"You Can't Prove a Negative"

I've always been annoyed when I hear this phrase.  People tend to claim it, loud and proud, in a very authoritative way, but it's never sat well with me.  Before now, I never put a whole lot of thought into it, but why can't you prove a negative?  Don't we do this all the time?  A little Googling makes me feel better.

The misconception appears to stem from inductive reasoning.

For example:
1)  We have searched extensively for Bigfoot
2)  We haven't found any evidence of Bigfoot
3)  Therefore, Bigfoot doesn't exist.

Using this kind of inductive reasoning, we have NOT sufficiently proven that Bigfoot doesn't exist.  We haven't looked everywhere at that exact same time, so he could still be hiding out there.  He PROBABLY doesn't exist, but we certainly haven't proven it.

But by the same reasoning, you can't prove POSITIVE statements, either.  The famous example is this:

1)  Every swan we've ever seen has been white
2)  Therefore all swans are white.

This held true for a long time, until black swans were discovered.

So are all swans either black or white?  Can you prove it?

Inductive reasoning provides a good framework for what is probably true (that the sun will come up tomorrow, that Bigfoot probably doesn't exist), but not always for proving them to be so (see black swans).

I think people tend to use the phrase "You can't prove a negative" with regard to this kind of reasoning.  Within this context, it's true.

A couple problems with the original statement:   For one thing, "You can't prove a negative" is itself a negative statement.  If you were to prove it to be true, it would no longer be true (Yeah, that was your mind that was just blown…)

And you can always prove a double negative.  If something is true, then it's always not not-true.  Try it out on anything that you can prove to be true!  For instance, I can prove that I'm not dead, for all practical, legal, and scientific purposes (I think I'll go for a walk…) 

And finally, I can prove that there are no hot air balloons in my section of cubicles this morning.  If there were a hot-air balloon in here, I would see it.  I don't see one, therefore I can conclude that there aren't any here.

1)  I don't see any hot-air balloons in cubicle
2)  If there was a hot-air balloon nearby, I would see it
3)  Therefore, there aren't any hot-air balloons in my cubicle

Of course, I could misunderstand what a hot-air balloon is, or where my cubicle is, or where I am, but for all practical definitions and to the extent that anyone can know and observe anything, I can prove that there aren't any here.

Proving something doesn't exist or isn't true isn't a logic problem, but a scoping problem.  It's very hard to prove that something isn't so, because it's very hard to observe all of the possible range at once.

"You can't prove a negative" is frequently used by Bigfoot lovers who want to use "You can't prove that A is false" as evidence that "A is true".  This is bad.  This is no.  Keep your ears open, and you may be surprised at how often this comes up.