[[https://plato.stanford.edu/entries/logic-justification/#GeaLog][Justification Logic (Stanford Encyclopedia of Philosophy)]]
Peter Geach proposed the axiom scheme ◊□X→□◊X◊◻X→◻◊X. When added to axiomatic S4S4 it yields an interesting logic known as S4.2S4.2. Semantically, Geach’s scheme imposes confluence on frames. That is, if two possible worlds, w1w1 and w2w2 are accessible from the same world w0w0, there is a common world w4w4 accessible from both w1w1 and w2w2. Geach’s scheme was generalized in Lemmon and Scott (1977) and a corresponding notation was introduced: Gk,l,m,nGk,l,m,n is the scheme ◊k□lX→□m◊nX◊k◻lX→◻m◊nX, where k,l,m,n≥0k,l,m,n≥0. Semantically these schemes correspond to generalized versions of confluence. Some people have begun referring to the schemes as Geach schemes, and we will follow this practice. More generally, we will call a modal logic a Geach logic if it can be axiomatized by adding a finite set of Geach schemes to KK.