Outer Limits of Reason (4 page)

• Woody Allen cheated on his metaphysics exam by looking into the soul of the boy sitting next to him.

• Steven Wright said he would kill for a Nobel Peace Prize.

• Groucho Marx didn't care to belong to any club that would have him as a member.

In all of these jokes, normal ideas are taken too far. Cheating on an exam, desiring a Nobel Peace Prize, or resigning your club membership in disgust are all common ideas. However, these great thinkers have taken these usual concepts where they do not belong: to the silly and ridiculous.

Even puns fall into this category. A pun is a joke where the meaning of a word or phrase is taken into an area where it was not intended:

• “Have you heard about the guy whose whole left side was cut off? He's all right now.”

• “I'm reading a book about antigravity. It's impossible to put down.”

• “Did you hear about the par-a-dox? . . . Doctor Shapiro and Doctor Miller.”

Groan! (Sorry. The only thing worse than a pun is an analysis of a pun. Let us move on.)

 

I close this introduction with a few questions about the nature of reason and its limitations. Read the book with these questions in mind. I return to these issues in the last chapter and perhaps get closer to the answers using some of the ideas presented in the book.

I would be remiss in writing a book titled
The Outer Limits of Reason
without giving a definition of
reason
. After all, how can we say something is beyond the limits of reason if we do not define reason? What is a reasonable process to determine facts? Are there different levels of reason? How do we draw the line between alchemy and chemistry? Between astrology and astronomy? Why are some actions deemed reasonable and others not? Why does it make sense to check your blood pressure while it is ludicrous to check your horoscope? What thought processes are reasonable and will avoid contradictions?

The
Oxford English Dictionary
gives sixteen classes of definitions for the word
reason
. The definition closest to the one we want is the following: “The power of the mind to think and form valid judgments by a process of logic; the mental faculty which is used in adapting thought or action to some end; the guiding principle of the mind in the process of thinking. Freq. contrasted with
will, imagination, passion,
etc. Often personified.” But this definition just raises more questions. What is a “valid judgment”? When is something a logical process as opposed to an illogical process? When is thinking part of the will and when is it reason? This definition is unsatisfying. Other purported definitions are not much better.

There is something self-referential in our entire enterprise. We are using reason to find limitations of reason. If reason is limited, how are we to use reason to discover those limitations? What are the limits to our limit-showing abilities?

Let's hold these questions in abeyance and return to them in
chapter 10
, when we conclude our explorations of the limits of reason.

Further Reading

Other books that discuss limitations of reason are Barrow 1999, Dewdney 2004, and Poundstone 1989. Sorensen 2003 is a wonderful history of paradoxes.

2

Language Paradoxes

What we cannot speak about we must pass over in silence.
¹

—Ludwig Wittgenstein (1889–1951), Proposition 7 of
Tractatus Logico-Philosophicus

After all, Mr. Wittgenstein manages to say a good deal about what cannot be said.

—Bertrand Russell (1872–1970), introduction to Wittgenstein's
Tractatus Logico-Philosophicus

Half the lies they tell about me aren't true.

—Yogi Berra
²

Rather than jumping headfirst into the limitations of reason, let us start by just getting our toes wet and examining the limitations of language. Language is a tool used to describe the world in which we live. However, don't confuse the map with the territory! There is one major difference between the world we live in and language: whereas the real world is free of contradictions, the man-made linguistic descriptions of that world can have contradictions.

In
section 2.1
, we encounter the famous liar paradox and its many variants. These are relatively easy puzzles that will get us started.
Section 2.2
contains a collection of self-referential paradoxes. I show that they all have the same form. In
section 2.3
we meet several paradoxes involving descriptions of numbers.

2.1  Liar! Liar!

A linguistic paradox is a phrase or sentence that contradicts itself. A baby version of a linguistic paradox is an oxymoron (from the Greek
oxys
“sharp” and
moros
“stupid”—together they mean “pointedly foolish” or “pointedly dull”). These are phrases, usually consisting of two words, that contradict each other. Some examples are “original copies,” “open secret,” “clearly confused,” “militant pacifist,” “larger half,” “alone together,” and my favorite, “act naturally.” Even though these phrases do not really make sense, we human beings have no problem using them in common everyday speech.

The classic example of a linguistic paradox is the famous
Epimenides paradox
. This dates back more than two and a half millennia to when Epimenides (600 BC), a philosopher and poet who lived in Crete, complained about his neighbors in a poem called
Cretica
. He wrote: “The Cretans, always liars, evil beasts, idle bellies!” This seems
³
paradoxical. If this statement is true, then since Epimenides is a Cretan, he is including himself as a liar and this line of the poem is false. In contrast, if it is false, then Epimenides is not a liar and the line is true.

There are many linguistic paradoxes similar to Epimenides' statement. The
liar paradox
is a simple sentence like

I am lying.

or

This sentence is false.

If these sentences are true, then they are false. Furthermore, if they are false, then they are true.

The liar paradox is found in many different forms. For example, we can denote a sentence
L
₁
and then say that
L
₁
asserts its own falsehood:

L
₁
: L
₁
is false.

Again, if
L
₁
is true, then it is false. And if
L
₁
is false, then it is true. Other variations of the liar paradox have sentences that are not directly self-referential. Consider the following two sentences:

L
₂
: L
₃
is false.

L
₃
: L
₂
is true.

