Quine's paradox
Quine's paradox is a paradox concerning truth values, attributed to W. V. O. Quine. It is a related problem to the liar paradox and purports to show that a sentence can be paradoxical even if it is not self-referring and does not use demonstratives or indexicals. The paradox can be expressed as follows:
- “yields falsehood when preceded by its quotation” yields falsehood when preceded by its quotation.
To understand the paradox, take each step that the sentence implies in turn:
- 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.
We now have returned to the original case. So this sentence asserts:
- “The sentence ““yields falsehood when preceded by its quotation” yields falsehood when preceded by its quotation.” is false.”
In other words, the sentence says that it is false, which is paradoxical.
Motivation
The liar paradox, "This 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". Forbidding this sort of self-reference, they decided, would be enough to solve the problem.
Quine's construction demonstrates that eliminating such direct self-reference is insufficient to resolve the paradox, and that the problem is intrinsic to the notion of sentences that discuss truth and falsity. 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 describe themselves, must contain this type of paradox.
Application
In Gödel, Escher, Bach: an Eternal Golden Braid, author Douglas Hofstadter uses the Quine sentence to demonstrate an indirect type of self-reference which he then shows to be a crucial component in the proof of Gödel's incompleteness theorems.