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The Collected Writings of John Maynard Keynes: Volume 8, A Treatise on Probability

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In this path breaking contribution to the logic of probability,Keynes showed how to adapt the work of George Boole for the purpose of estimating probabilities.For Keynes there are only two types of probability estimates,point estimates and interval estimates.Unfortunately,Keynes decided to call interval estimates "non-numerical"probabilities.His reasoning is really quite obvious.A precise estimate of probability used a single numeral for the point estimate.Therefore,an imprecise estimate of probability used two numerals to denote an interval(set).Thus, an interval estimate is not based on a single numeral but two. These types of probabilities are thus "non-numerical"because you are not using a single numeral.In 1922 and 1926,Frank Ramsey reviewed Keynes's book based on his reading of chapters 1-4 plus 3 pages from Part two and 4 pages from Part five.Keynes's discussion of non-numerical probabilities takes place in chapters 5,10,15 and 17.Keynes then applies his new approach to induction and analogy in chapters 20 and 22,using his concept of "finite probability"which applies to both precise numerical probabilities and imprecise non-numerical probabilities.All of Keynes's discoveries ,however,were ignored by the ignorant Ramsey.To this day(2004)one can regularly read about Keynes's "strange,mysterious,unfathomable,undefined"non-numerical probabilities in literally hundreds of economics and philosophy journal articles based on Ramsey's reviews.Keynes then showed that interval estimates,because they overlap,would very likely also,in many cases,be noncomparable and/or nonrankable if a decision maker used such order preserving operators like"greater than or equal to"or "less than or equal to".While this is quite obvious,it went completely over Ramsey's head. Keynes's second major advance was to create his "conventional coefficient of weight and risk", (...) The goal of the decision maker is to Maximize cA.This decision rule solves all of the paradoxes and anomalies that plague subjective expected utility theory.In Part 5 of his book Keynes showed how one could use Chebyshev's Inequality as a lower bound to the normal probability distributions overly precise point estimate . Part 5 of the Treatise also includes Keynes's advocacy of the Lexis Q test for stability of a statistical frequency[law of large numbers].This will then bring the reader back to Keynes's chapter 8 of the Treatise where he presents his own logical frequency interpretation of probability as a special case of his general logical approach to probability.