# The Unbridgeable Gulf Forever Separating A. Smith and J. M. Keynes from Bentham, Classical Economists, Neoclassical Economists, And ‘Modern’ Economists (Friedman, Becker, Stigler, Lucas, Sargent, Wallace, Muth, Kydland, Prescott, etc.): The Formal and Mathematical Concepts of Uncertainty, Weight of the Evidence, and Confidence

33 Pages Posted: 15 May 2017 Last revised: 20 May 2017

Date Written: May 13, 2017

### Abstract

It is logically impossible for Classical Economists, Neoclassical Economists, and “Modern” Economists (Irving Fisher, Friedman, Becker, Stigler, Lucas, Sargent, Wallace, Muth, Kydland, Prescott, etc.) to grasp Smithian and Keynesian economics because both Smith and Keynes explicitly incorporated a specific term, weight of the evidence, into their decision theories to deal with uncertainty besides an additive concept of probability. F. Knight’s distinction between risk and uncertainty is flawed because, although he tried, he failed to explicitly integrate such a term as the “weight of the evidence” into his theory of decision making in order to differentiate uncertainty from risk mathematically and logically. Knight needed an understanding of non-additivity. Integrating such a term into decision theory automatically creates a violation of the purely mathematical laws of the probability calculus because the probabilities can no longer be precise (additive), but must be modeled mathematically as interval valued probabilities (non additive). A concept of weight of evidence thus violates the additivity, complementarity, and linearity requirements needed for the application of either the frequency approach to probability, using objective, precise probabilities or the Subjective, Bayesian approach to probability of Ramsey, de Finetti, Savage, and M Friedman upon which Subjective Expected Utility is build, which requires precise, subjective probabilities.

George Boole explicitly provided the first explicit logical and mathematical foundation for the concept of upper and lower probabilities, or interval valued probability, in 1854 with the publication of The Laws of Thought in which he put forth the first explicit logical theory of probability in history. Keynes built on Boole’s foundation and based his logical theory of probability and decision theory on interval valued probability. However, this non-additive and nonlinear approach directly conflicts with the linearity and additivity constructs that underlie the subjective and frequency approaches to probability of I. Fisher, Friedman, Becker, Stigler, Lucas, Sargent, Wallace, Muth, Kydland, Prescott, etc. It is not possible for subjectivists to deal with weight because it would automatically make the subjective theory of probability and SEU theories very special theories that could never be more than a limiting case of the work of Adam Smith, George Boole, and J M Keynes.

Thus, the intellectual conflict over the role of the weight of the evidence and uncertainty variables in decision theory, which completely divided Smith from Cantillon and Bentham in the eighteenth century, also completely divided Keynes from the Classical and Neoclassical economists of the 20th century. Practically all of the “What did Keynes mean in the General Theory?” literature can be traced back to economists’ gross ignorance of the Keynesian concept of the weight of the evidence and its role in the General Theory. Once Keynes introduced the weight of the evidence variable into the General Theory on p.148, the probabilities(expectations) automatically become interval valued. Non-additivity and non-linearity become the general case, while classical and neoclassical theories based on Benthamite utilitarianism’s use of additive and linear precise probabilities, becomes a very special case of Keynes’s.

Keynes’s concept of liquidity preference is directly founded on the logical and mathematical analysis of the weight of the evidence that he incorporated into his conventional coefficient of weight and risk, c, in chapter 26 of the A Treatise on Probability. The conventional coefficient , which substitutes a mathematically more tractable analysis for the much more difficult Boolean interval valued approach, adapted with revisions by Keynes for use in Part II of the A Treatise on Probability, allows a reader to see the explicit role that weight plays in Keynes’s theory .Part II of the A underlies Part III and Part III underlies Part V of the A Treatise on Probability. Liquidity preference automatically brings in interval valued probability if w, the weight of the evidence, is less than 1.

Keynes’s analysis of imprecise probability, which occurs in chapter 29 the A Treatise on Probability, is based, as acknowledged by Edgeworth in 1922, on using Chebyshev’s inequality to form a lower bound with the upper bound being given by probability estimates using other known probability distributions, such as the Normal distribution. As more evidence is accumulated, improved estimates will start to approach the upper bound and become precise. Keynes’s analysis of indeterminate probabilities takes place in chapters 3, 5, 10, 15, 16, 17, 20, 22, and 26 of the A Treatise on Probability.

“Modern” Economists (Irving Fisher, Friedman Becker, Stigler, Lucas, Sargent, Wallace, Muth, Kydland, Prescott, etc.) can’t deal with Keynes’s general theory of decision making, which underpins the General Theory, because all of their theoretical approaches must use precise probabilities that are additive.

**Keywords:** Weight, Interval Valued Probability, Upper and Lower Probabilities, Nonnumerical Probabilites, Liquidity Preference

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

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