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In The Sign of Four, Holmes asks Watson: "How often have I said to you that when you have eliminated the impossible, whatever remains, however improbable, must be the truth?"

This may be valid in principle, but it certainly carries the risk of a fallacy: You might not have considered the truth in the first place, and if what remains in your range of vision after eliminating the impossible is very improbable you should continue searching rather than settle for it.

I'm sure there is some academic discussion of this, but under what label? Are there other examples of the same fallacy?

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    Rational Wiki attributes Holmes's quote to "The Adventure of the Blanched Soldier". See the link in Double Knot's answer. – Mark Andrews May 4 at 0:40
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    Well, the fallacy would not be in Sherlock Holmes line; that remains perfectly valid. The fallacy would be in the hybris of the person who did not carefully conduct an exhaustive search for alternatives. In order to use "whatever remains, however improbable, must be the truth" you must exhaust the space of possibilities first. If you didn't do that, you are not entitled to appeal to Sherlock Holmes! – cmaster - reinstate monica May 4 at 6:51
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    @cmaster-reinstatemonica Sherlock Holmes was that type of person; he is not entitled to appeal to himself. – wizzwizz4 May 4 at 16:27
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    I surely have thought something was impossible until it wasn't twitter.com/Bill_Gross/status/928354964933619719/photo/1 – GettnDer May 10 at 18:28

10 Answers 10

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There's a fallacy called Holmesian fallacy.

A Holmesian fallacy (also Sherlock Holmes fallacy or process of elimination fallacy) is a logical fallacy that occurs when some explanation is believed to be true on the basis that alternate explanations are impossible, yet not all alternate explanations have been ruled out.

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    There's ambguity which simultaneously allows the statement to be correct and for abuse. In particular, "what remains" is a set, the set-inverse of "the impossible". The truth is contained within this set. But this does not imply that every member of the set is truth. The use of a linking verb "be" instead of set terminology "contain, exist, forall" leads directly to the problem. – Ben Voigt May 4 at 22:28
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Holmes' advice is correct if and only if you assume a complete search was done to list all possibilities before starting the elimination process.

Note that Sherlock Holmes is both incredibly observant, and incredibly arrogant. I would consider it a matter of great writing for Sherlock to arrogantly assume that his superior observation skills somehow make him beyond reproach (which they don't), but you're asking about the cold logic of the advice and you are correct that if you take what Sherlock says to the letter, without adding anything implicit, he is not 100% correct, he's just "mostly right" (as it is fair to assume that Sherlock does a significantly extensive search of possibilities, just not a provably complete one).

This closely ties in to the general misconception that Sherlock Holmes is a master of deduction and knows truths others do not. He isn't. He uses abductive reasoning, which is "educated guessing". If you assume the following truth:

Thomas buys pizza when he burns his dinner.

  • A deductive reasoner would observe that "Thomas burnt his dinner, therefore he will buy a pizza."
  • An abductive reasoner, however, observes that "Thomas is at the pizzeria, therefore he must have burnt his dinner".
  • Related to the comment below, an inductive reasoner would observe that "Robert is at the pizzeria, so he must've burned his dinner, as all people get pizza after burning their dinner".

That is not ironclad logic, since we never said Thomas only eats pizza when he burns his dinner. But Holmes often uses a few abductive reasonings at the same time (he's in the pizzeria, the fire department was in his street, he was cooking the same thing that he burned the last time he tried to cook it) which would align with a single event (burning his dinner tonight), which does increase the likelihood of him being correct, but is still not an ironclad logical deduction.

In short, Holmes doesn't follow the path of ironclad logic, and therefore his advice isn't ironclad logic either. It's an approximation, one that plot-wise commonly yields the correct outcome.

