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Math Help - Epimenides “Paradox”: I am a liar

  1. #16
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    Re: Epimenides “Paradox”: I am a liar

    When you attempt to assign a T/F value to a statement you are treating it as a variable.

    “John went into town” can be True or False. (If John went into town, then….)

    “A liar went into town” is True or False depending on whether “liar” is True or False (assuming someone did indeed go into town).

    If x = liar:
    “A liar went into town” = G(x) = x
    “I am a liar” = F(x) = not x

    Comment
    We have been confusing semantics with “logic.” Semantically, “I am a liar” is either bad grammar or a limitation of language.

    liar: Someone who does not (verifiably) tell the truth.

    (an interesting discussion might be treating "logic" as the mathematics of functions of variables which can only have one of 2 (undefined) values, call them T/F.)

    EDIT: Compare:
    "He is a liar" = G(x) = x
    "I am a liar" = F(x) = not x
    Last edited by Hartlw; October 18th 2013 at 04:40 AM.
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  2. #17
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    Re: Epimenides “Paradox”: I am a liar

    Quote Originally Posted by Hartlw View Post
    When you attempt to assign a T/F value to a statement you are treating it as a variable.

    “John went into town” can be True or False. (If John went into town, then….)

    “A liar went into town” is True or False depending on whether “liar” is True or False (assuming someone did indeed go into town).

    If x = liar:
    “A liar went into town” = G(x) = x
    “I am a liar” = F(x) = not x

    Comment
    We have been confusing semantics with “logic.” Semantically, “I am a liar” is either bad grammar or a limitation of language.

    liar: Someone who does not (verifiably) tell the truth.

    (an interesting discussion might be treating "logic" as the mathematics of functions of variables which can only have one of 2 (undefined) values, call them T/F.)
    As you wish
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  3. #18
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    Re: Epimenides “Paradox”: I am a liar

    You forgot to reference the edit:

    EDIT: Compare:
    "He is a liar" = G(x) = x
    "I am a liar" = F(x) = not x
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  4. #19
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    Proof by assumption (circular reasoning)

    “I am a liar” as a language short-cut has already been discussed ( “I am a liar” except for this statement.). Just like (“this page intentionally left blank” except for this statement.). The short-cut is understood in commom usage. Any one who heard Epemenides statement knew exactly what he meant.

    However, the logical aspect hasn’t been exhausted.

    PROOF BY ASSUMPTION:

    Assume “I am Napoleon” is T. “I am Napoleon” can’t be F because it violates the assumption. Therefore “I am Napoleon” is T.

    Assume “I am a liar” is T. Then “I am a liar” can’t be F because it violates the assumption. Therefore “I am a liar” is T.

    The above arguments are logically disallowed: the assumption alone that something is True or False is not sufficient to determine its Truth or Falsity.

    Prove 2>1. Assume 2>1. Then 2>1 by assumption. (You need more than the assumption to draw a conclusion about 2>1).

    SUMMARY for “I am a liar”:
    As a language shortcut, it is perfectly clear.
    As a logical statement, there is not enough information to arrive at a conclusion (paradox).
    Last edited by Hartlw; November 5th 2013 at 06:37 AM.
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