Negative and positive claims are not artificial or equivalent, one is making a claim about positive existence, the other is making claims about lack of existenc

Negative and positive claims are not artificial or equivalent, one is making a claim about positive existence, the other is making claims about lack of existence. Of course we can manufacture the universal quantifier insofar as it's a mapping of the existential quantifier. That is to say, proving something is "not nonexistent" is not a statement that requires the universal quantifier, it's a claim that has simply mapped the equivalency of the existential quantifier, using the universal one. This is still only a statement about positive knowledge.