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DOI: 10.4230/LIPIcs.CSL.2026.19

A Unifying Conservation Theorem

Authors: Giulio Fellin

Published in: LIPIcs, Volume 363, 34th EACSL Annual Conference on Computer Science Logic (CSL 2026)

Abstract

The relationship between classical and constructive logics has long been illuminated by a series of conservation results, beginning with Kolmogorov’s negative translation and Glivenko’s double negation theorem, and later extended by Kuroda and Segerberg to first-order and minimal logics respectively. These results reveal how certain classical principles can be interpreted or recovered within weaker constructive frameworks, either via translations or through minimal extensions that satisfy specific logical properties. In this paper, we propose a unifying generalisation of these conservation theorems, that consolidates and expands the abstract methods introduced in earlier studies, offering a unified perspective on the interplay between classical provability and constructive reasoning.

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Giulio Fellin. A Unifying Conservation Theorem. In 34th EACSL Annual Conference on Computer Science Logic (CSL 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 363, pp. 19:1-19:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{fellin:LIPIcs.CSL.2026.19,
  author =	{Fellin, Giulio},
  title =	{{A Unifying Conservation Theorem}},
  booktitle =	{34th EACSL Annual Conference on Computer Science Logic (CSL 2026)},
  pages =	{19:1--19:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-411-6},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{363},
  editor =	{Guerrini, Stefano and K\"{o}nig, Barbara},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2026.19},
  URN =		{urn:nbn:de:0030-drops-254431},
  doi =		{10.4230/LIPIcs.CSL.2026.19},
  annote =	{Keywords: double negation, negative translation, conservation, minimal logic, Glivenko’s theorem}
}

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DOI: 10.4230/LIPIcs.TYPES.2021.7

Constructive Cut Elimination in Geometric Logic

Authors: Giulio Fellin, Sara Negri, and Eugenio Orlandelli

Published in: LIPIcs, Volume 239, 27th International Conference on Types for Proofs and Programs (TYPES 2021)

Abstract

A constructivisation of the cut-elimination proof for sequent calculi for classical and intuitionistic infinitary logic with geometric rules - given in earlier work by the second author - is presented. This is achieved through a procedure in which the non-constructive transfinite induction on the commutative sum of ordinals is replaced by two instances of Brouwer’s Bar Induction. Additionally, a proof of Barr’s Theorem for geometric theories that uses only constructively acceptable proof-theoretical tools is obtained.

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Giulio Fellin, Sara Negri, and Eugenio Orlandelli. Constructive Cut Elimination in Geometric Logic. In 27th International Conference on Types for Proofs and Programs (TYPES 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 239, pp. 7:1-7:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{fellin_et_al:LIPIcs.TYPES.2021.7,
  author =	{Fellin, Giulio and Negri, Sara and Orlandelli, Eugenio},
  title =	{{Constructive Cut Elimination in Geometric Logic}},
  booktitle =	{27th International Conference on Types for Proofs and Programs (TYPES 2021)},
  pages =	{7:1--7:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-254-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{239},
  editor =	{Basold, Henning and Cockx, Jesper and Ghilezan, Silvia},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2021.7},
  URN =		{urn:nbn:de:0030-drops-167763},
  doi =		{10.4230/LIPIcs.TYPES.2021.7},
  annote =	{Keywords: Geometric theories, sequent calculi, axioms-as-rules, infinitary logic, constructive cut elimination}
}

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Documents authored by Fellin, Giulio

A Unifying Conservation Theorem

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Constructive Cut Elimination in Geometric Logic

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