Archive for Mathematical Logic

, Volume 58, Issue 3–4, pp 359–385 | Cite as

Maehara-style modal nested calculi

  • Roman KuznetsEmail author
  • Lutz Straßburger
Open Access


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.


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 



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


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Authors and Affiliations

  1. 1.Technische Universität WienViennaAustria
  2. 2.Inria Saclay—Ile-de-France & LIX, Ecole PolytechniquePalaiseauFrance

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