It is often claimed, with a triumphant air of finality, that “you can’t prove a negative.” Along similar lines it is often said, as if it were an unquestionable truth acknowledged by all, that “absence of evidence is not evidence of absence.”

Both of these supposed rules are epistemic nonsense.

You can prove that something doesn’t exist, and absence of some kinds of evidence is in fact evidence of absence. These are straightforward consequences of elementary probability theory.

Let’s first dispense with some obvious straw men. One might say, “Of course you can prove a negative; for example, it’s easy to prove that there does not exist any real number whose square is negative.” But that is a proof about the world of mathematical abstractions; when people say that you can’t prove a negative, they have in minding proving nonexistence of some entity or phenomenon in the physical world. This is the more interesting case that I am addressing here.

The second straw man goes in the other direction: it is true, in a vacuous and utterly uninteresting sense, that you can’t prove nonexistence of a hypothesized phenomenon or entity with 100% certainty… but this is only true in the same sense that you can’t prove any claim about the physical world with 100% certainty. For example, I could claim that you are not actually reading these words, but are instead experiencing an elaborate hallucination as you drool in your padded cell in a mental institution. You cannot prove, with absolute certainty, that this is not the case. Absolute proof is reserved to the realm of mathematics only. In assessing claims about the physical world we are always working with imperfect information, and so the relevant question is not, can you prove that X is true, but rather, how probable is it that X is true?

So let’s agree that “prove,” in the context of assertions about the physical world, in practice means “demonstrate that a high degree of confidence is warranted.”

So how can you prove a negative, and how can you use absence of evidence as evidence of absence? The key lies in considering the probable consequences of a hypothesis. Consider the hypothesis, “There is a cat living in my apartment.” If this hypothesis is true, then I would expect to observe the following:

• Urine stains somewhere in the apartment; the cat has to pee sometime.
• Feces of a certain size appearing from time to time; the cat is going to have bowel movements.
• Food going missing; the cat has to eat sometime. Or, if the cat doesn’t eat, it’s going to die, and I expect the bad smell of a decomposing body to eventually become evident.
• Scratched up furniture or other items; it’s an established behavioral pattern of cats that they scratch things.
• Unexplained sounds of movement. It’s unlikely that the cat can be so thoroughly stealthy that, aside from never seeing it, I never even hear it.
• Loose animal hairs on the carpet or in other areas around the apartment.
• Sneezing, itching, and a runny nose even when I’m not suffering a cold and it’s not allergy season. (I’m allergic to cat dander.)

If I observe none of these expected consequences of a cat living in my apartment, I can be very confident that there is, in fact, no cat living in my apartment. I have proven a negative through a lack of evidence.

Notice the form of this logical rule: absence of expected evidence is evidence of absence. If I did not expect to have this evidence—say, because I haven’t even entered the apartment for the last three months—then its absence would be meaningless.

For an entertaining fictional illustration of this idea, let’s look at the Sherlock Holmes story, “Silver Blaze.” A race horse has been stolen and its trainer killed. Suspicion is laid upon a man named Fitzroy Simpson. Holmes argues that Simpson could have been present in the stables that night:

Gregory (Scotland Yard detective): “Is there any other point to which you would wish to draw my attention?”

Holmes: “To the curious incident of the dog in the night-time.”

Gregory: “The dog did nothing in the night-time.”

Holmes: “That was the curious incident.”

As Holmes later explained,

…a dog was kept in the stables, and yet, though some one had been in and fetched out a horse, he had not barked enough to arouse the two lads in the loft. Obviously the midnight visitor was some one whom the dog knew well.

(You can stop here if the above intuitive explanation satisfies you.) Here’s the math, for those who are interested:

$$ \frac{\Pr(A \mid \neg D, X)}{\Pr(\neg A \mid \neg D,X)} = \frac{\Pr(A \mid X)}{\Pr(\neg A \mid X)} \cdot \frac{\Pr(\neg D \mid A, X)}{\Pr(\neg D \mid \neg A, X)} $$

In the above equation,

• \(X\) stands for any relevant background information;
• \(A\) stands for the hypothesis and \(\neg A\) stands for its negation (the statement that the hypothesis is false);
• \(D\) stands for a datum that is not observed;
• \(\Pr(A \mid X)\) means the probability of hypothesis \(A\) given only the background information \(X\), and similarly for the other expressions of the same form.

In the example of the cat, for the specific expected evidence of urine stains, we would have the following:

• \(A\) means “there is a cat living in my apartment.”
• \(\neg A\) means “there is not a cat living in my apartment.”
• \(D\) means “I find urine stains in the apartment.”
• \(\neg D\) means “I do not find urine stains in the apartment.”
• \(X\) might stand for background information such as “I have never brought a cat into the apartment.”
• \(\Pr(A \mid \neg D, X)\) means “the probability that there is a cat living in my apartment, given that I find no urine stains in the apartment and I have never brought a cat into the apartment.”
• \(\Pr(\neg A \mid \neg D, X)\) means “the probability that there is not a cat living in my apartment, given that I find no urine stains in the apartment and I have never brought a cat into the apartment.”
• The expression $$ \frac{\Pr(A \mid \neg D, X)}{\Pr(\neg A \mid \neg D, X)} $$ is the odds in favor of there being a cat living in my apartment, given that I find no urine stains and have never brought a cat into the apartment.
• \(\Pr(A \mid X)\) means “the probability that there is a cat living in my apartment, given that I have never brought a cat into the apartment”.
• \(\Pr(\neg A \mid X)\) means “the probability that there is not a cat living in my apartment, given that I have never brought a cat into the apartment”.
• The expression $$ \frac{\Pr(A \mid X)}{\Pr(\neg A \mid X)} $$ is the odds in favor of there being a cat living in my apartment, given only the information that I have never brought a cat into the apartment.
• \(\Pr(D \mid A, X)\) means “the probability that I find urine stains in my apartment, if there is a cat living in my apartment and I have never brought a cat into my apartment.”
• \(\Pr(D \mid \neg A,X\) means “the probability that I find urine stains in my apartment, if there is not a cat living in my apartment and I have never brought a cat into my apartment.”
• The expression $$ \frac{\Pr(\neg D \mid A, X)}{\Pr(\neg D \mid \neg A, X)} $$ is the likelihood ratio for the (lack of) evidence; that is, the ratio of (a) the probability that I find no urine stains if there is a cat living in my apartment, versus (b) the probability that I find no urine stains if there is not a cat living in my apartment.

The important point is that $$ \Pr(\neg D \mid A, X) < \Pr(\neg D \mid \neg A, X). $$ That is, it is more probable that I find no urine stains if there is no cat living in my apartment than it is to find no urine stains if there is a cat living in my apartment. The fact that I do not find urine stains in my apartment multiplies the initial odds in favor of there being a cat living in my apartment by a number less than one, thus decreasing those odds. Each additional piece of expected evidence that I do not find further decreases the odds.