@BrendanMitchell Thank you, yes, I think my wording was incorrect.

@DavidMcKee Yes, this point is covered in the preface of my answer.

This assumption on the behavior of guard 2 means that if you have 2 known liars guarding a keypad with a 100 digit password, and you ask liar 1 "What would liar 2 say is the keypad password?" then liar 1 would be required to give you the exact 100 digit password, because any other answer is something liar 2 might possibly say. I think a more reasonable interpretation is that a liar may give any answer, provided the liar does not know it to be true.

To lie and to tell a falsehood are not the same thing. It is not guaranteed that every statement a liar makes will be false. Lying is about intent; does the liar believe what he is saying is true? If the liar believes X is true, and therefore says Y, but it turns out that Y is actually true, that's still compatible with being a liar.

Also if Guard 2 is the truthteller, then Guard 2 does not know which lie Guard 3 would give if questioned. Guard 3 could give either of the non-gold doors. So Guard 2 would be unable to give any answer at all, and Guard 1, being a liar and knowing Guard 2 cannot answer, could give any answer he likes.

@causative Regarding your first and second comments - With regard to logic problems, liars are equivalent to people who tell falsehoods. Regardless, a liar is someone who tells intentionally false statements so even if we take intent into account, the intent must be to tell a falsehood and not accidentally tell a truth through guessing.

@causative With regard to your third comment, I do take your point here. My idea was that Guard 1 cannot know what Guard 2 knows about Guard 3 so, from Guard 1's point of view, it could be that Guard 2 knows where Guard 3 will point. However, it's a little messy here.

In logic puzzles with liars it is normally clear what the liar must say in response to any question, e.g. they are true/false questions with only one right answer and the liar knows what the right answer is. The assumption you are making is not one I've ever seen in another logic puzzle.

@causative It's essentially the generalisation of the question asked in the regular knights and knaves, as seen here: puzzling.stackexchange.com/a/2190/18422

@hexomino The regular 2 door knights and knaves does not involve this ambiguity. The 2 door liar knows what the truth-teller will say, and vice versa. Consider that (with 3 doors, 2 liars, 1 truth-teller, gold at door 1) if you ask liar 1 what liar 2 would say, and liar 1 answers "Liar 2 would definitely say door 2," then liar 1 has told a lie, because liar 2 might say door 2, not definitely. Does the addition of the word "definitely" to liar 1's answer substantively change it? I don't think so. If Liar 1 omitted "definitely" then he's still making the same claim just without emphasis.

@causative But, for example, the original problem also contains this ambiguity under the hood. The liar here, for example, may not point to the opposite door and instead point to his own mouth. This is still a lie. How will the truthteller know he will not do this?

@hexomino That's a different ambiguity, and your proposed interpretation does not resolve it.

@causative I disagree with that last point, the ambiguities are very much related. Further to that, it may be the case, in your example, that Liar 2 will definitely say door 2 - maybe 2 is his favourite number and he doesn't waiver, who's to say otherwise. In that case, Liar 1 would be telling the truth.

@hexomino (1) No, they are different ambiguities. The 2 door ambiguity must be resolved by requiring both liars and truth-tellers to select an answer from a prescribed list. (2) If liar 1 does not know this tendency of liar 2, then he is still telling a lie if he says "definitely," because from his perspective it is not definite what liar 2 will do. If liar 1 does know this tendency of liar 2, then he may answer door 3 or door 1 as he pleases even by your interpretation of what a liar must say, also ruining your solution.

@causative You seem to be saying that liars in these logic problems can tell the truth, is that right? Because it may be the case that liar 1 says "definitely" without knowledge but this turns out to be true.

@hexomino No, the claim "Liar 2 will definitely say door 2" was both a lie and false, even if liar 2 does in fact choose door 2 according to some consistent preference Liar 1 was unaware of. Because what is definite in this context is only what is definite in Liar 1's knowledge, and it was not definite in Liar 1's knowledge that liar 2 would choose door 2, even if it was predetermined. (But also yes, liars can tell the truth if they think it to be a lie.)

@causative So the focus is on perspective, is that right? What is to stop a truthteller from telling a falsehood if they believe it to be true?

@hexomino Yes, a truthteller would tell a falsehood if they believe it to be true. Hasn't come up in this puzzle though.

@causative You seem to be introducing perspective into these problems which I have not seen before. My interpretation of these problems is that truthtellers tell truths and liars tell falsehoods. By introducing their beliefs, it seems that either party could potentially say anything (because they believe it to be so).

you need some assumptions about what either party knows, e.g. that they know the correct door

So what are we assuming each party knows in this case?

In logic puzzles, typically truthtellers do tell truths and liars do tell falsehoods, because the logic puzzles are carefully designed so that they are never asked a question where their personal uncertainty comes into play. This logic puzzle is not carefully designed in that way so we have to be more careful in what we call a lie, using a broader context. In broader context, a lie is when you say something you do not believe.

So if someone doesn't believe that the Earth is round, does that make "the Earth is round" a lie?

yes

That's a very strange perspective

no it's not

I would say truth is objective, regardless of opinion.

Actually I think in this puzzle the most reasonable interpretation is that we as the questioner cannot be sure what the guards know about each other's preferences. So if a plan of questioning would fail or succeed depending on what the guards know about each other, then that plan of questioning fails because we as the questioner can't be sure.

truth may be objective but to lie is to say what you do not believe

if you attempt to deceive by telling what you believe to be false, that's a lie, even if it turns out your statement was true by accident

Your last point about the questioner being unsure is fair and I take that. I guess from that point of view, any question that's not specifically related to what's behind the doors should be ruled out, right?

I don't think I agree with the general idea of lies. If somebody says the Earth is flat, I plainly tell them they are lying. However, I do accept that this is not a strange perspective and I'm sorry for saying that.