# On the Need for an Extensive Revision of the ‘Imprecise Probabilities’ Entry regarding Boole and Keynes in
*The Stanford Encyclopedia of Philosophy* (Spring, 2019 Edition) in the ‘ Supplement to Imprecise Probabilities-Historical appendix: Theories of Imprecise Belief.’

35 Pages Posted: 17 Dec 2019

Date Written: November 30, 2019

### Abstract

The "Supplement to Imprecise Probabilities-Historical appendix: Theories of imprecise belief " presents a severely inaccurate representation of Keynes’s contributions to imprecise probability. It also completely ignores the seminal, path breaking contributions to imprecise probability made by George Boole in his 1854 The Laws of Thought in chapters 16-21. The error is compounded with regard to Keynes because Keynes’s entire system of logical probability in the A Treatise on Probability is built on Boole’s exposition of lower (greatest lower bound)-upper (least upper bound) probabilities that Keynes used in Parts II and III of the A Treatise on Probability to develop his method of inexact measurement and approximation using interval valued probability.

It is in chapter 15 of the TP on pp. 161-163, as well as in chapter 16, 17, 20 and 22 of the TP, in a very detailed, mathematical analysis of his earlier, brief, graphical exposition in chapter III of the TP, that an analysis of imprecise probability was provided by Keynes. Keynes, through his adaptation of Boole’s original approach, that Keynes had adapted and improved upon in the same manner that Wilbraham had improved the Boolean approach in 1854, allowed for him to provide an explicit, imprecise approach to probability showing the importance of non additivity in 1921.

In 1986 and 1996,Theodore Hailparin showed decisively that the Boole-Keynes approach involved the use of an early version of linear programming techniques, using an initial, basic, feasible solution, involving primal and dual maximization and minimization problems with constraints that were both equalities (linear) and inequalities (nonlinear) [Keynes’s inequations].

Keynes had already solved the mystery of the diagram on page 39 of chapter III in 1921(Actually,the same analysis is provided by Keynes in the 1908 Cambridge Fellowship Dissertation. This Fellowship Dissertation would have earned Keynes a Ph.D in Applied Mathematics at any University in the world at that time if he had been interested in obtaining a Ph.D,which he was not) in chapter 15 with additional problems solved in chapters 16, 17, 20, and 22 of the TP.

Keynes’s analysis on pp.161-163 is straightforward if the reader of the TP has the requisite mathematical training and knowledge.It is highly likely that no economist ,who has written on Keynes’s TP,has such knowledge.

The presentation of Keynes’s theory in the appendix to the entry on’ Imprecise Probability’ is completely misleading. The failure to mention Boole’s path breaking work is a lacuna that can only be remedied by an extensive revision. Hopefully,such a revision will finally set right the historical record regarding the original and detailed mathematical and logical contributions made by Keynes in 1907,1908 and 1921, and by Boole in 1854.

**Keywords:** Keynes, Boole, Imprecise Probability, Upper-Lower Bounds, Inexact Measurement, Approximation, Part II of the TP

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

**Suggested Citation:**
Suggested Citation

*The Stanford Encyclopedia of Philosophy*(Spring, 2019 Edition) in the ‘ Supplement to Imprecise Probabilities-Historical appendix: Theories of Imprecise Belief.’ (November 30, 2019). Available at SSRN: https://ssrn.com/abstract=3495817 or http://dx.doi.org/10.2139/ssrn.3495817