There is a reason why existence can not be a predicate, namely:
- Let's prove that unicorns exist.
- It is sufficient to prove that there is an existing unicorn.
- There are two possibilities: either an existing unicorn exists or it does not.
- The second possibility is a contradiction: how could an existing unicorn not exist? That's the same as saying that a blue ball is not blue.
- So, unicorns exist.
This argument probably dates back to Kant and the ontological argument of God's existence. It shows that $\exists$ is not a "property" and should be treated in a special way.
My question is why we need $\forall$ as a special symbol. Are there any arguments like the one (for existence) that I mentioned?
I suspect that that text about unicorns could be rewritten using negations and $\forall$, but I don't exactly see how.