Unknown, I agree entirely with your comments about the distinction between the idealised calculable probabilities and the actual error prone human calculations of them.
Nominull, I think you are right that the problem feels somewhat paradoxical. Many things do when considering actual human rationality (a species of ‘bounded rationality’ rather than ideal rationality). However, there is no logical problem with what you are saying. For most real world claims, we cannot have justifiable degrees of beliefs greater than one minus a billionth. Moreover, I don’t have a justifiable degree of belief greater than one minus a billionth in my last statement being true (I’m pretty sure, but I could have made a mistake...). This lack of complete certainty about our lack of complete certainty is just one of the disadvantages of having resource bounds (time, memory, accuracy) on our reasoning. On a practical note, while we cannot completely correct ourselves, merely proposing a safe upper bound to confidence in typical situations, memorizing it as a simple number, and then using it in practice is fairly safe, and likely to improve our confidence estimates.
Unknown, I agree entirely with your comments about the distinction between the idealised calculable probabilities and the actual error prone human calculations of them.
Nominull, I think you are right that the problem feels somewhat paradoxical. Many things do when considering actual human rationality (a species of ‘bounded rationality’ rather than ideal rationality). However, there is no logical problem with what you are saying. For most real world claims, we cannot have justifiable degrees of beliefs greater than one minus a billionth. Moreover, I don’t have a justifiable degree of belief greater than one minus a billionth in my last statement being true (I’m pretty sure, but I could have made a mistake...). This lack of complete certainty about our lack of complete certainty is just one of the disadvantages of having resource bounds (time, memory, accuracy) on our reasoning. On a practical note, while we cannot completely correct ourselves, merely proposing a safe upper bound to confidence in typical situations, memorizing it as a simple number, and then using it in practice is fairly safe, and likely to improve our confidence estimates.
Toby.