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Not so fast, Sherlock

"When you have eliminated all which is impossible,
then whatever remains, however improbable, must be the truth."
In multiple Sherlock Holmes stories, by Sir Arthur Conan Doyle Note 1

Sherlock Holmes, by Sidney PagetI love the Sherlock Holmes stories, written by Sir Arthur Conan Doyle. I think I read all of them when I was a high school student (my father had the entire collection in a book).

I've already discussed at length one disagreement I have with Doyle's fictional detective, in my blog entry Dan and the moon. In that entry, I discussed how having "big picture" knowledge (such as knowing that the earth goes around the sun) can give you the ability to derive numerous useful facts that might otherwise appear to be entirely unconnected.

I also have issues with the Sherlock Holmes quote at the top of this entry. While logically true, it's seldom applicable in the real world, where it's rarely possible to fully enumerate all alternatives. There are always possibilities you haven't thought of, or don't know about, so it's seldom possible to use a process of elimination.

Enumeration of all possibilities can be done in mathematics, where one can frequently list all possible cases. For example, an integer (a whole number) must be either positive, negative, or zero. Another example: a number, by definition, must be either irrational or rational ("ratio-nal", that is, able to be expressed as a ratio of two integers). The simplest and most common proof that the square root of two is irrational proceeds by assuming it to be rational, and then showing that this leads to a contradiction. That possibility having been eliminated, the only remaining possibility, namely that the square root of two is irrational, must be the truth (Sherlock would be pleased). Note 2

In the real world, as opposed to mathematics, a list of all possibilities can also be attempted in some cases. In medicine, this is the starting point of a process called "differential diagnosis". Doctors are often called upon to attempt to find out what condition is causing a particular set of symptoms. They explicitly list the possibilities, and then order tests which are chosen to distinguish between them. I discussed one such case recently in my blog entry Ixodes scapularis, in which Margie's doctor needed to distinguish between Lyme disease and cellulitis.

This usually works because medicine has a great deal of experience with most common medical problems. But it sometimes fails for the very reason I'm discussing in this blog entry: it's really impossible to be sure that you have exhaustively listed all possibilities. For one thing, there are medical conditions which are extremely rare, and are apt to be missed by even very experienced clinicians. In fact, Lyme disease, discussed in the blog entry I just mentioned, was a condition that was often misdiagnosed in geographical areas where the disease was not very widespread. Now that it is becoming more common (unfortunately), it is less frequently missed. Note 3

Not only is it possible for a clinician to fail to list a rare possibility, but every now and then an entirely new disease pops up that's never been seen before. Thus when AIDS first began to appear, it took a long time for doctors to realize that they were dealing with something new. And while Lyme disease is now being more accurately diagnosed, it's becoming clear that other diseases are transmitted by the same tick bites, and the diagnostic picture is becoming even more complicated. These are not new diseases, but were not previously recognized. Differential diagnosis doesn't work when the problem is something that's not on the list at all.

Differential diagnosis generally works in medicine because it's a much studied field in which the most probable causes of a set of symptoms are fairly well known. But people often try to apply a process of elimination to questions which are much harder to pin down. In particular, Sherlock's reasoning (I might call it the "Holmsian Fallacy") is used to justify a great deal of pseudoscience.

Suppose someone sees a moving light in the sky which he interprets as an actual flying object. Since it's unidentified, it's called an Unidentified Flying Object, or UFO. UFO enthusiasts will sometimes list various possible explanations for the sighting. It might have been a secret military aircraft, or a drone, or an optical illusion, or the northern lights, or a meteorite, or the moon. Sometimes, all the listed possibilities can be eliminated one by one. The UFO enthusiast then might say, "Well then, it had to have been an alien craft of extraterrestrial origin. If that's not what it was, you tell me what you think it was."

But this is simply not a logical conclusion. The answers, "I don't know" or "Something I've been unable to think of yet" are both perfectly acceptable. It is not even remotely conceivable to enumerate all possibilities in this situation. And probabilistically, "something I've been unable to think of" is far more likely than "an alien craft of extraterrestrial origin".

A classic case of this error was made in the book Chariots of the Gods, by Erich von Daniken. Looking at the famous statues on Easter Island, he simply couldn't come up with any possible way that the people on the island could have moved them down from where they were carved, and erected them on the beaches, in the absence of any powered machinery. Thinking he had eliminated all other possibilities, he concluded that visitors from an alien population must have helped with the project.

