Quantum Logic

Origin. 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.

Models. Propositions about quantum systems. In classical physics, propositions form a Boolean algebra (distributive). In quantum mechanics, propositions correspond to subspaces of Hilbert space, forming an orthomodular lattice (non-distributive). "The particle has position in region A AND momentum in range B" may not be well-defined simultaneously.

Formalism.

Classical propositional lattice (Boolean algebra):

Meet (∧), join (∨), complement (¬)
Distributive: A ∧ (B ∨ C) = (A ∧ B) ∨ (A ∧ C)

Quantum propositional lattice (orthomodular lattice):

Meet (∧), join (∨), orthocomplement (⊥)
Orthomodular law: A ≤ B implies B = A ∨ (B ∧ A⊥)
Distributivity FAILS: A ∧ (B ∨ C) ≠ (A ∧ B) ∨ (A ∧ C) in general

Hilbert space realization:

Propositions = closed subspaces of Hilbert space H
A ∧ B = A ∩ B (intersection)
A ∨ B = closure of A + B (span)
A⊥ = orthogonal complement
A ≤ B means A ⊆ B

Compatibility: Two propositions A, B are compatible if they generate a Boolean subalgebra. Compatible = can be measured simultaneously. Incompatible propositions (position and momentum) don't distribute.

States: A state assigns probabilities to propositions. In Hilbert space: density operators ρ. P(A) = Tr(ρ Pₐ) where Pₐ is the projection onto subspace A.

Gleason's theorem: In dimension ≥ 3, every probability measure on the lattice of subspaces comes from a density operator. Constrains possible interpretations.

Symbols.

Symbol	Unicode	Name	Meaning
∧	U+2227	Meet	Quantum conjunction
∨	U+2228	Join	Quantum disjunction
⊥	U+22A5	Orthocomplement	Quantum negation
≤	U+2264	Subspace order	Implication
⊆	U+2286	Subset	Subspace inclusion
⊕	U+2295	Direct sum	Orthogonal subspaces
H	—	Hilbert space	State space
P	—	Projection	Observable for proposition
ρ	U+03C1	Density operator	Mixed state
Tr	—	Trace	For computing probabilities

Metatheory. Orthomodular lattices are not distributive, so classical inference rules fail. No material implication with usual properties. Kochen-Specker theorem: no non-contextual hidden variable assignment. The lattice is not uniquely determined by "logic" — it comes from quantum mechanics. Various attempts at quantum logic proof systems exist but none is canonical.

Applies to. Foundations of quantum mechanics. Quantum information theory. Interpreting quantum probability. Connections to effect algebras and operator algebras. Philosophical debates about logic and physics.

Limitations. Is it really "logic" or just algebra? No clear notion of proof or deduction. The structures come from physics, not logic. Practical quantum reasoning uses Hilbert space math, not quantum logic directly. Multiple competing formalizations. Doesn't handle dynamics (measurements, time evolution). Limited uptake even in quantum foundations community.