(Billy Joel, “She’s Always a Woman”)
If we agree that we have degrees of belief in any and all propositions, then, it is often argued (by Bayesians), that if your beliefs do not conform to the probability calculus, you are being incoherent, and will lose money for sure (by a clever enough bookie). We can accept the claim that, were we required to take bets on our degrees of belief, then given that we prefer not to lose, we would not accept bets that ensured our losing. But this is a tautology, as others have pointed out, and entails nothing about degree of belief assignments. “That an agent ought not to accept a set of wagers according to which she loses come what may, if she would prefer not to lose, is a matter of deductive logic and not a property of beliefs” (Bacchus, Kyburg, and Thalos 1990: 476).[i] Nor need coerced (or imaginary) betting rates actually measure an agent’s degrees of belief in the truth of scientific hypothesis..
Nowadays, surprisingly, most Bayesian philosophers seem to dismiss as irrelevant the variety of threats of being Dutch-booked. Confronted with counterexamples in which violating Bayes’s rule seems perfectly rational on intuitive grounds, Bayesians contort themselves into a great many knots in order to retain the underlying Bayesian philosophy while sacrificing updating rules, long held to be the very essence of Bayesian reasoning. To face contemporary positions squarely calls for rather imaginative deconstructions. I invite your deconstructions (to email@example.com) by April 23 (see So You Want to Do a Philosophical Analysis). Says Howson:
It used to be that frequentists and others who sounded the alarm about temporal incoherency were declared irrational. Now, it is the traditional insistence on updating by Bayes’s rule that was irrational all along. Continue reading