Do all things exist?
Nonexistence is the absence of existence, by definition. So, nonexistence does not exist. Therefore there is no such thing as nonexistence. To say that something does not exist thus seems to be a fallacy, since NOTHING does not exist. Everything, therefore, must exist.
Is this right? If not, what is wrong with the above argument?
Actually you can prove that something does not exist, both inductively and deductively. But the burden of proof is on the person who wants to assert that such and such is
the case. Take for example the theist. The theist would have to first prove that God exists, whereas the atheist does not have to prove the non-existence of God from the outset. Perhaps an atheist can give proofs for why it is impossible to prove the existence of God, but this does not imply that God does not exist. It simply means that we do not know.
Also it is not a fallacy to simply assert "It is not the case that Socrates existed in the year 2010." There is not a state-of-affairs in to which the proposition "Socrates existed in the year 2010." occrs and hence we can assert the denial of this, which is true. Now you may reply, "But Socrates could
have existed in the year 2010.", but this is a separate matter altogether.
"If nothing does not exist, then everything must exist" Seems dubious for a couple of reasons. 1) You are treating existence as a thing that has the property of non-existence. But if it is a thing, then is it really nothing? 2) In a modal context, there a number of possible worlds in which you or I do not exist, namely, in the worlds where your parents did not copulate and have you. So I would merely contingently exist, or possibly exist.
But there are even problems with this. If you assume that everything exists, then square circles must exist as well. But this creates a contradiction. Even if it is possible for a square circle to exist, you still have a contradiction, precisely because a square cannot be circle and a square at the same time (although it can be a rectangle).
Hope this helps.