Fallacy of eliminative induction

A probable explanation that was established by a process of inductive elimination is (erroneously) understood as deductively true.

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

“If one has ruled out all impossible cases, then whatever remains, no matter how improbable, must be the truth.”
  — Sherlock Holmes

This is the description of the method used by the literary detective Sherlock Holmes, as it was put into his mouth by the author 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.

• Sherlock Holmes fallacy

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

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 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.

This fallacy resembles that of “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: Modus Ponendo Tollens.

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.

• Affirming a disjunct
• Induction
• Negation of a conjunction

• Holmesian fallacy on Rational Wiki