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A Sherlock Holmes Logical Fallacy

We recently received the following question:

‘If you’ve eliminated all other possibilities whatever remains must be the truth,’ is the famous a quote from the brilliant, but fictional, detective Sherlock Holmes. It seems to be an inescapable statement of cold, hard logic but it is often used in movies or on TV as a fig leaf to cover a huge leap of logic on the part of the hero or as justification for why the answer to the mystery at hand must be ghosts, aliens or some other supernatural phenomena. Is the statement a logic fallacy, a practical impossibility or a viable method of investigation that is being misused?

Thanks for a consistently great show.


Gothenburg, Sweden

Thanks for the great question, Stuart. Like many points of logic there are many subtleties that need to be understood before they can be applied properly.

The actual quote is this:

“Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth.”

As stated this is an obvious point of logic – the truth must lie within the set of the possible, which is defined as everything that is not impossible. But Holmes (by which I me Sir Arthur Conan Doyle, the author of Sherlock Holmes) meant the statement as a practical rule of thumb. As an investigator you create a mental list of all potential explanations for a situation. You then systematically eliminate those explanations that you can demonstrate are impossible, either through logic or empirical evidence. Whatever you are left with is the solution – even if it may seem extremely improbable.

Holmes, the hyper-rationalist, was a genius at just this type of endeavor. He could make the connections necessary to eliminate possibilities. He also had the imagination to consider possible solutions that may at first seem entirely improbable – but if that is what you are left with then it must be true.

But Sherlock Holmes was working within a specific framework – a materialist, rational, scientific view of the world. Within that framework this process of elimination works well. In fact, I took a course on Sherlock Holmes in medical school, applying his investigative principles to medical diagnosis, which is a type of investigation.

For example, physicians will make a list of possible diagnoses – called a differential diagnosis. We then systematically eliminate possibilities until we are left with one diangosis that fits all the signs, symptoms, and laboratory results. Whatever remains must be the proper diagnosis, even if it is a rare disease or a very rare manifestation of a more common disease.

In practice this process does not always work because our knowledge is incomplete. Also, this process is often trumped by a more important clinical principle – risk vs benefit. Often the risk of doing diagnostic procedures to establish a specific diagnosis is not justified by a corresponding benefit (it won’t lead to a treatment), or sometime the pathway of maximal benefit vs risk involves giving a low-risk treatment to see if it works.

As an aside, the TV character House is a very Sherlock Holmes type character applied to medical diagnosis. The character is portrayed as a complete egotistical ass, partly because he pursues diagnosis even at the expense of other clinical and ethical principles, such as risk vs benefit or informed consent.

Getting back to Stuart’s question – problems arise when this very logical principle of investigation are applied without constraints. The logic breaks down in a world where one allows for the existence of magic. How, then, does one define possible vs impossible? Holmes clearly assumes magic does not exist, and Doyle places him in a world (the real world) where magic in fact does not exist. Therefore Holmes (very much unlike Scully from The X-files, who lives in a paranormal world) is never “baffled” when his rational explanations do not fit an irrational world he refuses to accept.

There is also a very practical consideration in applying this principle – how complete is your set of alternate explanations?  As I tell my students, if you fail to consider the proper diagnosis you will never make it (this is actualy only true for some diagnoses, for others may be made by doing screening tests, even if you are not looking specifically for the diagnosis you find), or if you prematurely limit the range of possible diagnoses you have no chance of finding the right diagnosis.

The abuses of this principle that Stuart refers to generally combine the above two failings – including “magical” explanations in the set of the possible, and failing to consider all possible explanations. Let’s take a very common example – the sighting of a UFO. Proponents of the ET hypothesis aften argue that the UFO could not be a plane, a balloon, a cloud, or the planet Venus, therefore it must be an ET craft.

The first problem with such arguments is that the list of possible explanations is too short. It should also consider more and even rare but mundane explanations, such as a satellite re-entry, an experimental aircraft, a hoax, or an optical illusion.

The logic also breaks down in concluding that the UFO (by which I simply mean a flying object that is truly unidentified) must be an ET craft, because that is all that remains. In addition to not being “all” that remains, it should not even be included in the list of known possibilities. The error here is including unknown or new phenomena on the list of the possible, which should only include established phenomena.

