Most non-classical logics are motivated by either philosophical considerations, such as the metaphysical status of the future, or technical concerns about the neutrality of the logical connectives or the notion of implication. One notable exception to this are the proposals to revise classical logic in the face of empirical results in quantum mechanics.
It is a consequence of quantum mechanics that some properties are complementary. For example, a particle measured to have a determinate momentum cannot simultaneously be measured to have a determinate position, and vice versa. It has been proposed, therefore, that any satisfactory logic for quantum mechanics must reject the laws of distribution:
((φ ∨ ψ) & (φ ∨ θ)) ⊃ (φ ∨ (ψ & θ))
((φ ∨ ψ) & θ) ⊃ ((φ & θ) ∨ (ψ & θ))
Thus, just because a particle is measured to have a determinate momentum (M1), and a probabilistic range of positions (P1 or P2 or P3 or …), it does not logically follow that the particle has any one combination of determinate momentum and position (M1 and P1) or (M1 and P2) or …, any one of which quantum mechanics tells us is impossible.
Such restrictions do not require a great deal of revision in either the semantics or natural deduction systems for the classical propositional calculus, although there have also been proposals to introduce three-valued truth tables, in which problematic conjunctions are evaluated as indeterminate. The main difficulty with quantum logic is that the proposed restrictions seem only to apply to a limited range of cases, as the laws of distribution will continue to hold for non-quantum phenomena. Quantum logic may therefore be better seen as merely a technical framework for accommodating quantum phenomena, rather than a global non-classical logic.
