The connections between Keynes's 1921 logic of probability and the Birkhoff-von Neumann 1936 logic of quantum mechanics

Abstract

Birkhoff and von Neumann make it clear what the goal of their October, 1936 paper, “The logic of quantum mechanics”, is: “The object of the present paper is to discover what logical structure one may hope to find in physical theories which, like quantum mechanics, do not conform to classical logic. Our main conclusion, based on admittedly heuristic arguments, is that one can reasonably expect to find a calculus of propositions which is formally indistinguishable from the calculus of linear subspaces with respect to set products, linear sums, and orthogonal complements-and resembles the usual calculus of propositions with respect to and, or, and not.” (Birkhoff and von Neumann, 1936, p.823; underline added).

Specifically,

“Up to now, we have only discussed formal features of logical structure which seem to be common to classical dynamics and the quantum theory. We now turn to the central difference between them-the distributive identity of the propositional calculus:

L6: a U (b ꓵ c) = (a U b) ꓵ (a U c) and a ꓵ (b U c) = (a ꓵ b) U (a ꓵ c)

which is a law in classical, but not in quantum mechanics.” (Birkhoff and von Neumann, 1936, p.830; underline added)

Continuing, Birkhoff and von Neumann comment:

“From a deeper mathematical viewpoint, L6 is the characteristic property of set-combination. More precisely, every "field" of sets is isomorphic with a Boolean algebra, and conversely.²¹ This throws new light on the well-known fact that the propositional calculi of classical mechanics are Boolean algebras.” (Birkhoff and von Neumann, 1936, p.831).

Birkhoff and von Neumann then show on p.831 that the distributive law breaks down in quantum mechanics and that the generalized distributive law of logic breaks down in the related field “in the quotient algebra of the field of Lebesgue measurable sets by the ideal of sets of Lebesgue measure 0, which is so fundamental in statistics and the formulation of the ergodic principle.” (ibid., p.831).

However, it must be noted that Birkhoff and von Neumann are talking about the Jevons-Pierce-Schroder interpretation (two -valued) of Boole’s original approach (four -valued), which Boole completely rejected. (See Hailperin, 1986, 1996), The reason for this rejection is that the Jevons-Peircean approach required strict complementation, which translated into additivity and linearity in probability assessments, as opposed to Boole’s realization that real world decision making required sub additivity and non-linearity, Thereafter, we will refer to the Jevons-Pierce-Schroder approach as Jevons’s approach to the interpretation of Boolean logic, which uses standard classical logic to underpin discussions of standard probability logic. Jevons’s approach removed the basic algebra which Boole combined with his new relational, propositional logic, so as to deal with indeterminate, imprecise probabilities, which are nonstandard. Jevons’s approach can’t deal with the non-standard probability logics used by Boole in chapters XVIII-XX of The Laws of Thought (1854) and by Keynes in chapters XII, XV-XVII of the A Treatise on Probability (1921) in order to deal with indeterminate, interval valued probabilities.