Liar paradox

In philosophy and logic, the liar paradox encompasses paradoxical statements such as:

I am lying now.

This statement is false.

To avoid having a sentence directly refer to its own truth value, one can also construct the paradox as follows:

The following sentence is true.
:The preceding sentence is false.

Eubulides of Miletus' words

The oldest version of the liar paradox is attributed to the Greek philosopher Eubulides of Miletus who lived in the fourth century B.C. Eubulides reportedly said:

A man says that he is lying. Is what he says true or false?

The Epimenides paradox

"Epimenides paradox" is often considered an equivalent or interchangeable term for "liar paradox" and it is also the kind of supposed "liar paradox" that is best known to the general public. However, an identification of the two is very questionable:

Epimenides was a sixth century BC philosopher-poet. Himself a Cretan, he reportedly wrote:

The Cretans are always liars (Bible, NT, Titus 1:12)

While Epimenides' words were stated substantially earlier than Eubulides', it is likely that Epimenides did not intend them to be understood as a kind of liar paradox. Little is known about the circumstances in which he made them, the original poems containing them have been lost and the only confirmed record of them is St. Paul quoting them in the Epistle to Titus (where they were arguably also not intended as a paradox). It was only much later that the aforementioned Bible quote was taken up again and referred to as the Epimenides paradox. It is not known (but very much in doubt) whether Eubulides knew of, or made reference to Epimenides' words in his original contemplation of the liar paradox. For these reasons, Eubulides is rightly currently credited as the oldest known source of a liar paradox.

Moreover, if Epimenides' words are simply false, then himself erring or lying does not make all of his fellow countrymen liars. A false statement of The Cretans are always liars. hence can remain false, because no proof exists that they really are liars. Epimenides' statement thus is not paradoxical if false. There are further reasons why the statement also is not necessarily paradoxical even if it is true (Cretans might sometimes, but not always, be liars). The liar paradox after Eubulides however is paradoxical per definitionem. (For more information see Epimenides paradox.)

A discussion of the liar paradox

The problem of the paradox is that it seems to show that our most cherished common beliefs about truth and falsity actually lead to a contradiction. Sentences can be constructed that are completely in accord with grammar and semantic rules that cannot consistently be assigned a truth value: Consider the simplest version of the paradox, the sentence This statement is false. If we assume that the statement is true, everything asserted in it must be true. However, because the statement asserts that it is itself false, it must be false. So assuming that it is true leads to the contradiction that it is true and false. OK, can we assume that it is false? No, that assumption also leads to contradiction: if the statement is false, then what it says about itself is not true. It says that it is false, so that must not be true. Hence, it is true. Under either assumption, we end up concluding that the statement is both true and false. But it has to be either true or false (or so our common intuitions lead us to think), hence there seems to be a contradiction at the heart of our beliefs about truth and falsity.

However, the fact that the Liar sentence 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 response to the paradox is, in effect, to reject one of our common beliefs about truth and falsity: the claim that every statement has to be one or the other. This common belief is called the Principle of Bivalence.

The proposal that the statement is neither true nor false 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 again has led some, notably Graham Priest, to posit that the statement is both true and false (see paraconsistent logic).

A. N. Prior claims that there is nothing paradoxical about
the Liar paradox. His claim (which he attributes to Charles S. Peirce and John Buridan) is that every statement 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 said to be equivalent to

This statement is true and this statement is false.

The latter is a simple contradiction of the
form "A and not A", and hence is false. There is no paradox because the claim that this two-conjunct Liar is false does not lead to a contradiction.

Does Prior's approach provide a solution to versions of the paradox that don't use direct self-reference? An example is the two-sentence version:

The next sentence is false.

The preceding sentence is true.

On Prior's analysis these would be equivalent to:

This whole sentence is true and the next sentence is false.

This whole sentence is true and the preceding sentence is true.

