Liar paradox
Encyclopedia
In philosophy
and logic
, the liar paradox or liar's paradox (pseudomenon in Ancient Greek
), is the statement "this sentence is false". Trying to assign to this statement a classical binary truth value leads to a contradiction (see paradox
).
If "this sentence is false" is true, then the sentence is false, which would in turn mean that it is actually true, but this would mean that it is false, and so on ad infinitum.
Similarly, if "this sentence is false" is false, then the sentence is true, which would in turn mean that it is actually false, but this would mean that it is true, and so on ad infinitum.
(circa 600 BC) has been suggested as an example of the liar paradox, but they are not logically equivalent. The fictional speaker 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.
It is unlikely that Epimenides intended his words to be understood as a kind of liar paradox, and they were probably only understood as such much later in history.
The oldest known version of the actual liar paradox is attributed to the Greek philosopher Eubulides of Miletus
who lived in the 4th century BC. It is very unlikely that he knew of Epimenides's words, even if they were intended as a paradox. 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:
In early Islamic tradition liar paradox was discussed for at least 5 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.
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:
If (A) is true, then "This statement is false" is true. Therefore (A) must be false. The hypothesis that (A) is true leads to the conclusion that (A) is false, a contradiction.
If (A) is false, then "This statement is false" is false. Therefore (A) must be true. The hypothesis that (A) is false leads to the conclusion that (A) is true, another contradiction. Either way, (A) 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:
If (B) is neither true nor false, then it must be not true. Since this is what (B) itself states, it means that (B) must be true. Since initially (B) was not true and is now true, another paradox arises.
Another reaction to the paradox of (A) 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:
If (C) is both true and false, then (C) is only false. But then, it is not true. Since initially (C) 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:
Assume (D1) is true. Then (D2) is true. This would mean that (D1) is false. Therefore (D1) is both true and false.
Assume (D1) is false. Then (D2) is false. This would mean that (D1) is true. Thus (D1) is both true and false. Either way, (D1) is both true and false - the same paradox as (A) above.
The multi-sentence version of the liar paradox generalizes to any circular sequence of such statements (wherein the last statement asserts the truth/falsity of the first statement), 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:
Assume (E1) is true. Then (E2) is false, which means (E3) is true, and hence (E1) is false, leading to a contradiction.
Assume (E1) is false. Then (E2) is true, which means (E3) is false, and hence (E1) is true. Either way, (E1) is both true and false - the same paradox as with (A) and (D1).
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 (or falsehood) of another sentence in the same language (or even of itself). To avoid self-contradiction, it is necessary when discussing truth values to envision levels of languages, each of which can predicate truth (or falsehood) 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.
asserts that there is nothing paradoxical about the liar paradox. His claim (which he attributes to Charles Sanders 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 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:
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.
But the claim that every statement is really a conjunction in which the first conjunct says "this statement is true" seems to run afoul of standard rules of propositional logic, especially the rule, sometimes called Conjunction Elimination, that from a conjunction any of the conjuncts can be derived. Thus, from, "This statement is true and this statement is false", it follows that "this statement is false" and so we have, once again, a paradoxical (and non-conjunctive) statement. It seems then that Prior's attempt at resolution requires either a whole new propositional logic or else the postulation that the "and" in, "This statement is true and this statement is false", is a special type of conjunctive for which Conjunction Elimination does not apply. But then we need, at least, an expansion of standard propositional logic to account for this new kind of "and".
argued that whether a sentence is paradoxical or not can depend upon contingent facts. If the only thing Smith says about Jones is
and Jones says only these three things about Smith:
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.
and John Etchemendy
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 situation semantics
, that the "denial liar" can be true without contradiction while the "negation liar" can be false without contradiction.
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 ex falso quodlibet, which asserts that any proposition can be deduced from a true 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 "ex falso quodlibet". Logics that reject "ex falso quodlibet" are called paraconsistent
.
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 (C) 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.
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 slightly 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 (when he was working on the proof of the incompleteness theorem) 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.
, GLaDOS
tries to use this paradox to defeat Wheatley
, on assumption that paradoxes take all of the CPU power to process. But, as Wheatley is programmed to be "the dumbest moron who ever lived", the paradox doesn't affect him.
Philosophy
Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...
and logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
, the liar paradox or liar's paradox (pseudomenon in Ancient Greek
Ancient Greek
Ancient Greek is the stage of the Greek language in the periods spanning the times c. 9th–6th centuries BC, , c. 5th–4th centuries BC , and the c. 3rd century BC – 6th century AD of ancient Greece and the ancient world; being predated in the 2nd millennium BC by Mycenaean Greek...
), is the statement "this sentence is false". Trying to assign to this statement a classical binary truth value leads to a contradiction (see paradox
Paradox
Similar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...
