Cox's Theorem Revisited

The assumptions needed to prove Cox’s Theorem are discussed and examined. Various sets of assumptions under which a Cox-style theorem can be proved are provided, although all are rather strong and, arguably, not natural. I recently wrote a paper (Halpern, 1999) casting doubt on how compelling a justification for probability is provided by Cox’s celebrated theorem (Cox, 1946). I have received (what seems to me, at least) a surprising amount of response to that article. Here I attempt to clarify the degree to which I think Cox’s theorem can be salvaged and respond to a glaring inaccuracy on my part pointed out by Snow (1998). (Fortunately, it is an inaccuracy that has no affect on either the correctness or the interpretation of the results of my paper.) I have tried to write this note with enough detail so that it can be read independently of my earlier paper, but I encourage the reader to consult the earlier paper as well as the two major sources it is based on (Cox, 1946; Paris, 1994), for further details and discussion. Here is the basic situation. Cox’s goal is to “try to show that . . . it is possible to derive the rules of probability from two quite primitive notions which are independent of the notion of ensemble and which . . . appeal rather immediately to common sense” (Cox, 1946). To that end, he starts with a function Bel that associates a real number with each pair (U, V ) of subsets of a domain W such that U 6= ∅. We write Bel(V |U) rather than Bel(U, V ), since we think of Bel(V |U) as the belief, credibility, or likelihood of V given U . Cox’s Theorem as informally understood, states that if Bel satisfies two very reasonable restrictions, then Bel must be isomorphic to a probability measure. The first one says that the belief in V complement (denoted V ) given U is a function of the belief in V given U ; the second says that the belief in V ∩V ′ given U is a function of the belief in V ′ given V ∩U and the belief in V given U . Formally, we assume that there are functions S : IR → IR and F : IR → IR such that A1. Bel(V |U) = S(Bel(V |U)) if U 6= ∅, for all U, V ⊆ W . A2. Bel(V ∩ V |U) = F (Bel(V |V ∩ U),Bel(V |U)) if V ∩ U 6= ∅, for all U, V, V ′ ⊆ W . If Bel is a probability measure, then we can take S(x) = 1− x and F (x, y) = xy. Before going on, notice that Cox’s result does not claim that Bel is a probability measure, just that it is isomorphic to a probability measure. Formally, this means that there is a continuous one-to-one onto function g : IR → IR such that g ◦ Bel is a probability measure on W , and g(Bel(V |U))× g(Bel(U)) = g(Bel(V ∩ U)) if U 6= ∅, (1) c ©1999 AI Access Foundation and Morgan Kaufmann Publishers. All rights reserved.