You Can Prove a Negative

by Steven D. Hales

This is an excerpt from the longer article at eSkeptic: Wednesday, December 5, 2007 and Think, vol 10, Summer 2005 (pp. 109-112). In the first part Hales shows how it is certainly possible to prove a negative for a particular form of argument in elementary logic. This excerpt focusses on the problem of inductive reasoning being unable to prove a negative beyond all shadow of doubt. This is most likely what scientists means by one cannot prove a negative (i.e., no space aliens have visited/are visiting Earth). This is where this excerpt will pick up Hales thoughts. Select the link to the full article to get the full context.


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.

So why is it that people insist that you can't prove a negative? I think it is the result of two things: (1) Disappointment that induction is not bulletproof, airtight, and infallible, and (2) A desperate desire to keep believing whatever one believes, even if all the evidence is against it. That's why people keep believing in alien abductions, even when flying saucers always turn out to be weather balloons, stealth jets, comets, or too much alcohol. You can't prove a negative! You can't prove that there are no alien abductions! Meaning: your argument against aliens is inductive, therefore not incontrovertible. Since I want to believe in aliens, I'm going to dismiss the argument no matter how overwhelming the evidence against aliens, and no matter how vanishingly small the chance of extraterrestrial abduction.

If we're going to dismiss inductive arguments because they produce conclusions that are probable but not definite, then we are in deep manure. Despite its fallibility, induction is vital in every aspect of our lives, from the mundane to the most sophisticated science. Without induction we know basically nothing about the world apart from our own immediate perceptions. So we'd better keep induction, warts and all, and use it to form negative beliefs as well as positive ones.

You can prove a negative — at least as much as you can prove anything at all.