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Intuitive counterexamples for constructive fallacies

  • James Lipton
  • Michael J. O'Donnell
Invited Papers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 841)

Abstract

Formal countermodels may be used to justify the unprovability of formulae in the Heyting calculus (the best accepted formal system for constructive reasoning), on the grounds that unprovable formulae are not constructively valid. We argue that the intuitive impact of such countermodels becomes more transparent and convincing as we move from Kripke/Beth models based on possible worlds, to Läuchli realizability models. We introduce a new semantics for constructive reasoning, called relational realizability, which strengthens further the intuitive impact of Läuchli realizability. But, none of these model theories provides countermodels with the compelling impact of classical truth-table countermodels for classically unprovable formulae.

We outline a proof that the Heyting calculus is sound for relational realizability, and conjecture that there is a constructive choice-free proof of completeness. In this respect, relational realizability improves the metamathematical constructivity of Läuchli realizability (which uses choice in two crucial ways to prove completeness) in the same sort of way Beth semantics improves Kripke semantics.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • James Lipton
    • 1
  • Michael J. O'Donnell
    • 2
  1. 1.Wesleyan UniversityUSA
  2. 2.The University of ChicagoUSA

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