# LEC2HPS.10EEE - Lecture#2 Human Problem Solving(Cont of...

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1 Lecture #2 Human Problem Solving (Cont. of Lec. #1) III. Taxonomy of Conceptual Products D. Logical Paradoxes E. Practical Problems F. Games

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2 D. Logical Paradoxes A logical paradox is an assertion or line of reasoning contrary to common sense. Type 1 . An assertion that seems true (false) but is actually false (true). Type 2 . A line of reasoning that seems impeccable but leads to a contradiction of logic. The term paradox may not have immediate connections to logic, e.g. a paradoxical personality
3 A Type 1 Example Consider the following assertion; “You can not draw a ‘perfect map’ of England in a London flat, but you might be able to do it in a New York City pad.” The statement seems false ( you either can do it in both places or you can not do it in either place), but there is an interesting sense in which the statement is true. Most logical paradoxes set in real world settings as opposed to a formal logic setting require a person to interpret things in a certain way to ‘get the point’.

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Solution Hint Hint 1- Now just what might be a ‘perfect map’? Hint2- Remember my discussion of the infinite reflection of a person in two mirrors. Of course you could argue as a matter of practical truth that a ‘perfect map could not be drawn anywhere. True, but is there any interesting sense of the problem that would make New York different than London. 4
Solution Now the difference between being in England and new York is that you would have to depict yourself on the map drawn in London. And what would you be doing? Drawing a map, and you would have to be on that, doing what? etc., etc. 5

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6 Self Reference Paradoxes There is a long history of logical paradoxes that result from the case where a logical arguments ‘talks about (references) itself.’ A philosopher from Crete in 6 BC named Epimenides exclaimed 'All Cretans are liars!' . If Epimenides' statement is true, then he is a liar and hence his statement is false. A contradiction. If Epimenides' statement is false, then it would be possible to find a Cretan who sometimes tells the truth----- Of course it wouldn’t be Epimenides
7 The Liar’s paradox So far we do not have a logical paradox, but a small change results in the so-called liar’s paradox. In 4 BC Eubulides of Miletus stated “ I am lying” If true, then he is lying, so his statement is false. If false, he isn’t lying, so the statement is true. In the New Testament of the Bible, Saint Paul warned, “One of themselves, even a prophet of their own, said that the Cretans are always liars.” Later A French philosopher wrote on an otherwise blank page, “ All statements on this page are false.”

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8 More on the Liar’s paradox Suppose the last sentence in today’s NYT is: ‘The last sentence in tomorrow's edition of The New York Times newspaper is true.’ There seems to be nothing paradoxical about this except tomorrow when you get the paper you see as the last sentence: ‘The last sentence in yesterday's edition of The New York Times newspaper is false.’ This is a bit like the map paradox.
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• Fall '10
• WilliamH.BATCHELDER
• Logic, thumb tack, spare thumb tack, Equal things times, times equal things

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