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    (Beyond _re_proach?)  I've also heard your second type called inductive reasoning. – gidds May 4 at 8:38
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    @gidds: Whoops, fixed. Also: "The difference between abductive reasoning and inductive reasoning is a subtle one; both use evidence to form guesses that are likely, but not guaranteed, to be true. However, abductive reasoning looks for cause-and-effect relationships, while induction seeks to determine general rules." (link). Sherlock clearly uses both, but the example I used is specifically abductive, not inductive (except arguably the notion that Thomas burned the same dinner twice in a row) – Flater May 4 at 8:43
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    Holmes also frequently assumes that people (and animals) are creatures of habit, so that any deviation from normal behavior has a cause that can be readily deduced (like the dog that didn't bark in "Silver Blaze" -- the perpetrator must have been someone the dog knew). – Barmar May 4 at 15:55
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    @Barmar: That is inductive reasoning - it's a generalized guess. Holmes had no way to know if the dog would bark at every stranger. Based on the limited evidence, it barked at the strangers it observed, and thus the assumption was made that the dog would bark at any stranger. – Flater May 4 at 16:26
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    I thought he was referring to the general behavior of dogs -- they tend to bark when a stranger is at a door. – Barmar May 4 at 16:28
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Deep down, mechanically, it's merely a false dilemma: assert that one of these options must be true and disprove all but one.

A traditional false dilemma is an attempt to bully and has only 2 options -- [thing-I-want-to-force-you-to-say] and [thing-you-would-never-choose]. But the "Far-fetched hypothesis" in RationalWiki is a nice example of how it works with more options. If I want to convince you aliens did something I'll disprove many alternate explanations before claiming aliens are the only other possibility.

Holmes seems different since he isn't trying to bully anyone -- he's sincerely trying to find the truth. If we think that matters -- we want to consider intent -- then "Holmes logic" is a type of false dilemma.

But to be fair to Mr. Conan Doyle, it's not a fallacy yet. Holmes isn't telling Lestrade "arrest that man. He's the only possibility". He's deciding what to investigate next. The slogan is more like "when the likely leads pan out, check out the unlikely ones". Probably good advice -- so many true crime stories begin with "we went back and looked at suspects the original investigation ignored".

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I think another point worth mentioning is that even if Holmes has correctly enumerated all possibilities, he is invoking the Law of the Excluded Middle[1]:

Either a proposition is true, or its negation is.

More specifically, I'd say he's applying Double Negation[2]:

If not (not A), then A.

If "A" is the remaining explanation, then by ruling out all the others, Holmes has claimed to have proven that "not A" cannot be true, therefore, A must be true.

These are usually considered valid laws of logic, not fallacies. However, not everyone agrees with them in all scenarios. For example, it's common in conversation for someone to say things like "I don't disagree", which has a different meaning than "I agree". Similarly, I think many of us would be uncomfortable convicting a person of murder even if it were proven that a human did it and no other human in the world could possibly have done it. You might not consider that exactly the same as presenting proof that this person did it.

[1] https://en.wikipedia.org/wiki/Law_of_excluded_middle

[2] https://en.wikipedia.org/wiki/Double_negation

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As you recognize yourself (and as many commentators repeat), Sherlock's reasoning is valid in principle but inapplicable unless you have actually considered all (relevant) possibilities. If you haven't, you are effectively making an argument from ignorance, which might be the fallacy you are looking for:

Argument from ignorance (from Latin: argumentum ad ignorantiam), also known as appeal to ignorance (in which ignorance represents "a lack of contrary evidence"), is a fallacy in informal logic. It asserts that a proposition is true because it has not yet been proven false or a proposition is false because it has not yet been proven true. This represents a type of false dichotomy in that it excludes the possibility that there may have been an insufficient investigation to prove that the proposition is either true or false.

The bottom line is that Sherlock's argument is just as strong as his argument for his investigation being thorough enough (which may be very strong indeed).

In the philosophy of science there is a corresponding problem with Inference to the best explanation, where you accept that a hypothesis is true on the basis that you cannot think of a better explanation. (And maybe you haven't thought of the right explanation; this is known as the bad lot objection or the problem of underconsideration. You'll find some discussion here, here, and here.)

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This is closely related to abduction or inference to the best explanation. See the Stanford Encyclopedia of Philosophy entry.

