Axiomatize validities for several confluence-based frame classes

Develop axiomatizations for the logics Log(Cuc), Log(Cfuc), Log(Cbuc), Log(Cduc), Log(Cfbuc), Log(Cfduc), Log(Cbduc), Log(Cfbduc), Log(Cquc), and Log(Csym∩Ctra), where each Log(C•) denotes the set of formulas valid on the corresponding class of frames defined by confluence combinations (upward, forward, backward, downward) or by the intersection of symmetric and transitive frames.

Background

The paper introduces a new semantics for intuitionistic modal logics with preorder ≤ and relation R, and defines several frame classes based on confluence conditions: forward, backward, downward, and upward confluence (and their quasi-variants).

While the authors provide finite axiomatizations for many such classes, they explicitly leave open axiomatizations for combinations involving upward confluence and for the intersection of symmetric and transitive frames.