Liar paradox
In philosophy and logic, the classical liar paradox or liar's paradox or antinomy of the liar is the statement of a liar that they are lying: for instance, declaring that "I am lying". If the liar is indeed lying, then the liar is telling the truth, which means the liar just lied. In "this sentence is a lie" the paradox is strengthened in order to make it amenable to more rigorous logical analysis. It is still generally called the "liar paradox" although abstraction is made precisely from the liar making the statement. Trying to assign to this statement, the strengthened liar, a classical binary truth value leads to a contradiction.
If "this sentence is false" is true, then it is false, but the sentence states that it is false, and if it is false, then it must be true, and so on.
Possible resolutions[edit]
Fuzzy logic[edit]
In fuzzy logic, the truth value of a statement can be any real number between 0 and 1 both inclusive, as opposed to Boolean logic, where the truth values may only be the integer values 0 or 1. In this system, the statement "This statement is false" is no longer paradoxical as it can be assigned a truth value of 0.5,[9][10] making it precisely half true and half false. A simplified explanation is shown below.
Let the truth value of the statement "This statement is false" be denoted by . The statement becomes
Applications[edit]
Gödel's first incompleteness theorem[edit]
Gödel's incompleteness theorems are two fundamental theorems of mathematical logic which state inherent limitations of sufficiently powerful 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". His proof showed that for any sufficiently powerful 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.[20]
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.
In popular culture[edit]
The liar paradox is occasionally used in fiction to shut down artificial intelligences, who are presented as being unable to process the sentence. In the 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 that the last thing I said was a lie, would you believe me?" BOSS tries to figure it out but cannot and eventually decides the question is irrelevant and summons security.
In the 2011 video game Portal 2, artificial intelligence GLaDOS attempts to use the "this sentence is false" paradox to kill another artificial intelligence, Wheatley. However, lacking the intelligence to realize the statement is a paradox, he simply responds, "Um, true. I'll go with true. There, that was easy." and is unaffected. Humorously, all other AIs present barring GLaDOS, all of which are significantly less sentient and lucid than both her and Wheatley, are still killed from hearing the paradox. However, GLaDOS later notes that she almost killed herself from her own attempt to kill Wheatley.
The Devo song, Enough Said, includes the lyrics The next thing I say to you will be true / The last thing I said was false.
In the seventh episode of Minecraft: Story Mode, titled "Access Denied", the main character Jesse and their friends are captured by a supercomputer named PAMA. After PAMA controls two of Jesse's friends, Jesse learns that PAMA stalls when processing and uses a paradox to confuse him and escape with their last friend. One of the paradoxes the player can make Jesse say is the liar paradox.
Rollins Band's 1994 song "Liar" alluded to the paradox when the narrator ends the song by stating "I'll lie again and again and I'll keep lying, I promise".
Robert Earl Keen's song "The Road Goes On and On" alludes to the paradox. The song is widely believed to be written as part of Keen's feud with Toby Keith, who is presumably the "liar" Keen refers to.[21]