  We show that the propositional system of a many-box model is always a
set-representable effect algebra. In particular cases of 2-box and 1-box models
it is an orthomodular poset and an orthomodular lattice respectively. We
discuss the relation of the obtained results with the so-called Local
Orthogonality principle. We argue that non-classical properties of box models
are the result of a dual enrichment of the set of states caused by the
impoverishment of the set of propositions. On the other hand, quantum
mechanical models always have more propositions as well as more states than the
classical ones. Consequently, we show that the box models cannot be considered
as generalizations of quantum mechanical models and seeking for additional
principles that could allow to "recover quantum correlations" in box models is,
at least from the fundamental point of view, pointless.
