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