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Black and white balls

That there is an out-and-out head-to-head conflict between Keynes and Ramsey—perhaps not fully appreciated by either—becomes clear in the fourth part of Ramsey's essay. He repeats there his claim that he cannot "see what these inconclusive logical relations can be or how they can justify partial beliefs" (Ramsey, 1931, p. 185). Now it is all very well for Ramsey modestly to admit that he sees no logical relation of probability such as the one that Keynes seeks to draw our attention to, but it is clear that Ramsey wants to go further than that. Ramsey wants to claim that there is no such relation.

The connection between partial belief and choice behavior, according to Ramsey, is provided by mathematical expectation and the Dutch Book Argument. That we do not find the argument entirely valid does not keep it from being suggestive, and of course there is a connection between mathematical expectation and behavior. Ramsey regards the two interpretations (partial belief and frequency) as "the objective and subjective aspects of the same inner meaning" (Ramsey, 1931, p. 188). Whatever this may mean, it is alleged to relieve us of the need for a principle of indifference—a principle not loved by Keynes, and whose mechanical imposition he inveighs against. It is worth noting, however, that, as Koopman showed, we do not need the full power of the Principle of Indifference to support the mathematics of the probability calculus in those cases in which there is an appropriate approximation.

What is much more to the point is Ramsey's frank avowal of subjectivism: he writes, "we do not regard it as belonging to formal logic to say what should be a man's expectation of drawing a white or a black ball from an urn; his original expectations may within the limits of consistency be any he likes" (Ramsey, 1931, p. 189).

It is true that Ramsey was thinking of the Principle of Indifference here, and emphasizing in his own mind the word "original." It might be thought that Ramsey is saying merely that logic can provide no a priori probability of drawing a white ball. But this is not what he says: what he says is that logic can be of no help, even, one supposes, in the face of a long sequence of draws of black balls. Ramsey does not use the words "prior" or "a priori." It is only given his "initial expectations" that he is bound to have certain others. "This is in line with ordinary formal logic, which does not criticize premises but merely declares that certain conclusions are the only ones consistent with them" (Ramsey, 1931, p. 189).

But though this view of uncertain inference is echoed today, for example, by Halpern (Fagin and Halpern, 1988) and Morgan (1998; 2000), it presupposes exactly that complete ordering of the probability manifold that Keynes (and Smith, and many others) was at pains to deny. Perhaps a man's rational expectation of drawing a black or a white ball is wholly indeterminate: the whole interval (0,1) or more generally, on Keynes's view, a member of the manifold that is comparable only to 0 and 1. To be sure, this does not solve the problem of induction (but Keynes was very skeptical of any mechanical solution to that problem) for if the initial probability is wholly indeterminate any application of Bayes's theorem will leave the conditional probability wholly indeterminate as well. Furthermore, if a man's initial expectation may be any he likes, he is surely free to adopt that initial expectation that leads to whatever expectation conditioned on the evidence he wants. P{he) = P(h A e)/P(h A e) + P(> h A e), so by manipulating the two

absolute probabilities in the denominator we can make the conditional probability have any value we want. One can no more admit a little subjectivity into these matters than one can be a little bit pregnant.

The issue that divides Ramsey and Keynes most deeply is the issue of objectivity. Keynes believes that there is an objective logic of nondemonstrative inference; Ramsey takes non-demonstrative inference to be arbitrary and subjective. For Keynes, even though he cannot spell out the conditions of partial validity for this sort of inference, there is a logical fact of the matter that we can often perceive, if only dimly. For Ramsey this is 'metaphysical moonshine' (Levi, 1986, p. 27; see also de Finetti, 1964). It may be conjectured that the plausibility of subjectivism was partly a matter of the popularity of Einstein's theory of relativity at the time. "The degree of a belief is in this respect like the time interval between two events; before Einstein it was supposed that all the ordinary ways of measuring a time interval would lead to the same result if properly performed. Einstein showed that this was not the case" (Ramsey, 1931, p. 167).

Whatever the source of Ramsey's relativism with regard to partial belief, it was clearly far-reaching—in fact more far-reaching than perhaps he understood himself. It is sometimes claimed that Keynes was led to revise his views by Ramsey's essay, but that is not clear. Keynes did write in Ramsey's death notice for The New Statesman and the Nation:

Ramsey argues, as against the view which I had put forward, that probability is concerned not with objective relations between propositions but (in some sense) with degrees of belief, and he succeeds in showing that the calculus of probabilities simply amounts to a set of rules for ensuring that the system of degrees of belief which we hold shall be a consistent system, thus the calculus of probabilities belongs to formal logic. But the basis of our degrees of belief . . . is part of our human outfit, ... So far I yield to Ramsey—I think he is right. But in attempting to distinguish "rational" degrees of belief from belief in general he was not yet, I think, quite successful.

(Keynes, 1972 [1933], pp. 338-9)

But this is not throwing in the towel. The issue remains: is there an objective difference between "rational" belief and belief in general? It seems to me that Keynes clearly still believes that there is.

Some writers—for example, I. J. Good (1950) and Harold Jeffreys (1954-55)—maintain that it was only briefly that Keynes flirted with objectivity in probability, and that he was primarily a subjectivist. Economists tend to be more balanced; for example, O'Donnell (1989) and Skidelsky (1983), though Carabelli (1988, ch. 3) is uncompromising in taking Keynes to have an objective view. Bateman (1987, p. 107) claims that in 1931 Keynes was willing "to accept a subjective epistemic theory." Runde, even in an article subtitled "In Defence of A Treatise on Probability," writes, "After all, he [Keynes] does accept Ramsey's demonstration that 'the calculus of probabilities simply amounts to a set of rules for ensuring that the system of degrees of belief which we hold shall be a consistent system'" (Runde, 1994, p. 115).2 This whole issue of subjectivity and objectivity is complicated by the ambiguity of the terms and also by the role that intuition played in Keynes's general epistemology. It is to this philosophical issue, though not to Keynes's treatment of it, that we next turn.

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