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Advances in Proof Theory pp 245–267Cite as

Birkhäuser

Intuitionistic Decision Procedures Since Gentzen

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Part of the book series: Progress in Computer Science and Applied Logic ((PCS,volume 28))

Abstract

Gentzen solved the decision problem for intuitionistic propositional logic in his doctoral thesis [31]; this paper reviews some of the subsequent progress. Solutions to the problem are of importance both for general philosophical reasons and because of their use in implementations of proof assistants (such as Coq [4], widely used in software verification) based on intuitionistic logic.

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Notes

  1. 1.

    A referee points out that this paper gave a classical provability interpretation of intuitionistic logic and an early syntactic proof of its faithfulness, a result conjectured by Gödel (and proved semantically by McKinsey and Tarski), as could usefully have been mentioned in [20].

  2. 2.

    The final “p” in the name indicates “propositional”.

  3. 3.

    He commented “Unfortunately, this method does not work for the second problem.”.

  4. 4.

    As established using a cut of the conclusion with \(D, D\!\rightarrow \!P \Longrightarrow P\) and a cut with \(C, D\!\rightarrow \!P, P\!\rightarrow \!B \Longrightarrow (C \!\rightarrow \!D) \!\rightarrow \!B\).

  5. 5.

    As realised by the author after too long a delay.

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Acknowledgments

Thanks are especially due to Gerhard Jäger and Helmut Schwichtenberg, whose scientific encouragement over the years has been substantial; and to Grisha Mints, now, alas, no longer with us, for helpful comments on historical matters—regrettably not all incorporated (thanks to a failure of technology).

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  1. University of St. Andrews, St. Andrews, UK

    Roy Dyckhoff

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Correspondence to Roy Dyckhoff .

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  1. Campus de Caparica DM FCT, Univ Nova de Lisboa, Caparica, Portugal

    Reinhard Kahle

  2. Inst.of Computer Sci. applied math., University of Bern, Bern, Switzerland

    Thomas Strahm

  3. Institute of Computer Science and A, University of Berne, Berne, Switzerland

    Thomas Studer

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Dyckhoff, R. (2016). Intuitionistic Decision Procedures Since Gentzen. In: Kahle, R., Strahm, T., Studer, T. (eds) Advances in Proof Theory. Progress in Computer Science and Applied Logic, vol 28. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-29198-7_6

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