For instance, how can I prove that the red ball isn't in my black box that holds 20 balls?
Well, if I take out every one of all 20 balls, and none of them are red, then I have just proven a negative.
Well, technically speaking, what you have done is prove a series of positives. After removing each ball, you can say "This ball is black." After the series, you can inductively reason that "All of the balls are black," and therefore deductively reason that none of the balls are red, with all of the problems that Karl Popper pointed out with induction inherent in your reasoning. But most hypotheses are based on sample sets well short of the entire set (and most scientific hypotheses actually investigate an infinite set), and so the statement that you cannot prove a negative is a good rule of thumb in almost all situations.
Indeed, but my reply was to a poster who claimed that it was "scientifically impossible" to ever prove a negative.
And that's simply not the case. In fact, anytime that you can exhaust all possibilities in a set, you *can* prove a negative.