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