Kochen-Specker theorem in the modal interpretation of quantum mechanics

According to the modal interpretation of quantum mechanics, subsystems of a quantum mechanical system have definite properties, the set of definite properties forming a partial Boolean algebra. It is shown that these partial Boolean algebras have no common extension (as a partial Boolean subalgebra of the properties of the total system) that is embeddable in a Boolean algebra. One has thus either to restrict the rules to preferred subsystems (Healey), or to advocate a shift in metaphysics (Dieks).