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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.

preprint2016arXivOpen access
Richard DeJongheKimberly FreyTom Imbo
math.LOmath-phmath.MPquant-ph
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Richard DeJongheKimberly FreyTom Imbo

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