You Can Prove a Negative
by Steven D. Hales
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 .
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.