You Can’t Prove a Negative

This claim, that “you can’t prove a negative” is a kind of urban legend of logic that floats around the internet.  It isn’t true, however.

First, and most obviously, the claim “you can’t prove a negative” is itself a negative claim, so if it were true, it could not be proven to be true, according to itself, and so anyone believing it would be without rational justification for believing it or asserting it.

Besides the fact the principle is self-defeating, it is also false, and obviously so.  “This sentence does not contain 900 words” or “This sentence does not contain an adverb” are both negative claims that can easily be demonstrated by careful inspection and basic counting. This applies to innumerable negative claims.  My cat, as I write this, is not downstairs.  I know this because my cat is here asleep beside me, upstairs.

Another common way to prove a negative claim true is to prove the contradictory of the negative claim, which will be positive, to be false. This can often be done by a reductio ad absurdum, a “reduction to absurdity,” a strategy which is often used in mathematical proof.  For example, the negative claim “there is no largest prime number” is proven by making the assumption that “there is a largest prime number” and inferring logical consequences until a contradiction is reached, which proves the assumed premise to be false, which in turn proves its (negative) contradictory true.¹

Another type of proof of a negative, one that works especially well with negative existential claims, is to prove that the claim contains a contradiction.  For example, the negative claim “A man taller than himself does not exist” is proven true by showing that “a man taller than himself” is a contradiction.  We can also prove “There does not exist a three-sided square” since a square is defined as a four-sided equilateral plane figure with four right angles.

Or consider the basic logical forms Modus Ponens and Modus Tollens.  Both can easily prove negative claims:

1. P ⇒ ¬Q
2. P
3. ∴ ¬Q

1. P ⇒ Q
2. ¬Q
3. ∴ ¬P

E.g.

1. If the President is in Washington, D.C., the President is not in Moscow.
2. The President is in Washington, D.C.
3. Therefore, the President is not in Moscow.

1. If the President is in the White House, the President is in Washington, D.C.
2. The President is not in Washington, D.C.
3. Therefore, the President is not in the White House.

I have no idea where this odd logical myth that “you can’t prove a negative” originated or why it is so widely believed and proclaimed, especially since it is so obviously false, and also self-defeating as a rational claim.

If I had to guess, I would say it probably gets its traction from the common perception that positive universal claims may be refuted by a single counterexample, e.g. “All swans are white” can be refuted by producing a single black swan.  But this has nothing logically to do with positive and negative.  A positive universal, an A-statement in traditional terminology, may be obverted into an E-statement, or universal negative: “All swans are white” is logically equivalent to “No swans are non-white,” a negative claim, which is equally refuted by producing a single counterexample, e.g. a black swan.

Every positive statement is logically equivalent to a negative statement, so if it were true that “you can’t prove a negative,” it would be true “you can’t prove a positive” and thus “you can’t prove anything”—which is false.  If any positive proposition whatever can be proven true, the logically equivalent proposition that it is not false can also be proven, which again, would be to prove a negative.

This silly “you can’t prove a negative” claim needs to die.

[ADDENDUM: See also “You Can’t Prove a Negative,” Part 2]

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1

Negative existential claim: “There is no largest prime number.”

Proof:

1. Suppose there is a largest prime number, P.
2. Let N be the number generated by multiplying all the primes from 1 to P and then adding 1. I.e. N = (2 x 3 x 5 x 7 x… x P) + 1
3. N is either prime or N is not prime.
4. N cannot be prime since N > P, and P is the largest prime number by 1.
5. ∴ N is not prime.
6. Since N is not prime it has a devisor besides 1 and N.
7. N cannot have a prime divisor between 1 and P, since this would leave a remainder of 1,   by the way N is constructed.
8. Since N is not prime, it must have a divisor D where D > P and D < N.
9. If D is a prime factor of N then since D > P, P  is not the largest prime number, contradictory to 1
10. If D is a non-prime factor of N, it will have a prime factor F > P since any number may be expressed uniquely as a product of prime numbers, none of which can be between 1 and P, since P is prime. So F is a prime number > P, which is contradictory to 1, since P is the largest prime number.
11. Or N has no factors between 1 and N, in which case N is prime, and since N > P, there is a contradiction with 1, since P is the largest prime number (and also a contradiction to 5).
12. ∴ There is no largest prime number P.

Q.E.D.