====== Fallacy of eliminative induction ======

A //probable// explanation that was established by a process of [[glossary:induction|inductive]] elimination is (erroneously) understood as [[glossary:deduction|deductively]] //true//.

The most famous example of this fallacy can be found in literature:

> “<u questionable "Fallacy of eliminative induction">If one has ruled out all impossible cases, then whatever remains, no matter how improbable, must be the truth.</u>”
>   — Sherlock Holmes

This is the description of the method used by the literary detective [[wp>Sherlock Holmes|Sherlock Holmes]], as it was put into his mouth by the author [[wp>Arthur Conan Doyle|Arthur Conan Doyle]]. 

This statement describes a method that is also known as “eliminative induction”. However, this does not lead to a solution that is //guaranteed// to be //true//, but at best to a //probable// one.

===== Other names =====

  * [[logic:induction:sherlock-holmes|Sherlock Holmes fallacy]]

===== Description =====

The underlying problem here is to mistake an [[glossary:abduction|abductive]] method (known as “eliminative induction”) for a [[glossary:deduction|deductive]] one, and as a consequence, ignoring the inherent //uncertainties// that such a method has.

==== Eliminative induction ====

The process of //eliminative induction// can be described as follows:

  * Either ''A'' is true, or ''B'', etc. … or ''N''.
  * ''A'' is //improbable//.
  * ''B'' is //improbable//.
  * … and so on.
  * ''N'' is by far the //most probable// or //least improbable// explanation.
  * Therefore, ''N'' is the //most likely// explanation.

This method allows to reduce a large number of //possible// explanations to one or a few //probable// ones.

However, one must be aware that there is only a //certain probability// that the conclusion is correct – and in particular one should be prepared to //revise// the conclusion if new information becomes available that changes any of the probabilities, or if new explanations (theories) are put forward that may be better than those already assessed.

Particularly in the case of social phenomena (as in [[wp>Forensic science|criminal forensics]]), even the first step, i.e. enumerating all possible options, can prove to be very difficult or even impossible. It is even more difficult to exclude all but one of them with any certainty. This is because the number of these possibilities and the associated assumptions and judgements very quickly become impossible to judge. Ultimately, the probabilities can therefore usually only be estimated with great uncertainty. 

The mistake, however, is to ignore this uncertainty and think that one has found a “guaranteed to be true” solution in this manner.

<aside info>
**Note:** As a character in a novel, Sherlock Holmes has the advantage that his author can ensure that the solution he finds actually turns out to be the correct one. In real life, this is not the case, meaning that the “Sherlock Holmes method” is not a recommended approach to truth-finding. 
</aside>

===== When are such conclusions valid? =====
==== Modus Ponendo Tollens ====

This fallacy resembles that of “[[logic:formal_fallacies:affirming_a_disjunct|affirming a disjunct]]” insofar as it can be valid if it is ensured that all of the possible cases are indeed covered.

In addition, all possibilities (except for one) must be formally excluded. In practice, this is only possible within formal systems such as mathematics or computer science, but certainly not with social issues or criminology in particular. 

For more information, see: <b maniculus "further information under">[[logic:inferences:modus_ponendo_tollens|Modus Ponendo Tollens]]</b>.

==== Complete knowledge ====

Outside of formal systems, there are only very few situations in which we can actually have //complete knowledge// of a situation. This applies in particular when this refers to a relatively small sample that we do indeed have good knowledge of.

For example, parents usually know their //children// quite well and can therefore draw eliminative conclusions based on this knowledge:

> Someone has munched on the cake that was intended for dessert.
> Child A is at school.
> Child B is currently at their grandmother's house.
> It follows from this: It must have been child C.

Here too, however, it could be argued that other, possibly rather unlikely, possibilities were not considered (it could have been someone else who tried the cake), which is why this would also only be a “most likely” correct conclusion.

===== See also =====

  * [[logic:formal_fallacies:affirming_a_disjunct|Affirming a disjunct]]
  * [[glossary:induction|Induction]]
  * [[logic:formal_fallacies:denying_a_conjunct|Negation of a conjunction]]

===== More information =====

  * [[https://rationalwiki.org/wiki/Holmesian_fallacy|Holmesian fallacy]] on //Rational Wiki//