Quine's paradox

Quine's paradox is a paradox concerning truth values, attributed to W.V.O. Quine. It is related to the liar paradox as a problem, and it purports to show that a sentence can be paradoxical even if it is not self-referring and does not use demonstratives or indexicals (i.e. it does not explicitly refer to itself). The paradox can be expressed as follows:

:“Yields falsehood when preceded by its quotation” yields falsehood when preceded by its quotation.

If the paradox is not clear, consider each part of the above description of the paradox incrementally:

:it = "yields falsehood when preceded by its quotation":its quotation = "“yields falsehood when preceded by its quotation”":it preceded by its quotation = "“yields falsehood when preceded by its quotation” yields falsehood when preceded by its quotation."

With these tools, we may now reconsider the description of the paradox. It can be seen to assert the following::The statement “"‘yields falsehood when preceded by its quotation’" yields falsehood when preceded by its quotation” is false.

In other words, the sentence implies that it is false, which is paradoxical - for if it is false, what it states is in fact true. Note, however, the type error: what is "quoted" is a string of words, whereas what "yields falsehood" is a proposition.

Motivation

The liar paradox ("This sentence is false", or "The next sentence is true. The previous sentence is false") demonstrates essential difficulties in assigning a truth value even to simple sentences. Many philosophers, attempting to explain the liar paradox, concluded that the problem was with the word "this". Once we properly understand this sort of self-reference, they claimed, the paradox no longer arises.

Quine's construction demonstrates that paradox of this kind arises independently of such direct self-reference. In fact, there is no way to eliminate the paradoxes short of a severe crippling of the language. Any system, such as English, that contains entities such as words or sentences that can be used to apply to themselves, must contain this type of paradox.

Application

In , author Douglas Hofstadter suggests that the Quine sentence in fact uses an indirect type of self-reference. He then shows that indirect self-reference is crucial in the proofs of Gödel's incompleteness theorems.

See also

* Grelling paradox
* Russell paradox

Bibliography

* Hofstadter, Douglas. (1979) "" New York: Basic Books.

* Quine, W. V. O. (1962) "The Ways of Paradox" reprinted in Quine (1966) "The Ways of Paradox and Other Essays" Cambridge: Harvard Univ. Press. pp. 1-21.