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Tuesday, October 4, 2011

Just Some Classic Hofstadter...

From chapter 17 of "The Mind's I"(1981) by Douglas Hofstadter and Daniel Dennett:

"One variant is: 'Thiss sentence contains threee errors.' On reading it, one's first reaction is, 'No, no - it contains two errors. Whoever wrote the sentence can't count.' At this point, some readers simply walk away scratching their heads and wondering why anyone would write such a pointless, false remark. Other readers make a connection between the sentence's apparent falsity and its message. They think to themselves, 'Oh, it made a third error after all - namely, in counting its own errors.' A second or two later, these readers do a double-take, when they realize that if you look at it that way, it seems to have correctly counted its errors, and is thus not false, hence contains only two errors, and... 'But...wait a minute. Hey! Hmm...' The mind flips back and forth a few times and savors the bizarre sensation of a sentence undermining itself by means of an interlevel contradiction, possibly on the purpose or interest of the idea, possibly on the cause or resolution of the paradox, possibly simply to another topic entirely."
And in some further self-referential fun Tanya Khovanova recently offered these two sets of sentences that come from David Bernstein (where a sentence and its 'negation' are either both true or both false):

This sentence contains five words.
It is not true that this sentence contains five words.

This sentence contains ten words.
It is not true that this sentence contains ten words.

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