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Remarks on Mathematical Logic: Paradoxes and Puzzles AAG 1.11.01 Selmer Bringsjord The Minds & Machines Laboratory Department of Philosophy, Psychology & Cognitive Science Department of Computer Science RPI Troy NY USA selmer@rpi.edu www.rpi.edu/~brings

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What is Logic? The science of reasoning. The only invincible subject there is. The basis for all things intellectual, not only the basis of mathematics, but also of engineering to computer science to philosophy. The most challenging subject there is. A key to riches. The key to divining the meaning of life (and other such big questions) One of two fundamental approaches to studying minds, and replicating/simulating minds in machines… … The thing many detectives have mastered – have you (as a New Yorker)?…

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Encapsulating the Minds & Machines Lab & Program

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Number Sense 9 111 4 7 3 5 Vicky’s secret number is inside the triangle. It is outside the square. It is greater than 7 but less than 10. Vicky’s secret number is… 9

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This sentence is false.

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The sentence below is false. The sentence above is true.

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We can handle “Liar-based” puzzles: You travel through space and arrive on the planet Trekia, upon which reside aliens each of which are in one of two cultures: the Larpals, who always lie, and the Tarsals who always tell the truth. You ask three aliens which culture they belong to. The first alien murmurs something you can’t make out. The second alien says: “It said it was a Larpal.” The third alien says to the second alien: “You are a liar!” To which culture does the third alien belong?

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Proof: A3 is either a Larpal or a Tarsal. If a Larpal, then A2 is a Tarsal, which implies that A1 said: “A1 is a Larpal,” i.e., “A1 is a liar.” But then if A1 is a liar, since on this assumption A1 said it is a liar, A1 is truth-teller. On the other hand, if A1 is a truth-teller, then since on this assumption A1 said it is a liar, A1 is a liar. In other words: A1 is a truth-teller if and only A1 is a liar – which is a contradiction! Ergo, A3 is not a Larpal and the other possibility must be right: A3 is a Tarsal.

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Simple Selection Task ET47 Suppose I claim that the following rule is true. If a card has a vowel on one side, it has an even number on the other side. Which card or cards should you turn over in order to try to decide Whether the rule is true or false?

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NYS 1 Given the statements a b b c a which one of the following statements must also be true? c b c h a none of the above

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NYS 2 Which one of the following statements is logically equivalent to the following statement: “If you are not part of the solution, then you are part of the problem.” If you are part of the solution, then you are not part of the problem. If you are not part of the problem, then you are part of the solution. If you are part of the problem, then you are not part of the solution. If you are not part of the problem, then you are not part of the solution.

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“NYS 3” Given the statements c c a a b b d (d e) which one of the following statements must also be true? c e h a all of the above

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J-L 1 Suppose that the following premise is true: If there is a king in the hand, then there is an ace in the hand, or else if there isn’t a king in the hand, then there is an ace. What can you infer from this premise? There is an ace in the hand. NO! In fact, what you can infer is that there isn’t an ace in the hand!

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The Dreadsbury Mansion Mystery Someone who lives in Dreadsbury Mansion killed Aunt Agatha. Agatha, the butler, and Charles live in Dreadsbury Mansion, and are the only people who live therein. A killer always hates his victim, and is never richer than his victim. Charles hates no one that Aunt Agatha hates. Agatha hates everyone except the butler. The butler hates everyone not richer than Aunt Agatha. The butler hates everyone Agatha hates. No one hates everyone. Agatha is not the butler. Now, given the above clues, there is a bit of disagreement between three (competent?) Norwegian detectives. Inspector Bjorn is sure that Charles didn’t do it. Is he right? Inspector Reidar is sure that it was a suicide. Is he right? Inspector Olaf is sure that the butler, despite conventional wisdom, is innocent. Is he right? Email your answer with supporting proof…

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Do my stuents at RPI really learn logic through paradoxes, and what kind of system would they use?

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Barber Paradox in Hyperproof In a certain village in England, there was a barber who claimed to shave all and only those men who did not shave themselves. Is this possible? What if the barber is a man? Let’s use –dodecs to represent men –tets to represent women –and Likes(x,y) to represent Shaves(x,y) –b to represent the barber Now let’s look at a model of our village…

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Last Brain Teaser Is the following assertion true or false? Prove that you are correct. There exists something which is such that if it’s a bird, then everything is a bird. Email your answer with supporting proof w/i -- for a free dinner at River Street Cafe

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