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DOI: 10.4230/LIPIcs.TYPES.2022.2

Classical Natural Deduction from Truth Tables

Authors: Herman Geuvers and Tonny Hurkens

Published in: LIPIcs, Volume 269, 28th International Conference on Types for Proofs and Programs (TYPES 2022)

Abstract

In earlier articles we have introduced truth table natural deduction which allows one to extract natural deduction rules for a propositional logic connective from its truth table definition. This works for both intuitionistic logic and classical logic. We have studied the proof theory of the intuitionistic rules in detail, giving rise to a general Kripke semantics and general proof term calculus with reduction rules that are strongly normalizing. In the present paper we study the classical rules and give a term interpretation to classical deductions with reduction rules. As a variation we define a multi-conclusion variant of the natural deduction rules as it simplifies the study of proof term reduction. We show that the reduction is normalizing and gives rise to the sub-formula property. We also compare the logical strength of the classical rules with the intuitionistic ones and we show that if one non-monotone connective is classical, then all connectives become classical.

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Herman Geuvers and Tonny Hurkens. Classical Natural Deduction from Truth Tables. In 28th International Conference on Types for Proofs and Programs (TYPES 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 269, pp. 2:1-2:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{geuvers_et_al:LIPIcs.TYPES.2022.2,
  author =	{Geuvers, Herman and Hurkens, Tonny},
  title =	{{Classical Natural Deduction from Truth Tables}},
  booktitle =	{28th International Conference on Types for Proofs and Programs (TYPES 2022)},
  pages =	{2:1--2:27},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-285-3},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{269},
  editor =	{Kesner, Delia and P\'{e}drot, Pierre-Marie},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2022.2},
  URN =		{urn:nbn:de:0030-drops-184450},
  doi =		{10.4230/LIPIcs.TYPES.2022.2},
  annote =	{Keywords: Natural deduction, classical proposition logic, multiple conclusion natural deduction, proof terms, formulas-as-types, proof normalization, subformula property, Curry-Howard isomorphism}
}

Document

DOI: 10.4230/LIPIcs.TYPES.2017.3

Proof Terms for Generalized Natural Deduction

Authors: Herman Geuvers and Tonny Hurkens

Published in: LIPIcs, Volume 104, 23rd International Conference on Types for Proofs and Programs (TYPES 2017)

Abstract

In previous work it has been shown how to generate natural deduction rules for propositional connectives from truth tables, both for classical and constructive logic. The present paper extends this for the constructive case with proof-terms, thereby extending the Curry-Howard isomorphism to these new connectives. A general notion of conversion of proofs is defined, both as a conversion of derivations and as a reduction of proof-terms. It is shown how the well-known rules for natural deduction (Gentzen, Prawitz) and general elimination rules (Schroeder-Heister, von Plato, and others), and their proof conversions can be found as instances. As an illustration of the power of the method, we give constructive rules for the nand logical operator (also called Sheffer stroke). As usual, conversions come in two flavours: either a detour conversion arising from a detour convertibility, where an introduction rule is immediately followed by an elimination rule, or a permutation conversion arising from an permutation convertibility, an elimination rule nested inside another elimination rule. In this paper, both are defined for the general setting, as conversions of derivations and as reductions of proof-terms. The properties of these are studied as proof-term reductions. As one of the main contributions it is proved that detour conversion is strongly normalizing and permutation conversion is strongly normalizing: no matter how one reduces, the process eventually terminates. Furthermore, the combination of the two conversions is shown to be weakly normalizing: one can always reduce away all convertibilities.

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Herman Geuvers and Tonny Hurkens. Proof Terms for Generalized Natural Deduction. In 23rd International Conference on Types for Proofs and Programs (TYPES 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 104, pp. 3:1-3:39, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{geuvers_et_al:LIPIcs.TYPES.2017.3,
  author =	{Geuvers, Herman and Hurkens, Tonny},
  title =	{{Proof Terms for Generalized Natural Deduction}},
  booktitle =	{23rd International Conference on Types for Proofs and Programs (TYPES 2017)},
  pages =	{3:1--3:39},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-071-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{104},
  editor =	{Abel, Andreas and Nordvall Forsberg, Fredrik and Kaposi, Ambrus},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2017.3},
  URN =		{urn:nbn:de:0030-drops-100519},
  doi =		{10.4230/LIPIcs.TYPES.2017.3},
  annote =	{Keywords: constructive logic, natural deduction, detour conversion, permutation conversion, normalization Curry-Howard isomorphism}
}

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Classical Natural Deduction from Truth Tables

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Proof Terms for Generalized Natural Deduction

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