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UnclGhost comments on The Parable of the Dagger - Less Wrong
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He did not assume either of the inscriptions were true. He assumed that each was either true or false.
He never assumed a correlation. He deduced a correlation. He was wrong because the deduction hinged on a false assumption.
Edit: Looking back on this, I guess he did assume a correlation. He implicitly assumed that the position of the dagger did not cause the liar paradox. This is still a lot less of an assumption than assuming that either inscription was true.
In the explanation for the puzzle this is adapted from (Puzzle 70 in What is the Name of this Book?, in the "Portia's Casket's" chapter), Raymond Smullyan raises both points: "The suitor should have realized that without any information given about the truth or falsity of the sentences, nor any information given about the relation of their truth-values, the sentences could say anything, and the object (portrait or dagger, as the case may be) could be anywhere. Good heavens, I can take any number of caskets that I please and put an object in one of them and then write any inscriptions at all on the lids; these sentences won't convey any information whatsoever. So Portia was not really lying; all she said was that the object in question was in one of the boxes, and in each case it really was. ... Another way to look at the matter is that the suitor's error was to assume that each of the statements was either true or false."
The given puzzle (the boxes are labeled "the portrait is not in here" and "exactly one of these two statements is true", where the portrait is the desired object, is contrasted with an earlier problem, where there are two boxes saying "the portrait is not in here" and "exactly one of these two boxes was labeled by someone who always tells the truth" (and it's given that the only other box-maker always lies). The distinction the author draws is that the second box in the earlier problem really does have to be true or false, since "it is a historic statement about the physical world", but there's no such guarantee with purely self-referential labels.
If one of the boxes says that exactly one of them was written by Alice, and you know from another source that Alice always tells the truth, Bob always lies, and both boxes were inscribed by one of them, and Alice and Bob never say anything self-referential, then this is correct.
If it says that one of the boxes was labelled by someone who always tells the truth, then it's not just talking about the person who labelled that box. It's also talking about every aspect of reality that they've ever referenced, and if they were the one to write that inscription, then it's self-referential.
Good point--in the original wording, it says it was inscribed by "Bellini", who is established earlier to always tell the truth.
In which case, if Bellini ever references anything self-referential, the idea that he always tells the truth is not a statement about the physical world. It's likely that the origin of the paradox is that the claim that Bellini always tells the truth and the rest of the scenario are contradictory.
I notice we're somehow not debating what Bellini always telling the truth means for the truth value of the inscribed text which may have had no meaning to him?