“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.

Actually, the Sherlock Holmes argument is valid.

Divide the world into two categories: the solution you’re looking at, and everything else. Call them ‘A’ and ‘Everything but A’ if you want. Or, we could just call them ‘A’ and ‘B’. It’s easier.

By definition, it must be A or B. It’s not B (that is, it’s not ‘Everything but A’. Therefore (no matter how improbable) it’s B.

You say Holmes can’t eliminate everything else. I think he would disagree. You can eliminte large numbers of things all at once:

– things that violate the laws of physics

– things that are logical impossibilities

– things that happened in other dimensions

– things that involve the living Elvis

– things that are not practical

Etc.

You *can* for all intents and purposes rule out everything else. Most everyday statements are exactly like that.

Eg. ‘London is in England’ – we’ve ruled out every other possible place, and so it must be there

Eg. ‘Hydrogen is the lightest atom’ – we’ve ruled out every other atom

etc.

Megan’s point is an excellent one, and hits on a really old problem with the technique of mathematical induction (http://en.wikipedia.org/wiki/Mathematical_induction), which is used extensively in formal logic and math. I am not going to elaborate on it here (although I am tempted), and will just informally summarize. Spose you had an infinite set of statements that were all disjunctive on to one another. In order to use the syllogism properly, you have to pull off statements one at a time (an (formal) example is at the end). If you have an infinite number of statements to pull off, you will never get to the end, and so will never be able to reach your desired conclusion.

This is a subtle point, but a really important one. The arguement is valid since IF you could prove that we had ruled out every solution but one THEN the thing that is left would have to be correct (by disjunctive syllogism). The conditional statement is fine, the difficulty is in actually proving the first part. Since we can never check an infinite set of possibilities to rule them out, the soundness of the arguement is indeterminite at best, since we can’t properly assert the truth of the first argument.

I suppose one could argue that the set of possibilities that is being checked is not actually infinite, but I’m kind of lazy, so I will leave that to others to be discussed.

The afore promised (formal) example:

from not A, not B, not C, A or (B or (C or D)), prove D:

1)not A {premise}

2)A or (B or (C or D)) {premise}

3)B or (C or D) {1,2, disjunctive syllogism}

4)not B {premise}

5)C or D {3,4, disjunctive syllogism}

6)not C {premise}

7)D {5,6, disjunctive syllogism}