beliefs are investigated in particular. The first is the triviality axiom, which says that one is certain (that is, has a belief of degree one) of one's own beliefs. The second is the averaging axiom, which states that a first order belief concerning some fact F is the average of all degrees of beliefs concerning F, weighted by the degrees given to them by the second order beliefs. It is shown that one whose beliefs satisfy the averaging axiom must be certain that his beliefs satisfy the triviality axiom.