There are many beings or existents, but they have something in common: they exist. Call what they have in common 'existence.' Now what is existence? And does existence itself exist in reality outside the mind?
For me, existence is that which makes concrete things be, outside their causes, outside the mind, outside of language and its logic, outside of the realm of mere possibility, and outside of nothing. Existence or Being is what makes beings be. I take this to imply that existence cannot be a concept under which existents fall or a property they instantiate.
For some, however, existence is a subjective concept, a concept dependent on minds such as ours. If so, the relation between the Moon, say, and existence is one of falling-under: the Moon falls under the subjective concept, existence, and the Moon exists in virtue of falling under this concept.
There are two obvious problems with this. The first is that the existence of the Moon cannot depend on the existence of minds like ours. The Moon existed before we existed, and were we not to exist, the Moon would still exist. The first point is temporal, the second modal: not every possible world in which the Moon exists is a world in which there are beings who deploy concepts. This two-pronged objection could be circumvented by maintaining that existence is an objective concept, or a property, one that does not depend on minds like ours. But then a different problem arises.
The problem is one of explanatory circularity. An individual x cannot fall under a concept C unless x exists. Thus the Moon falls under the concept natural satellite only if the Moon exists. The relation falling-under cannot obtain unless both of its relata exist. No problem arises in the case of the Moon's falling under natural satellite. But a problem arises if we suppose that existence is a first-level concept. One moves in an explanatory circle if one maintains both that (a) the Moon exists because it falls under the concept, existence, and (b) the concept, existence, has the Moon as an instance because the Moon exists. This objection is fatal to every theory of existence that conceives existence as abstractly common to existing things. The very existence of a thing cannot be its having a property or falling under a concept since it wouldn't be there at all if it didn't already (logically speaking) exist.
So I say: existence cannot be a first-level subjective or objective concept.
Followers of Frege appreciate that existence cannot be a first-level concept, but they make the mistake of conceiving of it as a second-level concept. They think of existence as a property of concepts, the property of being instantiated. Thus existence is not a property of cats, but a property of the concept cat, the property of being instantiated. This highly influential theory gets one thing right: it accommodates the insight that existence is no part of what a thing is. This insight is of course an old one. One finds it in Kant and before him in Aquinas, and before him in Avicenna to mention only three luminaries.
Kant made it clear that there is no quidditative difference – no difference as to quiddity or whatness – between a merely possible hundred dollars and an actually existing hundred dollars. And Aquinas was quite clear as to the difference between the questions Quid est? (What is it?) and An est? (Is it?) and the irreducibility of the latter to the former. Frege and his followers can be read as agreeing with Aquinas and Kant as to the negative thesis that existence is no part of what anything is. But there are serious problems with Frege's positive thesis that existence is a second-level concept. The positive thesis has the intolerable consequence of divorcing existence from the very things that primarily exist, namely, concrete individuals.
So I say: existence can be neither a first-level nor a second-level concept.
If you grasp this, then you are ready to tackle the problem of existence. If not, not.