How to Prove Higher Order Theorems in First Order Logic

In this paper we are interested in using a first order theorem prover to prove theorems that are formulated in some higher order logic. To this end we present translations of higher order logics into first order logic with flat sorts and equality and give a sufficient criterion for the soundness of these translations. In addition translations are introduced that are sound and complete with respect to L. Henkin's general model semantics. Our higher order logics are based on a restricted type structure in the sense of A. Church, they have typed function symbols and predicate symbols, but no sorts. Die Grenzen meiner Spache bedeuten die Grenzen meiner Welt. 1 Introduction First order logic is a powerful tool for expressing and proving mathematical facts. Nevertheless higher order expressions are often better suited for the representation of mathematics and in fact almost all mathematical text books rely on some higher order fragments for expres-siveness. In order to prove such theorems mechanically there are two options: either to have a theorem prover for higher order logic such as TPS [Andrews et al., 1990] or to translate the higher order constructs into corresponding first order expressions and to use a first order theorem prover. As important as the first development is-which may be the way of the future-we follow the second approach because strong first order theorem provers are available today. The Limitations of First Order Logic First order logic and the set theories of ZERMELO-FRAENKEL or VON NEUMANN-GODEL-BERNAYS have been developed for the formalization of mathematical concepts and for reasoning about them. Other approaches are RUSSEL'S ramified theory of types and CHURCH'S simple theory of types which formalize higher Why and How Translation Representing knowledge in an adequate way-adequate with respect to the naturalness of the representation of the object-is one thing, the other thing is to have an adequate and strong form of reasoning. If one uses higher Kerber 137