A very special island is inhabited only by knights and knaves. Knights always tell the truth, and knaves always lie. You meet 6 inhabitants: Michael, Abigail, Luke, William, Alexander, and Aria. As Michael put it, "Abigail is not a knight". In a statement by Abigail: "If Aria is a knight then Luke is a knight". As Luke put it, "Abigail is a knave or Michael is a knight". William expressed that Luke is not a knight. Alexander told you that If William is a knave then William is a knight. "Luke is a knight if and only if Alexander is a knight" - Aria. So who is a knight and who is a knave?
Michael is a knave, Abigail is a knight, Luke is a knave, William is a knight, Alexander is a knight, and Aria is a knave.
(1) Michael is a knave
(2) Abigail is a knight
(3) Luke is a knave
(4) William is a knight
(5) Alexander is a knight
(6) Aria is a knave
Assume Michael is a knight. No contradiction is found in their claim that Abigail is not a knight. Abigail cannot be a knight, because this would contradict the claim of Michael that Abigail is not a knight. Assume Abigail is a knave. No contradiction is found in their false claim that If Aria is a knight then Luke is a knight. Assume Aria is a knight. No contradiction is found in their claim that Luke is a knight if and only if Alexander is a knight. Luke cannot be a knight, because this would contradict the false claim of Abigail that If Aria is a knight then Luke is a knight. Luke cannot be a knave, because this would contradict the false claim of their own that Abigail is a knave or Michael is a knight. We have exhausted all possibilities for Luke, so let us go back and reconsider Aria. Aria cannot be a knave, because this would contradict the false claim of Abigail that If Aria is a knight then Luke is a knight. We have exhausted all possibilities for Aria and Abigail, so let us go back and reconsider Michael. Assume Michael is a knave. No contradiction is found in their false claim that Abigail is not a knight. Assume Abigail is a knight. No contradiction is found in their claim that If Aria is a knight then Luke is a knight. Assume Aria is a knight. No contradiction is found in their claim that Luke is a knight if and only if Alexander is a knight. Luke cannot be a knight, because this would contradict the claim of their own that Abigail is a knave or Michael is a knight. Luke cannot be a knave, because this would contradict the claim of Abigail that If Aria is a knight then Luke is a knight. We have exhausted all possibilities for Luke, so let us go back and reconsider Aria. Assume Aria is a knave. No contradiction is found in their false claim that Luke is a knight if and only if Alexander is a knight. Luke cannot be a knight, because this would contradict the claim of their own that Abigail is a knave or Michael is a knight. Assume Luke is a knave. No contradiction is found in their false claim that Abigail is a knave or Michael is a knight. Assume Alexander is a knight. No contradiction is found in their claim that If William is a knave then William is a knight. Assume William is a knight. No contradiction is found in their claim that Luke is not a knight.