We present the Higher-Order Skolem-Herbrand-Gödel Theorem for the Clausal Theory of Types. This is a form of Church's formulation of the Simple Theory of Types [27]. The Theorem uses Henkin's Completeness Theorem [87] for validity in general models.

We begin by reviewing automations of higher-order logic, its syntax using the simply typed λ-calculus, and Henkin-Andrews general model semantics [5, 87]. We introduce higher-order analogues of Herbrand interpretations called λ-models. We then define the Clausal Theory of Types as a sub-logic of the Simple Theory of Types with equality, and show that a higher-order Skolem-Herbrand-Gödel Theorem holds for it.

The Theorem leads to a partial decision procedure for testing the validity of formulas. It can be automated by generalizing resolution.

Automating Higher-Order Logic

Simple Type Theory derives from the Ramified Theory of Types and was intended to formalize mathematical reasoning. Church [27] presented a λ-calculus formulation of simple type theory and used it to prove the deduction theorem, Peano's postulates for arithmetic, and a formalization of definition by primitive recursion.

More recent applications of logics which are based on this higher-order logic are in the areas of theorem proving, hardware verification, programming language design, automating Zermelo-Fränkel set theory, natural language processing, and program transformation.

The theorem prover TPS [9, 10, 136], uses Church's formulation of higher-order logic. The most recent version, called TPS3, uses expansion tree proofs [134], Huet's higher-order unification procedure [97], and tactics [78, 150].