Question Can sentences refer to themselves? Take the "Liar" paradox: (1) This sentence is false. Does "This" really refer to the sentence I've labeled as (1)? Can sentences predicate properties to themselves in this way?

Accepted:
September 28, 2007

Comments

Your sentence (1) is often given as an example of a paradoxical sentence. The paradox arises from interpreting the phrase "this sentence" as referring to sentence (1) itself. If the sentence is true, then what it says is incorrect, so it is false. But if it is false, then what it says is correct, so it is true.

It is tempting to think that the problem is caused by the interpretation of the phrase "this sentence". I suspect that this is the motivation for your question. Can we avoid the problem by saying that the phrase "this sentence" can't be interpreted in this way? The answer is no: the paradox can be created without using this phrase.

To see how to do it, first note that it is possible for a sentence to talk about a linguistic expression. The usual way to accomplish this is to use quotation marks. For example, here is a sentence that talks about sentence (1):

(2) "This sentence is false" is a paradoxical sentence.

Of course, there's nothing paradoxical about sentence (2). Sentence (2) is true; it says that sentence (1) is paradoxical, and as we've seen, sentence (1) is paradoxical, so sentence (2) is true.

Here's another example of a sentence that uses quotation marks to talk about a certain linguistic expression:

(3) "is three words long" is three words long.

This sentence is also not paradoxical. It is false because the expression "is three words long" is actually four words long, but this doesn't lead to a paradox.

Now, sentence (3) has a very strange form. It consists of a certain phrase, "is three words long", preceded by the same phrase in quotation marks. So the following sentence is true:

(4) "is three words long" yields a falsehood when preceded by its quotation.

Sentence (4) refers to sentence (3) in a roundabout way, by describing the strange way it is constructed, and what it says about sentence (3) is that it is false. And since sentence (3) is false, what sentence (4) says is correct; in other words, sentence (4) is true.

Finally, consider this sentence:

(5) "yields a falsehood when preceded by its quotation" yields a falsehood when preceded by its quotation.

Just as sentence (4) refers, in a roundabout way, to sentence (3), sentence (5) refers, in a roundabout way, to ... sentence (5)! So sentence (5) refers to itself (without using the phrase "this sentence"), and what it says about itself is that it is false. Thus, sentence (5) is paradoxical, in the same way that sentence (1) is. Banning the use of the phrase "this sentence" will not enable you to escape paradox.

The idea behind the construction of sentence (5) is one of the main ideas in Godel's proof of his incompleteness theorem. Godel wasn't working with sentences in English; rather, he was working with sentences in the formal language of number theory. Sentence (5) was constructed by Quine, by taking the idea behind Godel's proof and applying it to English rather than the language of number theory.

Question Can sentences refer to themselves? Take the "Liar" paradox: (1) This sentence is false. Does "This" really refer to the sentence I've labeled as (1)? Can sentences predicate properties to themselves in this way? Accepted:September 28, 2007

Question Can sentences refer to themselves? Take the "Liar" paradox: (1) This sentence is false. Does "This" really refer to the sentence I've labeled as (1)? Can sentences predicate properties to themselves in this way?

Accepted:
September 28, 2007