A wider understanding of the fact that you can't prove a negative would, in my view, do a great deal to upgrade the public debate around science and technology.
As a journalist I have lost count of the number of times that people have demanded that a particular technology be "proven to do no harm". This is, of course, impossible, in just the same way that proving that there are no black swans is impossible. You can look for a black swan (harm) in various ways, but if you fail to find one that does not mean that none exists: absence of evidence is not evidence of absence.
All you can do is look again for harm, in a different way. If you still fail to find it after looking in all the ways you can possibly think of, the question is still open: "lack of evidence of harm" means both "safe as far as we can tell" and "we still can't be sure if it's safe or not".
Scientists are often accused of logic-chopping when they point this out. But it would be immensely helpful to public discourse if there was a wider understanding that you can show something is definitely dangerous, but you cannot show it is definitely safe.