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

TL;DR

This paper demonstrates that for a broad class of first-order algebraic logics, including quantum logic, completeness can be established using only irreducible truth-value algebras, simplifying the model theory.

Contribution

It generalizes the known Boolean algebra result to other algebraic semantics, showing irreducible algebras suffice for completeness proofs in various first-order logics.

Findings

Completeness holds using only irreducible truth-value algebras.

The result applies to first-order quantum logic.

Simplifies the model theory for algebraic logics.

Abstract

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.