# Can we establish the truth by eliminating the impossible

What conditions would have to hold for Sherlock Holmes' dictum, 'Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth', to be true ?

• Nov 12 '17 at 16:17
• Depends on if you're willing to make any simplifying assumptions. Are you? Nov 12 '17 at 16:28
• I take it the basic assumption is that the world or universe is such that every event or state of affairs is either necessary or impossible. I entirely agree that even in such a world or universe, eliminating the impossible, supposing that could be done, would leave a potential infinity of truths. Hardly helpful to an investigator. I think we can assume that whatever Holmes read at university it was not philosophy. Thanks for comments. Nov 12 '17 at 19:11

The short answer is "we can't." The problem is that without certain assumptions, the number of potential truths can be infinite, and even uncountably infinite. But if you are willing to make assumptions, then you can reduce the number of potential truths. We do this in science, which is really just a system of falsification. We keep finding out what isn't true. This gets us closer to the truth, but only infinitesimally closer. I go over this process in another answer.

Example

As an example of the aforementioned issue, consider the series {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, What is the next element in the series? Is it 13? It might be, but maybe I'm describing the hours on a clock, in which case the next number is 1. In fact, even if we restrict the answer to some sequence described by a polynomial, there are still an infinite number of answers: there are infinitely many polynomials that pass through a finite set of points.

So we can see that we would have to make some pretty extreme simplifications to restrict the number of potential truths to a finite number.

• Yeah, so you can give a true set: {13, 1, and all other possible solutions }, are true. Let that set be infinite. What is the problem? It's basically a trade off. To get exact and precise solutions, we make assumptions. Nov 12 '17 at 23:41
• I'm not sure I follow what you're saying. I am pointing out that even if we restrict our assumptions considerably, we can still end up with an infinite number of potential truths. Therefore process of elimination can be really problematic. You have to somehow get from an infinite number of potential answers don't to one. Nov 13 '17 at 11:34
• I agree, but you can just talk about "answer set", instead of "answer". The principle would hold. Nov 13 '17 at 12:41
• Define "answer set." Nov 13 '17 at 13:30
• The set of all the possible truths that can be established with certainty under the given set of assumptions. Nov 13 '17 at 16:46

In principle, Holmes was right. However, there are a few problems in practice.

One problem is that after you eliminate the impossible, there may be two possibilities left. Or hundred. Or infinitely many. So the result may be true but less useful than what you wanted.

Another problem is that you can never eliminate the impossible, you can only eliminate what you think is the impossible. What you thought was impossible may in fact only be very improbable. If you eliminated something that was very improbable, then what remains may not be just improbable, but actually impossible.

A third problem is that it may be difficult to determine exactly "what remains" means. You must be able to determine exactly what it is you eliminated, and to determine exactly what it is that remains.

Of course, we (or the detective) will sometimes run into situations where it can be established what is impossible, and where we can determine just one thing which remains, and in that case "what remains" is quite likely to be true.

The other answers raise the question of multiple possibilities. What I think is missing is that Conan Doyle puts this quote in Sherlock Holmes' mouth.

Specifically, only one possibility, the truth, will fit all of the facts. As such, once you have all the facts then the answer is evident. The key, of course, is to have all the facts and, for that, you'll need Sherlock Holmes' abilities. Otherwise you can only eliminate some of the impossibilities and that won't be sufficient.

Essentially, this is a form of humble bragging on Holmes' part: the dictum is true but only, in practice, for him.

Either the speed of light is finite or it isn't.

Once you have established that it is impossible that the speed of light be infinite, you can eliminate this impossibility. And once you have eliminated this impossibility, then what remains is that the speed of light is finite. Full marks for Doyle!

So, I guess, one condition, surely, is that we be logical to begin with.

Either A or B;

Not B;

Therefore, A.

So, apparently, "either" seems the key logical concept we need to master fully before we play Sherlock.

There are other conditions, presumably.

For example, we need to be able to identify impossibilities.

A state of affair S is possible if I don't know that it is not the case. As soon as I know that it isn't the case, it becomes an impossibility.

So, do we actually know that it is impossible that the speed of light be infinite? Me, I would say, I believe not.

Still, we can rephrase Doyle a bit:

Once you have eliminated what you believe is impossible, then you should accept whatever possibility remains, however awful it might be to contemplate.

I wonder what Hercule would have thought of that.

• Thanks for answer. Both clear and subtle. Best - Geoffrey Aug 26 '19 at 17:49

To the main question the answer is, in short, yes and no.

In metaphysics the principle method is abduction, defined by Peirce as 'inference to the best explanation'. This is exactly the method used by Holmes. You have a list of suspects and cross them off one at a time until there's one left, and if there are none let you know the list is incomplete. This is the only way to proceed in metaphysics.

But this method cannot establish truth, only the best or most reasonable explanation. Logic cannot prove facts about reality unless we can prove that reality obeys our puny rules. By reducing ideas to absurdity in the dialectic we can establish that only a fool would believe the universe is a pink cube but we cannot actually prove that it isn't on the basis of logic alone.

It's a hobby-horse for me since metaphysics is solvable by the methods of Holmes. I feel he should be a role-model for philosophers for his rigorous methods and also his confidence in his own reasoning.If we do not trust abduction then we can make no progress in metaphysics.

What conditions would have to hold...

One condition is this: that it is indeed possible to show that an event is impossible. To reach this conclusion, it is necessary to accept one or more of the laws of thought: Identity, that something is what it is; Non-contradiction, that nothing can both be and not be; and Excluded Middle, that everything must either be or not be.

Without such basic assumptions, everything becomes possible. Events cannot be eliminated as impossible. Holmes's case goes unsolved.

I would say absolutely yes. You can use this method to solve almost any problem. In some of the above examples, especially the one with numbers half of the problem was missing. In a case with Sherlock Holmes or anytime, you would employ this method would be to find the cause of a particular outcome. In the case of the answer with the numbers, there is no known outcome, of course, you would have infinite possibilities. This only works if you know the outcome. Then you can eliminate all the possibilities that could create that outcome until you're left with the possible no matter how impossible.