I'm having trouble with classic knights and knaves problems that use the wording "I could say" or something similar. I haven't seen this variant discussed elsewhere, although versions of this problem with this wording are not uncommon.
For example,
Knights always tell the truth, and knaves always lie.
Alice says "I could say Bob is a knight."
… [puzzle continues]
My interpretation is: if Alice is a Knave, and Bob is a Knight, then Alice cannot say Bob is a Knight, so she lies and says "I could say Bob is a Knight." If Alice is a Knight, and Bob is a Knight, then of course she can (truthfully) say "I could say Bob is a Knight." Therefore regardless of what Alice is, Bob must be a Knight.
Is this correct? If not, how should I think about this instead?
Best Answer
Your logic is good but you answered the wrong question. You answered: If Bob is a knight, can Alice say "I could say Bob is a knight"?
The question you should have answerr is: If Alice says "I could say Bob is a knight", what is Bob?
And the answer is a knight and the logic is much the same:
i) Exhaustive truth tables, Of the four option of Alice/Bob being knights/knave we have:
A knight/B knight: "I could say B knight" is TRUE. Possible.
A knight/B knave: "I could say B knight" is FALSE. Not possible.
A knave/B knight: "I could say B knight" is FALSE. possible.
A knave/B knight: "I could say B knight" is FALSE. Not possible.
So Bob is knight.
ii: Direct logic.
If Alice is a knave then "I could say Bob is knight" is false. So she can't say "Bob is a knight". A knave can say any lie and can't so any truth. So if she can't say "Bob is a knight" that would be because it is true. So Bob is a knight.
If Alice is a knight then "I could say Bob is a knight" is true. So she can say "Bob is a knight". But knights can only tell the truth so "Bob is a knight" must be true. So Bob is a knight.
iii: Generalize.
If anyone says "I could say $X$" then $X$ must be true.
Pf: A knight can only say "I could say $X$" if s/he could. And s/he could only say $X$ if it were true. So if a knight say "I could say $X$" then $X$ is true.
A knave could only say "I could say $X$" if s/he couldn't. And the only way s/he couldn't say $X$ is if it were true.
So if anyone says "I could say $X$" then $X$ is true.
So if Alice says "I could say Bob is a Knight" then... Bob is a knight.