A Road Map for Economists, Logicians, Philosophers, Mathematicians, Statisticians, Psychologists and Decision Theorists Seeking to Follow the Mathematical Structure of Keynes’s Approach to Specifying Lower and Upper Bounds for Probabilities in the A Treatise on Probability, 1921

This paper concentrates on Keynes’s solution to a version of Boole’s Challenge Problem of 1851. The problem is solved by Keynes mathematically at the end of Chapter 15 of the A Treatise on Probability, 1921. A study of this problem demonstrates Keynes’s understanding of Boole’s technique and shows that Keynes’s approach is a mathematical approach. The paper concentrates on presenting a step by step study of Keynes’s analysis of this problem. The paper demonstrates that Keynes’s “non-numerical” probabilities are identical to Boole’s constituents and that Keynes’s standard probabilities can be derived so as to specify upper and lower bounds for probabilities, i.e., intervals. It is demonstrated that Frank P Ramsey did not understand the concept of an interval estimate. His critique of Keynes's theory is based on a poor reading of chapter 3 of the A Treatise on Probability (TP, 1921). Ramsey completely overlooks Keynes’s worked out mathematical analysis in chapters 15 and 17 of the TP