Faced with engine problems, Ella Wright made an emergency landing on the beach of the island of Knights and Knaves. The island is inhabited by two distinct groups of people, knights and knaves. Knights always tell the truth and knaves always lie. Ellen decided that her best move was to reach the capital and call for service.
Walking from the beach, she came to an intersection, where she saw two men, A and B, working nearby. After hearing her story, A told Ellen, "The capital is in the mountains, or the road on the right goes to the capital." B then said, "The capital is in the mountains, and the road on the right goes to the capital." Then A looked up and said, "That man is a liar." Shrugging his shoulders, B then said, "If the capital is in the mountains, then the road to the right goes to the capital." Ellen then made a table on the back of her guidebook, thanked the two men, and walked down the road on the left. Did ELlen make the correct decision?
If Bs first statement is true, then As first statement has to be true as well, since the truth of P and Q implies that of P or Q. However one of them calls the other a liar; therefore A and B cannot both be knights. Therefore Bs first statement is false, i.e. B is a knave. Thus Bs second statement is also false. Hence, the capital is in the mountains but the road to the right does not go to the capital.