This is necessary because the list of possible new phenomena is theoretically infinite. Why favor ET craft often time-traveling psychic bigfeet? Or fairies from another dimension? Or a previously hidden race of inteligent dinosaurs who survived the extinction 65 million years ago. Or the act of a mischevious deity. Favoring one unknown explanation over another based solely on the absence of an established explanation is a logical fallacy we call the argument from ignorance.

So Sherlock Holmes’ principle needs to be clarified, in a way that was simply assumed by Holmes:

Within the set of known phenomena, once you have eliminated the impossible, whatever remains, no matter how improbable, must be true. If the entire set of known phenomena are eliminated as impossible, then the solution is simply uknown until a new phenomena that can serve as a solution is positively established.

That is a bit more cumbersome than Doyle’s poetic phrasing, but it is more complete. So before we can conclude that the earth is being visited by ET craft, we must completely eliminate all possible explanations (even the quirky and improbable ones) and then find evidence that points specifically to ET craft, rather than just assume that “unknown” means extraterrestrial.

10 comments to A Sherlock Holmes Logical Fallacy

  • just assume that “unknown” means extraterrestrial.

    Believers have taken the poor, defenseless “U” in UFO and made it synonymous with “alien.” Nevermind the wide range of ordinary flying objects that they can’t identify. If they don’t know what it is, it must be piloted by anal-probing extraterrestrials.

    It makes me so angry, I almost immediately flip my lid and start foaming at the mouth. It’s certainly not the best mode for calmly engaging a believer on the merits of the claim. Any advice on how to stay rational in that situation?

  • thatneilguy

    Isaac Asimov amended it like this:

    “As far as I am concerned, if, when everything impossible has been eliminated and what remains is supernatural, then someone is lying.”

  • Steve Page

    Any advice on how to stay rational in that situation?

    Practice. Oh, and try to remember that most people don’t try to be stupid; they’ve just been taught how to by a lazy media and a centuries-old culture that lionizes people who base their lives around faith instead of evidence.

  • Sherlock always says deduction when he is in fact doing induction. Tsk Tsk.

  • alexjbutterfield

    Hey, Mjhavok.

    I thought that was an interesting point so I looked it up.

    Deductive Reasoning – is reasoning which uses deductive arguments to move from given statements (premises), which are assumed to be true, to conclusions, which must be true if the premises are true.

    Inductive Reasoning – is the process of reasoning in which the premises of an argument are believed to support the conclusion but do not entail it.

    Neither of these suggest forming a list of all possible conclusions and then settling on one by process of elimination.

    However I found this definition on a blog post

    Deduction — [Element] — a process of thought that determines certainty — Deduction is the process of thought that arrives at a determination of what is by limiting out all that cannot be.

    I don’t know if they just made up this definition or not.

    Could you explain what you said? cause I’m getting confused.

  • DLC

    Holmes called it deduction because you start out with a list of possible explanations and deduct those which do not fit the known facts.

  • lurchwurm


    Deductive reasoning is the system of logic that allows one to determine what valid inferences one can make based upon on implicational and equivalence rules. Examples include things like modus ponus (if you know p and you know if p then q, then you know q). The validity of an argument is one of the main goals of deductive reasoning. As Steve was saying, there are numerous possibilities of any argument, and deductive reasoning allows one to know which arguments are valid regardless of all the possibilities.

  • Litch

    The comparison between House and Holmes isn’t lost with the writers of House — his street address is 221B, same as Sherlock’s.

  • landrew

    I don’t think the Sherlock Holmes quote is a fallacy unless you have decided that “absence of evidence is evidence of absence.” Carl Sagan said it wasn’t, but the above argument seems to be predicated on that assumption. The “U” in UFO does not mean alien, but it also doesn’t mean “nothing,” it’s an unknown variable (unless you are a hardboiled skeptic). Even Carl Sagan wasn’t a denialist skeptic, he admitted to to the possibility of hidden unknowns, such as the possibility of ET life, but he also said there was no evidence to support it’s existence. Big difference.

  • DLC

    The default position should be that the thing does not exist unless evidence for it can be shown. I am perfectly willing to admit the existence of extra-terrestrial visitors just as soon as I am provided with conclusive evidence that such exist. Until then, I have to take the position that they do not. Also (as Carl Sagan said) extraordinary claims demand extraordinary evidence.

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