Existence when treated as a predicate is a difficult concept to grasp. Clearly, “things” must either exist or not exist, this is the law of the excluded middle. The question remains whether non-existent things, which are denied yet conceived, also exist (leading to an obvious paradox).
To explain why this is an issue especially when we treat existence as a predicate,
I will give an example formalized using predicate logic.
The statement “Socrates is a man” can be stylized as follows:
However, in stating that “Socrates is man”,
we are also stating that “Socrates is” or in other words “Socrates exists”.
If we were to use the system above, the statement “Socrates exists”,
A problem arises however if we want to claim that “Socrates does not exist”.
We would write something like:
However, we pointed out earlier that having a construction of X(y) affirms the existence of y.
So in saying that “Socrates does not exist”, we are affirming his existence.
It is for this reason (and potentially others), that existence is not treated as a predicate in predicate logic.
Instead, we state that
∀(x): S(x) → M(x)
Which should be translated as “For all x (everything) if x is Socrates, then x is a man”.
Taking existence apart like this can avoid issues with affirming the existence of something by denying its existence.
The proposition “Socrates does not exist” would be formalized as:
For all x, x is NOT Socrates.
This solves our problems in the realm of logic, but fundamentally, our problem still persists. The proposition S still exists. For it to exist, we must have a concept of Socrates, which gets us back to the problem that: Things which are denied are still conceived (in the denial). I should add here that modal logic aims to solve this fundamental problem as well.
However, let us exit the realm of logic and look into why this problem is even problematic. Medieval Arabic philosophers were already discussing the importance of this issue. Say that non-existent things also exist, what implications does that have? Well, one issue scholars at the time raised is where these things are. If the non-existent-unicorn exists, then where is it? This become seven more problematic when talking about people who are yet to exist. If I talk about my daughter (who I hope one day to exist), the question may be asked: Where is she now? And what happens when she starts existing?
If we were to look at this from a pure materialistic perspective, we could say that all atoms (in the Greek sense of the word) that will one day constitute my daughter already exist. Her non-existence would merely amount to saying “There are as of yet no atoms that assumed the shape of my daughter”. The essence of “my daughter” is already conceptualized, her existence would be merely the affirmation of that essence. In other terms, one might say that my daughter exists only as a possibility, and of the “merely” possible, we say that they do not exist. Though, once that possibility moves to an actuality (once atoms are arranged as “my daughter”), we name it existence.