Maehara-style modal nested calculi


We develop multi-conclusion nested sequent calculi for the fifteen logics of the intuitionistic modal cube between IK and IS5. The proof of cut-free completeness for all logics is provided both syntactically via a Maehara-style translation and semantically by constructing an infinite birelational countermodel from a failed proof search. Interestingly, the Maehara-style translation for proving soundness syntactically fails due to the hierarchical structure of nested sequents. Consequently, we only provide the semantic proof of soundness. The countermodel construction used to prove completeness required a completely novel approach to deal with two independent sources of non-termination in the proof search present in the case of transitive and Euclidean logics.


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Open access funding provided by Austrian Science Fund (FWF). We would like to thank the anonymous reviewer for useful comments.

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Correspondence to Roman Kuznets.

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This work was partially supported by the ANR/FWF project “FISP” and by the PHC/BMWFW Amadeus project “Analytic Calculi for Modal Logics”. The first author was supported by the Austrian Science Fund (FWF) Grants P25417-G15 and S11405 (RiSE).

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Kuznets, R., Straßburger, L. Maehara-style modal nested calculi. Arch. Math. Logic 58, 359–385 (2019).

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  • Proof theory
  • Sequent calculus
  • Nested sequents
  • Modal logic
  • Intuitionistic logic
  • Cut elimination
  • Multiple conclusion
  • Intuitionistic modal logic

Mathematics Subject Classification

  • 03B45
  • 03B60
  • 03B62
  • 03B70
  • 03F03
  • 03F05
  • 03F07
  • 03F55