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Monday 24 December 2018

The Sherlock Holmes Fallacy

A topic came up the other day, and it prompted me to ask a question on Facebook regarding the text quoted in the header image on the right. It's everywhere. A quick google search for the first six words turns up almost sixty million results. It's a quite beautifully meretricious statement, taken as a simple truth by very many people, and it's found its way into the mouths of many actors, all of them playing characters seen as 'highly logical'. Most obviously, it's said by any actor who's
played Sherlock Holmes in The Sign of Four (including Peter Cushing, pictured), but it's also been uttered on at least two occasions by Spock in the various iterations of Star Trek. I thought it would be interesting to get a quick taste of what people thought about it.

I was pretty pleased by the responses, pretty much all of which highlighted problems with it. I began posting my promised response and, when I got about 270 words in with no end in sight, I decided it was worth giving it some room to breathe and some decent typesetting, so here we are.

In one of my earliest posts, in which I talked at length about how logic is employed in the sciences, I began with that quote, questioning why Holmes had said it. The obvious answer is that that was what was in the script, but the question of what was wrong with it wasn't actually dealt with explicitly in that post, because the quote wasn't really the point of the post, it was to simply a springboard for talking about different kinds of reasoning.

Here, I want to make explicit some of what was implicit about Holmes' statement in that post. As usual, I'll link to the earlier post at the bottom.

As we learned there, there are three types of reasoning, broadly speaking.

Deductive: True premises and valid reasoning lead to a conclusion that cannot be false.

Inductive: True premises and valid reasoning stand as strong evidence that the conclusion is true. 

Abductive: Reasoning in which observations are used to infer hypotheses.

It's worth taking a little look at that in propositional form because, when we look back at the statement, it will be instructive.

\(\dfrac {P \Leftarrow Q, Q} {\therefore P}\)

Proposition P is implied by consequence Q, Q is observed, therefore P*. All well and good. But what if this were a deduction? Well, now we have a problem, because the output of a deductive argument is a true statement, assuming soundness (valid reasoning from true premises). In other words, we're now taking our observation as confirmation that the Proposition P is true. That looks like this.

\(\dfrac {P \Rightarrow Q, Q} {\therefore P}\)

And there it is, our old friend 'affirming the consequent', the most ubiquitous formal fallacy in discourse.

Proposition P implies consequence Q. Q is observed, therefore P is true. My goto example is:

P1. All men are mortal.
P2. Hitler was mortal.
C. Therefore, Hitler was a man.

This might look fine on the surface, right until I tell you that Hitler was the name of my next-door neighbour's cat. Many things are mortal, and not all of them are men.

It can be tricky to spot this fallacy, usually because it's buried beneath a truckload of other fallacies, but this is far and away the most ubiquitous fallacy committed in discourse. It underpins every single bit of religious apologetic bar none. For some flavours of apologetic, such as presuppositionalism, this is the entire foundation.

It takes a lot of picking apart of the statement to extract all of this, but that's not the only problem, and others are much easier to treat. Let's fisk the statement so that we can highlight all of them.

"Once you have eliminated the impossible"

In Conan-Doyle's day, many of the things we take for granted today would have been considered impossible, and duly eliminated from the set of 'whatever remains'. Instant communication over vast distances was not thought possible (in fact, it still is, given the limitations of relativity, but we can communicate as close to instantly as makes no difference over distances we'd not have thought possible). Powered flight was thought impossible, and wasn't achieved until thirteen years after publication of The Sign of Four I with the first flight at Kitty Hawk. How do we definitively identify the set of 'impossible', when it's entirely probable that our set of the possible is at best incomplete? Even in principle, this would require omniscience which, as we've learned elsewhere, is a logical absurdity, breaching the absolute limits of epistemology. In short, it's impossible to eliminate the impossible (possibly)‡.

"whatever remains"

To the scientific or sceptical mindset, this statement is a trigger for a carillon of klaxons. It's a black swan statement of quite epic proportions. As we noted above, it's entirely probable that our understanding of what's possible is horribly incomplete. This means, of course, that we'll be excluding as impossible that which is perfectly possible, and including as possible that which turns out to be impossible. Neither of these sets can be robustly defined.

"however improbable"

There's a nice little trap in this for the unwary. It's certainly the case, as we've seen from a previous outing, that probabilities can be really tricky in guiding our thinking. The thrust of the allusion to probability in this instance is reasonable, because it's an enjoinder not to dismiss something just because the probability is low, but there are still problems with it, mostly arising from the 'highly improbable' being compared with some spurious notion of 'impossible'. 

There's something about probabilities, especially very tiny probabilities, that causes the mind of the apologist to go 'squeeeee!' even without understanding what probabilities are or how they work. The simple fact is that low probabilities aren't a problem. Events with low probability occur all the time. Pick a number. Whatever number you just chose has a probability, given no other information, that is as asymptotically close to zero probability as you can work with, because you just chose one of an infinite set.

"must be the truth"

And this is where the wheels really come off, and everything said above comes sharply into focus. This portion of the quotation is an expression of logical necessity, which is how we got from an abductive argument to a deductive conclusion.

Nits, crits and comments always welcome.

*The way I've presented this is simplified. In reality, this argument would be better cast as:
\(\dfrac {P \Leftarrow Q, Q}{\therefore \lnot (\lnot P)}\)

Which says that Proposition P is implied by consequence Q, Q is observed, therefore not not-P (it is not the case that P has been shown to be false). This doesn't materially affect the point, however, and we needn't be concerned with that complication here.

I made this up. My next-door neighbour's cat was called Himmler.

‡This is not always the case. For highly-well-defined sets, 

Deduction, Induction, Abduction and Fallacy: A more complete treatment of some of the ideas presented here. A detailed exposition of logic, particularly in how it's used in the sciences.

All Kinds of Everything: A debunking of the classic theological omnis, including omniscience.

Probably the Worst Argument in the World: A look at spurious probability calculations and large numbers in apologetics.

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