Higher order functions in first order logics

Most of approaches to extracting programs from (constructive) proofs use type theories. Usually it is argued that first order logic has many drawbacks to be used as a language for programming in logic. In particular, higher order functions are not directly expressible in first order logic. Here we show how to use proof schemes in first order logic for representing higher order functions. We generalize the semantics introduced in [Vor 90] to the proof schemes and show how it is related to extraction of higher order functions from proofs in first order logic.