).
If "this sentence is false" is true, then the sentence is false, which would in turn mean that it is actually true, but this would mean that it is false, and so on ad infinitum.
Similarly, if "this sentence is false" is false, then the sentence is true, which would in turn mean that it is actually false, but this would mean that it is true, and so on ad infinitum.
History
The Epimenides paradoxEpimenides paradox
The Epimenides paradox is a problem in logic. It is named after the Cretan philosopher Epimenides of Knossos , There is no single statement of the problem; a typical variation is given in the book Gödel, Escher, Bach, by Douglas Hofstadter:...
(circa 600 BC) has been suggested as an example of the liar paradox, but they are not logically equivalent. The fictional speaker Epimenides
Epimenides
Epimenides of Knossos was a semi-mythical 6th century BC Greek seer and philosopher-poet. While tending his father's sheep, he is said to have fallen asleep for fifty-seven years in a Cretan cave sacred to Zeus, after which he reportedly awoke with the gift of prophecy...
, 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.
It is unlikely that Epimenides intended his words to be understood as a kind of liar paradox, and they were probably only understood as such much later in history.
The oldest known version of the actual liar paradox is attributed to the Greek philosopher Eubulides of Miletus
Eubulides of Miletus
Eubulides of Miletus was a philosopher of the Megarian school, and a pupil of Euclid of Megara. He is famous for his paradoxes.-Life:Eubulides was a pupil of Euclid of Megara, the founder of the Megarian school. He was a contemporary of Aristotle, against whom he wrote with great bitterness...
who lived in the 4th century BC. It is very unlikely that he knew of Epimenides's words, even if they were intended as a paradox. 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
Jerome
Saint Jerome was a Roman Christian priest, confessor, theologian and historian, and who became a Doctor of the Church. He was the son of Eusebius, of the city of Stridon, which was on the border of Dalmatia and Pannonia...
in a sermon:
In early Islamic tradition liar paradox was discussed for at least 5 centuries
Liar paradox in early Islamic tradition
Many early Islamic philosophers and logicians discussed the liar paradox. Their work on the subject began in the 10th century and continued to Athīr al-Dīn al-Abharī and Nasir al-Din al-Tusi of the middle 13th century and beyond...
starting from late 9th century apparently without being influenced by any other tradition. Naṣīr al-Dīn al-Ṭūsī
Nasir al-Din al-Tusi
Khawaja Muḥammad ibn Muḥammad ibn Ḥasan Ṭūsī , better known as Naṣīr al-Dīn al-Ṭūsī , was a Persian polymath and prolific writer: an astronomer, biologist, chemist, mathematician, philosopher, physician, physicist, scientist, theologian and Marja Taqleed...
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 truthTruth
Truth has a variety of meanings, such as the state of being in accord with fact or reality. It can also mean having fidelity to an original or to a standard or ideal. In a common usage, it also means constancy or sincerity in action or character...
and falsity
Falsity
Falsity or falsehood is a perversion of truth originating in the deceitfulness of one party, and culminating in the damage of another party...
actually lead to a contradiction
Contradiction
In classical logic, a contradiction consists of a logical incompatibility between two or more propositions. It occurs when the propositions, taken together, yield two conclusions which form the logical, usually opposite inversions of each other...
. Sentences can be constructed that cannot consistently be assigned a truth value even though they are completely in accord with grammar
Grammar
In linguistics, grammar is the set of structural rules that govern the composition of clauses, phrases, and words in any given natural language. The term refers also to the study of such rules, and this field includes morphology, syntax, and phonology, often complemented by phonetics, semantics,...
and semantic rules.
The simplest version of the paradox is the sentence:
If (A) is true, then "This statement is false" is true. Therefore (A) must be false. The hypothesis that (A) is true leads to the conclusion that (A) is false, a contradiction.
If (A) is false, then "This statement is false" is false. Therefore (A) must be true. The hypothesis that (A) is false leads to the conclusion that (A) is true, another contradiction. Either way, (A) 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
Principle of bivalence
In logic, the semantic principle of bivalence states that every declarative sentence expressing a proposition has exactly one truth value, either true or false...
, 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:
If (B) is neither true nor false, then it must be not true. Since this is what (B) itself states, it means that (B) must be true. Since initially (B) was not true and is now true, another paradox arises.
Another reaction to the paradox of (A) is to posit, as Graham Priest
Graham Priest
Graham Priest is Boyce Gibson Professor of Philosophy at the University of Melbourne and Distinguished Professor of Philosophy at the CUNY Graduate Center, as well as a regular visitor at St. Andrews University. Priest is a fellow in residence at Ormond College. He was educated at the University...
has, that the statement is both true and false. Nevertheless, even Priest's analysis is susceptible to the following version of the liar:
If (C) is both true and false, then (C) is only false. But then, it is not true. Since initially (C) 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:
Assume (D1) is true. Then (D2) is true. This would mean that (D1) is false. Therefore (D1) is both true and false.
