"God doesn't exist", grammatically incorrect?

It's interesting that you've noticed that because it's similar to a common objection to the ontological argument for God's existence.

God is an object of the imagination even if God does not exist. It's difficult to account for that in conceptual terms, but a common description of the situation is that existence is not in itself a property of something. If you say "some stones are brown", that can be negated as "some stones are not brown", and both of them mean something, even if they were false. However, if you say "some brown stones don't exist", it sounds peculiar. This is because existence is not a predicate. In a way, the word "exists" is just a fancy "is".

There are ways round this, though. One is to describe existence in a different way, for example "be real" or "have an instance". So, these are apparently properties and also shared by all things which actually do exist.

Unicorns don't exist. That should clear up the question.

Also note, there is no "proper" English grammar. (source below) While prescribing a specific formulation of grammar is useful to students just learning how to use the language, ultimately only usage can say what is "proper". Usage based grammar is called "descriptive" (as opposed to prescriptive). The grammar police on this very forum could stand to recognize this point, and let things that are clear enough to understand go without comment.

The 'concept' of a God exists, this however does not mean that God is an actual reality.

If no concept of God existed, then we wouldn't have a religious forum, then of course we have different 'concepts' of God/s.

We can say a leprechaun doesn't exist.

We have a 'concept' of a leprechaun, but if you want to explain to me that a leprechaun does exist then be my guest.

"Doesn't something have to exist in order to not exist?"

The kind of "incorrectness" you should be referring to is logical incorrectness (or invalidity), not grammatical incorrectness. The statement uses perfectly acceptable english grammar for the formulation of a proposition.

Secondly, Gary did not run does not require that Gary exists, because if Gary does not exist, then the proposition "Gary did not run" is still true, infact, it's tautologically true if Gary simply does not exist, since Gary did not and never will run, thus the proposition is true for all time, making it tautological.

"Doesn't something have to exist in order to not exist?"

My god.

You've just proven the existence of not only God, but leprechauns, UFOs, the Flying Spaghetti Monster, and the Tooth Fairy - anything you can invent a word for must exist!

You must alert the press immediately!

Something can exist as a concept without existing in reality. So your statement would be that the concept of god, which exists, does not exist in reality.

The point is that there are two different types of existence here.

"Doesn't something have to exist in order not to exist?"

Answer: Something has to exist conceptually in order to say it doesn't exist in reality.

Purple hairy leprechauns with orange spots do not exist. I don't think that they need to have existed at one point to not exist now. They just never have and never will. Ergo your argument falls apart.

Almost everything on Y/A is grammatically incorrect.

In that case, unicorns and ogres also exist.

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"Doesn't something have to exist in order to not exist?"

What?

You are negating your own statement.

If God existed, he would exist.

But God does not, therefore he doesn't exist.