**J M Keynes’s A Treatise on Probability (1921): A Preparation Guide for Non –Mainstream/Heterodox Readers**

*(Pages: 95-106)*

Michael Emmett Brady*

*California State University, Dominguez Hills, College of Business Administration and Public Policy, Department of Operations Management 1000 Victoria St, Carson, California 90747.*

#### Abstract:

Extensive preparation is required in order for a potential reader to be able to successfully read and under-stand Keynes‘s analysis in the A Treatise on Probability, 1921. The crucial, prerequisite source is George Boole‘s 1854 The Laws of Thought .Boole, using his propositional logic, provided the first theoretical derivation of upper and lower probabilities in history on pp.265-268 of The Laws of Thought. Boole then extended his limits to least upper bounds and greatest lower bounds .Boole applied his approach to many problems on the next one hundred plus pages of The Laws of Thought (1854)… Keynes completely accepted Boole‘s theoretical result and built on Boole, although he incorrectly questioned Boole‘s technique in arriving at answers for some of his worked out problems. Boole also presented the first technically advanced logical theory of probability in history, using propositions and not events .Keynes built on Boole‘s logical foundation using propositions. However, it is practically impossible to understand the technical way in which Boole applied his pp.265- 268 result in his later chapters on probability ex-cept for simple problems. Wilbraham‘s 1854 article in the Philosophical Magazine provided a significantly improved exposition over Boole‘s, which Boole himself adopted in his later work after 1854. Theodore Hailperin (1986) showed even more clearly what it was that Boole was doing. Boole was applying linear programming techniques. Miller (2009) discovered a missing, important step that is necessary in Boole‘s own approach that also makes his so-lutions technique clear. However,it lacks the generality of Hailperin‘s approach based on linear programming. Of course, to read this material requires that the reader have at least a BS (BA) degree in either mathematics or statistics or their equivalent.The reader with these degrees, or the equivalent, is now prepared, not to read the A Treatise on Probability, but to read the two F. Y. Edgeworth reviews, the Bertrand Russell review, and the review by CD Broad in that precise order. Anyone attempting to read the A Treatise on Probability without having satisfied the above mentioned prerequisites might fail to understand Keynes‘s technical analysis. A common result will be to conclude that it is Keynes who must have been in error. This is the common conclusion arrived at by reviewers like H Jeffries (1931), I J Good (1962), H.Mellor (1995), R Monk (1991) and D Gillies (2003). The potential reader can then start to read the A Treatise on Probability with confidence. However, the first chapter to read is not chapter 1, but chapter 26.After reading chapter 26, the reader can then start on chapters 1-3.It is crucial to absorb Edgeworth‘s point in his reviews that Keynes‘s probabilities are always between two numbers, i.e., interval valued. Part II requires careful reading as regards Keynes‘s points about non additivity, i.e. the probabilities can‘t be added so as to sum to one on all occasions. This will be obvious to a reader who understands p.268 of The Laws of Thought. Part III of the A Treatise on Probability is built on an adapted problem from Boole analyzed by Keynes in Part II, Boole‘s problem X. Part V is built on Part III .Especially important is Keynes‘s work on Chebyshev‘s Inequality in Part V. Keynes tied his analysis of R, Least Risk, in Part IV in chapter 26 to his analysis in Part V on Chebyshev‘s Inequality .This allowed him to derive what may have been the first ―Safety First‖ approach in history.

#### Keywords:

Approximation, c coefficient, imprecise probabilities, indeterminate probabilities, interval valued probability, upper and lower probabilities, weight of the evidence,and uncertainty.