 In case we wanted a similar outcome for another modal logic, say S4, the set \(\Delta _{K}\) can be extended to a set \(\Delta _{S4}\) including as axioms the many-sorted abstract conditions for axioms \(T\) and 4 (i.e., formulas \(\MS(T)\ \)and  \(\MS(4)\)[23]).  In fact, we can prove that the first-order axioms for reflexivity and transitivity are equivalent to these many-sorted translated sentences.