Posted by: George | April 20, 2011

A Logic Paradox

I don’t know if Chris and captainfalcon remember this, but there was once a time when I was trying to recall a particular logical paradox that deals with self-referencing statements. We got into this discussion because of the book Godel, Escher and Bach. I ran across a similar paradox recently. Here it is.

If A is the set which contains all sets that do not contain themselves, then does A contain A?

Neither “yes” nor “no” work as answers.



  1. Bertrand Russell’s work; I believe it’s actually called Russell’s Paradox for those who want to look up it’s history.

    While on the topic of logical paradoxes, I’ll trot out my favorite, the Unexpected Hanging Paradox.

    “A judge tells a condemned prisoner that he will be hanged at noon on one weekday in the following week but that the execution will be a surprise to the prisoner. He will not know the day of the hanging until the executioner knocks on his cell door at noon that day.

    Having reflected on his sentence, the prisoner draws the conclusion that he will escape from the hanging. His reasoning is in several parts. He begins by concluding that the “surprise hanging” can’t be on Friday, as if he hasn’t been hanged by Thursday, there is only one day left – and so it won’t be a surprise if he’s hanged on Friday. Since the judge’s sentence stipulated that the hanging would be a surprise to him, he concludes it cannot occur on Friday.

    He then reasons that the surprise hanging cannot be on Thursday either, because Friday has already been eliminated and if he hasn’t been hanged by Wednesday night, the hanging must occur on Thursday, making a Thursday hanging not a surprise either. By similar reasoning he concludes that the hanging can also not occur on Wednesday, Tuesday or Monday. Joyfully he retires to his cell confident that the hanging will not occur at all.

    The next week, the executioner knocks on the prisoner’s door at noon on Wednesday — which, despite all the above, was an utter surprise to him. Everything the judge said came true.”

    Incidentally, “The Surprise Examination or Unexpected Hanging Paradox” by Timothy Y. Chow is a pretty good coverage for those with a mild interest in logic and mathematics and who have worked through G.E.B. but are not experts; a google search should turn up a .pdf to peruse if you’re curious.

  2. Russell’s Paradox is supposed to pwn Gottlieb Frege, but, if I ever knew how, I’ve forgotten.

  3. Apparently it destroys Frege’s Basic Law V (whose moniker gives it an ominous although superficial similarity to that most nefarious of all axioms, S5). (section 2.4.1 is the relevant portion)

  4. NPN(I will never read that)
    PN(I will never read that) [Axiom M]
    N(I will never read that) [Axiom S5]
    I will never read that [Axiom M]

  5. Hmm. The rest of the world calls “M” “T.” But it ought to be M. Therefore, by axiom T as applied to deontic logic, it is M.

  6. I believe the specific example which we settled on concerned autological vs. heterological* words (words that do and do not describe themselves, respectively):

    Short or staccato are autological while say long is heterological.

    The question was, if I recall correctly: “Is “heterological” itself heterological?” to which there is no definitive answer.

    *I am pretty we used a word that began with a “c” about which the internet seems strangely ignorant. Auto/heterological work equally well, especially since part of me thinks we just invented the word at the time (it was enroute to CJ’s, everyone was hungry and the navigator was taking an entirely planned detour through the less scenic locales in Falls Church).

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