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 materialin 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.
- Modal logic