My difficulty is that everything that I would call knowledge is like what you get when you look at a calculator display.
In some sense, sure. But you still have to use certain specific reasoning procedure to think about imperfection of knowledge-acquisition methods. That level where you just perform the algorithm is where logic resides. It's not clear to me how to merge these considerations seamlessly.
My guess was that your "logical knowledge" includes (in your terminology) the "moral arguments" that "the agent can prove" in the "theory it uses".
Yes. This theory can include tools for reasoning about observational and logical uncertainty, where logical uncertainty refers to inability to reach the conclusions (explore long enough proofs) rather than uncertainty about whether the reasoning apparatus would do something unintended.
Could you spell out how exactly the 99% correctness rate means that I've probably labeled things confusingly?
I referred to this statement you made:
I assume that, according to the agent's mathematical intuition, Omega is just as likely to offer the decision in a correct-calculator world as in the incorrect-calculator world.
It's not clear what the "Omega offers the decision in a correct-calculator world" event is, since we already know that Omega offers the decision in "even" worlds, in some of which "even" is correct, and in some of which it's not (as far as you know), and 99% of "even" worlds are the ones where calculator is correct, while you clearly assign 50% as probability of your event.
It's not clear what the "Omega offers the decision in a correct-calculator world" event is, since we already know that Omega offers the decision in "even" worlds, in some of which "even" is correct, and in some of which it's not (as far as you know), and 99% of "even" worlds are the ones where calculator is correct, while you clearly assign 50% as probability of your event.
When you speak of "worlds" here, do you mean the "world-programs" in the UDT1.1 formalism? If that is what you mean, then ...
Consider the following thought experiment ("Counterfactual Calculation"):
Should you write "even" on the counterfactual test sheet, given that you're 99% sure that the answer is "even"?
This thought experiment contrasts "logical knowledge" (the usual kind) and "observational knowledge" (what you get when you look at a calculator display). The kind of knowledge you obtain by observing things is not like the kind of knowledge you obtain by thinking yourself. What is the difference (if there actually is a difference)? Why does observational knowledge work in your own possible worlds, but not in counterfactuals? How much of logical knowledge is like observational knowledge, and what are the conditions of its applicability? Can things that we consider "logical knowledge" fail to apply to some counterfactuals?
(Updateless analysis would say "observational knowledge is not knowledge" or that it's knowledge only in the sense that you should bet a certain way. This doesn't analyze the intuition of knowing the result after looking at a calculator display. There is a very salient sense in which the result becomes known, and the purpose of this thought experiment is to explore some of counterintuitive properties of such knowledge.)