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It is well-known that a Hilbert-style deduction system for first-orderclassical logic is sound and complete for a model theory built using allBoolean algebras as truth-value algebras if and only if it is sound andcomplete for a model theory utilizing only irreducible Boolean algebras (whichare all isomorphic to the two-element Boolean algebra). In this paper, we provean analogous result for any first-order logic with an algebraic semanticssatisfying certain minimal assumptions, and we then apply our result tofirst-order quantum logic.

Irreducible Truth-Value Algebras Suffice for the Completeness of Many First-Order Algebraic Logics