Topics Logic

Question Is the Liar paradox, stated as "this sentence is false" false? The Liar surely means, "The proposition expressed by this sentence is false", but this implies that there is one and only one proposition contained within the sentence. If this is not the case then the whole statement is false because "The proposition" must pick out exactly one object. The direct proposition expressed is "This sentence is false", yet surely since the predicate "is false" applies to the sentence in question, "This sentence is false" is false is a proposition that is also logically entwined with the sentence. Since the sentence expresses two propositions, and not one, there is no object which corresponds to "The proposition expressed" and so the whole sentence becomes false.

Accepted:
November 10, 2005

Comments

There are two aspects of your argument that worry me:

1. If I understand your argument correctly, you are saying that the liar sentence expresses two different propositions, namely:
(a) This sentence is false.
(b) "This sentence is false" is false.
But don't (a) and (b) mean the same thing? Isn't that, in fact, what your argument shows? So is this really two different propositions?

2. You say that you think the liar sentence is false. But aren't you worried that that would make it true, since it asserts--correctly, in your view--that it is false?

By the way, a common first reaction to the liar paradox is to think that the problem is caused by the phrase "this sentence". There is a wonderful example, due to Quine, of a sentence that achieves the same effect as the sentence "this sentence is false" without using the phrase "this sentence". (The example makes use of the same mechanism for achieving self-reference that Godel used in his proof of the Incompleteness Theorem.) Here is Quine's example:

"yields a falsehood when preceded by its quotation" yields a falsehood when preceded by its quotation.

The subject of Quine's sentence is the phrase "yields a falsehood when preceded by its quotation". What the sentence says about this phrase is that if you precede it by its quotation, then the result will be a falsehood. But if you precede this phrase by its quotation, then the result is Quine's sentence itself! So the sentence asserts of itself that it is false, without using the phrase "this sentence" to refer to itself.