On the exclusion of the excluded middle law and quantum mechanics

We argue that limitations arise when using non-constructive mathematics (NCM) as a foundation for quantum mechanics, since the Boolean logical substrate of NCM fits a distributive propositional structure, but fits less naturally into the non-distributive lattice of quantum propositions. We distinguish three logical structures relevant to this discussion: Boolean algebras (classical, distributive, complemented), Heyting algebras (constructive, distributive, not necessarily complemented), and orthomodular lattices (quantum, non-distributive). The linear structure of quantum state space is compatible with NCM, but the propositional structure of quantum mechanics forms a non-distributive lattice rather than a Boolean algebra. Topos theory, one of the models of constructive mathematics, offers a way forward: the Döring–Isham framework provides an explicit bridge—the daseinization map—from quantum propositions to a topos whose internal logic is intuitionistic. Although some topoi carry Boolean logic, in general, topoi are not constrained to it. We argue that this bridged framework is more suitable for the mathematical foundation of quantum mechanics than one that retains the law of excluded middle (LEM) without such a bridge. Our discussion centers on the pivotal role of the LEM.