Holmes method weakness, trilemma fallacy, Jesus must have been God

 Holmes method weakness, trilemma fallacy, Jesus must have been God

 

THOUGHT OF THE DAY: "How often have I said to you that when you have eliminated the impossible, whatever remains, however improbable, must be the truth?" - Arthur Conan Doyle, from The Sign of the Four, ch 6. 

 

Horrible fallacy; Doyle should be ashamed. When you have nothing but improbables left, you don't know and should admit it.

 

I think you'd have gotten an argument out of Sherlock Holmes if you'd been standing in Baker Street.

The quote sounds rigorous, but it sneaks in a very strong assumption: that you've correctly identified all the possibilities and correctly eliminated every impossible one. In real life, that's often the weak link.

A more statistically minded version might be:

"When the likely explanations have failed, reconsider your assumptions, your evidence, and the possibility that you have overlooked an explanation."

Science is littered with examples where the "remaining explanation" turned out not to be true at all because someone had missed an alternative. The history of astronomy, medicine, and criminal investigations is full of discarded certainties.

That said, Holmes was often using the quote in a narrower sense. If you know a closed set of possibilities really is exhaustive, then eliminating all but one does leave the survivor, however unlikely. In formal logic, that's valid. The problem is that the real world almost never hands us an exhaustive list.

Your version reflects a probabilistic mindset:

"When all the explanations seem improbable, the correct conclusion may be that the evidence is insufficient."

That attitude tends to age well. One of the virtues of science is that "I don't know" is considered an acceptable answer, whereas detective fiction is under pressure to produce a culprit by the final chapter.

Doyle was writing mysteries, after all. If Holmes had concluded, "Watson, the posterior probabilities are poorly constrained and additional data are required," the stories might have sold fewer copies.

 

Well now, that possibilities may have been overlooked is just part of not knowing. I don't think Doyle was pandering, he was trying to make Holmes sound profound and himself didn't understand the logic.

That's certainly possible.

Doyle was a physician, not a logician, and many educated people then—as now—treated deduction as more powerful than it really is. The Holmes stories often blur three different things:

1. Deduction (certainty from premises).

2. Induction (generalizing from evidence).

3. Abduction (choosing the best explanation).

Holmes frequently calls what he is doing "deduction" when it is really abduction: inferring the explanation that best fits the facts. That can be brilliant reasoning, but it is never logically certain.

The famous quote sounds like deduction. In symbolic form:

• A is impossible.

• B is impossible.

• C is impossible.

• Therefore D is true.

That is perfectly valid if A, B, C, and D exhaust all possibilities.

But in practice Holmes almost never proves exhaustiveness. There may be an E, F, or G that nobody has considered. So the certainty vanishes.

What makes the quote memorable is that it flatters our desire for closure. Humans are uncomfortable with unresolved uncertainty. "The improbable explanation must be true" feels satisfying. "The evidence is insufficient to distinguish among several unlikely explanations" is often the more honest conclusion, but it doesn't have quite the same literary punch.

Interestingly, your objection resembles the attitude that became much more prominent in the twentieth century with statisticians and scientists. People like Fisher, Jeffreys, and later Bayesian thinkers were much more willing to say, "These hypotheses all fit poorly; confidence is not justified." The emphasis shifted from certainty to degrees of belief.

I suspect one reason Holmes remains appealing is that he inhabits a world where uncertainty can be conquered by intellect alone. Real life is messier. Quite often, after eliminating what seems impossible, what remains is not truth but a smaller set of unanswered questions.

Holmes is a good read, this is just a weak spot. It is like theories on the evolution of the Solar System, a lot of detail remains mysterious (although we can be highly confident about the general picture). This is not so much because we have no theories, but because we don't have the ability to determine which theory is right. This absence of firm understanding is not a reason to say it must have been a miracle.

Exactly. That's a good example because it separates having explanations from having evidence strong enough to choose among them.

For the Solar System, the broad framework is extraordinarily well supported: a collapsing molecular cloud, a protoplanetary disk, accretion, planetary migration, impacts, and so forth. But when you get down to details—how much Jupiter migrated, the exact sequence of events in the early bombardment period, why certain orbital features ended up exactly as they did—multiple models may remain viable.

The intellectually honest position is not:

"We don't know which model is correct, therefore Model A must be correct."

Nor is it:

"We don't know which model is correct, therefore a miracle occurred."

It is:

"The available evidence is insufficient to discriminate confidently among the remaining possibilities."

That sounds less dramatic than Holmes, but it is usually the stronger inference.

Historically, "God of the gaps" arguments often relied on the same mistake. A phenomenon lacked a satisfactory explanation, so a supernatural explanation was declared the winner by default. But an unexplained phenomenon is not evidence for any particular explanation. It is evidence that the explanation has not yet been established.

One of the things science has learned, often painfully, is that uncertainty is a genuine state of knowledge. There is no rule requiring reality to reveal a unique answer just because we are impatient for one.

In a way, your criticism of Holmes is that he skips too quickly from:

"I have ruled out several explanations"

to

"Therefore I know the answer."

Whereas many real problems end with:

"I have ruled out several explanations. The remaining candidates are all plausible to some degree. Further evidence is needed."

That's less satisfying in a detective story, but it is often exactly where a scientist finds themselves. The history of astronomy is full of such cases: competing explanations surviving for decades because the observations simply weren't yet good enough to settle the matter. The responsible conclusion was not certainty, but suspended judgment.

Like the preacher: "Jesus was either insane or a fraud or God."

Yes, that's a classic example of the same logical pattern.

The argument is often associated with C. S. Lewis and is sometimes called the "Lord, Liar, or Lunatic" trilemma. In simplified form:

• If Jesus claimed to be divine, then either:

  • he knew it was false (liar),

  • believed it falsely (lunatic),

  • or it was true (Lord).

• Therefore one of those three must be correct.

The most common criticism is not that the reasoning is invalid if those are truly the only possibilities, but that the list may not be exhaustive.

For example, critics have suggested additional possibilities:

• The Gospel accounts may not record Jesus's words exactly.

• Later followers may have interpreted or expanded claims over time.

• Statements may have been intended metaphorically or in ways modern readers misunderstand.

• The historical evidence may be too incomplete to support such a neat classification.

Whether any of those alternatives is persuasive is a separate question. The logical point is simply that the trilemma assumes the available options are complete. Once additional possibilities are admitted, the force of the conclusion weakens.

That's very similar to your objection to Holmes. The move from:

"I can think of only three explanations"

to

"Therefore one of those three must be true"

is often treated as stronger than it really is.

A lot of intellectual history consists of discovering a fourth option that nobody had considered. In science, philosophy, law, and religion, people frequently frame a question as a choice among a few possibilities, only to have later generations point out that the framing itself was too narrow.

Your underlying principle seems to be:

Before concluding that an improbable explanation is true, ask whether the list of explanations is complete and whether the evidence actually distinguishes among them.

That habit saves people from a surprising number of errors. It also leaves more questions unanswered—which can be frustrating, but is often the price of intellectual honesty.