It could be argued that one can consistently assign truth to the first sentence and falsehood to the second. The reasoning would be that there are four clauses here altogether. It is possible to consistently assign truth to both clauses of the first sentence, and the second clause of the second sentence, while assigning falsehood to the first clause of the second sentence. Thus while the sentences contradict each other, there is no paradox.

On the other hand, consider the two clause version:

The next clause is false and the preceding clause is true.

A Prior-like analysis would be:

This whole sentence is true and the next clause is false and the preceding clause is true.

But there is no way to consistently assign truth values to all three clauses, so we still have a paradox. The same point applies to If the next clause is false, then the preceding clause is true. What this suggests is that the Prior-like resolution of the two-sentence version is something of a fluke. His approach to the paradox doesn't, in general, handle versions of the paradox that lack direct self-reference.

Saul Kripke points out that whether or not a sentence is paradoxical can depend upon contingent facts. Suppose that the only thing Smith says about Jones is

A majority of what Jones says about me is false.

Now suppose that Jones says only these three things about Smith:

Smith is a big spender.

Smith is soft on crime.

Everything Smith says about me is true.

If the empirical facts are that Smith is a big spender but he is not soft on crime, then Smith's remark about Jones and Jones's last remark about Smith are both paradoxical. 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 do not have a truth value. Liar statements, and liar-like statements are ungrounded, and therefore have no truth value.

Jon Barwise and John Etchmendy propose that the Liar sentence (which they interpret as synonymous with the Strengthened Liar) is ambiguous. They base this conclusion on a distinction they make between a denial and a negation. If the Liar means It is not the case that this statement is true then it is denying itself. If it means This statement is not true then it is negating itself. They go on to argue, based on their theory of "situational semantics" that the "denial Liar" can be true without contradiction while the "negation Liar" can be false without contradiction.

G�del's theorem

The proof of G�del's incompleteness theorem uses self-referential statements that are similar to the statements at work in the Liar paradox.

In the context of a sufficiently strong axiomatic system A of arithmetic:

(1) This statement is not provable in A.

You will notice that (1) does not mention truth at all (only provability) but the parallel is clear. Suppose (1) is provable, then what it says of itself, that it is not provable, is true. But this conclusion is contrary to our supposition, so our supposition that (1) is provable must be false. Suppose the contrary that (1) is not provable, then what it says of itself is true, although we cannot prove it. Therefore, there is no proof that (1) is provable, and there is also no proof that its negation is provable (i.e., there is no proof that it is also unprovable). Whence, A is incomplete because it cannot prove all truths, namely, (1) and its negation. Statements like (1) are called undecidable. We take for granted that all provable statements are true, but G�del showed that the converse, that all true statements are provable in some one system is not the case. (This does not mean that all true statements are not provable in some system or other.)

Tarski's Theorem, closely related to G�del's Theorem, is a more direct application of the Liar Paradox, though there is no actual paradox involved; instead, the "paradox" simply demonstrates that all the true sentences of arithmetic are not arithmetically definable (or that arithmetic cannot define its own truth predicate; or that arithmetic is not "semantically closed").

References

Barwise, Jon and John Etchemendy 1987: The Liar. Oxford University Press.
Hughes, G.E., 1992. John Buridan on Self-Reference : Chapter Eight of Buridan's Sophismata, with a Translation, and Introduction, and a Philosophical Commentary, Cambridge University Press, ISBN 0521288649 (Buridan's detailed solution to a number of such paradoxes).
Kirkham, Richard 1992: Theories of Truth. Bradford Books. Chapter 9 is a very good discussion of the paradox.
Kripke, Saul 1975: "An Outline of a Theory of Truth" Journal of Philosophy 72:690-716.
Priest, Graham 1984: "The Logic of Paradox Revisited" Journal of Philosophical Logic 13:153-179.
Prior, A. N. 1976: Papers in Logic and Ethics. Duckworth.
Liar Paradox — at the Internet Encyclopedia of Philosophy
Smullyan, Raymond: What is the Name of this Book? (a collection of logic puzzles exploring this theme)

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