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Intuitionistic Gödel-Löb Logic, à la Simpson: Labelled Systems and Birelational Semantics

Authors Anupam Das, Iris van der Giessen, Sonia Marin

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Author Details

Anupam Das
  • University of Birmingham, UK
Iris van der Giessen
  • University of Birmingham, UK
Sonia Marin
  • University of Birmingham, UK


We would like to thank Marianna Girlando and Jan Rooduijn for several valuable discussions on the topic. We particularly thank Marianna for her input during our reading group on intuitionistic modal logic which led to the preliminary ideas on which part of this paper is built.

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Anupam Das, Iris van der Giessen, and Sonia Marin. Intuitionistic Gödel-Löb Logic, à la Simpson: Labelled Systems and Birelational Semantics. In 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 288, pp. 22:1-22:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


We derive an intuitionistic version of Gödel-Löb modal logic (GL) in the style of Simpson, via proof theoretic techniques. We recover a labelled system, ℓIGL, by restricting a non-wellfounded labelled system for GL to have only one formula on the right. The latter is obtained using techniques from cyclic proof theory, sidestepping the barrier that GL’s usual frame condition (converse well-foundedness) is not first-order definable. While existing intuitionistic versions of GL are typically defined over only the box (and not the diamond), our presentation includes both modalities. Our main result is that ℓIGL coincides with a corresponding semantic condition in birelational semantics: the composition of the modal relation and the intuitionistic relation is conversely well-founded. We call the resulting logic IGL. While the soundness direction is proved using standard ideas, the completeness direction is more complex and necessitates a detour through several intermediate characterisations of IGL.

Subject Classification

ACM Subject Classification
  • Theory of computation → Proof theory
  • provability logic
  • proof theory
  • intuitionistic modal logic
  • cyclic proofs
  • non-wellfounded proofs
  • proof search
  • cut-elimination
  • labelled sequents


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