# Keynes’s Major Result From Part II of the A Treatise on Probability Was That, Given That Numerical Probabilities Are Additive, Then Non Numerical Probabilities Must Be Non Additive: Non Additivity Is a Sufficient Condition for Some Degree of Uncertainty to Exist

43 Pages Posted: 23 Dec 2019 Last revised: 24 Dec 2019

Date Written: December 5, 2019

### Abstract

Keynes’s major accomplishment in Part II of the A Treatise on Probability (1921),which he also accomplished in the 1907 and 1908 Fellowship Dissertation versions, was to show that the addition property of the purely mathematical calculus of probability could only be operational in certain circumstances where the evidential weight of the argument, V(a/h) =degree w, was equal to a w of 1,where 0≤w≤1, so that the decision maker had 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, 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 the obvious fact that ordinal probabilities are not only not non additive, since they can’t sum to less than 1, but they also can’t be multiplied. Keynes then 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, of course, followed directly from pp.265-268 of Boole’s 1854 The Laws of Thought.

The adherents of all Heterodox, Post Keynesian, and Institutionalist schools of economics (Skidelsky, O’Donnell, Carabelli, Dow, Lawson, and Runde, for example. See the references) argue that Keynes’s non numerical probabilities were ordinal probabilities. However, this conclusion is an oxymoron, because ordinal probability can have absolutely nothing to do with questions concerned about additivity of probabilities summing to 1 or non additivity of probabilities not summing to one, but being less than 1, since, by definition, ordinal probabilities can’t be summed, added or multiplied. Numerical and ordinal probability can’t deal with non measurability, non comparability or incommensurability. Such problems require nonlinear and non additive approaches to measurement. Only Keynes’s and Boole’s initial ,imprecise approaches to probability, using interval valued probability, can do this. It should be obvious to a mathematically trained reader that Boole and Keynes are the founders of the imprecise probability approach to decision making.

Unfortunately, there are no economists or philosophers who covered Part II of the A Treatise on Probability in the 20th or 21st centuries. All readers, like Emile Borel, the French mathematician, skipped Part II of the A Treatise on Probability. This is why the obviously false claim, that Keynes’s non numerical probabilities had to be ordinal probabilities, which directly contradicts Keynes’s 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:** Additive, Non additive, precise, imprecise, Keynes, Boole, Hailperin, ordinal, interval valued probability

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

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