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# Probability values

Keynes repeatedly mentions "degrees" of rational belief, and acknowledges that degrees of belief are in some sense quantitative and "perhaps capable of measurement." However, he ends up by denying, in Chapter 3 of TP, that all probabilities are even capable of comparison. In reviewing arguments in favor of the measurability of probabilities, Keynes (interestingly, vis-a-vis Ramsey) mentions the practice of underwriters when they insure against almost any eventuality: "[b]ut this practice shows no more than that many probabilities are greater or less than some numerical measure, not that they themselves are numerically definite" (TP, p. 23).

Keynes also maintains that "there are some pairs of probabilities between the members of which no comparison of magnitude is possible . . . [nor is it] always possible to say that the degree of our rational belief in one conclusion is either equal to, greater than, or less than our degree of belief in another" (Keynes, 1973 , pp. 36-7). Indeed he indicates that "the closest analogy [to probability] is that of similarity" (TP, p. 39).

This argument suggests, as does the illustration in Keynes (TP, p. 39), reproduced below, that probabilities form a lattice structure (see Fig. 2.1; see also Kyburg, 2003, pp. 140-1). Upper and lower bounds for any probabilities exist, of course—namely, 0 and 1—but the question is whether the meet and join of any two probabilities exist. A definitive answer is hard to come by for Keynes, since he never seems to have put the question to himself. Nevertheless, the list of properties on page 41 of TP suggests that the answer is affirmative. Probabilities lie on paths, each of which runs from 0 to 1. Consider all the paths on which the probability p1 and p2 lie. If they have no points in common other than 0 and 1, then these are the meet and join of pi and pi-. Suppose there is no greatest lower bound, that is, that for every lower bound p such that p < p1 and p < p2, there is a greater lower bound p < p' < p1 and p' < p2. It is hard to imagine that Keynes should think it possible that there should be an unbounded sequence of ever greater lower bounds to p1 and p2, though it is, of course, mathematically possible.

An alternative, and perhaps more up to date, way of looking at probabilities is to think of them as intervals. This is the view of probability adopted in (Kyburg, 1961) and developed most clearly in Kyburg and Figure 2.1 Probability structures (from Keynes, TP, p. 42)

Teng (2001). The set of sub intervals of (0, 1) does form a lattice under the natural ordering (p, q) (r, s) if and only if every point in (p, q) is less than any point in (r, s). The meet of (p, q) and (r, s) is just the degenerate interval (min{r, p},min{r, p}), and similarly for the join. On this interpretation of the values of probability we have no difficulty in accommodating Keynes's graph. Furthermore, to the extent that probabilities are based on our knowledge of frequencies—and surely some probabilities are so based—it is natural to suppose that they are often interval valued, since our knowledge of frequencies is inevitably approximate.

In any event, it is clear that we can find a set of objects that has the structure that Keynes assigned to probabilities, and that this structure is consistent and coherent. What is curious is that the mathematician- philosopher, Frank Ramsey, paid no attention to this structure in his review of the Treatise (Ramsey, 1922), though he did attack the claims that some probabilities were incomparable and that some were nonnumerical.

One writer who did take Keynes's view of seriously was B. O. Koopman (1940a; 1940b; 1941). Koopman showed that if we focus on subsets of probabilities that can be approximated by a numerical net, as can the probabilities generated by well-tuned gambling apparatus, then the standard numerical calculus of probability can be obtained as a limiting approximation.

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