Von Daniken's book was full of nonsense like that, chosen to justify his alien visitors theory. In fact, researchers ultimately proposed numerous ways the Easter Island job could have been accomplished without power tools of any sort, although they're still arguing about how it was actually done. See, for example, here (with a video) and here (for some debate). But because von Daniken had been unable to think of alternative ideas himself, they didn't make it onto his list of possibilities.

Another place it's common to see the Holmsian fallacy is in the various "Intelligent Design" ("ID") arguments for the existence of God (sometimes called the "argument from design" or the "irreducible complexity" argument). Their fundamental claim is that since the universe, and in particular life, are complex, they therefore had to have been designed by an intelligent designer. The problem with this argument is that the "therefore" implies that all other possibilities can be eliminated. But perhaps the universe and life, in all their complexity, came about in some other way, not requiring a designer.

Proponents of Intelligent Design will sometimes then respond, "Exactly what other way do you propose?" As I noted above, the answers, "I don't know" or "something I've been unable to think of yet" are both perfectly acceptable. Who was able to think of Relativity before Einstein? Who was able to conceive of quantum mechanics before Planck, Bohr, Heisenberg, et al.? (actually, I still can't conceive of quantum mechanics.)

Remember, the same argument was made in the past for the manner in which complex life came about. Each of the animals, and man, were thought to be individual creations. But since the work of Charles Darwin, and all the research that has supported Darwin in the over 150 years since the publication of his book On the Origin of Species, it's now quite clear that complex life can evolve from simpler life in the absence of any designer. I strongly suspect that at some time in the future, mankind will figure out how the first simple life arose from nonliving molecules. There are already a host of theories as to how this might have happened, but the answer may still turn out to be "something we haven't thought of yet".

Sherlock's trains of deductions made for wonderful fiction, but fiction it was. Many of his deductions were highly probabilistic in nature, yet he never seemed to get any of them wrong. Similarly, the quote at the top of this entry implies an ability to list all the possibilities in any given situation, an ability which nobody has in real life. After you've listed possibilities A, B, C, and D, the answer very often turns out to be "E, none of the above".

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© 2013 Lawrence J. Krakauer   Click here to send me e-mail.
Originally posted August 1, 2013

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Footnotes (click [return to text] to go back to the footnote link)

Note 1:   Various versions of this occur in The Sign of the Four; The Adventures of Sherlock Holmes, "The Adventure of the Beryl Coronet"; The Memoirs of Sherlock Holmes, "Silver Blaze"; The Return of Sherlock Holmes, "The Adventure of the Priory School"; His Last Bow, "The Adventure of the Bruce-Partington Plans"; The Case-Book of Sherlock Holmes, "The Adventure of the Blanched Soldier". I got this list from a WikiQuote article, from which I also took the public domain Sidney Paget portrait.  [return to text]

Note 2:   Here's the proof: By definition, a rational number can be expressed as a fraction a/b, where a and b are integers. Such fractions can always be reduced to their lowest terms, so that a and b share no common factors - assume that has been done. Thus at least one of a and b must be odd, since if they were both even, they'd share the factor 2.

But if a/b = the square root of 2, then a2 = 2b2. Therefore a2 must be even. Because the square of an odd number is odd, that in turn implies that a is even. This means that b must be odd, because a/b is in lowest terms.

But since a is even (a multiple of 2), then a2 is a multiple of 4. If a2 is a multiple of 4 and a2 = 2b2, then 2b2 is a multiple of 4, and therefore b2 is even, so (as above), b is even as well.

So b is odd and even, a contradiction. Therefore the initial assumption that the square root of 2 can be expressed as a fraction must be false.

Having eliminated that possibility, the only remaining possibility, that the square root of 2 cannot be expressed as a fraction (that it's irrational) must be true.   [return to text]

Note 3:   Software for medical diagnosis can't, at this point in time, replace human diagnosis. But one area in which such programs are rather strong is in their ability to consider really obscure possibilities that even experienced clinicians are apt to miss. Thus, such programs can be useful in augmenting the judgment of human diagnosticians.   [return to text]

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