# A Demonstration of the Impossibility That Keynes’s Logical Theory of Probability Was Primarily an Ordinal Theory: Keynes’s C Coefficient of the a Treatise on Probability Is a Mathematical Translation of Interval Valued Probability into a Decision Weight Approach That Modifies the Additive, Linear Approach to Mathematical Probability

34 Pages Posted: 22 Apr 2019

See all articles by Michael Emmett Brady

California State University, Dominguez Hills

Date Written: March 23, 2019

### Abstract

It is straightforward to demonstrate that Keynes’s logical theory of probability could not primarily be an ordinal theory of probability. The demonstration is based on three main points made by Keynes.

Keynes’s first major point in the A Treatise on Probability covered Keynes’s extensive discussion of non additivity (the general inapplicability of the addition rule of the calculus of probabilities) in Part II of the A Treatise on Probability (1921), as well as his many worked out problems using interval valued probability in Part II of the A Treatise on Probability (1921). Any attempt to juxtapose ordinal probability and non additivity is an oxymoron because ordinal probability can’t be added or multiplied. Part II of the A Treatise on Probability has nothing to do with ordinal probability.

Keynes’s second major point was that the difficult topic of developing an interval valued approach to probability can be avoided by using decision weights that translate additive probability into non linear and non additive decision weights. Keynes’s conventional coefficient of weight and risk,c, was designed to allow his interval valued probability concept to be translated into non linear ,non additive coefficients or decision weights while starting the analysis using additive probability.It is mathematically and statistically impossible to do this with ordinal probability.

Keynes’s third major point in the A Treatise on Probability shows that Keynes’s discussion of statistical frequencies involves his use of interval valued probability through the use of upper and lower limits or bounds,an approach identical to that used by Keynes in Part II. None of Keynes’s results from Parts II ,III, IV, and V of the A Treatise on Probability using either interval valued probability or decision weights, first introduced by Keynes to deal with the problems of non linearity and non additivity are alluded to by any Post Keynesian or Fundamentalist Cambridge Keynesian in any article or book written by them over the last 40 years.

O’Donnell (2019) and Dow (2019) are have simply overlooked all of Keynes’s analysis in Part II, IV, and V of the TP dealing with measurement. O’Donnell (2019) and Dow (2019) rely completely on a very severe misinterpretation of pages 38-40 of the TP, as well as the diagram on page 39. The diagram on page 39 of the A Treatise on Probability illustrates the linear and additive probabilities of OAI with the nonlinear (quadratic) and non additive nature of the probabilities of all of the other paths. There is no theory of ordinal probability used, developed, deployed, or applied on pp.38-40 of the TP.

Keywords: inexact probability, approximation, non linearity, non additivity, interval valued probability, decision weights, ordinal probability

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

Suggested Citation

Brady, Michael Emmett, A Demonstration of the Impossibility That Keynes’s Logical Theory of Probability Was Primarily an Ordinal Theory: Keynes’s C Coefficient of the a Treatise on Probability Is a Mathematical Translation of Interval Valued Probability into a Decision Weight Approach That Modifies the Additive, Linear Approach to Mathematical Probability (March 23, 2019). Available at SSRN: https://ssrn.com/abstract=3358766 or http://dx.doi.org/10.2139/ssrn.3358766

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