Comparative indifference principle
Although the aforementioned principle of sufficient comparability is too strong, it is assumed in order to investigate the problem of initial probabilities. Hence there are two kinds of initial comparative judgements: those with the same evidence and those with the same hypothesis. Following Keynes (TP, ch. 4, s. 14), let us call the former 'preference judgements' and the latter 'relevance judgements'. When a relevance judgement is an equality, let us call it an 'irrelevance judgement'. And when a preference judgement is an equality, let us call it an 'indifference judgement':
The latter kind of judgement is very important for determining initial probabilities and so we shall investigate it further. Let us assume that c is constituted by the sentences cv..cN. To establish whether (8) is true or false, it is sufficient to analyse a series of more elementary irrelevance judgements of the form:
However, in general not all sentences of c are irrelevant for the hypotheses a and b. Let us order the sentences of c so that the first M (M < N) are relevant for a. That is:
Thereafter, let us order c according to the relevance for the hypothesis b. Let us assume that there are also M elements of c relevant for b - not necessarily the same as in the case of a. If these conditions hold then in order for the indifference judgement (8) to hold it is sufficient that for each ct (1 < i < M) relevant for a there is an element cj (1 < j < M) belonging to c - not necessarily different - which is equally relevant for b. The correspondence between ci and cj must be injective.
In other terms, for (8) to be true, it is sufficient that for each ct such that p(a/Ci) Ф p(a/~Cj ) there is a Cj (different for each different ct) such that:
To sum up, we have taken indifference judgements of type (8) back to more elementary judgements of type (9).
Moreover, hypotheses a and b must be indivisible.11 We say that hypothesis h is indivisible in the language Z when there are not two sentences к and m belonging to Z, such that h = к v m, and for both there is at least one relevant sentence in Z. That is, if h = к v m for each л: must be:
Therefore, it is possible to trace the notion of divisibility back to the evaluation of elementary judgements of irrelevance. It should be emphasized that this notion of indivisibility is efficacious only if the language Z is sufficiently rich. Indeed, if we artificially limit the language Z so that there are no sentences к and m such that h = к v m and (10) is violated but conceptually this division is possible, then the notion of indivisibility becomes useless.
Hence, it is possible to evaluate indifference judgements of the form (8) if it is possible to evaluate elementary judgements of type (9) and irrelevance judgements of type (10).
Judgements of form (8) are very important because they are the core of the celebrated 'principle of indifference', which in a context of comparative probability assumes the following form:
Comparative indifference principle: if there are reasons not to prefer any indivisible hypothesis at of a set a1...an, with respect to the available evidence b, then we can reasonably hold that p(a1/b) = p(a2/b) =...= p(an /b).
If a and b are indivisible, we avoid the well-known paradoxes arising from the unequal division of the possibilities space. In Chapter 4 of his TP, Keynes discusses this problem extensively. Of course, the language in which the sentences whose probability concerns us are to be expressed must be large enough to admit indivisible sentences that are actually equi-possible. Indeed, it is easy to introduce limitations in the expressive capacity of the language, so that it is not reasonable for indivisible hypotheses to be equiprobable.
As mentioned above, evaluations of type (8) are based on judgements of types (9) and (10). How are such evaluations possible? Keynes (TP, ch. 5, s. 5) presupposes an intrinsic capacity of human understanding to evaluate a priori elementary judgements of forms (9) and (10). In the next paragraph we follow a different track.
Following Strevens (1998), who unconsciously rediscovers some ideas12 expressed by von Kries (1886), and going against Keynes (TP, ch. 7, s. 8), who blames von Kries for 'physical bias', we state that elementary judgements of forms (9) and (10) could find their justification in the symmetrical character of the physical system being considered. As maintained by Franklin (2001), the fact that there are reasons for choosing initial probabilities does not mean that these reasons must be of a logical character, as believed by Keynes (TP), and above all by Carnap (1950). As shown by Festa (1993), it is possible to determine initial probabilities on the basis of the cognitive context in which we are operating. As mentioned above, and according to Verbraak (1990) and Castell (1998), the indifference judgements that appear in the principle of indifference do not have the following negative form: there are no reasons for choosing one hypothesis rather than another.
Instead, they have the following positive form: there are reasons for not choosing one hypothesis rather than another.
These reasons are based on the symmetry of the context of investigation. From this perspective, it is possible to talk about a true naturalization of the indifference principle. Paraphrasing a perceptive remark of Bartha and Johns (2001, p. 110), we can say that the principle of indifference is to the probability calculus as the red light district to our big cities; they have always been there and they will always be there, but they will never be altogether respectable.