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Meta-Logic

Meta-Logic Difficulty 7 10 questions by Marco DeLuca

Puzzles about reasoning about reasoning. What do others know? What can silence tell you? When does a statement about itself collapse into contradiction? These require thinking recursively about knowledge, belief, and inference.

Instructions Each puzzle involves reasoning about what others know, believe, or can deduce. Walk through the chain of inference step by step — who knows what, and when do they know it? Show your reasoning for full credit.
Question 1 of 10
A note reads: "This note is unreadable." If you can read the note, what does that imply?
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Question 2 of 10
Two perfectly rational players each secretly choose either Red or Blue. Payoffs: if both choose Red, each gets 3 points. If both choose Blue, each gets 2 points. If they choose different colors, both get 0. Both know the other is perfectly rational, and both know that both know this. What should they choose?
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Question 3 of 10
A sign says, "The sentence below is true." The sentence below says, "The sentence above is false." Can both sentences be assigned consistent truth values?
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Question 4 of 10
On an island, every inhabitant is either a truth-teller or a liar. You meet A and B. A says, "B is a liar." B says, "We are of opposite types." What are A and B?
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Question 5 of 10
Alice says, "Bob knows that I am lying." If Alice is telling the truth, what can you conclude?
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Question 6 of 10
A says, "Either I am a liar or B is a truth-teller." What can you conclude about A, regardless of B's type?
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Question 7 of 10
Three logicians walk into a bar. The bartender asks, "Do you all want something to drink?" The first logician says, "I don't know." The second says, "I don't know." The third says, "Yes." Why can the third logician answer definitively?
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Question 8 of 10
Three students are told that one scored highest, one middle, and one lowest on a test. Each knows their own score but not the others'. The teacher asks, "Can anyone determine whether they scored highest?" The first says no. The second says no. The third says yes. How does the third student know?
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Question 9 of 10
A judge tells three prisoners that one will be pardoned. Prisoner A asks the guard to name one prisoner (other than A) who will NOT be pardoned. The guard says, "B will not be pardoned." Should A now believe A's chances have changed from 1/3?
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Question 10 of 10
Three people are each wearing either a white or black hat. Everyone can see the others' hats but not their own. It is publicly announced that at least one hat is white. After one round where nobody can determine their hat color, one person then concludes their hat is white. What overall hat pattern makes this possible?
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