The dubious logic of Sherlock Holmes

I’ve been interrogating the contents of my bookshelves in case I get Zoomed by Owen Jones, and I’ve found a copy of A Study in Scarlet, the novella in which Sir Arthur Conan Doyle introduces us to the irritating character of Sherlock Holmes.

In the second chapter, Holmes explains to Watson how he knew on first meeting him that Watson had served as a medical doctor in Afghanistan:

“Here is a gentleman of a medical type, but with the air of a military man. Clearly an army doctor, then. He has just come for the tropics for his face is dark, and that is not the natural tint of his skin, for his wrists are fair… His left arm has been injured. He holds it in a stiff and unnatural manner. Where in the tropics could an English army doctor have seen such hardship and got his arm wounded? Clearly in Afghanistan”.

The title of that chapter is The Science of Deduction. It really shouldn’t be. In classical logic, a deductively valid argument is one in which it can’t be the case that the premises are true and the conclusion false. “All men smoke weed; Sherlock Holmes is a man; therefore, Sherlock Holmes smokes weed” is deductively valid. But Holmes did not smoke weed (he preferred opium in a seven per cent solution and did not, in fact, exist), and not all men smoke weed. The argument is therefore not sound, because to be sound as well as valid it is necessary also that the premises are true.

Holmes’s process of reasoning here, which is typical of how he operates, adds up to a chain of inferences which are probabilistic, not deductive. As a deductive argument it falls at the first hurdle: it may well be true that Watson is a doctor with the “air of a military man”. It doesn’t follow, deductively, that he is a medical doctor. It may be a reasonable inference, but it is not a logically valid one.

And that’s a good thing. The conclusion of a valid argument doesn’t add anything to what you already know, or think you know. They are, to use a contemporary analogy, like a computer model: you get out of them pretty much what you put in. If the “world’s first consulting detective” had insisted on cracking cases on the basis of purely deductive logic then he would rarely have put down his pipe and got out of his armchair.

Holmes’s methodology is a combination of two types of reasoning, induction (consisting of the inferential chains mentioned above) and abduction (roughly understood as “argument to the best explanation”). We argue that the sun will rise tomorrow on inductive grounds; theologians argue for the historicity of the resurrection on grounds of “best explanation”. In each case there is an element of epistemological risk — a lack of logical certainty. But an intellectual journey is like a life: without an element of risk it’s not worthwhile.

And, relatedly, except in the higher mathematical sciences, “proof” is not something that can usefully be insisted on, if you are interested in discovery. And even in those sciences, it is strange that such proof is available, a point made by the mathematician Eugene Wigner in his influential 1960 paper The Unreasonable Effectiveness of Mathematics in the Natural Sciences.

What Holmes is most interested in of course is not proof but truth. And despite the best efforts of the post-modern types, logicians still set much store by the concept of truth, and have developed multiple systems of logic which sit alongside those mentioned enough. There are logics which deal with issues around vagueness (is there a specific point at which it becomes true that someone is bald?); related logics which accept that there can be “degrees” of truth; there are “modal logics” which use the concept of possible worlds to give definitions of necessity and possibility (true in all possible worlds; true in some possible world, respectively).

There are even systems of “paraconsistent” logics which allow that contradictions can be true. (The most prominent defender of this counter-intuitive position is Graham Priest who, incidentally, has a short chapter on Holmes in his little primer, Logic.)

These were not intellectual developments that were available to Holmes, who did most of his sleuthing at around the time Bertrand Russell embarked on an ultimately forlorn attempt to reduce mathematics to logic. Logic itself decided to refuse the invitation (for reasons most famously given by the logician Kurt Godel). So, we can forgive Holmes a pass for his consistent misrepresentation of his own techniques. He was, in any case, too busy persecuting the mathematician Moriarty to worry too much about the pedantry of people like myself.

But we should understand that his “deductions” were no such thing and it’s surprising that with rare exceptions, such as A Scandal in Bohemia, he was right as often as he was.

And even in that case he failed only because his head was turned by a woman. How the “world’s most famous consulting detective” allowed that to happen is a three-pipe problem.