Higher Order Probabilities and Coherence
Soshichi Uchii · Philosophy of Science · 1973
It is well known that a degree-of-belief function P is coherent if and only if it satisfies the probability calculus. In this paper, we show that the notion of coherence can be extended to higher order probabilities such as P(P(h) = p) = q, and that a higher order degree-of-belief function P is coherent if and only if it satisfies the probability calculus plus the following axiom: P(h) = p iff P(P(h) = p) = 1. Also, a number of lemmata which extend an incomplete probability function to a complete one are established.