Proving negatives and burden of proof

The Skeptic's society has a weekly online newsletter called eSkeptic. A month ago, they had a great essay titled "You Can prove a Negative". It includes lots of things that I've tried to explain or was thinking of explaining in the future. All claims can be stated as negative claims. Absence of evidence is evidence of absence under certain circumstances. Inductive arguments do not imply their conclusions with certainty, but they still suffice as evidence. Inductive arguments are required to prove most positive claims as well as negative claims.

But I disagree on one point. Why do people think you can't prove a negative? Hales thinks that it's because of confirmation bias and disappointment in the uncertainty of induction. I think he missed an important reason: because it's true! Or at least partly true.

It's true that there is no fundamental difference between positive and negative claims. If we have a claim P, then there exists a negation of this claim which we might call Q. Q is equivalent to not-P (symbolized as ¬P or ~P) and P is equivalent to not-Q. P and Q are in a symmetrical relationship, and there is nothing in the logic to tell you that one claim is harder to prove than the other.

But in practice, this symmetry is often broken. In practice, one of these claims is what we call the null-hypothesis, and the other is not. The null-hypothesis states that there are no unknown processes at work. When we say "you can't prove a negative", the "negative" refers to the null-hypothesis. In practice, the null-hypothesis is not impossible to prove, but it is usually a little harder to prove.

For example, let's take the hypothesis that bigfoot exists. Let's assume that this hypothesis predicts approximately 1% chance of finding bigfoot behind any given tree. We look behind 300 trees. If the bigfoot hypothesis is true, then we have a 95% chance of finding him. If it's false, than we have a 0% chance of finding him. That means that if we don't find him, it's very unlikely that he exists. But if we do find him then that's some smoking-gun evidence for bigfoot (of course, I am making the spectacularly false assumption that our perceptions cannot be tricked). Either way, the evidence is fairly conclusive. But it would take hundreds of trees to prove the null-hypothesis, and only one to prove bigfoot (provided that we pick the right one). The null-hypothesis simply takes more work to prove.

Now, the reason I'm arguing about this asymmetry between "positive" and "negative" claims is not because I care about bigfoot or anything (seriously, bigfoot is boring). It's because the asymmetry is used as the basis for the concept of "burden of proof." Let's face it--proving stuff takes work. Proving the null-hypothesis takes more work than proving bigfoot. Therefore, the people who should be doing the work should be the bigfoot folks (known as cryptozoologists). They have the burden of proof.

The burden of proof concept is abused fairly often, I think because people detach it from its justifications. If you want to know when it's valid, you have to remember why it's valid!