University of Oxford, U.K., samson.abramsky@cs.ox.ac.uk
(joint work with Rui Soares Barbosa)
Kochen and Specker’s seminal work on contextuality used the formalism of partial boolean algebras. Unlike quantum logic in the sense of Birkhoff – von Neumann, partial boolean algebras only admit physically meaningful operations. We describe a refinement of current approaches to contextuality, in particular the sheaf-theoretic and graph-theoretic approaches, to incorporate partial boolean algebras. We discuss some striking and little-known results of Conway and Kochen (not the so-called “Free Will Theorem”!) in relation to this. We introduce a new axiom for partial boolean algebras, the Logical Exclusivity Axiom, and show that all probability models based on partial boolean algebras satisfying this axiom obey Specker’s Exclusivity Principle.