Liar paradox

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.

As opposed to the Epimenides paradox, this statement is indeed paradoxical: assuming that the statement is true, then it must be false; assuming it is false, then it is not false. No truth value can be consistently assigned to the statement.

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.

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.

If it is neither true nor false, then it is not true, which is what it says, hence it's true, etc. This has led some, notably Graham Priest, to posit that the statement is both true and false (see Paraconsistent Logic). Joachim Bromond (2002) has confuted this third truth value by means of a re-strengthened liar which says:

This statement is only false.

(Priest disagrees. See Priest, forthcoming)

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 sufficient to suggest that the liar does not depend upon self reference.

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" that is, "A and not A". But the latter can be consistently assumed to be false. Thus, the paradox — a seemingly harmless sentence leads to a contradiction regardless of whether we assume that it is true or assume that it is false — is resolved. But only at the price of concluding that the seemingly harmless sentence is a direct contradiction already. One might wonder if this is any improvement.

Saul Kripke proposes a solution in the following manner: If a statement's truth value is ultimately tied up in some evaluable fact about the world, call that statement "grounded." If not, call that statement "ungrounded." Ungrounded statements can not be evaluated for truth conditions. Liar statements, and liar-like statements are ungrounded, and therefore non-truth evaluable.

Gödel's Theorem

The proof of Gödel's incompleteness theorem uses self-referential statements that are similar to the Liar.

In the context of a sufficiently strong axiomatic system A:

A proof exists in A that this sentence is false.

If a proof exists using only the axioms in A that the statement is true, then this implies that there is also a proof that the statement is false. Conversely, if a proof exists in A that the statement is false, then this proof is an example showing that the statement is true. Thus, if a proof exists either way, the system is inconsistent, in that a single statement can be proven to be both true and 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.

References

Kirkham, Richard 1992: Theories of Truth. Bradford Books. Chapter 9 is a very good discussion of the paradox.