# The Roles of Keynes’s Logical Relations, V(a/h) =w and P(a/h) =alpha, in His Conventional Coefficient of Weight and Risk, C

26 Pages Posted: 19 Apr 2018 Last revised: 29 Apr 2018

See all articles by Michael Emmett Brady

California State University, Dominguez Hills

Date Written: April 19, 2018

### Abstract

Keynes’s chapter 26 analysis, contained in the A Treatise on Probability, is designed to give a reader of the A Treatise on Probability a command of the subjects of non additivity and non linearity, upon which Keynes’s interval valued approach to probability was constructed in chapters 3, 15-17, 20, 22, and 29, which he called approximation and modelled after the work of George Boole, without having to master the exceedingly difficult Part II of the A Treatise on Probability. Part II of the A Treatise on Probability was deliberately skipped by Raymond Pearl, Frank Ramsey, Ronald Fisher, Emile Borel, Francis Ysidro Edgeworth, Bruno de Finetti, Edwin Bidwell Wilson, and Richard Braithwaite in their reviews of the A Treatise on Probability. Practically all other readers of the A Treatise on Probability in the 20th and 21st centuries also skipped Part II, due to the extreme difficulties encountered in trying to follow Keynes’s analysis. The exceptions were Edgeworth, Russell, and Broad. Edgeworth and Russell mastered chapter 26 of the A Treatise on Probability. This allowed them to figure out that Keynes was using interval valued probability, which Keynes called “non numerical probabilities.” Both Russell and Edgeworth were attracted to the conventional coefficient of weight and risk, c.

The c coefficient integrates both of Keynes’s logical relations of probability and weight, P and V, in a mathematically specific manner. Keynes’s c is the first decision weight in history.

However, no logical relations appear in the c coefficient because they are not mathematical variables. You can’t normalize a logical relation. Thus, for example ,Keynes’s notation, P(a/h) = α, states that the degree of rational belief, given the probability relation P between a and h, is of degree α .h is the premises and a is the conclusion. c can then be normalized on the unit interval [0,1] to give α =p/(p plus q)[1-α=q/(p plus q)],where p is the probability of success and q is the probability of failure and 0≤ α ≤1.P(a/h) can’t be normalized. Similarly, V(a/h) denotes the Evidential Weight of the argument. V(a/h)=w states that the degree of evidential support for a, given h, is of degree w. w can then be normalized on the unit interval [0,1] to give w =K/(K plus I)[1-w=I/(K plus I)], where K is the amount of knowledge, data, information or evidence that is relevant and I is the amount of ignorance. Basic errors in both logic and mathematics result in the claim that the logical relation, V, can be normalized on the unit interval, that the logical relation V can be set equal to K/(K plus I), V=K/(K plus I), where K and I are mathematical variables, and that 0 ≤V ≤1. Such errors are the direct result of a complete failure to grasp the basics of Keynes’s c coefficient.

All heterodox economists have followed Runde (1990) in making the obvious mathematical errors that V=K/(K plus I) and that 0≤ V ≤1. It is completely unclear to this author how a logical relation can be restricted to the unit interval.

Keywords: Weight, Interval Valued Probability, C Coefficient, Logical Relations, P, V

JEL Classification: B10, B12, B14, B16, B20, B22

Suggested Citation

Brady, Michael Emmett, The Roles of Keynes’s Logical Relations, V(a/h) =w and P(a/h) =alpha, in His Conventional Coefficient of Weight and Risk, C (April 19, 2018). Available at SSRN: https://ssrn.com/abstract=3165298 or http://dx.doi.org/10.2139/ssrn.3165298