We introduce a new typed combinatory calculus with a type constructor that, to each type $$\sigma $$ σ , associates the star type $$\sigma ^*$$ σ ∗ of the nonempty finite subsets of elements of type $$\sigma $$ σ . We prove that this calculus enjoys the properties of strong normalization and confluence. With the aid of this star combinatory calculus, we define a functional interpretation of first-order predicate logic and prove a corresponding soundness theorem. It is seen that each theorem of classical first-order logic is connected with certain formulas which are tautological in character. As a corollary, we reprove Herbrand’s theorem on the extraction of terms from classically provable existential statements.
Archive for Mathematical Logic – Springer Journals
Published: May 19, 2017
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