On the algebraization of Henkin‐type second‐order logic

Abstract

AbstractThere is an extensive literature related to the algebraization of first‐order logic. But the algebraization of full second‐order logic, or Henkin‐type second‐order logic, has hardly been researched. The question arises: what kind of set algebra is the algebraic version of a Henkin‐type model of second‐order logic? The question is investigated within the framework of the theory of cylindric algebras. The answer is: a kind of cylindric‐relativized diagonal restricted set algebra. And the class of the subdirect products of these set algebras is the algebraization of Henkin‐type second‐order logic. It is proved that the algebraization of a complete calculus of the Henkin‐type second‐order logic is a class of a kind of diagonal restricted cylindric algebras. Furthermore, the connection with the non‐standard enlargements of standard complete second‐order structures is investigated.