In philosophy and logic, the liar paradox or liar's paradox is the statement "this sentence
is false". Trying to assign to this statement a classical binary truth value leads to a
paradoxical contradiction. If "this sentence is false" is true, then
the sentence is false, but then if "this sentence is false" is false, then the sentence is true,
and so on.
History The Epimenides paradox has been suggested
as an example of the liar paradox, but they are not logically equivalent. The semi-mythical
seer Epimenides, a Cretan, reportedly stated that "The Cretans are always liars." However,
Epimenides' statement that all Cretans are liars can be resolved as false, given that
he knows of at least one other Cretan who does not lie.
The paradox's name translates as pseudómenos lógos in Ancient Greek. One version of the
liar paradox is attributed to the Greek philosopher Eubulides of Miletus who lived in the 4th
century BC. Eubulides reportedly asked, "A man says that he is lying. Is what he says
true or false?" The paradox was once discussed by St. Jerome
in a sermon:
"I said in my alarm, 'Every man is a liar!' "(Psalms 116:11) Is David telling the truth
or is he lying? If it is true that every man is a liar, and David's statement, "Every man
is a liar" is true, then David also is lying; he, too, is a man. But if he, too, is lying,
his statement: "Every man is a liar," consequently is not true. Whatever way you turn the proposition,
the conclusion is a contradiction. Since David himself is a man, it follows that he also
is lying; but if he is lying because every man is a liar, his lying is of a different
sort."
In early Islamic tradition liar paradox was discussed for at least five centuries starting
from late 9th century apparently without being influenced by any other tradition. Naṣīr
al-Dīn al-Ṭūsī could have been the first logician to identify the liar paradox as self-referential.
Explanation of the paradox and variants The problem of the liar paradox is that it
seems to show that common beliefs about truth and falsity actually lead to a contradiction.
Sentences can be constructed that cannot consistently be assigned a truth value even though they
are completely in accord with grammar and semantic rules.
The simplest version of the paradox is the sentence:
This statement is false.
If is true, then "This statement is false" is true. Therefore must be false. The hypothesis
that is true leads to the conclusion that is false, a contradiction.
If is false, then "This statement is false" is false. Therefore must be true. The hypothesis
that is false leads to the conclusion that is true, another contradiction. Either way,
is both true and false, which is a paradox. However, 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, the rejection of
the claim that every statement has to be either true or false, also known as the principle
of bivalence, a concept related to the law of the excluded middle.
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 is neither true nor false, then it must be not true. Since this is what itself states,
it means that must be true. Since initially was not true and is now true, another paradox
arises. Another reaction to the paradox of is to posit,
as Graham Priest has, that the statement is both true and false. Nevertheless, even Priest's
analysis is susceptible to the following version of the liar:
This statement is only false.
If is both true and false, then is only false. But then, it is not true. Since initially
was true and is now not true, it is a paradox. There are also multi-sentence versions of
the liar paradox. The following is the two-sentence version:
The following statement is true.
The preceding statement is false.
Assume is true. Then is true. This would mean that is false. Therefore is both true and
false. Assume is false. Then is false. This would
mean that is true. Thus is both true and false. Either way, is both true and false - the same
paradox as above. The multi-sentence version of the liar paradox
generalizes to any circular sequence of such statements, provided there are an odd number
of statements asserting the falsity of their successor; the following is a three-sentence
version, with each statement asserting the falsity of its successor:
E2 is false.
E3 is false.
E1 is false.
Assume is true. Then is false, which means is true, and hence is false, leading to a
contradiction. Assume is false. Then is true, which means
is false, and hence is true. Either way, is both true and false - the same paradox as
with and. There are many other variants, and many complements,
possible. In normal sentence construction, the simplest version of the complement is
the sentence:
This statement is true.
If F is assumed to bear a truth value, then it presents the problem of determining the
object of that value. But, a simpler version is possible, by assuming that the single word
'true' bears a truth value. The analogue to the paradox is to assume that the single word
'false' likewise bears a truth value, namely that it is false. This reveals that the paradox
can be reduced to the mental act of assuming that the very idea of fallacy bears a truth
value, namely that the very idea of fallacy is false: an act of misrepresentation. So,
the symmetrical version of the paradox would be:
The following statement is false.
The preceding statement is false.
Possible resolutions Alfred Tarski
Alfred Tarski diagnosed the paradox as arising only in languages that are "semantically closed",
by which he meant a language in which it is possible for one sentence to predicate truth
of another sentence in the same language. To avoid self-contradiction, it is necessary
when discussing truth values to envision levels of languages, each of which can predicate
truth only of languages at a lower level. So, when one sentence refers to the truth-value
of another, it is semantically higher. The sentence referred to is part of the "object
language", while the referring sentence is considered to be a part of a "meta-language"
with respect to the object language. It is legitimate for sentences in "languages" higher
on the semantic hierarchy to refer to sentences lower in the "language" hierarchy, but not
the other way around. This prevents a system from becoming self-referential.
Arthur Prior Arthur Prior asserts that there is nothing
paradoxical about the liar paradox. His claim 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 equals 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 following two statements are equivalent:
This statement is false.
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 therefore no paradox because the claim that this two-conjunct Liar is false does
not lead to a contradiction. Eugene Mills and Neil Lefebvre and Melissa Schelein present
similar answers. Saul Kripke
Saul Kripke argued that whether a sentence is paradoxical or not can depend upon contingent
facts. If the only thing Smith says about Jones is
A majority of what Jones says about me is false.
and 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 Smith really is a big spender but is "not" soft on crime, then both Smith's remark about
Jones and Jones's last remark about Smith are 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, that statement is "grounded". If not, that
statement is "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 Etchemendy Jon Barwise and John Etchemendy propose that
the liar sentence 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 situation semantics, that
the "denial liar" can be true without contradiction while the "negation liar" can be false without
contradiction. Their 1987 book makes heavy use of non-well-founded set theory.
