Since Pascal’s Mugging is well known on LW, I won’t describe it at length. Suffice to say that a mugger tries to blackmail you by threatening enormous harm by a completely mysterious mechanism. If the harm is great enough, a sufficiently large threat eventually dominates doubts about the mechanism.

I have a reasonably simple solution to Pascal’s Mugging. In four steps, here it is:

The greater the harm, the more likely the mugger is trying to pick a greater threat than any competitor picks (we’ll call that maximizing).
As the amount of harm threatened gets larger, the probability that the mugger is maximizing approaches unity.
As the probability that the mugger is engaged in maximizing approaches unity, the likelihood that the mugger’s claim is true approaches zero.
The probability that a contrary claim is true—that contributing to the mugger will cause the feared calamity—exceeds the probability that the mugger’s claim is true when the probability that the mugger is maximizing increases sufficiently.

Pascal’s Mugging induces us to look at the likelihood of the claim in abstraction from the fact that the claim is made. The paradox can be solved by breaking the probability that the mugger’s claim is true into two parts: the probability of the claim itself (its simplicity) and the probability that the mugger is truthful. Even if the probability of magical harm doesn’t decrease when the amount of harm increases, the probability that the mugger is truthful decreases continuously as the amount of harm predicted increases.

Solving the paradox in Pascal’s Mugging depends on recognizing that, if the logic were sound, it would engage muggers in a game where they try to pick the highest practicable number to represent the amount of harm. But this means that the higher the number, the more likely they are to be playing this game (undermining the logic believed sound).

But solving Pascal’s Mugging also depends on recognizing that the evidence that the mugger is maximizing can lower the probability below that of the same harm when no mugger has claimed it. It involves recognizing that, when it is almost certain that the claim is motivated by something unrelated to the claim’s truth, the claim can become less believable than if it hadn’t been expressed. The mugger’s maximizing motivation is evidence against his claim.

If someone presents you with a number representing the amount of threatened harm 3^3^3..., continued as long as a computer can print out when the printer is allowed for run for, say, a decade, you should think this result less probable than if someone had never presented you with the tome. While people are more likely to be telling the truth than to be lying, if you are sufficiently sure they are lying, their testimony counts against their claim.

The proof is the same as the proof of the (also counter-intuitive) proposition that failure to find (some definite amount of) evidence for a theory constitutes negative evidence. The mugger has elicited your search for evidence, but because of the mugger’s clear interest in falsehood, you find that evidence wanting.