On widespread disagreement

In his 1978 book "What Is the Name of This Book?," Raymond M. Smullyan repeats a riddle from his childhood:

4. Whose Picture Am I Looking At?

This puzzle was extremely popular during my childhood, but today it seems less widely known.  The remarkable thing about this problem is that most people get the wrong answer but insist (despite all argument) that they are right.  I recall one occasion about 50 years ago when we had some company and had an argument about this problem which seemed to last hours, and in which those who had the right answer just could not convince the others that they were right.  The problem is this.

A man was looking at a portrait.  Someone asked him, "Whose picture are you looking at?"  He replied:  "Brothers and sisters have I none, but this man's father is my father's son." ("This man's father" means, of course, the father of the man in the picture.)

Whose picture was the man looking at?

Often, when I get in long arguments about politics or science or the internet or any number of other things that people strongly differ on, someone (usually someone butting in, who wasn't listening, but occasionally the person I'm arguing with) says, "There is no right answer, people have different opinions and that's that."

Now, these people are clearly wrong.  A system can be complicated, and it can be easy to come to incorrect conclusions when trying to understand that system.  People can over-or-under-emphasize the importance of certain details, or fail to imagine certain actors in the system complexly, or for any number of other reasons become firmly convinced that their conclusion is right, even if it's not.

Lately, it's been reminding me of this riddle.  I have a lot of trouble with this one.  I've known it for years, and I still have trouble holding the whole thing in my head firmly enough to produce the correct answer.  I might have even defended that wrong answer, when I first heard the riddle.

But the wrong answer is definitely wrong.  There can be no difference of opinion about it, only people getting it right or wrong.

In real life, problems are more complicated than that.  Some people may be more right or wrong than others, or disagree about how to act on the knowledge of the correct answer.  People may be bitterly divided over small or large issues.  But in real life, like in this riddle, even if the wrong answer is really persuasive, and has a lot of very vocal supporters, it's still wrong.

Here's the solution to the riddle:


From the book:

A remarkably large number of people arrive at the wrong answer that the man is looking at his own picture.  They put themselves in the place of the man looking at the picture, and reason as follows: "Since I have no brothers or sisters, then my father's son must be me.  Therefore I am looking at a picture of myself."

The first statement of this reasoning is absolutely correct; if I have neither brothers nor sisters, then my father's son is indeed myself.  But it doesn't follow that "myself" is the answer to the problem.  If the second clause of the problem had been, "this man is my father's son," then the answer to the problem would have been "myself."  But the problem didn't say that; it said "this man's father is my father's son."  From which it follows that this man's father is myself (since my father's son is myself).  Since this man's father is myself, then I am this man's father, hence this man must be my son.  Thus the correct answer to the problem is that the man is looking at a picture of his son.

If the skeptical reader is still not convinced (and I'm sure many of you are not!) it might help if you look at the matter a bit more graphically as follows:

(1) This man's father is my father's son.

Substituting the word "myself" for the more cumbersome phrase "my father's son" we get

(2) This man's father is myself.

Now are you convinced?