16 Logic as a Tool (k) ((¬p ∧ q ) → ¬r) → (¬q → ¬(p ∧ ¬r )) (l) ((p → r) ∨ (q → r)) → ((p ∨ q ) → r) (m) ((p → r) ∧ (q → r)) → ((p ∨ q ) → r) (n) p → ((q → r) → ((p → q ) → (p → r))) 1.1.10 Lastly, some logical puzzles3 On the remote planet Nologic there are two types of intelligent creatures: • truth-tellers, who always tell the truth, and • liars, who (you guessed it) always lie It is not possible to distinguish them by appearance, but only by the truth or falsity of the statements they make (a) A space traveler visited Nologic and met two inhabitants, P and Q He asked them: “Is any of you a liar?” “At least one of us is a liar”, replied P Can you find out what P and Q are? (b) Next, the stranger met two other inhabitants and asked one of them “Is any of you a liar?” He got a “yes” or “no” answer and from that answer was able to determine for each of them whether he is a liar or not What was the answer and what was the stranger’s conclusion? (c) Walking about Nologic, the stranger met two other inhabitants A and B and asked A, “Is any of you a truth-teller?” “If B is a liar, then I am a liar too”, replied A What are A and B? (d) The stranger went on and met three locals X, Y, and Z and asked X: “Are you a liar?” X answered something which the stranger did not hear, so he asked Y: “What did X say?” “He said that he is a liar”, replied Y Then Z added “Don’t believe Y, he is a liar” Can you identify all liars? (e∗) The stranger went on In the evening, he began to look for a shelter for the night, but was very cautious because he knew that some of the inhabitants were man-eaters and it was not possible to recognize them by appearance He met three inhabitants, C, D, and E He asked C, “How many of you are truth-tellers?” “Flam flim” answered C in her language “What did she say?” asked the stranger D “Just one”, replied D “Do not trust D, he is a liar Come with me, I’m not a man-eater” said E “No, come with me, I’m not a man-eater” countered D What should the stranger do? (f∗) The stranger decided to go back to the port where his spaceship was, but he got lost After a long walk he got to a fork of the road He knew that one of the two roads would take him to the spaceship port, but did not know which one Luckily, he saw two of the inhabitants, one on each road He had met them before so he knew that one of them was a liar and the other a truth-teller, but could not remember who was who Can the stranger ask just one question to either of these inhabitants in order to find out which is the correct road to the spaceship port? If so, what question should he ask? For many more puzzles of this type, I warmly recommend to the reader the marvellous logical puzzle books by Raymond Smullyan (1998, 2009a, b, 2011, 2013, 2014)