# Smullyan Logic Puzzle about Nelson Goodman Principle - Knights and Knaves

In Smullyan's "Logical Labyrinths" there is the following problem 1.11:

(INTRODUCING THE NELSON GOODMAN PRINCIPLE).
Suppose that you visit the Island of Knights and Knaves because you have
heard a rumor that there is gold buried there. You meet a native and you
wish to find out from him whether there really is gold there, but you
don't know whether he is a knight or a knave. You are allowed to ask him
only one question answerable by 'yes' or 'no'.

What question would you ask? (The answer involves an important principle
discovered by the philosopher Nelson Goodman.)


I'm somewhat stuck on the problem. I looked into Goodman's work and his work on the "Problem of Induction" which seems to be somewhat related to this puzzle while about induction and not deduction.

The premises are, that knights always tell the truth and knaves always lie.

Is the "Problem of Induction" the right place to look for a solution? Thank you.

## 2 Answers

No, Goodman's work on induction (though interesting) isn't relevant here.

It turns out that in 1931 Goodman published a knights-and-knaves sort of puzzle in the Boston Globe newspaper, and I think Smullyan's referring to that. [EDITED to add:] Or maybe Smullyan may have in mind a later trick Goodman came up with, published in his 1972 Problems and Projects, which has a technical advantage over other basically-equivalent versions of the trick.

The principle he has in mind is elementary (in the sense that it doesn't require any mathematical or philosophical fanciness; I don't claim it's easy to think of if you haven't seen something like it before) and is a very common tool in solving this sort of puzzle. I think it's only Smullyan who attaches Goodman's name to it.

As to what the principle is, you'll find out when you look a bit later in the book at Smullyan's answer to the question.

If you want a nice clear explicit statement of the principle, a Google search for <<smullyan nelson goodman>> turns up (at least for me) somewhere around result #5 a book with Smullyan's name in the title, edited by Fitting and Rayman, on pages 40-41 of which the principle is stated clearly. I am not giving the actual title of the book because it's a little spoilery for the knights-and-knaves problem you're trying to solve.

• When in a post about Smullyan someone does not give the title of a book, the obvious question is of course: What is the Name of this Book? Feb 24 '20 at 17:50
• @gareth-mccaughan Thank you for guiding me in the right direction! I was stuck(after actually making good progress through the first puzzles). I looked up the title by Fitting and Rayman :-) and will now try to solve it with that in mind. Feb 24 '20 at 19:03

After some trial and error, I think I got it now. [Edit: @LannyStrack provides below the "classic" answer to this puzzle which is more elegant, but I also like my solution and the symmetry in the question :-)]

Hint:

The title of the book which @gareth-mccaughan mentions above is 'Raymond Smullyan on Self Reference'.

Solution:

What I want to know from the person which gets asked - knight or knave - is, whether there is gold buried on the islands. I can get this info with the question: Is there gold and you are not a knave or isn't there gold and you are knave? For the case where there is gold: A knight would answer with yes, because the part to left of or (there is gold and your not a knave) would be true, so the whole sentence would be true. A knave would also answer with yes, because the sentence would evaluate to false for a knave, so a knave would lie and tell that it would be the case. For the case where there wouldn't be any gold: A knight would answer with no, because the sentence would evaluate to false. A knave would also answer with 'no', because for him the sentence would evaluate to true, so he would lie and say no. With that, if the person says yes I know there's gold, if the person says no I there isn't.

• I thought that the "classic" answer to this question was one using the following format: rot13("Vs V jrer gb nfx lbh vs gurer vf tbyq ba guvf vfynaq, jbhyq lbh fnl, 'lrf'?"). If you ask the question in this way, then rot13(vs gurer vf tbyq, n Xavtug naq n Xanir jbhyq obgu nafjre 'lrf', naq vs gurer vf ab tbyq, obgu jbhyq nafjre 'ab'.) Mar 6 '20 at 15:14
• @LannyStrack the "classic" answer is way more elegant. :-) Mar 7 '20 at 9:10