## Lying Sisters

Setup: A woman runs into a goup of three other women, one of which is her long lost sister. The first says "I am your long lost sister." The second says "No, I am your long lost sister." The third says "At least two of us are lying."

This is a paradox because if we assume the statement made by the third woman is true, then she is not lying, and so the other two must be lying, so the third woman must be the long lost sister. And if we assume that the statement made by the third woman is false, then the third woman is lying, and also the statment implies 0 or 1 woman are lying, so the other two woman must be telling the truth, but this is a contradiction since there is only one long lost sister. Therefore it seems that the third woman must be the long lost sister, but how can we know this if none of them are credible?

The problem here is that not all statements can be assigned a truth value. The fact that the statement is self referential is the property that allows it to seem like information can be obtained from it. But actually, since no woman can be trusted, there is nothing that they can say that allows us to determine which is the long lost sister. So suppose that the first woman is the long lost sister. Then the first woman is telling the truth and the second woman is lying. Then by removing these two cases from the statement of the third woman yields "At least 1 of the set {me} is lying." This is equivalent to the statement "I am lying." or "This statement is false." and this is exactly the form of statement that cannot be given a truth value. Similar in principle is Curry's Paradox