Peter Geach proposed the axiom scheme \({\lozenge}{\square}X{\rightarrow}{\square}{\lozenge}X\). When added to axiomatic \({\textsf{S4}}\) it yields an interesting logic known as \(\textsf{S4.2}\). Semantically, Geach’s scheme imposes confluence on frames. That is, if two possible worlds, \(w_1\) and \(w_2\) are accessible from the same world \(w_0\), there is a common world \(w_4\) accessible from both \(w_1\) and \(w_2\). Geach’s scheme was generalized in Lemmon and Scott (1977) and a corresponding notation was introduced: \({\textsf{G}}^{k,l,m,n}\) is the scheme \({\lozenge}^k{\square}^l X {\rightarrow}{\square}^m{\lozenge}^n X\), where \(k, l, m, n\geq 0\). 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 \({\textsf{K}}\). The original Geach scheme is \({\textsf{G}}^{1,1,1,1}\), but also note that \({\square}X{\rightarrow}X\) is \({\textsf{G}}^{0,1,0,0}\), \({\square}X{\rightarrow}{\square}{\square}X\) is \({\textsf{G}}^{0,1,2,0}\), \({\lozenge}X{\rightarrow}{\square}{\lozenge}X\) is \({\textsf{G}}^{1,0,1,1}\), and \(X{\rightarrow}{\square}{\lozenge}X\) is \({\textsf{G}}^{0,0,1,1}\), so Geach logics include the most common of the modal logics. Geach logics constitute an infinite family.