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    As it stands your post is more like a comment than an answer to the question. Please include more explanation. – sammy gerbil May 4 at 9:09
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tl;dr From a Bayesian perspective, we can take the Sherlock-Holmes method as being a reasonable approximation so long as inappropriate dismal of initially-improbable possibilities is avoided. This is, when Holmes says that

when you have eliminated the impossible, whatever remains, however improbable, must be the truth

, the "whatever remains" must still be significantly more probable than the sum of all initially-excluded possibilities.


Potentially, inappropriate approximation.

The Sherlock-Holmes method,

when you have eliminated the impossible, whatever remains, however improbable, must be the truth

can approximately work when used appropriately. Part of this would be not going too far with "however improbable".

The issue's that there're a lot of improbable explanations for stuff: for example, a lot of weird things could be due to time-travelers. But we usually exclude these for being so improbable as to be approximately impossible. Still, if we can show that every other possibility is at least as improbable, then suddenly we have to go back and reconsider including other possible explanations.

For example, let's say that there's a set of possible explanations which we tag with probability-weights:

Potential explanation Probability weight
Possibility 1 100
Possibility 2 95
Possibility 3 50
Possibility 4 25
[...] [...]
Possibility 100 0.1
Possibility 101 0.05
[...] [...]
Possibility 1000 0.0000001
Possibility 1001 0.00000000000000000000001
[...] [...]
ZORG, THE AWESOME,
TIME TRAVELER
FROM THE EMPIRE,
did it!
[very, very low weight]
[...] [...]

Now suppose that:

  1. the total weight of Possibilities 1 through 1000 is ~270;

  2. the total weight of Possibilities 1001 and above is ~0.000000000000000000000011 (this is, 10% larger than Possibility 1001 alone).

Then, the proposition that the truth is in Possibilities 1 through 1000, rather than anything else, is better than 99.999999999999%. Many folks would then approximate the odds of Possibilities 1001 and beyond as zero.

Now, suppose that further fact-finding can push the odds of all Possibilities 1 through 999 to being less-than 10-100, while Possibility 1000 has also become far less likely, down to less-than 10-25. This is,

Potential explanation Probability weight
Possibility 1 < 10-100
Possibility 2 < 10-100
Possibility 3 < 10-100
Possibility 4 < 10-100
[...] [...]
Possibility 100 < 10-100
Possibility 101 < 10-100
[...] [...]
Possibility 1000 10-25
Possibility 1001 10-23
[...] [...]
ZORG, THE AWESOME,
TIME TRAVELER
FROM THE EMPIRE,
did it!
[very, very low weight]
[...] [...]

If Sherlock-Holmes was only considering Possibilities 1 through 1000, under the prior argument that it was vastly unlikely for any other to be true, then Holmes would conclude that it's vastly likely for Possibility 1000 to be the case under the same reasoning (because the cumulative weight of Possibilities 1 through 999 is vastly smaller than the weight of Possibility 1000).

And that's the problem with "however improbable": it's not qualified with "so long as it's still sufficiently more probable than the set of all excluded possibilities".

Because, in this case, Possibility 1001 is now the most likely one, with a probability-weight 100 times greater than the conclusion of the Sherlock-Holmes's method, Possibility 1000. But, the Sherlock-Holmes method might fail to realize this because it implicitly excluded Possibility 1001 for being too unlikely at first.

This is, the situation's now:

Possibility set Odds
Possibility 1001 ~90.1%
Possibilities 1002 and above ~9.01%
Possibility 1000 ~0.901%
Possibilities 1 through 999 < 10-74

, making the conclusion that it must be Possibility 1000 pretty shaky.


Summary: Can't forget to reconsider initially-implausible-seeming possibilities.

When using the Sherlock-Holmes method, an investigator may (reasonably) exclude consideration of possibilities that seem truly unlikely. However, if the investigator then finds that all of the considered possibilities become significantly less likely, they mustn't forget to reconsider the initially-excluded set.

This is, when Holmes says

when you have eliminated the impossible, whatever remains, however improbable, must be the truth

, there's a limitation on "however improbable": if it becomes so improbable as to make it less likely than the exclusion-criteria initially used, then the original exclusion of other possibilities would generally need to be reconsidered.


