# Given Keynes's Definitions of Logical Probability and Evidential Weight, It Is Impossible for Keynes’s Approach to Measurement to Be an Ordinal Theory; His 'Non Numerical' Probabilities Must Be Based on Inexact and Imprecise Measurement Using Approximation Involving Boolean Upper and Lower Probability Bounds and Bounded Outcomes

30 Pages Posted: 7 Mar 2019

Date Written: February 16, 2019

### Abstract

Keynes spent all of his introductory Chapter One of the A Treatise on Probability emphasizing that his logical relation of probability (a relation of similarity – dissimilarity based on analogy) P(a/h)=α, where α came in degrees and 0≤α≤1. In chapter 6, Keynes introduced his logical relation of the evidential weight of the argument, V(a/h), but did not specify what it equaled, although he clearly stated that he would integrate both probability and weight together later. In chapter 26, Keynes specified that V(a/h)= w, where we came in degrees and 0≤w≤1. Keynes showed how to integrate both probability and weight together in his conventional coefficient of weight and risk, c, in chapter 26 of the A Treatise on Probability. Probability and weight were also integrated together by Keynes using the theory of inexact measurement or approximation based on intervals with upper and lower bounds or limits. Ordinal probability can do none of these things. Advocates of the claim that Keynes’s theory was an ordinal theory of probability concentrate exclusively on a misinterpreted diagram that appears near the end of chapter 3 of the A Treatise on Probability. Thus, while Keynes’s theory can easily deal with ordinal probability with the aid of the Principle of Indifference, it is not part of Keynes’s main or major analysis in the A Treatise on Probability, which was interval probability. The same conclusion holds for numerical probability and rational expectations. Rational expectations are a very special case that occurs if w=1.

The heterodox, institutionalist, or Post Keynesian approaches, involving irreducible, fundamental, or radical uncertainty, are a very, very special case that occurs only if w=0 in the very long run. Their argument, that measurement is not possible at all, be it exact or inexact, is rejected by Keynes in chapter 4 of the General Theory.

Thus, while Keynes’s theory can easily deal with ordinal probability, it is impossible for the ordinal probability to deal with inexact and imprecise interval-valued probability measurement. Keynes’s theory of measurement involves interval-valued probability in the A Treatise on Probability with the numerical and ordinal probability being very special cases of his general theory of interval-valued probability. Keynes’s approach to measurement in the General Theory also involved inexact and imprecise measurement as deployed by Keynes in chapter 4 of the General Theory. No ordinal probability approach to measurement is used by Keynes in the General Theory.

The great value of the 1936-1938 Keynes-Townshend exchanges is that they show that Keynes presented an approach to measurement that was imprecise and inexact in both the A Treatise on Probability and the General Theory.

**Keywords:** Measurement, Ordinal, Numerical, Interval, Exact Precise, Inexact, Imprecise, Keynes, a Treatise on Probability

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

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