We present systems of Natural Deduction based on Strict Implication for the main normal modal logics between K and S5. In this work we consider Strict Implication as the main modal operator, and establish a natural correspondence between Strict Implication and strict subproofs.
"Natural Deduction Based upon Strict Implication for Normal Modal Logics." Notre Dame J. Formal Logic 35 (4) 471 - 495, Fall 1994. https://doi.org/10.1305/ndjfl/1040408608