# 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

See all articles by Michael Emmett Brady

California State University, Dominguez Hills

Date Written: December 5, 2019

### Abstract

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

Suggested Citation

Brady, Michael Emmett, 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 (December 5, 2019). Available at SSRN: https://ssrn.com/abstract=3498783 or http://dx.doi.org/10.2139/ssrn.3498783

## Paper statistics 