Skip to main content

Subsection 1.2.5 Paradoxes

Some English declarative sentences don’t correspond to any logical statement (i.e., an expression with a truth value). The best examples of this are some of the classical paradoxes (self-contradictory claims).

Activity 1.2.16.

Consider the sentence, “This sentence is false.” Can we assign it a truth value?

  • Suppose we say that it is true. But that cannot be since it would then contradict itself.

  • So it must be false. But then it is telling the truth, which it cannot do since it is false.

The problem in the last example is that the sentence is self-referential. It describes itself.

Activity 1.2.17.

Self reference is also the root of the problem in the barber paradox: Imagine a (very) small town with exactly one male barber. And we’re told that the barber shaves all and only those men in town who do not shave themselves. Then consider the sentence, “The barber shaves himself.” Can we assign it a truth value?

  • Suppose we say that it is true. But that cannot be. We’ve been told that the barber only shaves the men who do not shave themselves.

  • So it must be false. But it cannot be false since, if the barber doesn’t shave himself, then we’re told that he must do exactly that.

Paradoxes such as these are interesting and have led to more sophisticated systems for reasoning, for example, with sets. But, fun as they are to think about, they’re rare. We’ll find that, for the most part, English declarative sentences correspond to statements with truth value.

A more serious problem for us will be that not all statements can be straightforwardly and usefully represented in the logical frameworks that we’re going to study. We’ll see why that is as we develop our formal systems.

Problems 1.2.18.

(a)

Consider the following two-part claim:

“The next statement is true. The previous statement is false.”

Which of these sentences is true about our claim:

  1. The only way to avoid a contradiction is for it to be true.

  2. The only way to avoid a contradiction is for it to be false.

  3. There is no way to avoid a contradiction.

Answer.
iii is correct
Solution.
Suppose that the first sentence is true. Then the second one must also be true. But, if it is, the first sentence is false. Okay, let’s see what happens if we assume that the first sentence is false. Then the second sentence must also be false. But if it is, the first sentence must be true. We get a contradiction either way. This claim is sometimes called the Card Paradox.

(b)

Try this one only if you like paradoxes. It’s called the Grelling–Nelson paradox. Assume that a short word is one with fewer than 8 letters. Anything else is long. We’ll say that:

  • An adjective is autological (“auto” means “same”) just in case it describes itself. For example, “short” is autological since it is a short word.

  • An adjective is heterological just in case it does not describe itself. For example, “long” is heterological since it’s not a long word.

Consider the sentence, “The word “heterological” is heterological.” Is it a statement? In other words, can we assign it a truth value?

  1. Yes, it is a statement.

  2. No, it isn’t a statement.

Answer.
ii is correct.
Solution.

We can’t assign this sentence a truth value:

  • Suppose that we say that it is true. In other words, as the sentence says, the word “heterological” is heterological.” So it does describe itself. But, by the definition of “heterological”, it cannot describe itself. Contradiction.

  • So it must be false. In that case, the word “heterological” is not heterological. That means that it is autological – it does describe itself. But for “heterological” to describe itself, it would have to actually be heterological. It isn’t (since the claim that it is so is false). Contradiction.