Inductive warrant and weight of argument
Let E be the information about the relative frequency of blacks and whites obtained in a sample with replacement from the urn confronting X in the toy example. What constitutes an inductive warrant for adding HB to K+E? K is the inquirer's initial state of full belief or background knowledge. K+E is the state of full belief obtained by adding E to K together with all the logical consequences. Remember that, relative to K+E, X is committed to being certain of the logical consequences of K and E while judging it a serious possibility with positive probability that HB is false. If there is an inductive warrant for expanding K+E by adding HB, it is a warrant for becoming certain that HB. After one becomes certain that HB, there is no point in inquiring further as to the truth or falsity of HB. As far as that issue is concerned, the weight of argument for or against HB has reached a maximum.
This line of thinking seems to be consonant with Keynes's own, although his remarks at most gesture in this direction. Keynes explicitly claimed that the weight of argument associated with the probability argument a/h is always equal to the weight of argument associated with ~a/h. For an argument is always as near proving or disproving a proposition as it is to disproving or proving its contradictory (TP, p. 84).
Thus, according to Keynes the weight of the argument a/h—that is, the weight of the argument in favor of a afforded by h—is equal to the weight of the argument ~a/h—that is, the weight of the argument against a afforded by h. This suggests that lurking behind weight of argument are two dual notions of positive warrant or support b(a/h) for a by h and negative warrant or support d(a/h) for a by h. Clearly, d(~a/h) = b(a/h).
The idea is that given the weight of the argument from h to a, that weight can support or "prove" to some degree that a or it can disprove it or it can do neither. When it supports a, it disproves ~a.
Given the pair of values d(a/h) and d(~a/h) [i.e., b(~a/h) and b(a/h)], the weight of argument for or against a given h is the maximum value in the pair.
What happens in cases where the weight of the argument in favor of a equals the weight of the argument in favor of ~a? Keynes introduced the idea of "nearness" to proving or to disproving and, hence, of positive and negative warrant. But he did not seem to consider explicitly the case where the negative and the positive warrants for proposition a relative to h are equal except insofar as it can be teased out of applications of Insufficient Reason to probability judgements. If the probability of a given h equals the probability of ~a/h, the weight of argument in favor of a and in favor of ~a relative to h ought to be the same (see TP, pp. 79-80).
Let h and the background knowledge entail that exactly one of a, b, and c is true and that the conditions for applying the Principle of Indifference obtain. Each of the propositions gets probability 1/3. According to Keynes (TP, pp. 78-9) the weights of the arguments from h to each of the three alternatives are equal. So are the weights of the arguments for each of the negations.
The circumstances just envisaged are precisely of the sort where the argument inferring a from h is "as near" proving a from h as the argument inferring ~a from h is near to disproving a from h.
Consequently, we cannot take a measure of proximity of the inference from h to a as proof of a to be the probability 1/3 of a (see Keynes, TP, p. 80). The proximity of the inference from h to ~a would then be 2/3. But the one inference is supposed to be as close to proof as the other is. We could take the proximity of the inference from h to a to a proof and the proximity of the inference from h to ~a to a proof to be Уг or we could take it to be any non-negative value we like including 0.
The important point is that, whatever value we take, the proximity of a given h to proof and the proximity of ~a given h to proof are both at a minimum. An increase in the proximity of a given h to proof corresponds to a decrease in the proximity of ~a given h to proof (that is, to an increase in the proximity of ~a given h to disproof) and vice versa. Keynes's own appeal to the notion of proximity to proof and disproof as characterizing weight of argument hints at this much.
This suggests that 0 is a convenient value to adopt for the minimum. And, of course, Ramsey's argument is most compelling precisely when the data h provide minimum proof for a and for ~h.
Once more, by Keynes's own principles, the weight of argument for a v b given h ought to be equal to the weight of argument for ~a v ~b given h. The proximity to full proof of the former ought to equal proximity to full disproof of the latter and vice versa.
Here it seems plausible to take a step beyond Keynes's explicit discussion. The proximity of a v b given h to full proof ought to be the minimum of the proximity of a given h to full proof and of b given h to full proof. Likewise the proximity of ~a v ~b given h to full proof ought to be equal to the minimum of a given h to full disproof and of b given h to full disproof. No argument can come closer to proving a v b from h than the argument from h to the conjunct that is least close to full proof.
These observations are based on very slender threads of textual evidence in Keynes. The reasoning I have sketched is based, nonetheless, on suggestions that are found in Keynes himself when focusing on the evaluation of the weights of different hypotheses given fixed evidence h.
This reasoning points to the idea that the b-functions or d-functions used to define weight of argument given fixed evidence h exhibit the properties of George L. S. Shackle's measures for potential surprise or disbelief and the dual notion of degree of belief (see Shackle, 1952; 1961). The formal properties of Shackle-type belief and disbelief parallel those I have sketched for proximity to proof and disproof. One can use the one measure or the other to represent weight of argument.
Space does not permit illustration of this understanding of weight of argument. Nonetheless, I suggest that the notion of weight of argument that Keynes was seeking might be interpreted by reference to the specificity of conclusions warranted with a Shackle degree of confidence relative to K (see Levi, 1967, 1984, 1996 and 2001 for elaboration).