“When you have eliminated the impossible, whatever left, however improbable, must be the truth.”
This is famously stated by the character of Sherlock Holmes, written by Sir Arthur Conan Doyle. Holmes is known for his investigational skills, and I can look at this quote and understand how it could be practically applied and effective. However, from a purely logical stand point, I also see flaws in this approach.
After sifting around on the internet for more information about this quote, and reading the opinions of several fellow bloggers, I began to see an obvious hole in the argument. However, it was only when I tried to fit it into a syllogism (using the disjunctive form) that I saw it exactly.
Essentially, the disjunctive syllogism of this quote would go like this.
Either solution A, B, C, D, or E is true.
Solution B is not true.
Solution C is not true.
Solution D is not true.
Solution E is not true.
Therefore, solution A is true.
This syllogism, on its own, is valid. If there is a finite number of possible solutions (five was an arbitrary number I chose for the purposes of this post) then the approach that “when you have eliminated the impossible, whatever left, however improbable, must be the truth” is not only valid, but sound. If there are a certain number of possibilities, and all but one are proven to be false, then the one remaining must be true. Makes sense.
Yet this is where the problems arise. I quickly realized that for myself to be able to put the quote into a formal syllogism, I would have to decide on a set number of possible solutions. Yet how can one ever determine that they have considered every single solution? One may go over the practical solutions, the ones which seem the most likely, but one cannot say without a shadow of doubt that they have considered and disproven every single possibility under the sun. Even if the person, in this case Sherlock Holmes, missed a single possibility, then he cannot say that solution A is true. What if he missed solution Z? As suggested by the quote itself, we are to believe that only ONE solution can be true. Following this rule, solution A and solution Z cannot both be true. In this way, the great detective would be committing a formal fallacy.
However, I am also not going to say that this means of coming to a conclusion is not useless. In many situations, the possible solutions an individual could come up with on his own could be broad enough to cover all the likely options, and once all the impossible solutions were ruled out, the one left over may have a very high chance of being “true”. Yet to say “when you have eliminated the impossible, whatever left, however improbable, must be the truth” is not valid. There is not a finite number of solutions, so not every solution but one can be ruled out, and that means that a single conclusion can never be reached.