A Natural Intuitionistic Modal Logic: Axiomatization and Bi-Nested Calculus

Authors Philippe Balbiani , Han Gao , Çiğdem Gencer , Nicola Olivetti

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Author Details

Philippe Balbiani
  • CNRS-INPT-UT3, IRIT, Toulouse, France
Han Gao
  • Aix-Marseille University, CNRS, LIS, Marseille, France
Çiğdem Gencer
  • CNRS-INPT-UT3, IRIT, Toulouse, France
Nicola Olivetti
  • Aix-Marseille University, CNRS, LIS, Marseille, France


This paper is originated from a discussion started by Anupam Das and Sonia Marin in the proof theory blog (see the link https://prooftheory.blog/2022/08/19/), we are grateful to them as well as all other contributors to the discussion. In particular, Example 24 was reported in the blog by Alex Simpson, who learnt it in 1996 by Carsten Grefe in a private communication. Example 55 was suggested first by Anupam Das and Sonia Marin in the blog. Finally we thank the reviewers for their very helpful criticisms and insightful remarks.

Cite AsGet BibTex

Philippe Balbiani, Han Gao, Çiğdem Gencer, and Nicola Olivetti. A Natural Intuitionistic Modal Logic: Axiomatization and Bi-Nested Calculus. In 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 288, pp. 13:1-13:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


We introduce FIK, a natural intuitionistic modal logic specified by Kripke models satisfying the condition of forward confluence. We give a complete Hilbert-style axiomatization of this logic and propose a bi-nested calculus for it. The calculus provides a decision procedure as well as a countermodel extraction: from any failed derivation of a given formula, we obtain by the calculus a finite countermodel of it directly.

Subject Classification

ACM Subject Classification
  • Theory of computation → Modal and temporal logics
  • Theory of computation → Proof theory
  • Intuitionistic Modal Logic
  • Axiomatization
  • Completeness
  • Sequent Calculus


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