Dialetheism Graham Priest and other logicians, including
J.C. Beall, and Bradley Armour-Garb have proposed that the liar sentence should be considered
to be both true and false, a point of view known as dialetheism. Dialetheism is the view
that there are true contradictions. Dialetheism raises its own problems. Chief among these
is that since dialetheism recognizes the liar paradox, an intrinsic contradiction, as being
true, it must discard the long-recognized principle of explosion, which asserts that
any proposition can be deduced from a contradiction, unless the dialetheist is willing to accept
trivialism - the view that all propositions are true. Since trivialism is an intuitively
false view, dialetheists nearly always reject the explosion principle. Logics that reject
it are called paraconsistent. Non-cognitivism
Andrew Irvine has argued in favour of a non-cognitivist solution to the paradox, suggesting that some
apparently well-formed sentences will turn out to be neither true nor false and that
"formal criteria alone will inevitably prove insufficient" for resolving the paradox.
Logical structure of the liar paradox For a better understanding of the liar paradox,
it is useful to write it down in a more formal way. If "this statement is false" is denoted
by A and its truth value is being sought, it is necessary to find a condition that restricts
the choice of possible truth values of A. Because A is self-referential it is possible
to give the condition by an equation. If some statement, B, is assumed to be false,
one writes, “B = false”. The statement that the statement B is false would be written
as “C = “B = false””. Now, the liar paradox can be expressed as the statement
A, that A is false: “A = “A = false””
This is an equation from which the truth value of A = "this statement is false" could hopefully
be obtained. In the boolean domain "A = false" is equivalent to "not A" and therefore the
equation is not solvable. This is the motivation for reinterpretation of A. The simplest logical
approach to make the equation solvable is the dialetheistic approach, in which case
the solution is A being both "true" and "false". Other resolutions mostly include some modifications
of the equation; Arthur Prior claims that the equation should be "A = 'A = false and
A = true'" and therefore A is false. In computational verb logic, the liar paradox is extended to
statement like, "I hear what he says; he says what I don't hear", where verb logic must
be used to resolve the paradox. Applications
Gödel's First Incompleteness Theorem Gödel's incompleteness theorems are two fundamental
theorems of mathematical logic which state inherent limitations of all but the most trivial
axiomatic systems for mathematics. The theorems were proven by Kurt Gödel in 1931, and are
important in the philosophy of mathematics. Roughly speaking, in proving the first incompleteness
theorem, Gödel used a modified version of the liar paradox, replacing "this sentence
is false" with "this sentence is not provable", called the "Gödel sentence G". Thus for a
theory "T", "G" is true, but not provable in "T". The analysis of the truth and provability
of "G" is a formalized version of the analysis of the truth of the liar sentence.
To prove the first incompleteness theorem, Gödel represented statements by numbers.
Then the theory at hand, which is assumed to prove certain facts about numbers, also
proves facts about its own statements. Questions about the provability of statements are represented
as questions about the properties of numbers, which would be decidable by the theory if
it were complete. In these terms, the Gödel sentence states that no natural number exists
with a certain, strange property. A number with this property would encode a proof of
the inconsistency of the theory. If there were such a number then the theory would be
inconsistent, contrary to the consistency hypothesis. So, under the assumption that
the theory is consistent, there is no such number.
It is not possible to replace "not provable" with "false" in a Gödel sentence because
the predicate "Q is the Gödel number of a false formula" cannot be represented as a
formula of arithmetic. This result, known as Tarski's undefinability theorem, was discovered
independently by Gödel and by Alfred Tarski. George Boolos has since sketched an alternative
proof of the first incompleteness theorem that uses Berry's paradox rather than the
liar paradox to construct a true but unprovable formula.
In popular culture The liar paradox is occasionally used in fiction
to shut down artificial intelligences, who are presented as being unable to process the
sentence. In Star Trek: The Original Series episode I, Mudd, the Liar paradox is used
by Captain Kirk and Harry Mudd to confuse and ultimately disable an android holding
them captive. In the 1973 Doctor Who serial The Green Death, the Doctor temporarily stumps
the insane computer BOSS by asking it "If I were to tell you that the next thing I say
would be true, but the last thing I said was a lie, would you believe me?" However BOSS
eventually decides the question is irrelevant and summons security. In the 2011 videogame
Portal 2, GLaDOS attempts to use the "this sentence is false" paradox to defeat the naïve
artificial intelligence Wheatley, but, lacking the intelligence to realize the statement
a paradox, he simply responds, "Um, true. I'll go with true. There, that was easy."
and is unaffected. The second book in Emily Rodda's Deltora Quest
series, The Lake of Tears has the main character, Lief, forced to answer a riddle correctly
or be killed by the guardian of a bridge. When Lief answers the trick riddle wrongly,
he confronts the guardian with his treachery. The guardian answers with another riddle,
telling Lief to make a statement; if false, he will kill Lief by chopping off his head;
if true, he will strangle Lief. Lief replies, "You will cut off my head." As the guardian
was cursed to his fate by the evil sorceress Thaegan 'until truth and lies become one',
the paradox allows him to revert to his original form: an eagle.
See also Card paradox
Epimenides paradox Insolubilia
List of paradoxes Pinocchio paradox
Quine's paradox Socratic paradox
Yablo's paradox Notes
References
External links Liar Paradox entry by Bradley Dowden in the
Internet Encyclopedia of Philosophy Liar Paradox entry by J C Beall and Michael
Glanzberg in the Stanford Encyclopedia of Philosophy