Location: Chapter 8, solution to Problem 2. In particular, the seventh paragraph on the page, lines 5-7.
... or he could be a knight who truthfully claimed that the fountain was on the island and who also claimed that the fountain was not on the island. ...
It should become
... or he could be a knight who truthfully claimed that the fountain was not on the island and who also never claimed that the fountain was not on the island. ...
The line of reasoning presented in the book requires the corrections given above in order to comply with the subcase of \(B_2\) in which \(B\) is a knight.
About a different claim
In this Reddit discussion (dating back to 2014), doubts are raised over an important part of the solution: namely, it is argued that the reasoning in the Case \(A_2\) last paragraph of page 68 ("In this case, here is how Reynolds would reason: ...") is flawed.
More precisely, the discussion goes, how could (the hypothetical, case-\(A_2\)) Reynolds include in his reasoning the fact that the fountain is not on the island, a fact not yet available to him at the time (remember the question asked by Craig is "do you know whether you would have been able, at the time, to draw a conclusion on the fountain, had the other person answered instead?")
But this objection seems questionable: imagine, at present time (the time when Craig and Reynolds discuss), Craig having models of Reynolds-on-the-island in mind. One of these models is the Reynolds of case \(A_2\), who, after having gotten the second answer, knows the fountain is not on the island and is able to reason as reported in the paragraph ("The island is not here, both \(A\) and \(B\) are knights: had \(B\) answered, he would have given a negative answer. At the time, just knowing that \(B\) gave such an answer, I would have had no chance at finding out the answer. So I am going to tell Craig that yes, I know whether I would've been able to find out -- indeed, I would not have been able to.").
The rest of the reasoning, once this step is cleared, proceeds as reported in the book. For the time being, the claim found in the thread is considered invalid, but obviously the case would be immediately re-opened as soon as compelling evidence is presented.