Note: This is still probabilistic.

Appropriately used, the Sherlock-Holmes method could still fail to produce absolute proof of the conclusion. For example, it'd generally fail to exclude the possibility that an invisible-pink-unicorn is using hypnosis to alter investigators' perceptions.

However, it's presumably the case that Holmes would've understood that the method still rested on some probabilistic assumptions, e.g. in deciding the threshold for something being "impossible".

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  • Random curiosity: Would anyone be skeptical of the above description on the basis that the possibility-space might be uncountably infinite and therefore non-enumerable? This is, that the possibility-space might be uncountable? – Nat May 5 at 23:43
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I don't think there's a fallacy once you rule out the impossible however improbable it is it must be the truth.

Modal Logic at its best.

Improbable means: "Unlikely".

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If you read French, may I offer this hilarious excerpt from "Le Péril Bleu" (a 1911 proto-SciFi novel by M. Renard, I have no idea whether it has ever been translated), as an illustration (quick summary at the bottom):

Là, Tiburce s’enfonça dans un canapé, croisa les jambes, fixa un coin du plafond, se rongea quelque peu les ongles et débita d’une voix rapide et négligente ; aigre et blanche – de cette voix, enfin, que l’acteur Gémier prêtait au personnage de Sherlock Holmes :

– Monsieur, vous possédez un chien de la race dite « griffon Boulet à poils durs ». Et ce chien d’arrêt, vous en faites un toutou d’appartement. Car vous n’êtes pas chasseur. Pas chasseur, mais pianiste. Très bon pianiste, même ; ou du moins vous croyez l’être. J’ajouterai que vous avez servi dans la cavalerie, que vous portez à l’ordinaire un monocle, et qu’un de vos passe-temps favoris est le tir à la cible. Chut ! taisez-vous, prière de ne pas m’interrompre.

Et, sans cesser de regarder en l’air, il continua : – Le bas de votre pantalon est couvert de poils. Or, ces poils ne peuvent appartenir qu’à un chien de l’espèce précitée ou à une chèvre Mais il n’entre pas dans nos mœurs de faire coucher les chèvres sur nos pieds. Donc… Concluez vous-même. D’autre part, je sais que vos occupations ne vous laissent pas le loisir de chasser, et j’en déduis que votre chien, malgré sa nature, est un chien d’appartement, par destination. Vous jouez au piano ; oui. En vous donnant la main, j’ai reconnu au bout de vos doigts les callosités professionnelles des pianistes. Elles m’ont révélé que vous jouez même très fréquemment. Or, un homme de votre âge et de votre intelligence ne saurait montrer tant d’assiduité dans l’exercice d’un art aussi délicat que s’il y est excellent ou s’il croit y exceller. À cause d’Ingres et de son violon, je n’ose affirmer votre talent de pianiste, en dépit de votre génie d’astronome. Vous avez servi dans la cavalerie, car vous marchez les jambes écartées et vous descendez les escaliers comme si vous redoutiez d’accrocher vos éperons aux degrés. Donc, vous avez l’habitude du cheval. Et c’est une habitude qui date de loin, car on ne vous voit jamais cavalcader à Paris. Votre jeunesse humble et studieuse ne vous ayant pas permis l’équitation, il faut par conséquent que vous ayez chevauché les destriers du gouvernement. Silence, je vous prie. Vous portez un monocle. Parfaitement. J’ai découvert sa trace au pli de votre orbite ou à la carabine, car votre œil gauche a coutume de se fermer pour viser : il est un peu plus petit que l’autre, et les plis de la ride nommée « patte-d’oie » sont plus accusés à gauche qu’à droite. Comme vous ne chassez pas, il s’ensuit que vous pratiquez le tir à la cible. C’est tout. J’ai dit.

– Si vous n’êtes pas content avec cela ! s’écria Garan sur un ton moqueur.

Mais M. Le Tellier n’était pas disposé à la plaisanterie. Sans dire un mot, il tira de l’ombre, sous le bureau, une chancelière en peau de bique et la jeta au milieu de la pièce.

