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]

_______________________________________________________________

1  Negative 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.

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6 comments on “You Can’t Prove a Negative

  1. viredaedae says:

    I’m guessing the logic may come from the unique application of materialism and inductive reasoning; if the only reliable epistemology is science and inductive reasoning, and inductive reasoning cannot prove a negative, then you universally cannot prove a negtive.

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  2. TKDB says:

    I think, in short, this idea can more or less be traced back to scientism. As robalspaugh notes, it’s based on the limitations of induction, which is the primary tool of the scientific method. If you hold the scientific method to be the sole means of obtaining truth, its particular limitations are of course generalized into universal epistemological rules within your worldview.

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  3. robalspaugh says:

    My guess on its origin: you cannot prove a (certain kind of) universal negative inductively. “No swans are red” cannot be proven by an endless parade of swans.

    Just a guess.

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    • Eve Keneinan says:

      I suspect existential claims have a role. “There are no unicorns” cannot be proven empirically, since it would require total, simultaneous experience of all parts of space and time—depending on how “unicorn” is defined. But this seems confused.

      For example,

      1. if “unicorn” is used to translate the Greek μονόκερως, as in the King James Version, while μονόκερως was used to translate the Hebrew רְאֵם (reym) in the Septuagint, then there are (or were) “unicorns,” a now extinct type of wild ox.

      2. if “unicorn” means a magical beast, then there are no unicorns because there are no magical beasts, because there is no such thing as magic.

      I suspect it stems from the philosophical rejection of essences, such that people think they cannot rule out such things as “No dog has ever given birth to a cat” without ’empirically checking’ (so to speak) every given dog birth ever. Of course the person who thinks in this way is also confused, since such terms as “dog” “cat” and “birth” will also be without foundation: “None of the fictive category I call ‘dog’ as generalization of a number of sensibly similar animals I have experienced has ever done the fictive category of act I call ‘birth’ as a generalization of a number of similar sense experiences with respect to the fictive category I call ‘cat’ as a generalization of a number of sensibly similar animal experiences.” (Of course one would also have to ‘translate’ each noun or kind in the translation—especially, as Russell emphasizes, the term “similar.”)

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  4. I believe the saying “You can’t prove a negative” comes up during debates over evidentiary burdens in the court of law, commonly referred to as “Innocent until proven guilty”. What people usually mean by “You can’t prove a negative” is that you can usually not account for all the time you spend beyond a reasonable doubt, therefore making the legal system “Guilty until proven innocent” an inherently unjust system considering it’ll inherently convict innocent people or at least convict more innocent people than the alternative.

    I believe this misconception probably has it’s roots in the shortening of people’s attentions spans. People now have to say things quickly and sometimes in less than 144 characters. Nuance is dead in today’s society.

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    • Eve Keneinan says:

      “Innocent until proven guilty” is an appropriate and excellent principle in jurisprudence.

      “Juris-Prudence” itself is a compound of “right” and “judgment”—the principle recognizes the human situation that even though a given person is binarily either guilty or not-guilty, it is often the case that we cannot establish which one is the case on the basis of available evidence.

      It weighs in the ADDITIONAL factor of “it is worse to convict an innocent person as guilty than for a guilty person to be acquitted.” But this is an ethical judgment and not a general principle of logic. It is not the case that, in all cases, it is better to take a true proposition to be not-true, than to take a false proposition to be true. To take an obvious example, if Columbus had not had the false belief that earth was much smaller than it actually is, he would not have attempted to reach India via a Westward crossing, and would not have discovered the Americas.

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