In this fuzzy form of what "positive" and "negative" are, there is a symmetry that allows all arguments you have for one "side" to be easily flipped. Each positive is somebody elses negative. Not guilty? Not innocent? With that line of reasong you could not ever prove anything since it's "the negative" of something else.

Mathematical logic has a solution to this though. You may want to check out constructive logic. That's where the statement "you can't prove a negative" is reasonable but has very specific meaning.