**Algebraic Logic, I**

**Quantifier
Theories and Completeness Theorems**

**Zhaohua**** Luo**

**(12/28/2012)**

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. |

Preliminary version
(**12/28/2012**)

COMMENTS TO: zluo@algebraic.net