INCLUSION CRITERIA
An entry belongs in this subdivision if and only if it is complete for orthomodular lattices, or an ortholattice variety, in which distribution fails but orthocomplementation and the orthomodular law hold.
Required: At least one of the following:
- A completeness theorem with respect to orthomodular or ortho lattices
- A semantics over the closed subspaces of a Hilbert space with orthocomplementation
- A consequence relation validating orthomodularity while failing distribution
Not sufficient: Distributive many-valued matrices (Many-Valued). A modal operator formalizing quantum measurement or information over a classical base (Modal, or Applications/Quantum for the computational systems). A quantum programming or Hoare logic, which reasons about quantum programs rather than in the orthomodular propositional calculus (Applications/Quantum).
Boundary: The pure non-distributive propositional calculus belongs here; the applied quantum logics—quantum computation, information, programming, temporal, and modal systems—belong in Applications/Quantum. Orthomodular lattices studied as algebraic structures for their own sake belong in Metatheory.