Comparing John von Neumann's assessment of Keynes's logical theory of probability with that of F P Ramsey

Abstract

It is extremely important to recognize that von Neumann clearly and concisely recognized that Keynes’ theory of logical probability was a nonstandard logic, while the two valued, “Boolean logic “of Jevons, Peirce and Schroder was a standard logic that was in conflict with the original logic and algebra of George Boole when it came to applications to probability. The Jevons-Peirce -Schroder(J-P-S) two valued “Boolean logic” REQUIRES linearity and additivity when applied to probability. This means that it can’t deal with insufficiencies/deficiencies in the availability/completeness of the data or what Boole called indeterminate probabilities. Thus, the (J-P-S)” Boolean logic “is a standard logic, while Boole’s original The Laws of Thought (1854) logic is a non-standard logic. Keynes’s A Treatise on Probability was directly based on Boole’s The Laws of Thought, and hence is a nonstandard logic. Thus, von Neumann is giving Keynes too much credit, as Keynes’s nonstandard logic of 1921 is built directly on Boole’s nonstandard logic of 1854. Keynes was also well aware of Boole’s later, much easier, applications of indeterminate probabilities using Henry Wilbraham’s approach (Keynes, 1921, p. 161).

Theodore Hailperin (1986, 1996) had already grasped what Boole’s nonstandard approach was, which Boole modified shortly after The Laws of Thought was published in 1854 by switching to Henry Wilbraham’s littoral’s approach, which allowed one to bypass Boole’s four valued logic, while obtaining the same results in a much easier manner.

Given Keynes’s application of Boole’s relational, propositional, logic, which was the foundation for his logical theory of probability, von Neumann’s understanding of Keynes’s work is complete and correct. It contrasts with Ramsey’s understanding of Keynes’s work which is nil.