ORTHOMODULAR LATTICES
The logics complete for orthomodular lattices, in which the distributive law fails while the weaker orthomodular law survives. Consequence is preservation of designated values over the lattice of a Hilbert space's closed subspaces, the motivating model.
The subdivision centers on quantum logic, the logic of experimental propositions about a quantum system, where the meet and join of propositions are the intersection and closed span of subspaces and complementation is orthocomplementation. Distribution fails because a superposition may lie in the span of two subspaces without lying in either, which is the algebraic trace of complementarity. Orthomodularity is to this subdivision what non-Booleanness is to Heyting and non-classicality is to Many-Valued: the single structural fact that individuates it.