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Intuitive counterexamples for constructive fallacies

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Mathematical Foundations of Computer Science 1994 (MFCS 1994)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 841))

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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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Authors and Affiliations

  1. Wesleyan University, USA

    James Lipton

  2. The University of Chicago, USA

    Michael J. O'Donnell

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  1. James Lipton
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  2. Michael J. O'Donnell
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Igor Prívara Branislav Rovan Peter Ruzička

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© 1994 Springer-Verlag Berlin Heidelberg

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Lipton, J., O'Donnell, M.J. (1994). Intuitive counterexamples for constructive fallacies. In: Prívara, I., Rovan, B., Ruzička, P. (eds) Mathematical Foundations of Computer Science 1994. MFCS 1994. Lecture Notes in Computer Science, vol 841. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-58338-6_61

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  • DOI: https://doi.org/10.1007/3-540-58338-6_61

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-58338-7

  • Online ISBN: 978-3-540-48663-3

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