You Can Prove a Negative
BY STEVEN D. HALES
Steven D. Hales is a Professorial Fellow at the University of London and the University of Edinburgh. He is a past winner of Bloomsburg University’s Outstanding Teaching Award, and has published numerous books and articles.
A PRINCIPLE OF FOLK LOGIC is that you can’t prove a negative. Skeptics and scientists routinely concede the point in debates about the possible existence of everything from Big Foot and Loch Ness to aliens and even God. In a recent television interview on Comedy Central’s The Colbert Report, for example, Skeptic publisher Michael Shermer admitted as much when Stephen Colbert pressed him on the point when discussing Weapons of Mass Destruction, the comedian adding that once it is admitted that scientists cannot prove the nonexistence of a thing, then belief in anything is possible. Even Richard Dawkins writes in The God Delusion that “you cannot prove God’s non-existence is accepted and trivial, if only in the sense that we can never absolutely prove the non-existence of anything.”
There is one big problem with this. Among professional logicians, guess how many think that you can’t prove a negative? That’s right, zero. Yes, Virginia, you can prove a negative, and it’s easy, too. For one thing, a real, actual law of logic is a negative, namely the law of non-contradiction. This law states that that a proposition cannot be both true and not true. Nothing is both true and false. Furthermore, you can prove this law. It can be formally derived from the empty set using provably valid rules of inference. (I’ll spare you the boring details). One of the laws of logic is a provable negative. Wait … this means we’ve just proven that it is not the case that one of the laws of logic is that you can’t prove a negative. So we’ve proven yet another negative! In fact, “you can’t prove a negative” is a negative — so if you could prove it true, it wouldn’t be true! Uh-oh.
Not only that, but any claim can be expressed as a negative, thanks to the rule of double negation. This rule states that any proposition P is logically equivalent to not-not-P. So pick anything you think you can prove. Think you can prove your own existence? At least to your own satisfaction? Then, using the exact same reasoning, plus the little step of double negation, you can prove that you are not nonexistent. Congratulations, you’ve just proven a negative. The beautiful part is that you can do this trick with absolutely any proposition whatsoever. Prove P is true and you can prove that P is not false.
You can easily construct a valid deductive argument with all true premises that yields the conclusion that there are no unicorns. Here’s one, using the valid inference procedure of modus tollens (Latin for “mode that affirms by denying”):
If unicorns had existed, then there is evidence in the fossil record.
There is no evidence of unicorns in the fossil record.
Therefore, unicorns never existed.
Someone might object that that was a bit too fast — after all, I didn’t prove that the two premises were true. I just asserted that they were true. Well, that’s right. However, it would be a grievous mistake to insist that someone prove all the premises of any argument they might give. Here’s why. The only way to prove, say, that there is no evidence of unicorns in the fossil record, is by giving an argument to that conclusion. Of course one would then have to prove the premises of that argument by giving further arguments, and then prove the premises of those further arguments, ad infinitum. Which premises we should take on credit and which need payment up front is a matter of long and involved debate among epistemologists. But one thing is certain: if proving things requires that an infinite number of premises get proved first, we’re not going to prove much of anything at all, positive or negative.
Maybe people mean that no inductive argument will conclusively, indubitably prove a negative proposition beyond all shadow of a doubt. For example, suppose someone argues that we’ve scoured the world for Bigfoot, found no credible evidence of Bigfoot’s existence, and therefore there is no Bigfoot. This is a classic inductive argument. A Sasquatch defender can always rejoin that Bigfoot is reclusive, and might just be hiding in that next stand of trees. You can’t prove he’s not! (until the search of that tree stand comes up empty too). The problem here isn’t that inductive arguments won’t give us certainty about negative claims (like the nonexistence of Bigfoot), but that inductive arguments won’t give us certainty about anything at all, positive or negative. All observed swans are white, therefore all swans are white looked like a pretty good inductive argument until black swans were discovered in Australia.
The very nature of an inductive argument is to make a conclusion probable, but not certain, given the truth of the premises. That is just what an inductive argument is. We’d better not dismiss induction because we’re not getting certainty out of it, though. Why do you think that the sun will rise tomorrow? Not because of observation (you can’t observe the future!), but because that’s what it has always done in the past. Why do you think that if you turn on the kitchen tap that water will come out instead of chocolate? Why do you think you’ll find your house where you last left it? Again, because that’s the way things have always been in the past. In other words, we use inferences — induction — from past experiences in every aspect of our lives. As Bertrand Russell once pointed out, the chicken who expects to be fed when he sees the farmer approaching, since that is what had always happened in the past, is in for a big surprise when instead of receiving dinner, he becomes dinner. But if the chicken had rejected inductive reasoning altogether, then every appearance of the farmer would be a surprise.