I realize I’m a couple weeks behind and nobody might read this, but this is false (e.g., if I flip two fair + independent coins and A = “first coin is heads” and B = “second coin is heads”, P(B) = 1⁄2, but P(B|A) x P(A) = (1/2)x(1/2) = 1⁄4). I think what you mean is P(A and B) = P(B|A) x P(A).
(unless I’m missing some context and we are assuming B is a subset of A, but if so I’m not sure why we are assuming that)
At the recent Boston SSC meetup, I was asked what rationality skill I would most like everyone to have. Not sure I actually endorse this as the best, but one idea I had was that I would like everyone to be more aware that the things they believe might be false. The person who asked me the question reflected this answer back as ‘skepticism’, which didn’t seem right to me, but the conversation moved on before I could figure out why.
After some thought, I think I know what the distinction I was trying to draw is. I think of the central activity pointed at by the word ‘skepticism’ as being skepticism of new thoughts coming in from other people. I have a pretty strong impulse towards this kind of skepticism, and I think it’s sometimes helpful, but easy to take too far.
Skepticism towards things already in my head feels like a different move to me—I would more likely call it something like ‘careful thinking’. In my terminology, skepticism happens on the way in (tending to reject new ideas when I hear them), whereas careful thinking happens on the way out (when I’m constructing an argument, re-evaluating the truth of each claim I am making).
Doing more careful thinking also has its downsides; it makes me worse at the kind of discussions that require me to present a coherent argument in real time, and it makes me more likely to get sidetracked when I am trying to construct an argument or model. But I think these are relatively minor compared to the downsides of increasing skepticism—I find that if I’m not careful not to be too skeptical, I’ll tend to reject new ideas without actually considering them at all. Often I’ve realized much later that I should have taken more seriously arguments I initially rejected. (Of course, doing without either skepticism or careful thinking would save me a lot of time, but the downsides of that are pretty obvious).
I think at the right levels, these two skills can trade off against each other a bit—if you are more careful about which arguments you accept in the first place, you can relax a little about re-evaluating them in the moment. Or if you know you will re-evaluate your beliefs before doing anything important with them, maybe you can feel freer to accept ideas you’re not sure about. But if you are too skeptical of new ideas, you might actually end up with lower-quality beliefs and would need to do more careful thinking to compensate.
Anyway, I think both of these are important, but skepticism has serious downsides at high levels and many (if not most) people already have too much. I don’t think the world would be better if most people were better at rejecting new ideas. I do think the world would be better if most people were more inclined to re-evaluate their old ideas.