2013/01/04 by Luo, Zhaohua
#FOS: Computer and information sciences #Logic in Computer Science (cs.LO)
paper · doi:10.48550/arxiv.1301.0667
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.