Frege holds that existence cannot be said to be a first-order predicate for the following reasons. If existence is to be considered to be a first-order predicate, it follows that in statements denying the existence of certain objects, the absurdity ensues of stating something along the following lines: there is an A such that A doesn't exist. The question is simply left to be asked, "how can there be an 'A' such that it doesn't exist?" This led Frege to believe that the 'A' which is being spoken about is itself only a concept, and existence is only a second-order predicate, as speaking of concepts rather than actual objects. This analysis thus is taken to defeat the traditional doctrine of the real distinction between essence and existence, which asserted that existence is said of objects.
But does Frege's anaylsis not assume that the objects in an existential context exist already? For example, Frege holds that there is a contradiction in saying 'there is an A such that A doesn't exist'. But this is only because Frege assumes that what is meant by 'there is an A' is existentially biased, so as to preemptively assert the actual existence of the object that's actual existence is the very thing in question. Could one instead hold that the object in question is itself existentially neutral? It could be argued that A being a concept or being actually existent is thus not evident in the simple thought of A itself, but rather determined in complcated thoughts about A, which is comprised of predicates therein, of which it seems existence must be said to a predicate to the object in question. The statement could thus be amended as follows: 'there is an A, such that A is existentially neutral, which in fact does not exist'. This would seem to avoid the criticism of Frege, so long as I understand Frege correctly.