Assume (D1) is false. Then (D2) is false. This would mean that (D1) is true. Thus (D1) is both true and false. Either way, (D1) is both true and false - the same paradox as (A) above.
The multi-sentence version of the liar paradox generalizes to any circular sequence of such statements (wherein the last statement asserts the truth/falsity of the first statement), 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:
Assume (E1) is true. Then (E2) is false, which means (E3) is true, and hence (E1) is false, leading to a contradiction.
Assume (E1) is false. Then (E2) is true, which means (E3) is false, and hence (E1) is true. Either way, (E1) is both true and false - the same paradox as with (A) and (D1).
Non-paradoxes
Not all seemingly self-contradictory statements are liar paradoxes. The statement "I always lie" is often considered to be a version of the liar paradox, but is not actually paradoxical. It could be the case (and in reality almost certainly is) that the statement itself is a lie, because the speaker sometimes tells the truth, and this interpretation does not lead to a contradiction (unlike "I am lying"). The belief that this is a paradox results from a false dichotomy—that either the speaker always lies, or always tells the truth—when it is possible that the speaker occasionally does both.Alfred Tarski
Alfred TarskiAlfred Tarski
Alfred Tarski was a Polish logician and mathematician. Educated at the University of Warsaw and a member of the Lwow-Warsaw School of Logic and the Warsaw School of Mathematics and philosophy, he emigrated to the USA in 1939, and taught and carried out research in mathematics at the University of...
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 (or falsehood) of another sentence in the same language (or even of itself). To avoid self-contradiction, it is necessary when discussing truth values to envision levels of languages, each of which can predicate truth (or falsehood) 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 PriorArthur Prior
Arthur Norman Prior was a noted logician and philosopher. Prior founded tense logic, now also known as temporal logic, and made important contributions to intensional logic, particularly in Prior .-Biography:Prior was entirely educated in New Zealand, where he was fortunate to have come under the...
asserts that there is nothing paradoxical about the liar paradox. His claim (which he attributes to Charles Sanders 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 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:
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.
But the claim that every statement is really a conjunction in which the first conjunct says "this statement is true" seems to run afoul of standard rules of propositional logic, especially the rule, sometimes called Conjunction Elimination, that from a conjunction any of the conjuncts can be derived. Thus, from, "This statement is true and this statement is false", it follows that "this statement is false" and so we have, once again, a paradoxical (and non-conjunctive) statement. It seems then that Prior's attempt at resolution requires either a whole new propositional logic or else the postulation that the "and" in, "This statement is true and this statement is false", is a special type of conjunctive for which Conjunction Elimination does not apply. But then we need, at least, an expansion of standard propositional logic to account for this new kind of "and".
Saul Kripke
Saul KripkeSaul Kripke
Saul Aaron Kripke is an American philosopher and logician. He is a professor emeritus at Princeton and teaches as a Distinguished Professor of Philosophy at the CUNY Graduate Center...
argued that whether a sentence is paradoxical or not can depend upon contingent facts. If the only thing Smith says about Jones is
and Jones says only these three things about Smith:
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.
Barwise and Etchemendy
Jon BarwiseJon Barwise
Kenneth Jon Barwise was an American mathematician, philosopher and logician who proposed some fundamental revisions to the way that logic is understood and used....
and John Etchemendy
John Etchemendy
John W. Etchemendy and of Basque descent is Stanford University's twelfth and current Provost. He succeeded John L. Hennessy to the post on September 1, 2000....
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 situation semantics
Situation semantics
Situation semantics, pioneered by Jon Barwise and John Perry in the early 1980s, attempts to provide a solid theoretical foundation for reasoning about common-sense and real world situations, typically in the context of theoretical linguistics, philosophy, or applied natural language...
, that the "denial liar" can be true without contradiction while the "negation liar" can be false without contradiction.
Dialetheism
Graham PriestGraham Priest
Graham Priest is Boyce Gibson Professor of Philosophy at the University of Melbourne and Distinguished Professor of Philosophy at the CUNY Graduate Center, as well as a regular visitor at St. Andrews University. Priest is a fellow in residence at Ormond College. He was educated at the University...
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
Dialetheism is the view that some statements can be both true and false simultaneously. More precisely, it is the belief that there can be a true statement whose negation is also true...
