Liar paradox Guide, Meaning , Facts, Information and Description
The liar paradox is a concept from the fields of philosophy and logic. It refers to paradoxical statements such as:
- "I am lying now."
- "This statement is false."
- "The following sentence is true.
- The preceding sentence is false."
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2 The Epimenides paradox 3 A discussion of the liar paradox 4 Gödel's Theorem 5 References |
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:
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 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. The liar paradox after Eubulides however is paradoxical per definitionem. (For more information see Epimenides paradox.)
However, the fact 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 response to the paradox is, in effect, to reject one 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.
Some philosophers hold that even this conclusion that the statement be neither true nor false leads to a contradiction: The statement claims to be false, but is not, so it claims a falsehood and therefore is false.
Still, the debate over the possibility of the statement being neither true nor false has given rise to the following, strengthened version of the paradox:
This again 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:
Furthermore, there is 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.
It is 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 hypothesi 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 hold that there is nothing "paradoxical" about
the Liar paradox. Their 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 a simple contradiction of the
form "A and not A", and hence is false, but is not a paradox —
there is no circularity. Of course, this is no solution to versions of the paradox that don't use direct self-reference, such as the two-sentence version:
"The next sentence is false"
"The preceding sentence is true"
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 contradition: 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.
If it is neither true nor false, then it is not true, which is what it says, hence it's true, etc.
(Priest disagrees with this.)
Saul Kripke points out that whether or not a sentence is paradoxical can be a function of 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" and "Everything Smith says about me is true". If the empircal facts are that Smith is a big spender and 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 proposed a solution that exploits recognition of the context within which an assertion is made. Barwise argued that careful consideration of the context, in his terminology, the situation, of the statement does indeed yield a contradiction if the statement is assumed true. However, Barwise argued that similar careful examination of the context when the statement is assumed false yields no such contradiction and the falsity of the statement is thereby demonstrated.
The proof of Gödel'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:
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.
This is an Article on Liar paradox. Page Contains Information, Facts Details or Explanation Guide About Liar paradox Gödel's Theorem
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.References