Keynes’s Major Result from Part II of the A Treatise on Probability was that Non Numerical Probabilities must be Non Additive

(Pages: 01-07)

Michael Emmett Brady*

California State University, Dominguez Hills, College of Business Administration and Public Policy, Department of Operations Management, 1000 East Victoria St, Carson, California 90747, US

Abstract:

Keynes’s major accomplishment in Part II of the A Treatise on Probability (1921) was to show that the addition property of the mathematical calculus of probability could only be operationalized in certain circumstances where the evidential weight of the argument, V (a/h) = w, was equal to a w of 1, where 0≤w≤1, so that the decision maker had a complete data/evidence set. All relevant information or evidence had be known before any decision had to be made. Thus, all numerical probabilities are additive ,so that they would sum to 1.However,Keynes argued that there were also non numerical probabilities ,which were non additive because of the existence of missing ,relevant data or evidence, that would not sum to 1.By far the most important case was sub additive, as opposed to super additive, probabilities, which would sum to less than 1.Note that ordinal probabilities can’t be non-additive, since they can’t sum to less than 1 . Keynes showed that non numerical probabilities , which were non additive because they summed to less than 1, had to be interval valued probabilities with a lower (greatest lower bound)and a upper(least upper bound) bound. This followed from Boole’s 1854 The Laws of Thought (Boole, 1854, pp.265-268).

The adherents of the Heterodox ,Post Keynesian, and Institutionalist schools of economics such as Skidelsky (1983,1992,2010), O’Donnell (2003,2012,2014), Carabelli (1985,1988,1994,1995,2003,2009), Dow (2003), Lawson(1985), and Runde (1994,2003), for example, all argue that Keynes’s non numerical probabilities were ordinal probabilities. However, this conclusion can’t be correct, because ordinal probability has nothing to do with the question of whether probabilities are additive or non-additive. Numerical and ordinal probability can’t deal with nonmeasurability, non-comparability or incommensurability. Such problems require nonlinear and non-additive approaches to measurement. Keynes’s and Boole’s initial, imprecise approaches to probability, using interval valued probability, can do this. Boole and Keynes should be recognized as the founders of the imprecise probability approach to decision making.

It is unclear whether any economists or philosophers covered Part II of the A Treatise on Probability in the 20th or 21st centuries. It appears that readers, like Emile Borel, the French mathematician, skipped Part II of the A Treatise on Probability. This is why the claim, that Keynes’s non numerical probabilities had to be ordinal probabilities, which directly contradicts Keynes’s technical position that they are non-additive, continues to be generally accepted nearly one hundred years after the publication of Keynes’s A Treatise on Probability. 

Keywords:

Non-additive, Imprecise probability, Non-linear, Interval valued probability, Lower bound, Upper bound, Indeterminate.