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?
Best Answer
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».