Here's a challenge for you: Prove that Bigfoot does not exist.
Or Santa Claus. Or Zeus. But you can always play the Lightgigantic game and say that just because you can't see Santa's workshop at the north pole doesn't mean it doesn't exist - you simply aren't in the right frame of mind to perceive it.
When "other ways of knowing" actually provide something worth knowing, I'll pay attention.
So lets disambiguate the claim "you cannot prove a negative":
Disambiguation A: There is no negative such that you can prove it.
Disambiguation B: There exists a negative such that you cannot prove it.
A is, of course, what people mean when they say "You cannot prove a negative". B is trivially true and uninteresting. So asking me to prove that bigfoot/santa/zeus/whatever doesn't exist does
nothing to prove your point. Just because there are
some negatives that you cannot prove, does not imply that
no negatives can be proven.
Here's a few negatives that can be proven. I'll post them in premise/conclusion argument forms, and I challenge you to show me how my proofs do not work
without resorting to standard skeptical arguments/reasoning that would, if applied, invalidate any form of a posteriori knowledge. Because if the whole "can't prove a negative" slogan is to be interesting or meaningful it has to be something other than standard skeptical posturings.
Argument 1:
1) Barack Obama is married.
2) If 1), then Barack Obama is not a bachelor.
So, 3) Barack Obama is not a Bachelor.
We've proven a negative! Namely, that Barack Obama is not a bachelor. You can see how any number of arguments of this form can be drawn up, taking advantage of the simple fact that so many things can be defined in terms of the negation of other things. In this example, being married entails not being a bachelor.
So maybe the holder of the "can't prove a negative" position is thinking that the slogan only applies to negative existential statements. Well there's no dice there, either:
Argument 2:
1) There exists an x such that x is Barack Obama and x is married.
2) For any x, if x is married then x is not a bachelor.
3) So (subconclusion), there exists an x such that x is Barack Obama, and x is not a bachelor.
So 4), There exists an x such that x is not a bachelor.
Clearly, proving a negative holds up just fine under existential quantification.
I'll restate my point: the "you can't prove a negative" slogan seems vacuous to me, and I don't see how it is anything more than standard skeptical posturings. Those same posturings, of course, rule out any form of
a posteriori knowledge under the slogan-holders ludicrously strict standard of proof, yet most of the slogan-holders do not seem to realize this fact.