The Logic Puzzles Of Raymond Smullyan; Updated With Solution

Update: Scroll to the bottom of this post to see the solution to Smullyan’s logic puzzle discussed below.

Raymond Smullyan has written many books. What is the Name of This Book?, published in 1978, is a collection of logic puzzles and paradoxes that culminate in a development of Gödel‘s incompleteness theorem. The first page of Chapter 1 contains a story which must certainly have played a role in Smullyan becoming a logician and philosopher. I quote from his book:

My introduction to logic was at the age of six. It happened this way: On April 1, 1925, I was sick in bed with grippe, or flu, or something. In the morning my brother Emile (ten years my senior) came into my bedroom and said: “Well, Raymond, today is April Fool’s Day, and I will fool you as you have never been fooled before!” I waited all day long for him to fool me, but he didn’t. Late that night, my mother asked me, “Why don’t you go to sleep?” I replied, “I’m waiting for Emile to fool me.” My mother turned to Emile and said, “Emile, will you please fool the child!” Emile then turned to me, and the following dialogue ensued:

Emile: So, you expected me to fool you, didn’t you?
Raymond: Yes.
Emile: But I didn’t, did I?
Raymond: No.
Emile: But you expected me to, didn’t you?
Raymond: Yes.
Emile: So I fooled you, didn’t I!

Well, I recall laying in bed long after the lights were turned out wondering whether or not I had really been fooled. On the one hand, if I wasn’t fooled, then I did not get what I expected, hence I was fooled. (This was Emile’s argument.) But with equal reason it can be said that if I was fooled, then I did get what I expected, so then, in what sense was I fooled? So was I fooled, or wasn’t I?

Smullyan goes on to say that this puzzle is one of the themes of his book.

Has anyone ever been fooled this well, this inspirationally, on April Fool’s Day? My children have pulled a few pranks on me over the years, and my brother is legendary for having fooled a number of people (including my wife a couple of times) in classic fashion (particularly by telephone; his gift for voice impressions is a great aid), but Smullyan’s story is the best one I’ve ever heard of.

One of the many memorable puzzles in Smullyan’s book is the following one: A man is looking at a painting. Referring to the person in the painting, he says:

Brothers and sisters have I none,
But this man’s father is my father’s son.

Smullyan states that most people get the puzzle wrong, yet are convinced that they are right, and I experience the same thing. You can find the solution all over the internet, but I’ll give Smullyan’s elegant and clear solution tomorrow in an update at the end of this post.

Gödel’s incompleteness theorem demolished Hilbert‘s program of axiomatizing all of mathematics. Bertrand Russell and his mentor Alfred North Whitehead had apparently displeased the math gods when they chose the pretentious title Principia Mathematica (echoing Newton‘s great work) for their attempt at axiomatizing all of mathematics, and Gödel’s work was their “punishment!”

Smullyan’s book is written for lay people, and is thoroughly delightful.

Update: Here is Smullyan’s solution to this logic puzzle, discussed earlier; I have paraphrased Smullyan’s solution rather than give it verbatim.

First, Smullyan notes that most people think that the man is looking at a portrait of himself. This is because of the phrase “my father’s son;” people focus their attention on this part of the statement, and correctly determine that “my father’s son” must be the person looking at the portrait, because the person looking at the portrait has no brothers or sisters. However, this is only a fragment of the complete statement, and it’s incorrect to conclude that the man is looking at a portrait of himself.

A good problem-solving strategy is to break the problem down into parts, understand each part, and then put the parts together. The missing piece of this strategy, for most people, is that they have not put the parts together.

One way of implementing this strategy is to place parenthesis around part of the statement, to isolate one part of the problem, as follows:

Brothers and sisters have I none,
But this man’s father is (my father’s son).

Now almost everyone can determine, as we explained above, that (my father’s son) is the man who is looking at the portrait. Let’s adopt the perspective of the man looking at the portrait; then, to him, the phrase (my father’s son) can be replaced by (me). Let’s make this replacement:

Brothers and sisters have I none,
But this man’s father is (me).

Now, having made this simplification, continue with the reasoning. Another way of saying that “this man’s father is me” is “this is my son.” The man is looking at a portrait of his son. This is the end of the paraphrase of Smullyan’s solution.

If this doesn’t seem natural to you (to each his own, so having alternative strategies is helpful, both to teachers and students), then another strategy is to draw a diagram. For example, we might draw a fragment of a family tree, just a single branch, where parents are above their children. For example, if M is the mother of C, then the diagram would look like:

M
C

OK, let X represent the person in the portrait, let M represent the man looking at the portrait, and let F represent the father of M. Then as we start reading the couplet, we would almost certainly pause at the parenthesis in the second line, saying, “Hold on a second, let me figure out this part first. So, attending to the phrase in parentheses, we would draw:

F
M

That is, we would realize that “my father’s son” represents “me,” that is, the man looking at the portrait. Then, we would return to the first part of the sentence to complete the diagram as follows:

F
M
X

Then we could check the diagram by repeating the couplet again, walking ourselves up and down the diagram.

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About Santo D'Agostino

I have taught mathematics and physics since the mid 1980s. I have also been a textbook writer/editor since then. Currently I am working independently on a number of writing and education projects while teaching physics at my local university. I love math and physics, and love teaching and writing about them. My blog also discusses education, science, environment, etc. https://qedinsight.wordpress.com Further resources, and online tutoring, can be found at my other site http://www.qedinfinity.com
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