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	Rho / Density operator	Mixed state
Tr	—	Trace	For computing probabilities
∩	U+2229	Intersection	Meet of subspaces
¬	U+00AC	Not / Complement	Boolean negation in the classical case
≠	U+2260	Not equal	Failure of distribution
⋆	U+22C6	Star operator	Involution on a division ring
ℝ	U+211D	Double-struck R / Reals	One of the three Solèr division rings
ℂ	U+2102	Double-struck C / Complexes	The physical case
ℍ	U+210D	Double-struck H / Quaternions	The remaining Solèr case
L(H)	—	Subspace lattice	Closed subspaces of H, ordered by inclusion
MO2	—	Chinese lantern	Smallest non-Boolean orthomodular lattice
O6	—	Benzene ring	Smallest non-orthomodular ortholattice

Metatheory. The closed subspaces L(H) of a complex Hilbert space form a complete, atomic, orthomodular lattice satisfying the covering law; modular iff H is finite-dimensional, Boolean iff dim H ≤ 1. For ortholattices, distributive implies modular implies orthomodular, both strictly; an ortholattice is orthomodular iff it omits the benzene ring O6, and an orthomodular lattice is Boolean iff every pair of its elements commutes, MO2 being the smallest counterexample. The Foulis–Holland theorem recovers classical inference locally: if one of a, b, c commutes with the other two, the sublattice they generate is distributive. By Piron's theorem (1964), an irreducible complete atomic orthomodular lattice with the covering law and dimension at least 4 is the subspace lattice of a Hermitian space over a ⋆-division ring; Solèr's theorem (1995) adds that an infinite orthonormal sequence forces that ring to be ℝ, ℂ, or ℍ and the space to be a genuine Hilbert space. Gleason's theorem (1957) requires dim H ≥ 3 and then forces every measure on L(H) to be Tr(ρ ·); Kochen–Specker (1967) follows, ruling out two-valued homomorphisms there.

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. There is no well-behaved implication: five polynomial candidates satisfy a → b = 1 iff a ≤ b, none has all the classical properties, and for the Sasaki hook import-export, transitivity, and contraposition all fail, so the deduction theorem is not available in general. Non-distributivity blocks proof by cases and reasoning under a disjunction. Gleason's dimension hypothesis excludes dim 2, where dispersion-free states do exist and the qubit escapes both Gleason and Kochen–Specker — precisely where quantum computing works. Piron's theorem needs irreducibility, completeness, atomicity, the covering law, and dimension at least 4, and yields only a Hermitian space over an unspecified ⋆-division ring; Solèr narrows the ring to ℝ, ℂ, ℍ, and even then the lattice does not select ℂ. Orthomodular lattices have no general tensor product, so composite systems are not a lattice construction, and the formalism has no dynamics: evolution and state update are imposed from outside.