# Keynes’s Application of Inexact Measurement and Approximation in Chapter 15 of the A Treatise on Probability Directly Conflicts with R. O’Donnell’s Claims in His Chapter 3 concerning Keynes’s Approach to Measurement in His 1989 Book, 'Keynes, Philosophy, Economics, and Politics'

14 Pages Posted: 5 Jun 2020

Date Written: May 11, 2020

### Abstract

The claim that Keynes’s non numerical probabilities are ordinal probabilities was shown to be mathematically impossible by Keynes himself in Part II in chapter 15 of the A Treatise on Probability(1921) on pp.160-163 and in chapter 17 on pp.186-194, since Keynes’s non numerical probabilities are identical to Boole’s constituent probabilities. Keynes improved on Boole’s technique and was able to solve Boolean problems much quicker than it took Boole to solve the problems.Part II of the A Treatise on Probability is nearly identical to the analysis provided in his two Cambridge University Fellowships in 1907 and 1908.

R. O’Donnell(1989, p.60) attempted to analyze a part of page 160 of the A Treatise on Probability that dealt with Keynes’s inexact measurement and approximation approach using interval probability, but failed to comprehend that the discussion directly contradicts his claims concerning ordinal probability made earlier in his chapter 3 on pp.50-59. His claim that Keynes’s nonnumerical probabilities are ordinal probabilities that can be placed within upper and lower bounds is simply a mathematical error.

O’Donnell’s citation of the relevant quote on p.160 of chapter 15 of the A Treatise on Probability leaves out Keynes’s italics emphasis on the word “between”, as well as Keynes’s discussion of the concept of standard probabilities. O’Donnell fails to inform the reader that Keynes provides a detailed mathematical analysis of standard probabilities starting from the last half of page 161 and continuing on pp.162-163.This analysis directly refutes O’Donnell’s claims on pp.50-59 that Keynes’s non numerical probabilities are ordinal.

It is even stranger that O’ Donnell recognizes that Keynes’s position was that his method of approximation and inexact measurement was “…frequently adopted in common discourse” in real life.O’Donnell’s admission that inexact measurement and approximation is “ of frequent occurrence in ordinary life”(O’Donnell, p.60) directly negates O’ Donnell’s unsupportable claims about ordinal probability once it is realized that Keynes’s non numerical probabilities are identical to Boole’s constituent probabilities in the mathematical derivation of standard probabilities, which are interval valued probabilities.

All of O’Donnell’s results in his 1989 book(his 1982 dissertation) are based on the claim that Keynes’s nonnumerical probabilities are ordinal. An examination of p.60 of his book leads to the strange conclusion that Keynes built his logical theory of probability on ordinal probability while, at the same time, arguing that the use of interval valued probability was a frequent occurrence in ordinary life.

None of the major results in O’Donnell’s dissertation are left standing due to his complete and total failure to recognize that Keynes’s non numerical probabilities can’t possibly be ordinal probabilities, given that Keynes’s approach comes directly from George Boole, who not only is not mentioned in O’Donnell’s dissertation (1982) or book(1989), but not mentioned by O’Donnell in any publication in his life.

**Keywords:** interval valued probability, non numerical probability ,constituent probability, standard probability, approximation, inexact measurement, non additive probability

**JEL Classification:** B10, B12, B14, B16, B18, B20

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