Actually, You Can Prove a Negative

Related to: evidence of absence. If you do not understand why absence of evidence in favor of X is evidence against X, you will not agree with this post.

You have heard the phrase, "You can't prove a negative." I usually hear it in defense of religious belief, using some form of the argument, "You can't prove there is no god, therefore it is okay for me to believe in god, neener-neener." Even more bizarre is that I hear it second-most among atheists who think the supposed impossibility of a disproof of God helps their case. At any rate it has always struck me as trivially false. Negative statements are by definition the negation of a positive statement, so negative statements can be proven exactly to the degree that positive statements can be disproven. For example, you can disprove the statement [There is a pyramid on Mars] by thoroughly searching Mars and finding no pyramids. You have thus proven the negation: ¬[There is a pyramid on Mars].

Others have proven that you can prove a negative:
All of these pages make the same points I just did, unfortunately not in so few words. So I am not making any original claims here.

And yet the statement persists, so either I was missing some more reasonable interpretation, or this statement is useless and wrong.

One charitable way to interpret it is that it's a way of saying you can't have infinite certainty in a negative proposition. This, however, is just a special case of the general principle that you can't have infinite certainty in anything at all. It's impossible, you see, because that amounts to having an infinite amount of evidence, which is impossible by the laws of thermodynamics. This usage of "You can't prove a negative," is useless, even though it is trivially true.

The last straw was catching Daniel Dennett saying it at 2:15 in this interview. He is using it in the infinite certainty sense, the true-but-useless sense that will cause confusion. If you say you can't prove a negative because you can't have infinite certainty in one, and you omit that you can't have infinite certainty in positive statements either (remember, they are isomorphic), then you will either be or appear to be insufficiently confident in the latter.

Anyway, I asked around.

I asked Wikipedia, who answered that James Randi seems to be responsible for the popularization of this bastardious phrase. I was sad to learn that his use of the phrase is merely an equivocation, since to him, no amount of evidence against a hypothesis counts as disproof. No, that's not good enough.

One interpretation is, "You cannot prove a categorical negative against a sufficiently determined opponent." What this means is that if you assert in a debate that your opponent does not own any invisible unicorns, he will find a way to equivocate on "invisible unicorn" such that he "has" one. Note the equivocation on "has". This usage is of no use; it is false. What is actually being said here is that you can't disprove unfalsifiable theories, which aren't the same as negative statements.

Says John Beshir: "There are a lot of claims which are hard to prove impossible given only the basic assumptions people have. This does not mean the probability of them being true is non-negligible. Negatives quite often are examples of this- thus why 'you can't prove a negative' is quite common." I found this intriguing, but he didn't give any examples. And physics has proven many things impossible. So much for that.

In conclusion, the phrase, "You can't prove a negative," is always either a dishonest debate tactic, needlessly confusing, or wrong.

Don't use it.