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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?

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.

Not quite (but close). Your logic is backwards: you've shown that if Bob is a Knight, then Alice can say "I could say Bob is a Knight". You need the opposite implication.

Knights always tell the truth, and knaves always lie.

Alice says "I could say Bob is a knight."

Case $1$: Alice is a knight.

If Alice is a knight then she has to always say the truth, thus the statement from Alice that "I could say Bob is a knight" will be a true statement. Assume on contrary that Bob is a knave, then Alice being a knight cannot say that Bob is a knight, since in that case, Alice's statement that "I could say Bob is a knight" will be false. So, $fbox{Bob needs to be a knight}$.

Case $1$: Alice is a knave.

Alice now speaks only false. Thus Alice telling "I could say Bob is a knight", implies that she actually cannot say the 'fact' that Bob is a knight (Alice can't speak the truth). Also, if we assume that Bob is knave, then Alice being a false-speaker can say that Bob is a knight. In that case, if Alice does actually say "I could say Bob is a knight", makes a true statement from Alice, resulting contradiction. So, $fbox{Bob needs to be a knight}$.

When solving these Knights and Knaves puzzles, I often use biconditionals to represent the given information.

For example, if you know that Alice and bob are from the Island of Knights and Knaves, and Alice says: "Bob is a knave", then I represent this as:

$A leftrightarrow neg B$

where of course $A$ means that Alice is a knight and $B$ means that Bob is a knight.

The biconditional captures the fact that if Alice is a knight, then whatever Alice says is true, and if whatever Alice says is true, then Alice is a knight.

OK, so now let's consider what happens when Alice says: " I could say that Bob is a knight"

Well, you can use the very same technique: Alice is a knight if and only if what Alice is saying is true ... bit this time, Alice is saying something about what she could be saying about Bob ... and since knights and knaves only say what they could be saying (that is, they will never say anything that they could not be saying), we might as well interpret this as Alice actually saying this something about Bob. Hence, we can translate what Alice is saying as $A leftrightarrow B$, and plugging this into the general formula, we thus get:

$A leftrightarrow (A leftrightarrow B)$

Ok, and now comes the really cool part: at this point, we can do some very simple algebra to see if something interesting can be inferred. And the biconditional has some very nice properties.

For example, sice the biconditional is associative, we can rewrite the above expressin as:

$(A leftrightarrow A) leftrightarrow B$

OK, but of course $A leftrightarrow A$ is a tautology, and hence we get:

$top leftrightarrow B$

And finally, since the $top$ works as a kind of identity element for the biconditional, we obtain:

$B$

And so there you go: Bob has to be a knight!

There are some more useful algebraic properties regarding the biconditional, and once you get the hang of those, you can solve these Knights and Knaves puzzles very, very quickly!