Or can you?
Steven Hales, professor of Philosophy at Bloomsburg University has some interesting things to say on the topic:
A real, actual law of logic is a negative, namely the law of non-contradiction. This law states that that a proposition cannot be both true and not true. Nothing is both true and false. Furthermore, you can prove this law. It can be formally derived from the empty set using provably valid rules of inference.” (Italics in original)
Any claim can be expressed as a negative, thanks to the rule of double negation. This rule states that any proposition P is logically equivalent to not-not-P. … Think you can prove your own existence? At least to your own satisfaction? Then, using the exact same reasoning, plus the little step of double negation, you can prove that you aren’t nonexistent. Congratulations, you’ve just proven a negative. The beautiful part is that you can do this trick with absolutely any proposition whatsoever. Prove P is true and you can prove that P is not false.” (Italics in original)
Dr. Hales also makes a nice distinction between deductive arguments and inductive arguments as it relates to negative claims, which might be the basis for the idea that some think a negative can not be proven.
Here is another philosopher, Dr. Craig, discussing the topic with a member of the audience in a debate with Dr. John Shook:
Another philosopher, Dr. Stephen Law, shares an interesting observation, “Notice, for a start, that ‘You cannot prove a negative’ is itself a negative.”
A law of logic, double negation, and a self-reference problem for the statement itself … it sure looks like a negative can be proven.