Quantum logics with the existence property

Aquantum logic (-orthocomplete orthomodular poset L with a convex, unital, and separating set of states) is said to have theexistence property if the expectation functionals onlin() associated with the bounded observables of L form a vector space. Classical quantum logics as well as the Hilbert space logics of traditional quantum mechanics have this property. We show that, if a quantum logic satisfies certain conditions in addition to having property E, then the number of its blocks (maximal classical subsystems) must either be one (classical logics) or uncountable (as in Hilbert space logics).Part of this work was done while the author was a visitor at the Department of Mathematics and Computer Science of the University of Denver, Denver, Colorado.