Assessing the philosophical issues
One of the problems, of course, arises whenever we speak, on the one hand, of the normative, and, on the other, of the subjective. Of course our knowledge is subjective, in the sense that it belongs to a subject. On most views it is individuals who have knowledge. On the other hand, most of us feel that there is a real if fuzzy distinction between rationality and irrationality, and it was to this feeling that Keynes alluded in his remarks on Ramsey's view of probability. The man who has sampled an urn, in the most scientific manner, and drawn a great many black balls and no white balls, would still be entitled, on Ramsey's view, to be quite sure that the next ball he draws will be white.
These two issues may be usefully distinguished. It seems clear, as Carabelli (1988, ch. 3) emphasizes, that in the Treatise Keynes was more attuned to the intuitionism and focus on ordinary discourse of G. E. Moore than to the atomism of Russell and the early Wittgenstein. Moore argued, for example, that the good was an unanalyzable simple property, and therefore subject to direct perception. This did not render the perception of the good "subjective" in the pejorative sense of whimsical or arbitrary or "relative." No more did the fact that the probability relation was simple and unanalyzable and therefore indefinable render it arbitrary for Keynes.
Now, accounting for such relations and properties, and especially for our knowledge of them, is another matter, and much of what Ramsey said could be construed as casting doubt on the reality of indefinable relations that cannot be perceived. It is this that led Ramsey to the view that it did not belong to formal logic "to say what a man's expectation of drawing a black ball" should be. This is what divides them: Keynes sees an indefinable logical relation where Ramsey sees nothing.
It is perhaps the current atmosphere of relativism that makes it hard to come to a judgement about this issue. After all, if a man cannot "see" that in itself pleasure is better than pain, what arguments can sway him? If the tortoise cannot "see" that modus ponens is truth preserving, to what argument form can Achilles' turn?
We cannot even easily say that the burden of proof should be on the believer: it is as easy to fail to see the moons of Jupiter as it is to succeed in seeing the aura of the saintly. Many writers now, in philosophy and in computer science as well as in economics, agree with Ramsey's position that, once a person's degrees of belief satisfy the probability calculus, there is no more than can be asked of him. The role of probability is analogous to the role of logic: logic embodies our demand that a person's qualitative beliefs be logically consistent; probability embodies our demand that the degrees of a person's beliefs be consistent in the broader sense captured by the probability calculus. (This is to leave aside such issues as finite or countable additivity and the like.)
The one chink in the armor is that these "degrees of belief" do not seem to exist. When I introspect, I can no more find those degrees of belief than Ramsey was able to find the indefinable probability relations to which Keynes referred. We can solicit bits of behavior that can be characterized by numbers; some degree of measurement appears to be possible. If I offer you better odds than (say) 6:5 on heads on the toss of a coin, you'll probably take the bet. But a real valued function for a field of alternatives is less plausible. Sometimes it has been suggested that the agent could be "forced" to post odds and make book. Alternatively, it has been suggested that the agent should be forced to choose between finely distinguished alternatives. We have already mentioned that Koopman adopted the idea of throwing a grid over the only partly ordered degrees of probability of Keynes in order to make a connection to ordinary probability theory.
But none of this suggests that people's beliefs come in real values. If intuition suggests anything—the very same intuition that made Ramsey doubt the existence of Keynes's indefinable logical relations—it suggests that there are some relations of probability that are extremely indeterminate. What is the probability that the next species to be discovered will be aquatic as opposed to terrestrial? It might be possible for a biologist to come up with a reasonable constrained interval, but for most of us the answer would be "I haven't the faintest idea," or the whole open interval (0, 1).
Keynes indeed found the vagueness of our assessments of evidence the explanation for our interest in money: "our desire to hold money as a store of wealth is a barometer of the degree of distrust of our own calculations and conventions concerning the future. . . . The possession of actual money lulls our disquietude" (quoted in Carabelli, 1988, p. 169).
But Keynes's real interest was economics and social welfare. He took economics to be a matter of intuitive imagination and practical judgement. He did not devote himself to logic itself other than in the brief foray into probability. It is thus not surprising that Keynes's system of epistemology and logic contained certain unclearnesses and ambiguities.
One of these, which typically remains unremarked on by economists writing on Keynes, is the status of our "probable knowledge." Suppose that e is our total body of knowledge, and that h is some proposition that interests us. If h is "the next toss of this coin will land heads," and e is ordinary knowledge, we would suppose that h/e is close to a half (but not precisely a half!). But Keynes also considers cases in which h corresponds, for example, to "the conclusions of The Origin of the Species" (TP, p. 118). In these cases the proposition that is rendered probable by the evidence is accepted. It is accepted because, although the evidence does not entail it—it is not deducible from the evidence—the evidence does render it highly probable.
