# On J M Keynes's Original Contributions to Decision Making Under Uncertainty: Indeterminate, Interval Valued Probabilities in Part II of the a Treatise on Probability and Imprecise, Interval Valued Probabilities in Part V of the a Treatise on Probability

29 Pages Posted: 15 Nov 2017 Last revised: 17 Nov 2017

Date Written: November 11, 2017

### Abstract

All of the logical, statistical and mathematical foundations for Keynes’s work on uncertainty in the General Theory and his 1937 Quarterly Journal of Economics reply article can be found in Parts II and V of Keynes’s A Treatise on Probability.

Keynes, building on the original, mathematically developed, logical theory of probability first put forth by George Boole in his The Laws of Thought in 1854, demonstrated how to construct upper and lower probabilities that would define an interval valued probability. Keynes dealt with indeterminate probabilities first. Indeterminate probabilities are interval valued probabilities where the range between the upper and lower bounds or limits will not diminish as new additional data or evidence is examined because there will always be important relevant evidence that will never be attained and is permanently missing or unavailable when a decision or assessment must be made. The great American mathematician, Edwin Bidwell Wilson, acknowledged this begrudgingly in 1934 in his JASA article titled, “A Problem of Boole’s.”

Keynes also developed the first approach to “Safety First” by showing how to use Chebyshev’s Inequality to provide a lower bound or limit. This imprecise lower bound or limit would be based on the initial, small amount of data or information that was available to the decision maker at the start of his assessment of a problem. However, over time, as more and more relevant data and information became available, the decision maker would be able to make use of more accurate and reliable statistical methods related to the Normal distribution. The Normal distribution estimate would then serve as an upper bound. The range between the lower and upper bounds would then diminish as larger and larger samples were used. Eventually, a precise probability estimate would result. The lower bound would be relevant at the initial stages of a estimation process. Keynes’s views on imprecise probability estimates, in cases where significant, additional data and information can be obtained over time and a decision to act can be postponed indefinitely, imply that eventually a precise probability would be able to be calculated that was stable.

When Keynes defined uncertainty to be an inverse function of the weight of the evidence, he automatically supported his GT with a foundation consisting of the mathematical and logical analysis contained in Parts II and V of the TP on indeterminate probabilities, imprecise probabilities, the weight of the evidence, and the conventional coefficient of weight and risk. Unfortunately, 81 years after Keynes wrote the GT and 96 years after Keynes published his TP, economists are still unable to read Parts II and V of the TP. Economists, like the French mathematician, Emile Borel, and the American mathematician, Edwin Wilson, found that Parts II and V of the TP were just too overwhelming for them.

**Keywords:** Uncertainty Risk Keynesian Indeterminate Probabilities, Keynesian Imprecise Probabilities, Weight Of The Evidence

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

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