Geodesic
Higher-level Sequent-systems for Intuitionistic Modal Logic
Publications de l'Institut Mathématique, _N_S_39 (1986) no. 53, p. 3 .

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This paper presents higher-level sequent-systems for intuitionistic analogues of $S5$ and $S4$. As in [3] rules for modal constants involve sequents of level 2, i.e., sequents having collections of ordinary sequents of level 1 on the left and right of the turnstile. Starting from a canonical higher-level sequent formulation of $S5$, the restriction of sequents of level 2 to those with the single-conclusion property produces $S4$, without changing anything else. A similar restriction on sequents of level 1 produces Heyting $S5$, and if this restriction is made on sequents of both level 1 and 2, we obtain Heyting $S4$. The paper contains a brief discussion of Kripke-style models for the intuitionistic propositional modal logics in question.
Classification : 03B45 
@article{PIM_1986_N_S_39_53_a0,
     author = {Kosta Do\v{s}en},
     title = {Higher-level {Sequent-systems} for {Intuitionistic} {Modal} {Logic}},
     journal = {Publications de l'Institut Math\'ematique},
     pages = {3 },
     publisher = {mathdoc},
     volume = {_N_S_39},
     number = {53},
     year = {1986},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PIM_1986_N_S_39_53_a0/}
}
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Kosta Došen. Higher-level Sequent-systems for Intuitionistic Modal Logic. Publications de l'Institut Mathématique, _N_S_39 (1986) no. 53, p. 3 . http://geodesic.mathdoc.fr/item/PIM_1986_N_S_39_53_a0/