There is an important issue here, one focused on by Carnap (1968). When we say we "accept" the conclusions of Darwin's work, what we mean is ambiguous and loose, but what we are accepting may be one of two quite distinct things: that these conclusions, relative to the evidence we have supporting them, are very highly probable; or we may simply be accepting these conclusions. Hempel (1962) noted the distinction between accepting a conclusion because the evidence for it was very great and accepting a probability conclusion that assigned a probability to a conclusion. In some cases it may not be clear what is intended. Thus "Between 400 and 500 of the next thousand tosses of this fair coin will land heads" is sometimes what we want to conclude, and sometimes we want to conclude rather "The probability is very high that between 400 and 500 of the next thousand tosses of this fair coin will land heads." Surely we want to say that Darwin's basic conclusions are acceptable, not, except in the context of a discussion of scientific method, that they are "probable." Carnap's view, however, was that we are speaking loosely only when we talk of "acceptance" of general theories.
As a matter of speculation, I suspect that both Ramsey and Keynes supposed that in some sense one could "accept" conclusions that were rendered probable enough by the evidence, in some informal sense of
"probable." Although I take this to be a serious epistemological question for philosophy, it is surely not a question that bothers practical people. What separated Ramsey and Keynes very sharply was the issue concerning the nature of probability: was it a psychological property related to actual beliefs? Or was it a logical relation between a proposition and the body of evidence bearing on it to which the actual beliefs of any agent having that evidence should conform?
Although some feel that Keynes gave in on this question in the 1931 notice (Keynes, 1972), the issue does not seem to be settled. First of all, Keynes never mentioned his thesis that probabilities are only partially ordered, and so those probabilities to which Ramsey's arguments apply may constitute a small fraction of probabilities. Second, of the (possibly) small fraction of probabilities to which Keynes acknowledged that Ramsey's arguments did apply, Keynes felt that there were more rational constraints than were captured by mere obedience to the probability calculus.
In a sense, this is an unsatisfying conclusion. That Keynes did not roll over and give in doesn't mean that he was right; and that Ramsey did not succeed in persuading Keynes does not mean that Ramsey was wrong. Nor is this an empty issue—it is an issue of great importance not only in philosophy but in artificial intelligence, in statistical inference, in practical deliberation, and in the social sciences.
It is important in philosophy because, if Ramsey is right, the whole logic of rational belief is captured by the probability calculus; otherwise all is permitted. On the other hand, many people think that the evidence renders certain beliefs irrational.
The issue is important in artificial intelligence for the same reason: are there constraints that degrees of belief should satisfy? Or is one coherent distribution as good as another?
The question of objectivity is very important in statistical inference: if there are no objective constraints, it is hard to know how differences of opinion regarding statistical conclusions can be resolved.
My own conviction is that Keynes was right—that probability is best construed as a logical relation. I think of my own efforts to characterize probability along these lines as in Keynes's tradition, though my view of "logic" is rather more conventional than Keynes's. In any event, the large issue, which I commend to your reflection, is suggested by Ramsey: Is it the business of logic to constrain (at all, since we should not dismiss the possibility that probabilities are partially ordered) the belief of a man in drawing black ball from an urn (given any amount of evidence)? Put otherwise: are there degrees of belief that are irrational in themselves given the available evidence, or can degrees of belief be irrational only in relation to other degrees of belief? Allow any admixture of logic in answering these questions—for example, that it is irrational for a man who has conducted a thousand random draws from a bag, obtained only white balls, and is willing to offer ten to one that the next draw will produce a black ball—and you are in conflict with the common and vocal subjectivist interpretation of probability.
If Ramsey is right, then the rational agent has (ideally) a precise realvalued degree of belief in any proposition, though that degree of belief need not be determined on any objective grounds. If that is the case, then perhaps we should not suppose that "high probability" warrants acceptance. If general conclusions are accepted at all, it is for reasons of computational convenience, simplicity, and so on. On the other hand, if Keynes is right, and there is an objective (logical) truth of the matter about probability, it is hard to see how that can fail to involve objective general knowledge—that is, how it can fail to be based on some sort of rational acceptance. For example, what can be the objective basis of assigning a probability of about a half to heads on the toss of a coin, if not the empirical and corrigible generalization that in general coins land heads about half the time? Of course, this acceptance would be non-monotonic: further evidence could lead us to withdraw our acceptance of that generalization, since further evidence could lower its probability. We could learn that coins of this specific kind land heads more than three quarters of the time.
This issue of non-monotonic inference or inductive inference raises a whole new collection of problems; and it is true that Keynes did not get very far with these problems, although he devoted many pages of the Treatise to them. On the other hand, these are problems that most practical people seem to suppose can be solved somehow or other. They are certainly not problems on which we should turn our backs. To examine them closely, however, is a big job, and a job for another occasion.