README⤓ .txt 2026-07-17T121634.146 000000000000792 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.
Effect Algebras⤓ .md 2026-07-15T055427.000 000000000025696 Foulis and Bennett introduced effect algebras (1994) to generalize quantum logic and probability. Abstracts the structure of quantum effects (positive operators ≤ I). Connects quantum mechanics, fuzzy logic, and generalized probability. Part of the "quantum structures" tradition alongside orthomodular lattices.
Partial Boolean Algebras⤓ .md 2026-07-17T121634.146 000000000000928 Simon Kochen and E. P. Specker, "Logical structures arising in quantum theory" (1965) and "The problem of hidden variables in quantum mechanics" (1967), where the algebra and the theorem named for them appear together. Revived by Abramsky and Barbosa, "The logic of contextuality" (2021), as the natural setting for contextuality.
Quantum Logic⤓ .md 2026-07-15T054402.000 000000000027432 Garrett Birkhoff and John von Neumann proposed quantum logic (1936) based on the lattice structure of closed subspaces in Hilbert space. The distributive law fails for quantum propositions. Developed further by Mackey, Jauch, Piron, and others. Debated whether it's a genuine "logic" or just the algebra of quantum observables.
Sasaki Hook⤓ .md 2026-07-17T121634.146 000000000000808 Usa Sasaki, "Orthocomplemented lattices satisfying the exchange axiom" (1954), where the projection appeared; Finch (1970) and Mittelstaedt (1972) proposed it as the quantum conditional; Gary Hardegree, "The conditional in quantum logic" (1975) and "Material implication in orthomodular lattices" (1979), gave the argument that it is the one. Herman, Marsden, and Piziak (1975) classified the alternatives.
CRITERIA⤓ .txt 2026-07-17T121634.146 000000000000808 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).