Spinks introduces implicative BCSK-algebras, expanding implicative BCK-algebras with an additional binary operation. Subdirectly irreducible implicative BCSK-algebras can be viewed as flat posets with two operations coinciding only in the 1- and 2-element cases, each, in the latter case, giving the two-valued implication truth-function. We introduce the resulting logic (for the general case) in terms of matrix methodology in Section 1, showing how to reformulate the matrix semantics as a Kripke-style possible worlds semantics, thereby displaying the distinction between the two implications in the more familiar language of modal logic. In Sections 2 and 3 we study, from this perspective, the fragments obtained by taking the two implications separately, and--after a digression (in Section 4) on the intuitionistic analogue of the material in Section 3--consider them together in Section 5, closing with a discussion in Section 6 of issues in the theory of logical rules. Some material is treated in three appendices to prevent Sections 1-6 from becoming overly distended.
"An Intriguing Logic with Two Implicational Connectives." Notre Dame J. Formal Logic 41 (1) 1 - 40, 2000. https://doi.org/10.1305/ndjfl/1027953481