A herbrandized functional interpretation of classical first-order logic

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.

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Correspondence to Fernando Ferreira.

Additional information

Both authors acknowledge Centro de Matemática, Aplicações Fundamentais e Investigação Operacional (Universidade de Lisboa) and the associated support of Fundação para a Ciência e a Tecnologia (FCT) [UID/MAT/04561/2013]. The second author is also grateful to FCT [UID/CEC/00408/2013 and Grant SFRH/BPD/93278/2013] and to Large-Scale Informatics Systems Laboratory (Universidade de Lisboa).

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Ferreira, F., Ferreira, G. A herbrandized functional interpretation of classical first-order logic. Arch. Math. Logic 56, 523–539 (2017). https://doi.org/10.1007/s00153-017-0555-6

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Keywords

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

Mathematics Subject Classification

  • 03F10
  • 03B10
  • 03B40
  • 03B15