Quine's paradox
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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 uses indirect self-reference.
- “Yields falsehood when preceded by its quotation” yields falsehood when preceded by its quotation.
What is the sentence talking about?
Let's take the step that the sentence implies.
- 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. This is a paradox: If it's true, it's false, and if it's false, it's true.
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". Fobidding 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. For details, see Grelling paradox, Russell paradox, and other related matters.