Partial Boolean Algebras

Origin. 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.

Models. The rival diagnosis to orthomodularity. Birkhoff and von Neumann kept the lattice operations total and weakened distribution; Kochen and Specker say the operations on non-commuting propositions are meaningless, so the algebra should be partial — a family of Boolean algebras glued where they overlap, with no join defined between incompatible propositions. Quantum logic is then not a non-classical logic but a classical logic that fails to be globally defined.

Formalism.

Partial Boolean algebra: (A, ⊙, ¬, ∧, ∨, 0, 1) with ⊙ a reflexive, symmetric commeasurability relation. ∧ and ∨ are defined only on ⊙-related pairs. Every set of pairwise ⊙-related elements lies in a Boolean subalgebra of A. Locally Boolean; globally not a Boolean algebra at all.

The quantum example: A = the closed subspaces of a Hilbert space H. a ⊙ b iff the projections commute. Each maximal commeasurable family is a Boolean algebra — a context, a measurement one could actually perform.

Two-valued homomorphism: h : A → 2 preserving the operations where defined. A hidden-variable assignment is exactly such an h: a simultaneous truth value for every proposition, consistent inside every context.

Kochen–Specker theorem (1967): For dim(H) ≥ 3, the partial Boolean algebra of subspaces of H admits no two-valued homomorphism. Equivalently: no embedding into a Boolean algebra. The original proof uses 117 vectors in ℝ³; Peres (1991) reduced it to 33, Conway and Kochen to 31. There is no consistent global assignment, though every context has one.

Contextuality: The failure is not about any single measurement. Each context is classical. What fails is gluing the contexts, which is a sheaf-theoretic obstruction (Abramsky–Brandenburger 2011). The Kochen–Specker theorem is a cohomological non-triviality result in disguise.

Contrast with orthomodular lattices: OML totalize ∨ on non-commuting a, b; accept the failure of distribution. pBA refuse the totalization; accept partiality. Kochen and Specker's objection: a ∨ b for incompatible a, b denotes nothing, and the OML answers a question that cannot be asked.

Symbols.

Metatheory. Kochen–Specker is the strongest no-go theorem in the quantum-logic literature and the one that survives the collapse of the programme: it does not depend on locality, on probability, or on any interpretation, only on the partial algebra's structure. That it can be recast sheaf-theoretically (Abramsky–Brandenburger) relocates the whole subject: contextuality is the failure of a family of local sections to glue, which is the same obstruction that appears in databases, in relational query answering, and in constraint satisfaction. The partial-algebra formulation is what makes that visible, and the orthomodular formulation is what hides it.

Applies to. Hidden-variable no-go arguments. Contextuality and its resource theory. Quantum foundations. The sheaf-theoretic account of contextuality, where pBAs are the fibres. Relational databases and constraint satisfaction, where the same gluing failure recurs without quantum mechanics.

Limitations. The category of partial Boolean algebras is badly behaved: coproducts and free objects are awkward, and the theory never developed the algebraic depth orthomodular lattices did, which is why the OML programme won the textbooks despite the better diagnosis. Partiality makes the logic hard to state as a logic — there is no obvious consequence relation when the connectives are undefined on most pairs. And the theorem needs dim ≥ 3, so the qubit case, which is where quantum computing lives, escapes it.

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