You wish to marry the daughter of the king of the Island of Knights, Knaves, and Normals.

logic/smullyan/marry.non.normal New

New entry, edited from material collected for the archive and not yet in a released version.

Problem

You wish to marry the daughter of the king of the Island of Knights,
Knaves, and Normals.  He will only allow his daughter to marry someone
who is *not* a normal.  How many sentences must you utter to convince
the king that you are not a normal?

Knights can only tell the truth, knaves can only lie, normals can do
either.

[ Editor: this is a puzzle from R. Smullyan's _What Is the Name of This
Book?_, as paraphrased from memory by Jonathan Haas.  The puzzle before
it has a king who will only allow his daughter to marry a normal;
Smullyan's answer there is one sentence: "I am a knave." ]

Show solution

Smullyan's answer is "Can't be done. Since a normal can say *anything*,
anything you can say could also have been said by a normal. Better find
a different king's daughter to marry."

I disagree. The flaw in his answer is the statement, "A normal can say
*anything*." That's not true; a normal can only lie or tell the truth.
This is, after all, a *logical* island. Presumably, neither a knight,
nor a knave, nor a normal, could utter the Liar Paradox "This sentence
is false."

And therein lies the key to convincing the king that you are not a
normal.  I came up with the following three ways to do it, which have a
nice symmetry, IMHO:

If you are a knight, to affirmatively convince the king of that:

    "If I am normal, this sentence is false."

If you are a knave, to affirmatively convince the king of that:

    "I am normal, and this sentence is false."

If you aren't a normal, to affirmatively convince the king of that
*without* letting on whether you're a knight or a knave:

    "I am normal if and only if this sentence is false."

I wrote to Smullyan about it, never got a reply.

-- Jonathan Haas <positron@quip.eecs.umich.edu>