An Analysis of Edwin B. Wilson's Secret, Second Review of the 'A Treatise on Probability' in 1934: How It Demonstrated That Keynes's Theory of Probability Was an Interval Valued Approach to Probability and Not an Ordinal Theory

Abstract

Wilson admitted in private correspondence in 1923 with Francis Ysidro Edgeworth that he should not have attempted his review of the A Treatise on Probability (1921) in 1923.The reason Wilson gave to Edgeworth in his correspondence was exactly the same as the reason Edgeworth gave Wilson as the reason Edgeworth was writing to Wilson-a lack of understanding about what Keynes was doing in Part II of the A Treatise on Probability. What Keynes had done, of course, was revolutionary. Keynes had built on the earlier, logical theory of Boole. Keynes had erected a general theory of probability where numerical probabilities appeared as a special case when the weight of the evidence, w, defined on the unit interval [0,1], where a w=1 defined a complete evidence set upon which to base a decision maker’s estimates of probability, equaled, approximated or approached 1 w measured the completeness of the evidence. If the weight of the evidence equaled 1 and the decision maker had linear probability preferences, then precise, exact, numerical probabilities could be calculated.

Keynes ‘s axiom system in Part II of the A Treatise on Probability was set up to deal with both interval valued probabilities, which are non additive, and numerical (cardinal) probabilities, which are additive. Non additive probabilities are not subject to the mathematical laws of the probability calculus, the addition rule and the multiplication rule. Keynes used interval valued estimates to deal with the problems of uncertainty and ignorance, where missing relevant information, data, knowledge, or evidence meant that a decision maker could not use numerical probabilities because Keynes’s weight of the evidence variable, w, was less than 1. Only in the case of a complete information set could numerical probabilities or a probability distribution be specified.

In 1934,Wilson published a note titled, Boole’s Challenge Problem,in the Journal of the American Statistical Association. It is actually a disguised, camouflaged second review of J M Keynes’s A Treatise on Probability that deals with Keynes’s Part II again. This time, however, Wilson finally gets the technical analysis right.

Wilson’s note, although he himself had no idea about why dealing with conditions of uncertainty and ignorance was relevant when making a decision, totally and completely destroys the logical and intellectual foundations erected by G. L. S. Shackle and Paul Davidson ,as well as Jochen Runde, Shiela Dow, Rod O’Donnell, Tony Lawson, Anna Carabelli, Robert Skidelsky, Gay Meeks,Bradley Bateman, Donald Gillies ,and many other Post Keynesians, Institutionalists, and Keynesian Fundamentalists, who have claimed, on the basis of papers published by Frank Ramsey and Richard Braithwaite, for nearly 40 years that J M Keynes’s theory in the A Treatise on Probability is an ordinal theory that could only be applied some of the time or had to be interpreted as a relative frequency theory to make any sense out of it (Davis, Winslow).

The two books that supposedly claim to deal with Keynes’s approach to uncertainty, The Philosophy of Keynes’ Economics: Probability, Uncertainty, and Convention, 2003, edited by Runde and Mizuhara,and Fundamental Uncertainty, 2011, edited by Marzetti Dall’Aste Brandolini and Scazzieri contain no mention of either George Boole or interval valued probability.Thus,they have nothing to say about Keynes’s interval valued approach to probability.

Keywords: indeterminate probability, interval valued probability, Boole's Challenge problem, J M Keynes's approximation

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