Intuitionistic modal logic made explicit. Marti, M. & Studer, T. IfCoLog Journal of Logics and their Applications, 3(5):877-901, 2016.
Intuitionistic modal logic made explicit [pdf]Paper  abstract   bibtex   
The logic of proofs of Heyting arithmetic includes explicit justifications for all admissible rules of intuitionistic logic in order to satisfy completeness with respect to provability semantics. We study the justification logic iJT4, which does not have these additional justification terms. We establish that iJT4 is complete with respect to modular models, which provide a Kripke-style semantics, and that there is a realization of intuitionistic S4 into iJT4. Hence iJT4 can be seen as an explicit version of intuitionistic S4.
@article {mast16,
	title = {Intuitionistic modal logic made explicit},
	year = {2016},
	journal = {IfCoLog Journal of Logics and their Applications},
	volume = {3},
	number = {5},
	pages = {877-901},
	url = {2016/mast16.pdf},
	author = {Michel Marti and Thomas Studer},
	abstract = {The logic of proofs of Heyting arithmetic includes explicit
	justifications for all admissible rules of intuitionistic logic in order to
	satisfy completeness with respect to provability semantics. We study the
	justification logic iJT4, which does not have these additional justification
	terms. We establish that iJT4 is complete with respect to modular models,
	which provide a Kripke-style semantics, and that there is a realization
	of intuitionistic S4 into iJT4. Hence iJT4 can be seen as an explicit
	version of intuitionistic S4.}
}
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