Problem solution from Brilliant

Question

The setup is: androids - always lie, humans - always tell the truth, jokers - can say both true and false statements.

There are two people Taj and Zahra. Zahra says: "We are both androids". The solution from brilliant says that because that this statement implies that Zahra is an android and because an android can't say that they are an android, she must be a joker.

I'm not exactly sure why that is the case. If the Taj is human or a joker, then the statement is still false, because they are not both androids, so I think that Zahra could still be an android because she is not saying a true statement. Can someone clarify this?

Answer 1

Well spotted Sergiusz! I’m from Brilliant and you’re right, the original solution is incorrect. Thank you for pointing that out to us!

I'll explain whether Zahra could be a human, android, or joker here.

Human: If Zahra were human she'd need to tell the truth. But she says "We are both androids", which would be a lie in that case. Hence she isn't a human.

Android: If Zahra were an android she'd need to lie. As you pointed out, her statement "We are both androids" is false as long as either Zahra or Taj isn't an android. So as long as Taj isn't an android, Zahra could be an android. The valid solution for this problem would be Taj = Joker and Zahra = Android.

Joker: If Zahra were a joker then she can either say the truth or lie so she can be a joker. The two valid solutions in this case are Taj = Joker and Zahra = Joker or Taj = Android and Zahra = Joker . So Zahra can be either an android or a joker, which is contrary to our original solution.

We’ve changed the problem now to ask if Zahra is necessarily a joker, and the answer is "no". Thanks again for finding this error!

Answer 2

You are right. If, for example, Taj is a human and Zahra is an android, then the statement is false. Then Zahra lied, which agrees with the fact that Zahra is an android. So this option cannot be excluded.

Answer 3

The thing to realize is that "she says (Z and T are androids)" is different from "(she says Z is an android) and (she says T is an android)". In the latter case, if she is an android, then she is lying twice and none of them is an android. In the former case, she is lying only once and at least one of them is not an android.

For example if I say «I'm Gribouillis and I'm the Pope», then I'm lying. On the other hand if I say «I'm Gribouillis» and then «I'm the Pope», I told the truth once, hence I'm not a android.

It is a confusion based on the logical imperfections of natural language. Suppose Zhara is an android, then the sentence «Zhara says P» is equivalent to «P is false». Thus «Zhara says (P and Q)» is equivalent to «(P and Q) is false». On the other hand «Zhara says P and Zhara says Q» is equivalent to «P is false and Q is false». The two propositions are different because «(P and Q) is false» is equivalent to «P is false or Q is false».

Answer 4

You say:

" I'm not exactly sure why that is the case. If the Taj is human or a joker, then the statement is still false, because they are not both androids, (...) "

Right.

You say: "so I think that Zahra could still be an android because she is not saying a true statement. Can someone clarify this?"

The whole sentence is false because an android cannot say true statements. Zahra can't be a human because she would state so. Suppose she is an android: In that case, she is correct defining itself as an android, in contradiction to the setup, but we do not have enough information about Taj. It makes the option "Android" not possible. Finally, fot the "Joker" option, it can say he is an android without further consequences.

Hence, Brilliant is right.