There are clear cases in which 'exist(s)' functions as a second-level predicate, a predicate of properties or concepts or propositional functions or cognate items, and not as a predicate of individuals. The affirmative general existential 'Horses exist,' for example, is best understood as making an instantiation claim: 'The concept horse is instantiated.' Accordingly, the sentence does not predicate existence of individual horses; it predicates instantiation of the concept horse.
This sort of analysis is well-nigh mandatory in the case of negative general existentials such as 'Flying horses do not exist.' Here we have a true sentence that cannot possibly be about flying horses for the simple reason that there aren't any. (One can make a Meinongian move here, but if possible we should try to get by without doing so.) On a reasonable parsing, 'Flying horses do not exist' is about the concept flying horse, and says of this concept that it has no instances.
But what about singular existentials? Negative singular existentials like 'Pegasus does not exist' pose no problem. We may analyze it as, 'It is not the case that there exists an x such that x is the winged horse of Greek mythology.' Or we can take a page from Quine and say that nothing pegasizes. What we have done in effect is to treat the singular term 'Pegasus' as a predicate and read the sentence as a denial that this predicate applies to anything.
Problems arise, however, with affirmative singular existentials such as 'I exist' and with sentences like 'I might not have existed' which are naturally read as presupposing the meaningfulness of 'I exist' and thus of uses of 'exists' as a first-level predicate. Thus, 'I might not have existed' is construable in terms of the operator 'It might not have been the case that ____' operating upon 'I exist.'
C.J.F. Williams, following in the footsteps of Frege, maintains the draconian thesis that all meaningful uses of 'exist(s)' are second-level. He must therefore supply an analysis of the true sentence 'I might not have existed' that does not require the meaningfulness of 'I exist.' His suggestion is that
. . . my assertion that I might not have existed is the assertion
that there is some property . . . essential to me, which I alone
possess, and which might never have been uniquely instantiated . .
(What is Existence?, Oxford 1981, p. 104)
Williams is suggesting that for each individual x there is a property H such that (i) H is essential to x in the sense that x cannot exist except as instantiating H; and (ii) H, if instantiated, is instantiated by exactly one individual. Accordingly, to say that x might not have existed is to say that H might not have been instantiated. And to say that x exists is to say that H is instantiated.
This analysis will work only if the right properties are available. What is needed are essentially individuating properties. Suppose Ed is the fastest marathoner. Being the fastest marathoner distinguishes Ed from everything else, but it does not individuate him since it is not bound up with Ed's identity that he be the fastest marathoner. Ed can be Ed without being the fastest marathoner. So Ed's existence cannot be equivalent to, let alone idenctical with, the instantiation of the property of being the fastest marathoner since this is an accidental property of anything that possesses it, whereas the existence of an individual must be essential to it. After all, without existence a thing is nothing at all!
On the other hand, Ed's existence is not equivalent to his instantiation of any old essential property such as being human since numerous individuals possess the property whereas the existence of an individual is unique to it.
What is needed is a property that Ed alone has and that Ed alone has in every possible world in which he exists. Such a property will be essentially individuating: it will individuate Ed in every possible world in which Ed exists, one of these being the actual world.
Williams suggests the property of having sprung from sperm cell S and ovum O. Presumably Ed could not have existed without this origin, and anything possessing this origin is Ed. The idea, then, is that the existence of Ed is the instantiation of this property.
The property in question, however, is one that Michael Loux would call 'impure': it makes essential reference to an individual or individuals, in this case to S and O. Since S and O each exist, the question arises as to how their existence is to be analyzed.
For an analysis like that of Williams to work, what is needed is a property that does not refer to or presuppose any existing individual, a property that somehow captures the haecceity of Ed but without presupposing the existence of an individual. If there were such a haecceity property H, then one could say that Ed's existence just is H's being instantiated.
But as I argue in tedious detail in A Paradigm Theory of Existence and in this post such haecceity properties are creatures of darkness. That is one of the reasons I reject Frege-style theories of existence.
Existence, real pound-the-table existence, belongs to individuals. The attempt to 'kick it upstairs' and make it a property of properties or concepts or propositional functions is completely wrongheaded, pace such luminaries as Frege, Russell, and their epigoni.