Outer Limits of Reason (5 page)

Who shaves the barber?

He is a villager and so if he does not shave himself, must go to the barber. But he
is
the barber and so he shaves himself. If he does shave himself, then, since he is the barber, he goes to the barber and does not shave himself.
⁴

We might envision the barber paradox with
figure 2.1
. We split the set of villagers into two parts and look to see if the barber is on the right or left.

In contrast to the liar paradox, the barber paradox has a simple solution: the village described simply does not exist. It cannot exist because there is a contradiction inherent in its description. Our description entails a contradiction with the barber. Since the real world cannot have contradictions, the village does not really exist. There are many other villages in the Austrian Alps, but they have different setups. They might have two barbers that shave each other; they might have a female barber that does not shave; they might have long-haired hippie types who do not go to any barber regardless of need. These descriptions of other villages are totally legitimate; no contradictions result from them. But the village Russell described cannot exist.

Another clever paradox deals with adjectives in English and is called the
heterological paradox
or
Grelling's paradox
. Consider the word
English
.
English
is an English word. In contrast,
French
is not a French word (it is an English word). Let us look at some other adjectives and see how they relate to themselves:
polysyllabic
is polysyllabic.
Monosyllabic
is not monosyllabic.
Pentasyllabic
(made of five syllables) is pentasyllabic.
Misspelled
is not misspelled.
Adjectival
is adjectival.
Female
is not female.
Awkwardnessfull
is awkwardnessfull.
Unpronounceable
is not unpronounceable. In effect, we have two groups of adjectives: those that describe themselves and those that do not. All adjectives that describe themselves are called
autological
(from the Greek
auto
meaning “self” or “one's own” and
logos
meaning “word,” “speech,” or “reason”) or
homological
. In contrast, all adjectives that do not describe themselves are called
heterological
(from the Greek
heteros
meaning “other” or “different”). So we have that
English
,
polysyllabic
,
adjectival
, and so on are all autological. In contrast,
French
,
monosyllabic
,
unpronounceable
, and so forth are all heterological. With these two categories set up, we now pose the following question:

Is
heterological
heterological?

Let us say that
heterological
is heterological. Just as

English
is English ⇒
English
is autological,

so too

heterological
is heterological ⇒
heterological
is autological

and hence
heterological
is not heterological. In contrast, if we take the opposite view and say that
heterological
is not heterological, then just as we saw that

French
is not French ⇒
French
is heterological,

so too

heterological
is not heterological ⇒
heterological
is heterological.

We have come to the conclusion that
heterological
is heterological if and only if it is not heterological. Buzz! This is a contradiction and troublesome.

We can again envision this self-referential paradox as
figure 2.2
.

This paradox also seems to have a simple solution: there is no word
heterological
, or if the word does exist, it has no meaning. We saw that if one defines
heterological
, then we come to a contradiction. This is similar to saying that the village in the barber paradox does not exist.

However, we cannot simply solve all problems by waving our hand and declaring that the word
heterological
does not exist or has no meaning. The problem is too deeply rooted in the very nature of language. Rather than dealing with the word
heterological
, consider the related adjective phrase “not true of itself.” Simply ask if the phrase “not true of itself” is true of itself. It is true if and only if it is not true. Are we simply to posit that “not true of itself” is not a legitimate adjectival phrase? There are no problems with any of the words in the phrase. There is nothing about the phrase that is weird like the word
heterological
. Nevertheless, we come to a contradiction if we use it.

The
reference-book paradox
is very similar to the heterological paradox. A reference book is a book that lists books in different categories. There are many reference books that list books of many different types. There are reference books that list antique books, anthropology books, books about Norwegian fauna, and so on. Certain reference books list themselves. For example, if one were to publish a reference book of all books published, that reference book would contain itself. There are also certain reference books that would not list themselves. For example, a reference book on Norwegian fauna would not list itself. Consider the reference book that lists all reference books that do not list themselves. Now ask yourself the following simple question: Does this book list itself? With a little thought, it is easy to see that this book lists itself if and only if it does not list itself. We conclude that no such reference book with such a rule for its content can exist. (I leave to the reader the task of drawing a diagram similar to figures
2.1
and
2.2
for this paradox.)

Bertrand Russell used the barber paradox to explain a more serious paradox called
Russell's paradox
. This is more abstract than the other self-referential paradoxes we saw and is worth pondering. Consider different sets or collections of objects. Some sets just contain elements and some sets contain other sets. For example, one can look at a school as a set containing different grades, where each grade is the set of students in the grade. Some sets even contain copies of themselves. The set of all sets described in this book contains itself. The set of all sets with more than five elements contains itself. There are, of course, many sets that do not contain themselves. For instance, consider the set of all red apples. This does not contain itself since a red apple is not a set. Russell would like us to consider the set R of all sets that do not contain themselves. Now pose the following question:

Does R contain itself?

If R does contain itself, then, by definition of what belongs to R, it is not contained in R. If, on the other hand, R does not contain itself, then it satisfies the requirement of belonging to R and is contained in R. We have a contradiction. This can be visualized in
figure 2.3
.

This paradox is usually “solved” by positing that the collection R does not exist—that is, that the collection of all sets that do not contain themselves is not a legitimate set. And if you do deal with this illegitimate collection, you are going beyond the bounds of reason. Why should one not deal with this collection R? It has a perfectly good description of what its members are. It certainly looks like a legitimate collection. Nevertheless, we must restrict ourselves in order to steer clear of contradictions. The obvious (and seemingly reasonable) notion that for every clearly stated description there is a collection of those things that satisfy that description is no longer obvious (or reasonable). For the clearly stated description of “red things,” there is a nice collection of all red things. However, for the seemingly clear description of “all sets that do not contain themselves,” there is no collection with this property. We must adjust our conception of what is obvious.
⁵

Russell's paradox should be contrasted with the other paradoxes. There are simple solutions to the barber paradox and the reference-book paradox: those physical objects simply do not exist. And there is a simple solution to the heterological paradox: human language is full of contradictions and meaningless words. We are, however, up against a wall with Russell's paradox. It is hard to say that the set R simply does not exist. Why not? It is a well-defined idea. A collection is not a physical object, nor is it a human-made object. It is simply an idea. And yet this seemingly innocuous idea takes us out of the bounds of reason.

The liar paradox was summarized by one sentence:

This sentence is false.

It can also be summarized by the following description:

The sentence that denies itself.

Similarly, the other four self-referential paradoxes can be summarized by the following four descriptions:

• “The villager who shaves everyone who does not shave themselves.”

• “The word that describes all words that do not describe themselves.”

• “The reference book that lists all books that do not list themselves.”

• “The set that contains all sets that do not contain themselves.”

As you can see, all these descriptions have the exact same structure (as do figures
2.1
through
2.3
). Every time there is self-reference, there are possibilities for contradictions. Such contradictions will have to be avoided and will require a limitation. We explore such limitations throughout the book.

Before moving on to the next section, there is an interesting result that demands further thought. One might think that every language paradox has some form of self-reference. That is, there must be some chain of reasoning that is circular and returns to where it started. This was the common belief until Stephen Yablo came up with a clever paradox called
Yablo's paradox
. Consider the following infinite sequence of sentences:

K
₁
K
_i
is false for all
i
>
1

K
₂
K
_i
is false for all
i
>
2

K
₃
K
_i
is false for all
i
>
3