Unified quantum logic based on unified operations of implication is formulated as an axiomatic calculus. Soundness and completeness are demonstrated using standard algebraic techniques. An embedding of quantum logic into a new modal system is carried out and discussed.

Matthew Spinks [35] 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 §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 §§2 and 3 we study, from this perspective, the fragments obtained by taking the two implications separately, and – after a digression (in §4) on the intuitionistic analogue of the material in §3 – consider them together in §5, closing with a discussion in §6 of issues in the theory of logical rules. Some material is treated in three appendices to prevent §§1–6 from becoming overly distended. (shrink)