 To generalize, for every formula \(\varphi\) of FOTL, its \(E\)-relativization in the extended language can be obtained by replacing every occurrence of \(\forall x\) in \(\varphi\) by “\(\forall x (E(x) \to ...)\)” and every occurrence of \(\exists x\) by “\(\exists x (E(x) \land ...)\)”. Then the following holds: for any FOTL sentence \(\varphi\) not containing \(E\), \(\varphi\) is valid in the varying domain semantics if and only if its \(E\)-relativization is valid in the constant domain semantics. The question that remains, from a philosophical point of view, is whether existence is a legitimate predicate. An axiomatic system for the minimal FOTL with existence predicate can be obtained along the lines of free logic, as outlined in the supplementary document: