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benelliott comments on Confidence levels inside and outside an argument - Less Wrong
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Thank-you for expressing my worry in much better terms than I managed to. If you like, I'll link to your comment in my top-level comment.
I still don't know why everyone thinks this is the problem of induction. You can certainly have an agent which is Bayesian but doesn't use induction (the prior which assigns equal probability to all possible sequences of observation is non-inductive). I'm not sure if you can have a non-Bayesian that uses induction, because I'm very confused about the whole subject of ideal non-Bayesian agents, but it seems like you probably could.
Interesting that Bayesian updating seems to be flawed if an only if you assign non-zero probability to the claim that is flawed. If I was feeling mischievous I would compare it to a religion, it works so long as you have absolute faith, but if you doubt even for a moment it doesn't.
Feel free! I am all for increasing the number of minds churning away at this problem - the more Bayesians that are trying to find a way to justify Bayesian methods, the higher the probability that a correct justification will occur. Assuming we can weed out the motivated or biased justifications.
It's similar to Hume's philosophical problem of induction (here and here specifically). Induction in this sense is contrasted with deduction - you could certainly have a Bayesian agent which doesn't use induction (never draws a generalisation from specific observations) but I think it would necessarily be less efficient and less effective than a Bayesian agent that did.