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A brief meditation on formal systems and lying goblins

by mprast on 3/6/25, 10:00 PM with 29 comments

  • by schoen on 3/7/25, 1:58 AM

    An oddity here (that I think Smullyan is often careful about when introducing his knight and knave puzzles) is that the goblins in the story appeared to agree with each other about the surrounding context (that there is one liar and one truthteller, that there is one door that should be taken, etc.). They didn't contradict each other about that!

    Smullyan's liars normally lie about everything in every statement, so an official Smullyan liar would not agree that there is one liar and one truthteller, that there is one safe door and one unsafe door, and so on.

    I just watched the original scene, and the two goblins seem to agree with each other about all of that stuff! How confusing.

  • by ziofill on 3/6/25, 11:39 PM

    There is another version of this puzzle with a third goblin who flips a coin and depending on the outcome he will lie or tell the truth. The player is allowed 3 yes/no questions and the objective is to assign the three identities unambiguously.

    I can post the solution in 24h. Have fun! ^^

  • by photonthug on 3/6/25, 11:25 PM

  • by thaumasiotes on 3/6/25, 10:37 PM

    I see this problem differently, as mainly exhibiting a problem that cannot be analyzed without self-reference. Self-reference causes problems.

    The more obvious solution that the postscript hints at is the question "what would you say if I asked you whether [this door] leads to the castle?". Here, we immediately cancel out the influence of whatever goblin we're speaking to, and we get the right answer, whereas in the movie's solution we immediately incorporate the influence of the lying goblin and get the wrong answer.

    But the movie's solution is better for the movie, because "what would you say if I asked you X?" is a normal English way to ask "X", and in this case, where the two questions are different, the audience is guaranteed to be confused.

  • by Jtsummers on 3/7/25, 5:02 PM

    In some of the Smullyan books he extends the knights and knaves puzzles to incorporate beliefs with sane and insane variants. This is common in his Transylvania puzzles, where vampires always lie, humans always tell the truth, the sane believe true things, and the insane believe false things.

    The sane human and insane vampire always tell the truth, even though it's not the vampire's intent to tell the truth. Meanwhile, the insane human always makes false statements though their intent is to tell the truth (and they do, they tell you what they believe to be true).

  • by evil_genius on 3/7/25, 6:42 AM

    If you find logic puzzles interesting, take a look at "Games for Your Mind: The History and Future of Logic Puzzles" by Jason Rosenhouse. There's a whole chapter on Smullyan and his Knights and Knaves problems and is a generally good guide for getting into formal logic.

    The part I enjoyed the most in the book was "The Empuzzlement of Gödel's Theorems" that uses a twist with Knights and Knaves.

    https://www.goodreads.com/book/show/53232141-games-for-your-...

  • by KSteffensen on 3/7/25, 9:51 AM

    My preferred solution to this logic problem:

    https://www.giantitp.com/comics/oots0327.html