# A Guide to Help Prepare Readers for J M Keynes's A Treatise on Probability (1921)

63 Pages Posted: 7 Jun 2017 Last revised: 18 Jun 2017

Date Written: June 7, 2017

### Abstract

Extensive preparation is required in order for a potential reader to be able to successfully read and understand Keynes’s analysis in his A Treatise on Probability, 1921. The crucial, prerequisite source is George Boole’s 1854 The Laws of Thought. For instance, 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 then applied his approach to many problems on the next 100 plus pages of The Laws of Thought (1854). Boole’s work appeared 86 years before Koopman’s similar results were published in 1940. 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 likewise 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 except 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. In 1986, Theodore Hailperin 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 everything crystal clear, but lacks the generality of Hailperin’s approach based on linear programming techniques.

Of course, to read this material requires that the reader have, at a bare minimum, a BS (BA) degree in either mathematics or statistics, as well as BA degrees in philosophy and economics, 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.

Thus, anyone attempting to read the A Treatise on Probability without having satisfied the above mentioned prerequisites will certainly fail to understand Keynes’s technical analysis. A common result will be to conclude that Keynes must be in error. This is the H. Jeffreys, R. Braithwaite, I. J. Good, H.Mellor, R. Monk and D. Gillies conclusion.

After having accomplished all of the above, 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 ties his analysis of R, Least Risk, in Part IV in chapter 26 to his analysis in Part V on Chebyshev’s Inequality in order to derive the first “Safety First” theory in history.

After reading the A Treatise on Probability, the reader can now, if he chooses to, take a look at the many reviews of the A Treatise on Probability that are essentially nonsense. These reviews would include the two F. Ramsey reviews, the three R. Braithwaite reviews, the two Harold Jeffreys reviews, the Ronald Fisher review, the Arne Fisher review, the Raymond Pearl review, the first E. B. Wilson review, and the R. von Mises review.

The reader will now find it straightforward to read the General Theory because the General Theory is based on a direct connection between Keynes’s Theory of Liquidity Preference and the rate of interest, which is based directly on chapter 26 of the A Treatise on Probability concept of evidential weight, especially Keynes’s coefficient analysis dealing with W, the weight of the evidence. Keynes’s definition in the General Theory of uncertainty on p. 148 in chapter 12 was that uncertainty is an inverse function of the weight of the evidence.

Keynes’s original IS-LM, four simultaneous equation model of 1933-1934, has an explicit independent variable, W, in all of the equations that is called the “state of the news”. It appears in both of Keynes’s IS and LM equations. W, the “state of the news”, is related to his W variable, the weight of the evidence, from chapter 26 of the A Treatise on Probability. It accounts for changes in how the degree of confidence will be assessed as new additional, relevant evidence (news) occurs. The decision maker then changes his probabilistic expectations of the future because, as David Champernowne stated clearly in his 1936 review, there would be changes in the “nervousness” of the decision maker about the “uncertainty” of his future expectations of profit.

**Keywords:** weight, interval vaued probability, upper and lower probabilities, nonnumerical probabilites, liquidity preference

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

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