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 Cayleys Completeness Theorem and Godels 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.
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