– Voici le griffon Boulet à poils durs, fit-il.

Puis il ouvrit une armoire, et montrant sa machine à écrire : – Voici le piano.

D’un tiroir il sortit sa loupe d’horloger, l’encastra sous son arcade sourcilière droite, et ajouta d’une voix coupante :

– Voici le monocle.

Enfin il produisit une photographie qui le représentait dans la posture de son état : l’œil droit à l’oculaire d’une lunette méridienne et l’œil gauche fermé, ainsi qu’il arrive à tous les astronomes pendant leurs observations.

– Et voici la carabine ou le pistolet, dit-il avec un sifflement irrité. Quant à la cavalerie, je ne sais ce que vous voulez dire. Il se peut que j’aie les jambes en manches de veste, mais je ne suis jamais monté à cheval. À présent, mon jeune ami, permettez-moi de vous déclarer que, pour faire le jocrisse, vous avez mal choisi votre heure et votre lieu ; et que, s’il était de tradition de se servir des serins pour tirer des auspices, vous seriez un oiseau de bien mauvais augure. C’est tout. J’ai dit.

Essentially, the character Tiburce poses as a "Sherlockian" applying the Master's reasoning, and gives a wonderful demonstration of that using - in pastiche style- Sherlock's logic ("you have animal hair on your feet, it must be either a goat or a dog, since you are an interior person and are unlikely to keep a goat as a pet I conclude that you own a dog"). Of course, all of this is wrong, and for every point there is an alternative - and true- explanation that the other character, Le Tellier, points out ("I sit on my desk with a fur blanket on my lap, made of goat skin").

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    I'm sorry, but most of us don't read French, and I'm fairly sure that Google Translate will not provide us with the nuance that we'd need to truly appreciate the joke. Could you summarise and explain how it relates to the OP, please? – Chris the Hairy One May 5 at 9:46
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    Essentially, the character Tiburce poses as a "Sherlockian" applying the Master's reasoning, and gives a wonderful demonstration of that using - in pastiche style- Sherlock's logic ("you have animal hair on your feet, it must be either a goat or a dog, since you are an interior person and are unlikely to keep a goat as a pet I conclude that you own a dog"). Of course, all of this is wrong, and for every point there is an alternative - and true- explanation that the other character, Le Tellier, points out ("I sit on my desk with a fur blanket on my lap"). – jfmoyen May 5 at 12:10
  • Please add your summary to the answer. – RonJohn May 5 at 14:10
  • Was the blanket made from dog fur or goat fur? – Buh Buh May 5 at 14:44
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I acknowledge the following is not "in universe" with the human intent of the posted question, and so may not be satisfying. But maybe folks here are better than the Tolkien and Star Trek people. (Yes, go ahead and try to explain the truth about Ferengi, for example, being fit INTO the ST milieu, when it turns out they want absolutely nothing but "in universe" "information.")

What Holmes says isn't meant to be logic so:

It's not actually a fallacy (or even "not a fallacy").

It is a human statement made for a point, not for analysis or proof of anything at all. I'll explain in a second, but think in terms of a painting of a clock: it is NEVER "right"... not "twice a day" or at any time ever. Because it is NOT a clock at all. (For that matter, even a broken actual clock is not right "twice a day" since it is no longer performing any clock functions.)

People fall in love with their ideas, and do the opposite as well. An investigator, for example, might decide the "ultra-precise slipping of the knife between ribs" REQUIRES the attacker to be someone with such precise knowledge AND skills and so be perhaps a doctor or special forces trainee. He might charge such a man, if one is ready to hand, and fail to properly consider other possibilities (especially to not put much work into them, if considered at all). He got the idea in his head that it had to be the butler, so the butler it is. Or the black man with a height between 4' and 8'.

The Holmes statement is not meant to prove or disprove any fact or possible fact. It is intended to be a strong argument against fixating on some other idea that isn't even possible. It is his reason/rationale for choosing other ideas to investigate, or in general, not stopping (successful or failed at that moment) investigating.

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