# The Relevance Quotient: Keynes and Carnap

Domenico Costantini and Ubaldo Garibaldi

## Introduction

The well-known, but not widely read, book of John Maynard Keynes, *A Treatise on Probability* (hereafter *TP)* (Keynes, 1973a [1921])—essentially written more than ten years before it was published in 1921 by Macmillan & and Co. in London—consists of five parts. Part II, entitled "Fundamental Theorems," comprises Chapters 10-17. In the introductory chapter of this part the author says

In Part I we have been occupied with the epistemology of our subject, that is to say, with what we know about the characteristics and the justification of probable knowledge. In Part II I pass to its formal logic. I am not certain of how much positive value this Part will prove to the reader. My object in it is to show that [...] we can deduce by rigorous methods out of simple and precise definitions the usually accepted results.

(TP, p. 125)

Here the influence of Bertrand Russell is clear, as the author himself acknowledges at the beginning of Chapter 10. After having stated the axioms of probability and recalled some basic theorems of necessary (logical) inference, Keynes in Chapter 14 of *TP,* titled "The Fundamental Theorems of Probable Inference," gives the proofs of what he calls the "most fundamental theorems of Probability" *(TP,* p. 158). He proves the sum and product rules (after having defined the notions of irrelevance and independence), theorems on relevance, Bayes's theorem (which he calls the "inverse principle", and some theorems on the combination of premises. In section 8 (the final one) he introduces the notion of coefficient of influence or coefficient of dependence. We focus on this coefficient in the present chapter. After sketching a brief history of this notion, we consider a condition of invariance defined in term of a coefficient very close to Keynes's. Then we show some consequences of this condition in both inductive logic and statistical mechanics, and we present two economic applications of the condition of invariance. Finally, we make some comments on probability in economics.