Truthtellers and liars

The way we can prove, that this situation is impossible is by assuming one of the persons is a truthteller, and then proving that will contradict the other statements. But then also assume that person is a liar, and if it still contradicts the other statements, we know this situation is impossible on this island.

Let's take a look at Anne.

Assume Anne is a truthteller:

• If Bob is then a truthteller, then that means Carl is a truthteller. But that would mean both Anne, Bob and Carl would be truthtellers, which would contradict Bobs statement.
• So Bob cannot be a truthteller when Anne is a truthteller.
• Therefore Bob is a liar. That means that both Anne and Carl tells the truth, since else Bob wouldn't be lying.
• So we assume Carls statement is true. Since we know Anne is a truthteller, that must mean that David is a liar.
• But then Davids statement is false, but that's a contradiction because if Carl is telling the truth, then Davids statement is always true.

So Anne cannot be a truthteller.

Assume Anne is a liar.

• That automatically means, that Bob is a truthteller and Carl is a liar, else Annes statement is true.
• Bob must tell truth, and his statement also makes sense since Anne is a liar.
• Since Carl must be a liar, then David must also be a liar, else Carl would tell the truth.
• But looking at Davids statement, it is true that if Carl is a liar (which he is) means that Anne or Bob is a liar (because Anne is a liar). So he is a liar telling the truth, which is a contradiction.

So Anne cannot be a liar.

But if Anne cannot be a liar nor a truthteller, this means that the premise of this island is false and therefore this situation cannot be possible.

Anne : If Bob is a truthteller, then Carl is also a truthteller.
Bob : Anne or Carl is a liar.
Carl : David is a liar if and only if Anna is a truthteller.
David : If Carl is liar, then that means Anne or Bob is a liar.

Looking at the IF parts of their statements, David is automatically a truthteller when Carl is also one AND also Anne is automatically a truthteller when Bob is not.

If Carl is a truthteller, then Anne and David are of different 'tribes', and thus, combining this with our if-part conclusions, it follows that David is the truthteller and Anne the liar, which in turn makes Bob another truthteller but all of these are inconsistent with Anne's supposedly false statement as naturally logical to be spoken by a liar. So, we know now that Carl can't be a truthteller here.

If Carl is a liar, then Bob must be a truthteller and in turn, Anne another liar. Also, if Carl is a liar, then Anne and David are of the same 'tribe', so David must be the third liar, but David's true statement of Carl and Anne being liars didn't match his supposedly lying personality. So somehow, Carl can't be a liar here either.

The island must be a lie since this situation is an impossible one.