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Connecting formal semantics to constructive intuitions

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Constructivity in Computer Science (Constructivity in CS 1991)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 613))

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Abstract

We use formal semantic analysis to generate intuitive confidence that the Heyting Calculus is an appropriate system of deduction for constructive reasoning. Well-known modal semantic formalisms have been defined by Kripke and Beth, but these have no formal concepts corresponding to constructions, and shed little intuitive light on the meanings of formulae. In particular, the well-known completeness proofs for these semantics do not generate confidence in the sufficiency of the Heyting Calculus, since we have no reason to believe that every intuitively constructive truth is valid in the formal semantics.

Läuchli has proved completeness for a realizability semantics with formal concepts analogous to constructions, but the analogy is inherently inexact. We argue that, in spite of this inexactness, every intuitively constructive truth is valid in Läuchli semantics, and therefore the Heyting Calculus is powerful enough to prove all constructive truths. Our argument is based on the postulate that a uniformly constructible object must be communicable in spite of imprecision in our language, and we show how the permutations in Läuchli's semantics represent conceivable mprecision in a language, independently of the particular structure of the language.

We look at some of the details of a generalization of Läuchli's proof of completeness for the propositional part of the Heyting Calculus, in order to expose the required model constructions and the constructive content of the result. We discuss the reasons why Läuchli's completeness results on the predicate calculus are not constructive.

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Authors and Affiliations

  1. The University of Chicago, USA

    Stuart A. Kurtz & Michael J. O'Donnell

  2. Stanford University, USA

    John C. Mitchell

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  1. Stuart A. Kurtz
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  2. John C. Mitchell
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  3. Michael J. O'Donnell
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J. Paul Myers Jr. Michael J. O'Donnell

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© 1992 Springer-Verlag Berlin Heidelberg

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Kurtz, S.A., Mitchell, J.C., O'Donnell, M.J. (1992). Connecting formal semantics to constructive intuitions. In: Myers, J.P., O'Donnell, M.J. (eds) Constructivity in Computer Science. Constructivity in CS 1991. Lecture Notes in Computer Science, vol 613. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0021079

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  • DOI: https://doi.org/10.1007/BFb0021079

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  • Print ISBN: 978-3-540-55631-2

  • Online ISBN: 978-3-540-47265-0

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