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You may be very confused as I was by something Kyburg says on page 168. He says:

"His preference ranking may be incoherent in a more sophisticated way: for example, he may prefer A if state X obtains and B if state not-X obtains to A if X, and he may prefer A if X to B if not-X. This set of preferences is incoherent because there is no assignment of probabilities and desirabilities that wilI lead to mathematical expectations that conform to this ranking. (Suppose that the probability of X is p; then the value of the first ranked alternative is p times the value of the second ranked alternative plus (1 - p) times the value of the third ranked alternative; the value of the mixed alternative must be between the values of the pure alternatives, since 0 < p < 1; this contra- dicts our original stipulation.)"

What is supposed to be incoherent is U(A if X, B if ~X) > U(A if X) and U(A if X) > U(B if ~X). But there is nothing wrong with this at all. In fact, the first inequality requires U(B) < 0 and given that, lots of values for A and p are just fine to get the second inequality (for example, any positive value for A). But if you read the description, it is clear that Kyburg meant for the first gamble to be a mixture of the latter two. But the first gamble is a mixture of A and B. So if we assume that his description of the incoherence is correct, this is what he means: U(A if X, B if ~X) > U(A) and U(A) > U(B). This is indeed incoherent in the technical sense. Think of it like this: if you prefer maybe A, maybe B to A for sure, you must like B better than A. But this is precisely what the second claim says is false.

--And I suppose I might as well mention that the answer to the birthday problem is 23.