A quick note: someone whipped out this old bit of folk-logic that “you can’t prove a negative” at me earlier today. This statement shows up an awful lot in all sorts of debates, but despite its folklore position as some sort of rule of elementary logic, no logician ever has actually proposed it.
And that’s because it is pretty clearly not true, after all, it contains within itself its own handy refutation: “you can’t prove a negative” is itself a negative statement, so if you can’t prove a negative, you can’t prove that “you can’t prove a negative.”
But of course there are plenty of examples of negative statements that people can prove. One of the real elementary laws of logic is that any proposition P is identical to the negation of its negation, that is, to not-not-P. So if you can prove a positive statement, then you also prove a negative statement which is equivalent to it. (If Descartes could prove that he existed with cogito, ergo sum, he could also thus prove that he wasn’t nonexistent.)