The question of whether or not one can prove a universal negative usually comes into question for the atheist in proving that God does not exist. (I’m using atheism in the sense that an atheist believes that God does not exist). The question is can an atheist prove the non-existence of a being (a universal negative)? Yes, he can. Usually you’ll hear arguments against atheism that suggest it’s untenable because it’s impossible to prove a universal negative (that’s really a bad argument, no one should ever use it).
What if I say that there are no tyrannosaurus rex living on earth today? There are certainly some questions like, “Are there any polka-dotted geese that exist?” Well, if that may be difficult to disprove universally since I would need exhaustive knowledge of the universe. So, okay, I cannot disprove that universal claim. So what now?
It’s actually quite simple to prove a universal negative or the non-existence of anything. All you would need to do is demonstrate a logical contradiction within the universal claim. So, for the atheist, I believe his best argument is to demonstrate the logical impossibility of God (to prove a universal negative). How would he do this? I don’t know, as a theist I don’t believe any atheist can prove an inherent contradiction in the existence of God, that’s his task, not mine.