If
L
₂
is true, then
L
₃
is false, which would mean that “
L
₂
is true” is false and hence
L
₂
is false. In contrast, if
L
₂
is false, then
L
₃
is true and
L
₃
asserts that
L
₂
is true. Buzz! That's a contradiction.

It is important to note that just because sentences refer to themselves and their falsehoods does not mean there is a contradiction. Consider these two sentences:

L
₄
: L
₅
is false.

L
₅
: L
₄
is false.

Let's assume that
L
₄
is false. Then
L
₅
is true and
L
₄
is false. Similarly, if you start with the premise that
L
₄
is true, you get that
L
₅
is false, and hence
L
₄
is true. Neither assumption leads you to a contradiction.

There are many other forms of the liar paradox:

•
The only underlined sentence on this page is a total lie.

•
The boldface sentence on this page is a blatant falsehood.

• The sentence after the boldface sentence on this page is not true.

Are they true or false?

The liar paradox has been around for over 2,500 years and philosophers have devised many different ways of avoiding such contradictions. Some philosophers try to avoid these linguistic paradoxes by saying that the liar sentences are neither true nor false. After all, not every sentence is true or false. Questions such as “Your place or mine?” and commands such as “Go directly to jail!” are neither true nor false. One usually thinks of declarative sentences like “Snow is white” as either true or false, but the liar sentences show that there are some declarative sentences that are also neither true nor false.

There are those who say that the sentence “This sentence is false” is not even grammatically correct. After all, what does “This sentence” refer to? If it refers to something, we should be able to replace “This sentence” with whatever it refers to. Let's give it a try:

“This sentence is false” is false.

This is grammatically correct and it might be true or false. But it is not self-referential and not equivalent to the original liar sentence. This is similar to the sentence

“This sentence is false” has four words.

which is true, while

“This sentence is false” has five words.

is false. It would be nice to have a grammatically correct English sentence that is a self-referential paradox. W. V. O. Quine came up with a clever way around these problems. Consider the following
Quine's sentence
:

“Yields falsehood when preceded by its quotation”

yields falsehood when preceded by its quotation.

First notice that this is a legitimate English sentence. The subject is the phrase in quote marks and the verb is
yields
. Now, let us ask ourselves if it is true. If it is true, then when you attach the subject to the rest of the sentence, as we did, we get falsehood. So the sentence is false. In contrast, what if the sentence is false? That means that when you attach the subject to the sentence, you do not get a falsehood; rather, you get a true sentence. So if you assume that Quine's sentence is false, you derive that it is true. This is a grammatically correct English sentence that is self-contradictory.

Another potential solution to paradoxical sentences is to restrict language so as to avoid such sentences. Some have said that language should be stratified into different levels. They have declared that sentences cannot talk about other sentences of their own level or higher. For example, at the lowest level there will be sentences like “Grass is green” and “My pen is blue.” The next level will be sentences about sentences on the lowest level. So we might have

“Grass is green” is an obvious sentence.

or

“My pen is blue” has four words in it.

One goes on to higher-level phrases like

“‘My pen is blue' has four words in it” is a dumb fact.

By restricting the types of sentences, we will be avoiding sentences of the form

The sentence in italics on this page is grammatically correct.

This is a sentence dealing with itself and hence is a sentence on its own level. It is declared not kosher—that is, not a legitimate part of language. Every sentence is only permitted to talk about sentences that are “below” it. If a sentence does talk about a sentence that is on its own level, that sentence is proclaimed meaningless. This stratification will ensure that there are no self-references and hence no contradictions. With such restrictions in place, linguists are fairly certain that they have banned most paradoxical linguistic sentences. However, this solution is somewhat artificial. Common human language has always dealt with some type of self- reference without problem:

• Someone says, “Oh! I am groggy today and I do not know what I am talking about.” Is he aware of saying this sentence?

• Carly Simon sings a song with the lyrics “You're so vain, you probably think this song is about you.” But this song
is
about him!

• “Every rule has an exception except one rule: this one.”

• “Never say ‘never'!”

• “The only rule is that there is no rule.”

In all of these cases—and many more—human language is violating the restriction of only dealing with sentences that are “below” it. In each case, a sentence discusses itself. And yet, somehow, all these examples are a legitimate part of human language.

Another possible solution to paradoxical sentences was mentioned in
chapter 1
, namely, human language is a product of the human mind and, as such, subject to contradictions. Human language is not a perfect system that is free of discrepancies (in contrast to perfect systems like mathematics, science, logic, and the physical universe). Rather, we should simply accept the fact that human language is faulty and has contradictions. This seems reasonable to me.

2.2  Self-Referential Paradoxes

The cause of the problem with the liar paradox is that language can be used to describe language. In particular, one can have a sentence that discusses its own truthfulness. This ability of language to describe language is a form of self-reference. Paradoxes that arise from such self-reference are the subject of this section. While these paradoxes are not linguistic paradoxes per se, they are similar to the liar paradox and will help us understand the true nature of self-reference.

The British philosopher Bertrand Russell described a delightful little paradox that has come to be known as the
barber paradox.
Imagine a small isolated village in the Austrian Alps that has only one barber. Some villagers shave themselves and some go to the barber. Everyone in the village abides by the following rule: all those who do not shave themselves must go to the only barber and all those who do shave themselves do not go to the barber. This seems like a pretty innocuous rule. After all, if they can save some money by shaving themselves, why go to the barber? And if they go to the barber, why shave themselves? Now, simply ask yourself: