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lukstafi comments on Counterfactual Calculation and Observational Knowledge - Less Wrong
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Indeed. Consider a variant of the thought experiment where in the "actual" world you used a very reliable process, that's only wrong 1 time in a trillion, while in the counterfactual you're offered to control, you know only of an old calculator that is wrong 1 time in 10, and indicated a different answer from what you worked out. Updateless analysis says that you still have to go with old calculator's result.
Knowledge seems to apply only to the event that produced it, even "logical" knowledge. Even if you prove something, you can't be absolutely sure, so in the counterfactual you trust an old calculator instead of your proof. This would actually be a good variant of this thought experiment ("Counterfactual Proof"), interesting in its own right, by showing that "logical knowledge" has the same limitations, and perhaps further highlighting the nature of these limitations.
Pardon me if I repeat someone. Q causes the answer of the calculator, so if we set calculator's answer counterfactually we lose dependency between Q and the calculator, and so we don't have any knowledge of the counterfactual Q. Whereas if we had a formula R of comparable logical complexity to Q, drawn from a class of formula pairs with 90% correlation of values, then the dependency is bidirectional and counterfactually setting R we gain the knowledge about the counterfactual Q. Does "in the counterfactual you trust an old calculator instead of your proof" mean that you don't agree (with this analysis)? (I have the impression that the problem statement drifted somewhat from "counterfactual" to a more "conditional" interpretation where we don't sever any dependencies.)