From the coefficient of influence to the relevance quotient

First of all, we would like to discuss briefly the question of the paternity of the notion of coefficient of influence. Rudolf Carnap, who in 1950 (Carnap, 1950, s. 66) gave to the "coefficient of influence" the name of "relevance quotient," wrote: "Keynes gives a definition of the relevance quotient [...] and a series of theorems on this concept, based upon unpublished notes by W. E. Johnson" (Carnap, 1950, p. 357).

This sentence is obscure. In fact, regarding the coefficient of influence it is not clear whether either the definition or the related theorems are of William Ernst Johnson. On this respect, Keynes is rather clearer. In fact, he explicitly says (Keynes, 1973a [1921], p. 150 footnote)

The substance of the propositions (41) to (49) below [theorems regarding the coefficient of influence] (Johnson, 1924; 1932) is derived in its entirely from his [Johnson's] notes—the exposition only is mine.

Thus Keynes explicitly recognizes the theorems are from Johnson but this does not regard the definition of the quotient. In this respect he says (TP, p. 151):

It is first of all necessary to introduce a new symbol. Let us write

We may call {a^{h}b} the coefficient of influence of b upon a on hypothesis h [...]; we may also call {a^{h}b} the coefficient of dependence between a and b on hypothesis h.

The symbolism of Keynes is cumbersome essentially because he did not use any notation for the probability function, as is clearly shown by (1). In the next section, when we set out the definition in modern symbols, we will make the meaning of (1) clear. At present we note, first, that, with regard to his Definition XV, Keynes speaks of a new symbol; second, that, in saying that propositions (41) to (49) are from Johnson, he implicitly denies that the definition was suggested by this author. Hence we are entitled to suppose that the formal statement of the notion of coefficient of influence is his own. We stress that in no way this diminishes the great intellectual debt Keynes owes to Johnson. To underline this debt we recall the dedication that begins Irving John Good's The Estimation of Probabilities: "This monograph is dedicated to William Ernst Johnson the teacher of John Maynard Keynes and Harold Jeffrey" (Good, 1965).

To the best of our knowledge nobody used the coefficient of influence until Carnap began his work on inductive logic. This author stressed the importance of the study of the relevance of a sentence i upon an hypothesis h given evidence e. Carnap singled out two ways of expressing the relevance: one is the relevance quotient, ^{c(h}'^{e л i)}; the other is

c(h, e)

the relevance difference c(h, e л i ) — c(h, e). These notions are introduced by means of the quantitative concept of confirmation c(h, e), to be read: the degree of confirmation of h with respect to e, which is the probability function Carnap used in his studies on the foundation of probability. After having stated a few theorems about the relevance quotient, Carnap built up his theory of relevance in terms of the relevance difference, thus neglecting the coefficient of influence. However, before moving on to the theory of relevance, referring to this coefficient, Carnap wrote

The concept explained will hardly be used in the remainder of this book. It has been represented in this section in order to call attention to an interesting concept which deserves further investigation.

(Carnap, 1950, p. 360)

But this investigation was not performed in the following 30 years. Once more the coefficient of influence, in the meantime turned into the relevance quotient, was neglected. At the end of the 1970s we have used this notion to introduce a condition capable of solving a problem Carnap left open. We shall deal extensively with this problem after specifying the context in which we work.