a lattice reinterpretation of Keynes-Ramsey dispute

[Marcello Basili]

Dear all a just published paper by me on Keynes-Ramsey dispute about probability and belief. Luca and I (Mathematical Social Sciences (2026) and Fuzzy Sets and Systems (2026)) introduce a new representation of uncertainty through interval probability measures and define the notion of sub-weak complementation as a part of standard set complementation. This new theory allows a clear and coherent algebraic definition of Keynesian uncertrainty that solves Ramsey challenge.

Abstract. This paper aims to formalize the abstract algebraic difference between Keynes's and Ramsey's theories of probability. Drawing on the foundational paper of Birkhoff and von Newmann (1936) on quantum mechanics, the algebraic-axiomatic properties underpinning the relation between belief and probability in Keynesian and Ramseyan theories are identified. The paper demonstrates that a specific class of abstract algebras - bounded distributive lattice - can represent Keynes's problem while sharing key properties with traditional Ramsey's probability theory. By introducing the notion of an interval probability measure and assuming a model of uncertainty, Keynesian uncertain beliefs can be represented as isomorphic probability intervals. This provides a coherent resolution to Ramsey's long-standing challenge and reconciles the two approaches within a unified algebraic framework.

https://link.springer.com/article/10.1007/s11238-026-10139-2

Ciao

Marcello