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. A fortiori: 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. A partial Boolean algebra embeds into a Boolean algebra iff its two-valued homomorphisms h : A → 2 separate its elements, and any embedding yields at least one such homomorphism, so the existence of non-contextual hidden variables is an algebraic embedding question, not an interpretive one. Kochen and Specker (1967) answered it negatively for the subspace algebra of any Hilbert space of dimension at least 3; the theorem is a corollary of Gleason (1957), since a two-valued homomorphism would be a dispersion-free measure and Gleason's theorem admits none in those dimensions, but the Kochen–Specker proof is finitary and constructive where Gleason's is not. The result is independent of locality, of probability, and of any interpretation, resting only on the ⊙-structure. Every context — every maximal ⊙-clique — is a Boolean algebra with all the two-valued homomorphisms one could want, so the obstruction is entirely in the gluing, which Abramsky and Brandenburger (2011) identified as the non-existence of a global section of a presheaf of local assignments. Abramsky and Barbosa (2021) rebuilt the algebraic theory around this, giving pBAs a category, a tensor, and a logical exclusivity principle.

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. Partiality makes this hard to state as a logic: with ∧ and ∨ undefined on most pairs there is no total consequence relation, no implication, and no deduction theorem, and the equational methods that apply to varieties do not apply to partial algebras at all. Coproducts and free objects are awkward, which is why the theory never acquired the algebraic depth of the orthomodular programme despite the better diagnosis. Commeasurability is reflexive and symmetric but not transitive, so contexts overlap without composing and there is no way to reason across them. The dimension hypothesis is essential: in dimension 2 two-valued homomorphisms exist and non-contextual hidden variables are available, so the qubit — where quantum computing lives — escapes the theorem. And the formalism commits to the claim that a ∨ b is meaningless for incompatible a and b, which is an interpretive stance, not a derived result.

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