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Liar's Paradox

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"This sentence is a lie."

A presumed liar gives a statement inferring that they are currently lying: "I am lying", "this is a lie". However in doing so they create a paradox, as under the assumption that they are lying, that means what they just said is a lie and therefore are in fact telling the truth. While the assumption that they are truthful means they are once again lying... and the cycle keeps on going.

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This is known as the Liar's Paradox. A variant can be done with two people, where Alice states Bob is a liar, while Bob states Alice is telling the truth. Be careful about phrasing this. A variant like "everything I say is a lie" is not a paradox (rather, it means that not everything I say is a lie, but this particular statement is; which is logically sound).

Its most famous early formulation is ascribed to the Greek philosopher Epimenides, who came from Crete and said "All Cretans are liars". This version has a logical out (if it is untrue, then the implication is not "All Cretans tell the truth" but "Not all Cretans are liars but this one is"). The stronger version was formulated by the Greek philosopher Eubulides of Miletus, and indepently by the Indian philosopher Bhartrhari.

In early works it was often used as a Mind Screw for readers and viewers. However as it became popularized in fiction, it became the standard-issue Logic Bomb for heroes to use against computers and robots due to the endless paradox loop.

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Most instances of the Liar's Paradox involve either a Self-Proclaimed Liar or a Knights and Knaves problem.

Contrast Beware the Honest Ones and Crying Wolf. See also That Liar Lies.


Examples

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     Fan Works 
  • Plan 7 of 9 from Outer Space. Captain Proton tries the old "the next thing I say is a lie" trick on a Killer Robot only to have it smack him in the chops.
    "I'M NOT FALLING FOR THAT ONE! AND DON'T ASK ME TO CALCULATE THE VALUE OF PI EITHER, BECAUSE I'M NO GOOD AT MATH!"

     Folklore  
  • An old joke involves an explorer captured by natives and told that if his next sentence is true, he'll be thrown to the snakes, if false, he'll be thrown to the lions. Naturally, he predicts his death by lions and is let go.

     Literature  
  • In the second Deltora Quest book, "Lake of Tears", when Lief gets given a riddle by a Bridge Guardian and tricked into the wrong answer, the guardian presents another game to decide his method of death. Lief will say a statement, if it was true then the guardian would strangle Lief, if it was false he would cut his head off with his sword. Lief opts to Taking A Third Option and responds with the statement "You will cut off my head." It's then revealed the guardian was cursed by Thaegan the sorcerer to guard the bridge "until truth and lies become one", so Lief's liar's paradox sets him free and he reverts to the form of an eagle.
  • In Small Gods: One of the Ephebian philosophers gets into trouble when he tries to discuss this paradox with his fellows. Discworld intellects being what they are, Xeno takes offense:
    Xeno: He bloody well accused me of slander!
    Ibid: I didn't!
    Xeno: You did! You did! Tell 'em what you said!
    Ibid: Look, I merely suggested, to indicate the nature of paradox, right, that if Xeno the Ephebian said "All Ephebians are liars—"
    Xeno: See? See? He did it again!
    Ibid: —no, no, listen, listen... then, since Xeno is himself an Ephebian, this would mean that he himself is a liar and therefore—
    • Also in Discworld, the literal-minded omnicidal Auditors of Reality eventually deal with paradoxes by deciding that all statements are either (1) true, (2) false, or (3) bloody stupid.
  • Parodied and subverted in the Eighth Doctor Adventures novel Frontier Worlds, in which the Doctor tries the Liar's Paradox on a security robot, which simply snaps, "Get off with you. You'll be asking me to calculate pi next," and keeps attacking him.
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     Live Action TV  
  • In Star Trek: The Original Series, during the episode "I, Mudd", Captain Kirk and Harry Mudd use a liar's paradox to set off a Logic Bomb in an android holding them captive.
  • In Doctor Who, during the story "The Green Death", the Third Doctor manages to stump an insane computer called BOSS with the question "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 after some looping, BOSS decides the question was irrelevant.

     Music  
  • From Devo's "Enough Said"
    The next thing I say to you will be true
    The last thing I said was false
    Remember to do nothing when you don't know what to do
  • The 1990 song "All Men Are Liars" sung by Nick Lowe (a man, obviously) informs the listener emphatically that "all men are liars, and that's the truth".
  • "Opposites Day" by Lloyd Cole, the two-sentence variant ("The next line is the truth| The last line was a lie" and later "The next line is a lie|This one is the truth"), both being equivalent to the standard paradox.

     Video Games  
  • Knights of the Old Republic II: The Sith Lords: On Korriban, you find a computer terminal that has a test for trainee Sith which has the question "Which of these is not a paradox?" One of the answer choices is "I always lie." It's also the correct answer thanks to Exact Words (and if the player thinks in terms of first order /predicate logic instead of propositional logic). The negation of the statement isn't "I never lie" (which is the case in propositional logic and which would cause the paradox), but "I don't always lie," (in first order logic, trying to negate sentence with universal quantifier will result in negative sentence with existential quantifier. In this case, "always" is universal quantifier) which doesn't tell us anything about whether the subject is currently lying or not, and if the subject ever told the truth even once (which is likely), then the statement is just plain false, not a paradox.
  • In a Minecraft: Story Mode episode, "Access Denied", Jesse and several others are captured by a supercomputer named PAMA. Jesse notices that PAMA stalls whenever processing so he uses a paradox to loop it. One of the options to choose from is a Liar's Paradox.
  • In Portal 2, GLaDOS is afraid of the terrible effects this paradox can have in artificial intelligence. In fact, she uses this against Wheatley in an attempt to drive him insane. Unfortunately for her, he's too dumb to understand the paradox whereas even his "frankenturrets" are not.
  • Poker Night 2: Claptrap has a conversation about these with GlaDOS and Sam, and ends up falling into one of these after Sam gives an example.

    Webcomics 
  • Sam Starfall of Freefall boasts that he's the greatest liar in the world in the Monday 2 November 1998 strip. When Helix queries how that claim can be verified, Sam's response is, "You'll just have to trust me."

     Western Animation  
  • In The Simpsons episode "Itchy & Scratchy Land", Lisa tries to use this on a group of murderous robots in Itchy and Scratchy Land. Turns out their AI isn't complex enough to recognize the paradox, but Homer shows all the signs of suffering from the Logic Bomb.

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