This may not be exactly an answer, but it is too long for a comment.
We say things like:
- Black swans exist.
- Ghosts do not exist.
- Finland exists.
- God exists.
- God does not exist.
- My belief in God exists.
- Emerging properties do exist.
- Honesty doesn't exist (anymore).
If someone tells us that black swanns don't exist, we will usually tell them that they can be easily found in zoologics, or show them a picture of black swanns. If someone tells us that Finland does not exist, we may show them a map, or a travel agency's fares for travels to Helsinki. Is someone doubts the existence of emerging properties, we might point to the fact that while a plane can fly, no separate piece of a plane can. Or that while Boris Johnson is for Brexit, the cells on his liver, toe, or even brain, do not know anything about the issue.
We would, in fewer words, propose a "verificationist" approach to the subject: "you think X doesn't exist? Here is an instance of X: a black swan, a post card just coming from Finland, the emerging property of Mr. Johnson's support for Brexit."
Things get more complicated if someone tells us that ghosts or unicorns do exist. A verificationist approach seems insufficient, because we would have to search for either ghosts or unicorns in the whole universe. "Proving" in a verificationist way that there are no unicorns in the Isle of Man may be not too difficult, but our interlocutor may argue that "perhaps they exist in Alpha Centauri system, who knows". The obvious retort is that the burden of proof lies on the person making such claim. But this is not a very good answer; the person may well tell us that "I am not making any claim at all, except the very limited one that you cannot prove that unicorns do not exist in Alpha Centauri, or, indeed, in any place that you cannot directly inspect." A better line of argument is to show that unicorns are material impossibilities: horses have evolved in certain ways, that preclude the existence of horned horses; and the evolution of horses in other planets is even more impossible, for the natural history of another planet is necessarily different from Earth's own natural history.
But things get complicated again if we allow the interlocutor to move the goal posts (for instance, if he points us to a rhynoceros, and tells us "here, the unicorn; I told you it existed". Of course, the rhynoceros is not a horned horse, and the definition of unicorn requires exactly this). This is relevant for the discussion of the existence of God, for one of the argumentative tactics of theists is to get their interlocutors into admitting that they cannot prove that some "superior" entity doesn't exist (that is what the traditional arguments for the existence of God do), and then trying to confuse us into a quite different idea - that this "superior" entity, a) created the world, b) is omnipotent, c) is omniscient, and d) is benevolent towards humans (or even more, maintains a "personal relationship" with some or even all humans). The goal post is moved; while the theist originally intends to make us believe in a being with all the properties from a) to d), he then gets us to admit that we cannot prove the inexistence of a being that has at most the property a). The proper response, of course, is that property d) is incompatible with property b), or at least with properties b) and c) combined.
Anyway, what we seem to be discussing is a property independent of the person, or people, talking about it. If black swans exist, they exist whether or not I believe in them, see them, like them, and whether I am looking at them, awake, aware of their "existence", or even alive. This independence we use to call, rightly or - more probably - wrongly, "objectivity". Existence is "objective". Which brings up the issue of
While we can verify that black swans exist, or conclude that unicorns must not exist because their existence would contradict what we know about the structure of the universe, or that God doesn't exist because its description is self-contradictory, a sentence that refers to the psyche of an individual is much problematic. The faithful in question could be lying, or could have not given enough thought to the issue of God to really hold a significant faith - and we may not have any actual means to confirm or dismiss the claim. Besides, this seems to in some way break the idea that existence is "objective". Whether someone's faith in God exists or not, that certainly depends, at least partially, on that person and her beliefs - and we can't demonstrate that such existence is self-contradictory; while the belief in itself can be absurd, people do often believe in impossible things. In that sence, it would seem that the beliefs of a person - be them religious or not - do not "exist" in the same way that Finland exists. We can perhaps assert that certain symptoms of the existence of the belief exist, and that they may indicate the existence of the belief (whenever John is going to cross the street, he looks both sides first, which indicates that he may believe that cars exist and are dangerous) - but is that the same thing?
This lack of objectivity is, of course, part of the problem with
Here the person uttering the sentence is admiting to being dishonest (which brings the question, why would we believe what they are telling us, but that is a digression). But is that person being honest about her own dishonesty, and how can they assess the existence of honesty in other people? This brings on a very interesting issue: that of statements that imply the impossibility of any verification. Any person that tries to answer to this with a sentence by asserting their own honesty will probably be told that this in fact "proves" the inexistence of honesty, as their answer is itself one more instance of lying.
Similarly, the assertion of the existence of invisible green pixies: "of course you cannot see them; as I told you, they are invisible".