A Comparison of J. M. Keynes’s Logical Approach to Probability and Any ‘Objective Bayesian’ Approach to Probability Needs to Incorporate All Five Parts of Keynes’s a Treatise on Probability, Not Just Part I

Philosophers, historians, economists, decision theorists, and psychologists have been repeating a very severe error of omission for nearly a hundred years that was originally made by the French mathematician Emile Borel in his 1924 review of the A Treatise on Probability, 1921. Borel decided to skip Parts II through V of the A treatise on Probability. He explicitly apologized to Keynes at the beginning of his review for his decision involved in skipping Part II, acknowledging to Keynes, correctly, that Part II was the most important part of the A Treatise on Probability.

Borel’s acknowledgment and apology are, in fact, an understatement, because without an understanding of Part II,it is impossible to understand Keynes’s theory of decision making and the role played by that theory in the General Theory(1936). This all comes out in the Keynes-Townshend exchanges of 1937 and 1938, where Keynes makes it crystal clear to Townshend that his theory of liquidity preference is built on his non numerical probabilities, which a reading of Part II makes clear are interval valued probabilities, each with an upper bound and a lower bound. These probabilities are non additive. Keynes’s definition of uncertainty on page 148 of chapter 12 in footnote 1 defines uncertainty as an inverse function of Keynes’s evidential weight of the argument, defined on the unit interval between 0 and 1. Any probability with a w < 1 is an interval valued probability that is non additive. The only way to discuss Keynesian uncertainty is by non additive, interval valued probability or by decision weights like Keynes’s c coefficient.

D. P. Rowbottom attempts a defense of Keynes’s position against J. Williamson’s intellectual attacks which I view as correct. However, Rowbottom badly handicaps himself by providing a defense of Keynes’s position that is limited to the use of Part I of the A Treatise on Probability. Rowbottom could have presented an overwhelming counter argument against Williamson if he had understood Keynes’s concepts of interval valued, non additive theory of imprecise probability from Part II of the A Treatise on Probability, Keynes’s finite probabilities from Part III, Keynes’s decision weight translation of imprecise probability in chapter 26 of Part IV and Keynes’s inexact, approximation approach to statistics in Part V that Keynes combined with his application of Chebyshev’s Inequality for establishing the lower bound of a probability estimate.

Starting with the 1940 work of Koopman and continuing through the work of,for example H. Kyburg,Jr.,I. Levi, I. J. Good,and then on to the work of for example, B.Weatherson, D. Rowbottom, B. Hill, S. Bradley and practically all other academics who have written on Keynes and imprecise probability, the exact same error of omission has kept on repeating itself over and over again for a 100 years.