Knights, Knaves, Jokers

It is clear that both islanders cannot be knights, since that would make both of their statements false. Suppose that Anne is a knight and Dovi is a knave. Then Anne's statement is true. However, Dovi's statement is also true. A conditional statement is only false when the first proposition is true while the second proposition is false. In this case, Dovi can only lie, so this situation cannot occur. Likewise, we cannot have the situation in which Anne is a knave and Dovi is a knight. It is for the same reason that neither islander can be a liar, since it is impossible for either person's statement to be a lie. In fact, the only possible situation that works is if both islanders are jokers. In this case, they are jokers that are choosing to tell the truth.

Anne says, "If I am a knight, then Dovi is a knave."

Dovi says, "If I am a knight, then Anne is a knave."

How many of them are knights?

Their statements are of the same structure using the same form of if ~, then ~. If this kind of statement is a lie, then the speaker must be a Knight and the other person cannot be a knave, but it's a contradiction since Knights can't lie. Lying using this kind of statement is impossible, so we cannot have a knave between the two (since both used the same structure and form), therefore having a Knight among them is also impossible (since a Knight speaker would need a knave partner). In conclusion, both of Anne and Dovi are neither knaves nor Knights, so both must have been truthful 🃏 Jokers.