Keynesian Uncertainty Can Only Be Represented by Imprecise, Non Additive, Interval Valued Probability or Decision Weights Like Keynes’s C: Ordinal Probability Can’t Represent Keynesian Uncertainty

Abstract

J M Keynes’s two logical relations of rational degree of probability, α, 0≤α≤1 and Evidential Weight of the Argument, w, 0≤w≤1, where w measures the degree of completeness of the evidence, can’t be represented or associated with ordinal probability, although Keynes’s theory of probability can easily deal with ordinal probability with the aid of Keynes’s principle of indifference if symmetries are present. α can be, in some limited instances, represented by a numerical, precise, definite, exact, additive probability if, and only if, w=1, although, in general, for w<1, it must be represented by an non additive interval estimate of probability or by a decision weight, like Keynes’s original, path breaking innovation of his conventional coefficient, c. Nowhere in Boole’s 1854 The Laws of Thought is any concept of ordinal probability discussed analyzed or applied in any detail. This is because ordinal probability can never deal with overlapping estimates of probability, which creates problems of non comparability, non measurability or incommensurability that Boole and Keynes solved with interval valued probability.