Statistically, the Truth Is Disappointing
We have beliefs. Sometimes, we reflect on them. We think, ‘Sure, I believe X, but is X really true?’. In this sort of situation, given some reasonable assumptions, we can prove that the truth will be disappointing.
More precisely, we can prove:
Noticing that you want X to be true should make you think X is less likely to be true, and noticing that you don’t want X to be true should make you think X is more likely to be true.
More precisely still, let ‘True’ stand for X being true, ‘Believe’ stand for you currently believing X, and ‘Want’ stand for you wanting X to be true. Then we can prove:
Here’s one set of assumptions that will do the trick.
1. Wanting doesn’t affect truth.
First assumption: wanting X to be true doesn’t change the probability that X is true.
2. Wanting makes you more likely to believe.
Second assumption: wanting X to be true makes it more likely that you believe X to be true. Potential causes are wishful thinking and motivated reasoning.
3. Wanting works just as well on falsehoods.
Third assumption: the boost that wanting gives to belief is at least as big when X is false as when X is true.
This seems reasonable, because the processes by which wanting produces belief don’t consult the world. Wishful thinking and motivated reasoning push you toward believing X in the same way whether X is true or false. There’s no reason to expect the boost to be bigger when X is true.
If anything, we should expect the boost to be bigger when X is false. When X is true, the evidence is already in its favor, so wanting has less room to make a difference. When X is false, the evidence is against it, so wanting has more room to make a difference.
4. Absent wanting, belief tracks truth.
Fourth assumption: conditional on you not wanting X to be true, you’re more likely to believe X if it’s true than if it’s false.
This condition is weak. It just says that, with wanting out of the picture, your beliefs track the truth at least a little.
Proof
The idea behind the proof is simple. Your believing X is evidence that X is true because ‘X is true’ is a good explanation for your belief. But if you want X to be true, that suggests a rival explanation: you just believe X to be true because you want it to be true. The availability of this rival explanation means that your belief in X is no longer such good evidence for the truth of X.
The proof itself is pretty mundane, so I let Claude Fable write it
out in a
footnote.First, wanting produces false belief. By Assumption 2, you’re more
likely to believe X if you want it to be true than if you don’t. By
Assumption 1, wanting X tells us nothing about whether X is true, so
this overall boost is a weighted average of two conditional boosts: the
boost when X is true and the boost when X is false. By Assumption 3, the
boost when X is false is at least as big as the boost when X is true.
And a weighted average of two numbers can only be positive if the bigger
of the two is positive. So the boost when X is false must be
positive:
Conclusion
Statistically, the truth is disappointing. If you want X to be true, that should make you think X is less likely to be true. If you don’t want X to be true, that should make you think X is more likely to be true.