On Ramsey’s Rejection of Keynes’s Mathematical Lattice Structure and Interval Valued Probabilities in the A Treatise on Probability
Authors: Brady Michael Emmett 
Published: July 2024

Henry E. Kyburg was able to demonstrate that Keynes had a qualitative, graphical understanding of interval probability based on his careful analysis of Keynes’s diagram on page 39 of the A Treatise on Probability. Kyburg showed in four different papers, published in 1995,1999,2003 and 2010, that Keynes’s diagram on page 39 is a mathematical lattice structure encompassing interval valued probability. However, Kyburg rejected any conclusion that Keynes had provided a mathematical, quantitative theory of interval valued probability. Kyburg did recognize that Keynes had an intuitive understanding about the nature of interval valued probability, but that the best that Keynes had been able to accomplish in this regard in his A Treatise on Probability was to offer some hints, ideas, or suggestions. This is the same conclusion put forth by all members of ISIPTA since 1999. Of course, Keynes had provided a complete mathematical, quantitative theory, in Parts II and III of the A Treatise on Probability, of interval valued probability in chapters XV, XVI, XVII, XX, and XXII. Keynes’s interval valued theory was based on Boole’s original theory of interval valued probability that Kyburg and all members of ISIPTA have overlooked. Kyburg makes a very interesting point about the deficiencies of Ramsey, as regards Keynes’s graphical presentation (see Keynes, 1921, p.161, ft.2) that were based on Boole’s approach. Boole’s mathematical, lattice structure of upper and lower probabilities, that Boole demonstrated had least and greatest limits, narrowest limit, maximum limit, highest, inferior numerical limit, highest minor limit, greatest minor numerical limit, etc. (Boole,1854,pp.288,293,305- 313,317-324)all involve a mathematical lattice structure .All of these terms mean that Boole is solving for a greatest lower bound and/or a least upper bound with his solutions methods, which involved using second order, quadratic equations and third order ,cubic equations. Such greatest lower bounds and least upper bounds automatically specify a mathematical lattice structure to Boole’s partial orderings of probabilities: “What is curious is that the mathematician -philosopher, Frank Ramsey, paid no attention to this structure in his review of the Treatise (Ramsey, 1922) …” (Kyburg,2010, p.26). Actually, Kyburg must have known that Ramsey had never made any comment on Keynes’s graphical demonstration on page 39 of Keynes’s A Treatise on Probability in any publication in his life. The reason is very simple. Ramsey knew that the mathematical lattice demonstration on p.39 showed that his theory of exact and precise, additive probability, illustrated by Keynes with the linear line OAI, is a very, very special case of Keynes’s general theory. Therefore, Ramsey had to reject the foundation for the construction of any type of mathematical lattice structure, which was Keynes’s Boolean relational, propositional logic. Kyburg recognized that Ramsey’s criticism was directed at this foundation: “That there is an out-and-out conflict between Keynes and Ramsey ….becomes clear in the fourth part of Ramsey’s essay…Now it is all very well for Ramsey modestly to admit that he sees no logical relation of probability such as the one that Keynes seeks to draw our attention to[author’s note-Kyburg has completely overlooked that such an objective , logical probability relation was first specified by Boole, not Keynes],but wants to go further than that. It is clear that Ramsey wants to claim that there is no such relation.” (Kyburg, 2010, p.29; italics is Kyburg’s). Ramsey, in fact, did claim this. I have noted many, many times in the papers that I have written on this topic over the last 15 years that are available at SSRN, Researchgate and Academia, that Keynes, if he had actually ever thought himself threatened by Ramsey’s claims, could have, either with or without Bertrand Russell’s assistance ,completely and totally crushed Ramsey intellectually by simply making the following hypothetical one sentence statement at any Apostles meeting from 1922 to 1929: “My objective, logical probability relations are identical to Boole’s logical, probability relations as discussed in chapter I and XVI of his The Laws of Thought (1854).” That, of course, would have been the beginning of the end of Ramsey’s academic career