Algebraic Logic, I Quantifier Theories and Completeness Theorems

TL;DR

This paper introduces a new algebraic framework called quantifier theories for first-order logic, simplifying proofs of fundamental completeness theorems and providing a clear treatment of ultraproducts.

Contribution

It proposes the notion of quantifier theories as functors from monads to Boolean algebras, offering a novel algebraic approach to first-order logic and its completeness theorems.

Findings

Simplified proofs of Cayley's and Godel's completeness theorems.

Introduction of quantifier theories as algebraic structures.

Clear treatment of ultraproducts of models.

Abstract

Algebraic logic studies algebraic theories related to proposition and first-order logic. A new algebraic approach to first-order logic is sketched in this paper. We introduce the notion of a quantifier theory, which is a functor from the category of a monad of sets to the category of Boolean algebras, together with a uniquely determined system of quantifiers. A striking feature of this approach is that Cayley's Completeness Theorem and Godel's Completeness Theorem can be stated and proved in a much simpler fashion for quantifier theories. Both theorems are due to Halmos for polyadic algebras. We also present a simple transparent treatment of ultraproducts of models of a quantifier theory.