Irreducible Truth-Value Algebras Suffice for the Completeness of Many   First-Order Algebraic Logics
  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. 
Richard DeJonghe, Kimberly Frey, and Tom Imbo
math.LO math-ph math.MP quant-ph