. 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 ex falso quodlibet, which asserts that any proposition can be deduced from a true 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 "ex falso quodlibet". Logics that reject "ex falso quodlibet" are called paraconsistent
Paraconsistent logic
A paraconsistent logic is a logical system that attempts to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing paraconsistent systems of logic.Inconsistency-tolerant logics have been...
.
Chris Langan
Chris Langan in his work The Theory of Theories states: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-referentialSelf-reference
Self-reference occurs in natural or formal languages when a sentence or formula refers to itself. The reference may be expressed either directly—through some intermediate sentence or formula—or by means of some encoding...
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 (C) 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
Boolean domain
In mathematics and abstract algebra, a Boolean domain is a set consisting of exactly two elements whose interpretations include false and true...
"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
Arthur Prior
Arthur Norman Prior was a noted logician and philosopher. Prior founded tense logic, now also known as temporal logic, and made important contributions to intensional logic, particularly in Prior .-Biography:Prior was entirely educated in New Zealand, where he was fortunate to have come under the...
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.
Gödel's First Incompleteness Theorem
Gödel's incompleteness theoremsGödel's incompleteness theorems
Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of...
are two fundamental theorems of mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
which state inherent limitations of all but the most trivial axiomatic systems for mathematics. The theorems were proven by Kurt Gödel
Kurt Gödel
Kurt Friedrich Gödel was an Austrian logician, mathematician and philosopher. Later in his life he emigrated to the United States to escape the effects of World War II. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the...
in 1931, and are important in the philosophy of mathematics. Roughly speaking, in proving the first incompleteness theorem
Gödel's incompleteness theorems
Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of...
, Gödel used a slightly 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 (when he was working on the proof of the incompleteness theorem) and by Alfred Tarski
Alfred Tarski
Alfred Tarski was a Polish logician and mathematician. Educated at the University of Warsaw and a member of the Lwow-Warsaw School of Logic and the Warsaw School of Mathematics and philosophy, he emigrated to the USA in 1939, and taught and carried out research in mathematics at the University of...
.
George Boolos
George Boolos
George Stephen Boolos was a philosopher and a mathematical logician who taught at the Massachusetts Institute of Technology.- Life :...
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
In Portal 2Portal 2
Portal 2 is a first-person puzzle-platform video game developed and published by Valve Corporation. The sequel to the 2007 video game Portal, it was announced on March 5, 2010, following a week-long alternate reality game based on new patches to the original game...
, GLaDOS
GLaDOS
GLaDOS, short for Genetic Lifeform and Disk Operating System, is a fictional artificially intelligent computer system in Valve Software's Half-Life video game series and the main antagonist in the video games Portal and Portal 2. She was created by Erik Wolpaw and Kim Swift and is voiced by Ellen...
tries to use this paradox to defeat Wheatley
Wheatley (Portal)
Wheatley is a fictional artificial character in the 2011 video game Portal 2. He is voiced by British comedian Stephen Merchant and created in part by Portal 2s designer Erik Wolpaw. To date, he only appears in Portal 2....
, on assumption that paradoxes take all of the CPU power to process. But, as Wheatley is programmed to be "the dumbest moron who ever lived", the paradox doesn't affect him.
See also
- Card paradoxCard paradoxThe card paradox is a non-self-referential variant of the liar paradox constructed by Philip Jourdain. It is also known as the postcard paradox, Jourdain paradox or Jourdain's paradox.- The paradox :...
- InsolubiliaInsolubiliaIn the Middle Ages, variations on the liar paradox were studied under the name of insolubilia .Although the liar paradox was well known in antiquity, interest seems to have lapsed until the twelfth century, when it appears to have been reinvented independently of ancient authors...
- List of paradoxes
- Pinocchio paradoxPinocchio paradoxThe Pinocchio paradox arises when Pinocchio says: "My nose grows now", and is a version of the liar paradox. In philosophy and logic, the liar paradox consists of the statement "This sentence is false." Any attempts to assign a classical binary truth value to this statement leads to a...
- Quine's paradoxQuine's ParadoxQuine's paradox is a paradox concerning truth values, attributed to Willard Van Orman Quine. It is related to the liar paradox as a problem, and it purports to show that a sentence can be paradoxical even if it is not self-referring and does not use demonstratives or indexicals...
- Socratic paradoxSocratic paradoxThe phrase Socratic paradox can refer to two separate things.The more common usage refers to an object or idea whose very existence, or acknowledgment, is a paradox. Its name is derived from a quote of Socrates from the Republic, where he says, "I know nothing at all." The question that arises is...
- Yablo's paradox
- Epimenides paradoxEpimenides paradoxThe Epimenides paradox is a problem in logic. It is named after the Cretan philosopher Epimenides of Knossos , There is no single statement of the problem; a typical variation is given in the book Gödel, Escher, Bach, by Douglas Hofstadter:...