Archive for Mathematical Logic

, Volume 56, Issue 5–6, pp 523–539 | Cite as

A herbrandized functional interpretation of classical first-order logic

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Abstract

We introduce a new typed combinatory calculus with a type constructor that, to each type \(\sigma \), associates the star type \(\sigma ^*\) of the nonempty finite subsets of elements of type \(\sigma \). 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.

Keywords

Functional interpretations First-order logic Star combinatory calculus Finite sets Tautologies Herbrand’s theorem 

Mathematics Subject Classification

03F10 03B10 03B40 03B15 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Departamento de MatemáticaFaculdade de Ciências da Universidade de LisboaLisbonPortugal

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