Absence of Expected Evidence is Evidence of Absence

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.

 

Spiritual Witness

Many people base their religious beliefs on some sort of “spiritual witness”—an epiphany of some kind, a “burning in the bosom,” a feeling of peace, a strong positive feeling that comes over them. The “testimonies” that religious believers give can be quite moving, evoking a deep emotional response in even the most cynical of persons.

Now I ask my Mormon friends and family, are these feelings, these emotional responses, a reliable basis for discerning the truth? I would argue that they are not. Internal emotions and feelings may be useful data for understanding your own self, but they are not evidence about the external world. Least of all do they provide a reliable guide to questions of cosmology… and a belief in an omnipotent Creator is no less than a belief about the ultimate nature and origin of the cosmos itself.

Why would you expect mere feelings to reliably guide you to the truth about any aspect of the external world? Because other people told you that they could? What makes you think these people are correct? Because of warm feelings you have towards these people? Can you see the circular logic?

Let’s take a look at some “testimonies” I have pulled off of the web:

It is difficult to describe to someone who has never felt it how the Gospel can change and improve one’s life. But learning the Gospel changed me totally. I now have no doubt about our purpose in this world and that I am following the right path, I have a certainty I never knew before, and a peace that goes with it.

“No doubt.” “I have a certainty.” If a person were to use these words in any other context, lacking a scrap of evidence for their belief, we would call them delusional.

I still remember sitting alone, reading the Book of Mormon, looking for errors, and questioning. The more I read, the more I became convinced that this book could only have one source, God. I was reading about God’s mercy and His willingness to forgive any sin… and I began to weep. I cried from the depth of my soul. I cried for my past ignorance and in joy of finally finding the truth.

So… assuming that a God exists, how does this woman know what a book that comes from God reads like, in comparison to texts written by mere human beings? Is she such an expert in divine versus secular textual classification that she can discern without a doubt which is which? Oh, but the Spirit bore testimony to her, you say. And how can she, with any reliability, distinguish this hypothetical testimony of the Spirit from an emotional response caused by a desire to find absolution for past wrongs she may have committed?

“What am I doing here?” Dear God. I am here because I believe in you, because I believe in the compelling and majestic words of the Book of Mormon, and because I believe in the Prophethood of your servant Joseph Smith. I know in my heart my decision is the right one. Please give me the courage to carry on with this new self and new life, that I may serve you well with a strong faith.

“I believe.” Based on what evidence? Warm fuzzy feelings? “I know in my heart.” Which is just a way of saying, “I want this to be true but I don’t actually know.”

I read the whole thing through in one sitting. I don’t think I even changed position…
As I read a thought began to form and then started going through my head over and over and over: “Oh my God! This is from God!” It was like being slammed in the head with a brick or a hard plank of wood. I was stunned. It was real… It was direct revelation— it really was the Word of God. Literally. Oh my God! This really IS from God!

A lot of thoughts form in my head. Some of them stay there for quite a long while. That, however, is not enough for me to conclude that the thought is correct.

Now, I know what my Mormon friends and family are thinking: I am hard-hearted and stiff-necked; I have shut myself off from the still, small voice of the Spirit. But I ask you to do some serious soul-searching of your own, and ask yourself: are these people really, truly justified in their belief, a belief that is unsupported by hard evidence but arises, by their own admission, from nothing more than a positive emotional response? Be honest with yourself. Don’t give the answer that you feel obligated to give. Don’t shy away from the answer that you fear to acknowledge, the answer that you fear would make you an unrighteous person.

Suppose now that these are the testimonies, not of LDS converts, but of converts to another religion altogether. Would you still agree that their belief is justified, based on the “spiritual witness” they have received? Perhaps I just substituted a few words here and there to make them look like LDS testimonies, say, changing “Islam” to “the Gospel,” “the Qur’an” to “the Book of Mormon,” and “Muhammed” to “Joseph Smith”. Does your answer change? Should it?