Philosophical Theories of Probability

Rating:
Summary: Keynes's Ch.3 of the TP(1921)is an introduction only.
Review: Gillies(G)does a very good job covering the subjective,frequency,classical and propensity theories of applied probability.His discussion of his co-authored intersubjective approach,which is to be intermediate between subjective and logical theories of probability ,is interesting but not convincing to this reader.G fails in his exposition of logical theories in general and Keynes's logical theory specifically.G bases his discussion of Keynes's approach on two chapters of the A Treatise on Probability(1921)[TP],chapters 3 and 4 plus the error filled F.Ramsey reviews of 1922 and 1926,as well as Keynes's 1931 4-page review of Ramsey's published works.G thus commits the same error made by Ramsey in his two reviews.Ramsey completely confuses the meaning of "non-numerical probabilities" as well as the meaning of nonmeasurable.Keynes meant for chapter 3 of the TP to be an introduction aimed at interested readers with little or no training in mathematics,logic or statistics.A CAREFUL reading lends no support to Ramsey.On p.34,paragraph 3,Keynes clearly states that by nonmeasurable he means not by "any numerical relation".Thus most probabilities are "non-numerical",which means "not by a single numeral or number".It is in chapters 5,10,15,16 and 17 of the TP that Keynes explains his statements on pages 31-32 of the TP about "numerical limits" and "...since the probability lies between(emphasis added by Keynes)two numerical measures."For Keynes ,ALL probabilities are either precise point estimates(numerical probabilities)or non-numerical interval(set)estimates which are imprecise.Imprecise probabilities are nonmeasurable because they are non-numerical.They are non-numerical because two numbers,not one,are required in order to specify the probability relationship.Consider the two following interval(set)estimates[.4,.6] and [.5,.7].They are Keynesian non-numerical probabilities.They are noncomparable,nonrankable and incommensurable.On pages 135-138 of the TP,Keynes provides a generalized axiomatic analysis of the addition and multiplication rules for probability[conjunction and disjunction]that apply to both numerical and non-numerical probabilities.The axioms operate on sets of propositions.Keynes makes valuable extentions to these axioms in chapters 8 and 15.Unfortunately,neither Ramsey nor Gillies read any of these chapters.All of Gillies'references in this book are to chapters 3 and 4.Given that Keynes builds his entire approach to induction and analogy,in Part 3 of the TP ,on both numerical and non-numerical probabilities,Gillies needs to rewrite those parts of his book dealing with Keynes's logical theory of probability.