The liar paradox, attributed to the Greek philosopher Eubulides of Miletus who lived in the fourth century B.C, is the paradoxical statement
- I am lying now.
- This statement is false.
Even the conclusion that the statement is neither true nor false leads to a contradiction: the statement claims to be false, but isn't, so it claims a falsehood and is therefore false.
To avoid having a sentence refer to its own truth value, one can also construct the paradox
- The following sentence is true.
- The preceding sentence is false.
The proof of Gödel's incompleteness theorem essentially consists of a formally correct formulation of a variation of this paradox in the context of a sufficiently strong axiomatic system A:
- A proof exists in A that this sentence is false.
On the other hand, if there exists no proof in A of the statement either way, then no contradiction arises. The system A is called incomplete in this case: there exists a statement which can neither be proven nor disproven in A.
Similarly, by using the statement "No proof exists in A that this statement is true", we can see that in a consistent system there are statements that are "clearly" true, which cannot be proven to be so in A.
That A can be shown to be true if it is false and false if it is true has led some to conclude that it is neither true nor false. This has given rise to the following, strengthened version of the paradox:
- This statement is not true.
- This statement is only false.
Then there's Yablo's version of the paradox. Consider a list of sentences which is infinitely long in both directions. The sentences all say the same thing: All of the subsequent statements are false. Pick one statement at random. So it's true if all of the subsequent statements are false. But if all of the subsequent statements are false, then what they say is indeed the case: they say that all of the statements subsequent to them are false, and ex hyposthesi they are false. That contradiction means that the picked statement should be false, but its selection was arbitrary, implying all the statements must be false; again this leads to their description of subsequent statements being true. So like the liar, they're true if they're false and false if they're true, yet no propositions predicate falsity of themselves. This is sufficent to suggest that the liar does not depend upon self reference.
Consider for a moment the opposite of the liar:
- This statement is true.
There are some people who insist that there is nothing "paradoxical" about the Liar paradox. The claim is that every statement necessarily includes an implicit assertion of its own truth. Thus, for example, the statement "It is true that two plus two equals four" contains no more information than the statement "two plus two is four", because the phrase "it is true that..." is always implicitly there. And in the self-referential spirit of the Liar Paradox, the phrase "it is true that..." is equivalent to "this whole statement is true and ...". Thus the statement "this statement is false" is assumed by those who hold this position to be equivalent to "(implicitly) this statement is true and (explicitly) this statement is false", which is false because "A and not A" is necessarily false, but it is not paradoxical that "A and not A" implies A, and also implies "not A". The people who accept this argument see it as completely compelling, and believe that someday everyone will wake up and see the validity of the argument. Nevertheless, those who accept this argument are currently in the minority.