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# Dutch Book

The cornerstone of Ramsey's theory is the Dutch Book Argument. This is an argument to the effect that, if an agent has degrees of belief that do not conform to the axioms of the probability calculus, he can have a book made against him, that is, be subjected to a set of bets that entail a certain loss to him. Although not all "Bayesians" accept this argument (Howson and Urbach,1993, do not, for example), many do, and it provides the strongest motivation for many philosophers to become subjectivists.

But let us begin at the beginning.

The subject of our inquiry is the logic of partial belief, and I do not think we can carry it far unless we have at least an approximate notion of what partial belief is, and how, if at all, it can be measured. . . . It is not enough to measure probability; in order to apportion correctly our belief to the probability we must also be able to measure our belief.

(Ramsey, 1931, p. 166)

Note that here, at the beginning of this section, Ramsey has already committed himself, without visible argument, to the falsity of Keynes's view that "degrees" of belief are no more than partially ordered. Ramsey has now given himself the problem of "measuring" degrees of belief. He sets about solving it in a strictly behavioristic way: my degree of belief in a proposition can be measured by its causal efficacy in making decisions. "[T]his way of measuring beliefs . . . allows validity to betting as a means of measuring beliefs" (Ramsey, 1931, p. 176).

This may provide an "approximate" notion of partial belief, but there is nothing approximate about the corresponding notion of degree. There is exactly one real number, on Ramsey's construction, that corresponds to the odds at which the agent would be willing to accept a bet either on or against a given proposition.

This begs the question against Keynes's claim that the manifold of probabilities is richer than the set of real numbers between 0 and 1. If we leave that issue to one side, there are two objections to the claim that the person whose degrees of belief are not "coherent" (that is, do not satisfy the axioms of probability) will be in a position to have a sure-loss book made against him. First of all, whatever the agent's degrees of belief may be, it requires only prowess in deduction for him to decline to commit himself to a sure loss. I may believe in heads to the degree 2/3 and in tails to the degree 2/3; on Ramsey's behavioristic view this means that I am willing to offer 2:1 odds on tails, and also willing to offer 2:l odds on heads. But unless I am very confused about the nature of coins, or deductively incompetent, I will decline to make both bets simultaneously.

Second, a more natural, but still behavioristic, way of evaluating degrees of belief is to take the least odds at which the agent would accept a bet against a given proposition to be an indication of his degree of belief in it. Thus, that I would accept a bet against heads if offered odds of 2:1 shows that my degree of belief in heads is at least 1/3. If the least odds at which I would accept a bet against tails were also 2:1, then my belief in heads could be characterized by the interval (1/3, 2/3). This idea has been exploited by C. A. B. Smith to develop an interval valued approach to Bayesian probability (Smith, 1961; 1965). Note again that these intervals fit precisely the structure that Keynes gives to the manifold of probabilities.

So far, then, Ramsey is not apparently at loggerheads with Keynes; there is a dispute about the structure of the probability manifold; and there is the undisputed fact that if we are to assess the rationality of a given "degree" of belief in the manifold, for a given agent, we must have some way of getting into the agent's mind, either by offering alternatives or by a "psychogalvanometer" (Ramsey, 1931, p. 161). In fact, one might think that Ramsey was simply seeking a more thorough foundation for the laws of probability, "which we have proved to be necessarily true of any consistent set of degrees or belief" (Ramsey, 1931, p. 182). This certainly sounds like logic, although the gloss, "If anyone's mental condition violated these laws, . . . [h]e could have a book made against him by a cunning better and would then stand to lose in any event" (Ramsey, 1931, p. 182), is purely pragmatic in tone.

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