5.5352 Equally, one has wanted to express “There are no things” by “~ (Ex) . x = x”. But even if this were a proposition, — would it not also be true, if indeed “There were things”, but these were not identical with themselves?

Odd. The supposition that they might not be identical with themselves is nonsense (see 5.5303), at least roughly speaking. But then the symbolic notation is nonsense, and does not mean “There are no things.” Why can’t it mean that? Well, we have removed the equals sign from the notation, as unnecessary, so it doesn’t mean anything. Should we bring it back? No, for the reasons given before (redundancy). What we want here, after all, is a sign for existence, not identity.