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You can't prove a negative because of Hume's "problem of induction", which boils down to this: we don't know that the laws of physics are constant throughout space-time. This is where the idea came from that "all the swans ever observed are white" cannot prove that "no black swans exist".

You should read some Hume and Ayers and try this essay again. It doesn't hold together that well, and this is extremely well-trodden ground in the philosophical world (including your specific premises, though they don't make a lot of sense).

And, as far as your suggestion that nothing can be proved, this is a core topic of almost all philosophical work. It's been argued every possible way, including that nothing is real, only thoughts are real, etc. The umbrella term is ontology.

That was a shitty argument. It already failed to establish what is commonly refered to by "negative" in this context. You could criticise the term, that it is too general to hint at a wrongly infered statement in a special case. But the author does something else without really establishing anything. Going to dig in my garden now.