DOI: 10.36347/sjebm.2024.v11i07.004

Pages: 207-213

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 ....