And yet, we are making positive claims about it. I object.
We
aren't making positive claims about it- its a second-order claim. That's pretty much the difference between first and second order claims.
Second-order logic - Wikipedia, the free encyclopedia
Second-order predicate - Wikipedia, the free encyclopedia
First order claims are about individuals- if we analyze "the King of France doesn't exist" as a first order claim, we end up with this-
Dx= X doesn't exist
k= King of France
"The King of France doesn't exist" = Dk
This is what you're talking about. And this amounts to the claim that there exists an
object or
individual, k, that doesn't exist. This is contradictory, as I've noted.
But if "the King of France doesn't exist" is a second-order claim, what we are saying is that the function or predicate "X is the King of France" fails to be instantiated or satisfied, or, formally-
Kx= X is the King of France
"The King of France doesn't exist" =(~∃x)Kx
Colloquially, "there is no X such that X is the King of France".
On the second-order analysis, which is free of contradiction and apparently more faithful to the meaning of the English statement "there is no King of France", none of what you're saying is applicable.