Confusing Metaphors With Mathematics in Chapter 6 of the A Treatise on Probability, When Analyzing Keynes’s Modeling of the Evidential Weight of the Argument V(a/h ), Leads to confusion: If V(a/h)=w, As Defined in Chapter 26, Where 0≤w≤1, Then It Is Mathematically Impossible That Keynes’s Weight Is Monotonically Increasing
23 Pages Posted: 13 Nov 2018 Last revised: 28 Dec 2019
Date Written: October 21, 2018
Philosophers and economists continue to erroneously claim that Keynes’s Evidential Weight of the Argument, V(a/h) and/or his degree of the weight of the evidence, w, is always increasing monotonically over time based on a reading of a metaphor Keynes used in chapter 6 to help explain why the Evidential Weight of the Argument is different from probability. The Evidential Weight of the Argument will increase if there is additional relevant evidence, data, information or knowledge. It is also the case that we know more if we later discover additional relevant evidence that establishes that we, in fact, did not know what we thought we knew .What is required to actually measure weight mathematically ,as opposed to logical considerations, is an index that is normalized on the unit interval between 0 and 1 that is identical to the normalization of probability on the unit interval between 0 and 1 so that one can talk about degrees. Any attempt to discuss Keynes’s concept of weight is an exercise in futility unless the correct mathematical structure has been provided first so that one can talk about degrees of probability and degrees of knowledge and ignorance. Such a structure was not provided by Keynes in Chapter 6 of the A Treatise on Probability. However, Keynes did create such a structure in chapter 26 of the A Treatise on Probability. First, Keynes had to define V(a/h) to be equal to some type of degree. Keynes’s designated this variable as w, where w is normalized on the unit interval and represents the degree of completeness of the relevant evidence. Therefore, V(a/h) = w performs the same task for Keynes as α does for his other logical relation, P(a/h) =α ,0≤α≤1.
W measures the degree of the completeness of the relevant evidence, so that V(a/h)= w .Keynes then combines P and V in chapter 26 into his “conventional coefficient of weight and risk, c”, where c is also normalized on the unit interval between 0 and 1 for the specific form chosen by Keynes that integrates both non additivity, sub additivity, and non-linearity in the decision weights chosen by Keynes, which were 2w/(1 plus w) and 1/(1 plus q),respectively. It is mathematically impossible for w to be monotonically increasing in the decision weight, w, chosen by Keynes to illustrate his concepts on non-additivity and non-linearity, where w=K/(K plus I),where K denotes knowledge and I denotes Ignorance. Thus, α =p/(p plus q) and 1-α =q/(p plus q) while w= K/(K plus I) and 1-w=I/(K plus I).
The major error committed in all journal articles and books published over the last sixty years that discuss Keynesian weight is the complete misinterpretation of the following quote from page 77 of the A Treatise on Probability:
“The fundamental distinction of this chapter may be briefly repeated. One argument has more weight than another if it is based upon a greater amount of relevant evidence; but it is not always, or even generally, possible to say of two sets of propositions that one set embodies more evidence than the other. It has a greater probability than another if the balance in its favor, of what evidence there is, is greater than the balance in favor of the argument with which we compare it; but it is not always, or even generally, possible to say that the balance in the one case is greater than the balance in the other. The weight, to speak metaphorically, measures the sum of the favorable and unfavorable evidence, the probability measures the difference.”
Keynes is talking, as he himself so clearly stated, metaphorically. However, just as Adam Smith’s purely metaphorical statements about an Invisible Hand, as discussed by Gavin Kennedy for over 20 years, have been grossly misunderstood, so were Keynes’s statements.
Current assessment of Keynes’s weight concept is erroneously based on a metaphor Keynes used before he presented the mathematical analysis in chapter 26 of the TP on page 77 of chapter 6. Keynes’s chapter 26 is the omega while chapter 6 is the alpha. It is impossible to fully grasp Keynes’s weight of the evidence concept without a detailed study of chapter 26 of the TP.
Keywords: Chapter 26 Chapter 6 Edgeworth, Russell, Interval Valued Probability, Interval Valued Weight, W, C Coefficient
JEL Classification: B10, B12, B14, B16, B20, B22
Suggested Citation: Suggested Citation