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The liar's paradox

  1. Aug 21, 2003 #1
    The honest can't say :"I am a liar" , because he can't lie.

    The liar can't say :"I am a liar", because he can't say the truth.

    When someone in this boolean universe (where each person can be
    honest XOR liar) uses the words "All people are liars"
    he is included too.

    It means that no one in this boolean universe can say those words,

    Therefore, there is no paradox.



    What do you think ?
     
    Last edited by a moderator: Aug 24, 2003
  2. jcsd
  3. Aug 21, 2003 #2
    Isn't that the paradox? The paradox is that no one can truthfully say that he is a liar. You cannot tell the truth and lie at the same time. If someone says, "I am lying to you right now," then the statement he just made is false. But if the statement is false, then he is not lying and thus the statement is true. But if the statement is true...
    The problem with this statement is that it has the unfortunate consequence p<=>~p, which of course is impossible.
     
    Last edited: Aug 21, 2003
  4. Aug 21, 2003 #3
    Let "liar" mean "consistent liar", and "antiliar" mean "consistent NON-liar".

    If these categories (liars and antiliars) are instantiated, then members of these classes can't say this. But if they can, then the contradictions arise. In a logically consistent domain, this leads to the conclusion that these categories must be vacuous and all people lie sometimes and don't lie other times. That is interesting.

    Also note that "lying" denotes more than just telling untruths. There are the dimensions of knowledge and intention to deceive to be considered. To lie, one must know the truth and must intend to deceive the hearer. Are there limits here?
     
  5. Aug 21, 2003 #4

    Tom Mattson

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    "I am a liar".

    Nope, looks like I can say them.

    It is a paradox because it is a statement to which it is impossible to assign a truth value (that's what a paradox is). When I say, "I am lying" it means that it is the case that the statement I am making is false, which means I am not lying. Truth value: F

    Which means it is the case that the statement I am making is true, which means that I am lying. Truth value: T

    Which means that it is the case that the statement I am making is false, which means that I am not lying. Truth value: F
    .
    .
    .
    ...ad infinitum
     
  6. Aug 22, 2003 #5
    Hi Tom,


    Not to be able to say it means: "When I sey" = {}

    Therefore there is no paradox.
     
  7. Aug 22, 2003 #6
    Hi quartodeciman,

    To be a lair is first of all not to contradict your own property, so every layer knows exactly what is the truth.

    The truth for a layer is like death.

    Therefore, no one can hear a dead layer ( dead layer = {} ).

    The same logic holds for the honest.
     
    Last edited by a moderator: Aug 22, 2003
  8. Aug 22, 2003 #7
    Hi StephenPrivitera ,



    Because the statment = {} , there is no paradox.
     
  9. Aug 22, 2003 #8

    Tom Mattson

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    I don't know what you mean by {}, but I think you are getting hung up on an irrelevant detail. The liar's paradox is not about liars, but about self reference.

    Formal systems (such as logic) break down under self reference, or when statements refer directly to their own truth value. "Honesty" has nothing to do with it. You could just as well restate the paradox as follows:

    This sentence is false.

    So, it is the case that the above sentence is false. Truth value: F.
    So, it is not the case that the above sentence is false. Truth value: T.
    So, it is the case that the above sentence is false: Truth value: F.
    .
    .
    .
    ...ad infinitum

    The paradox is here to stay.
     
  10. Aug 22, 2003 #9
    Hi Tom,

    {} = the empty set (see ZF set theory) = content does not exist = 0

    Let X = This sentence

    0 = false = {} = content does not exist
    1 = truth = {X} = content exists

    If X refers ot itself (directly or indirectly) as false, then X does not exist in any stage.

    Therefore, there is no paradox.

    The paradox arises because we force something that does not exist,
    to act as if it exists.

    And then on top of this act of forcing, we build our artificial paradox.
     
    Last edited by a moderator: Aug 22, 2003
  11. Aug 22, 2003 #10

    Tom Mattson

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    OK

    How did you make this association? More specifically, what does the truth value have to do with existence of the statement?

    It looks to me more like the paradox is real and that your solution is artificial and forced. But maybe you can clear things up with your explanation of why "false" implies "does not exist".
     
  12. Aug 22, 2003 #11
    Hi Tom,

    Part 1:
    ---------

    not = ~

    0 = false-condition = {} = content does not exist

    1 = truth-condition = ~{} = content exists

    Let X = some ~{}



    ------------------------------------------------------------------------------

    Part 2:
    ---------


    {} = the empty set


    not = ~


    false-condition = {} = set's content ~exists = 0

    truth-condition = ~{} = set's content exists = 1


    false sentence exists = {sentence} = set's content exists = 1

    truth sentence exists = {sentence} = set's content exists = 1



    We must not mix between boolean-condition and {sentence}



    truth-condition operation on sentences:
    ------------------------------------------------------------------------------
    truth-condition on {sentence} implies ~{} = set's content exists = 1
    ------------------------------------------------------------------------------


    false-condition operation on sentences:
    ------------------------------------------------------------------------------
    false-condition on {sentence} implies {} = set's content ~exists = 0
    ------------------------------------------------------------------------------




    false-condition operation on conditions:
    ------------------------------------------------------------------------------
    false-condition on false-condition implies ~{} = set's content exists = 1

    false-condition on truth-condition implies {} = set's content ~exists = 0
    ------------------------------------------------------------------------------


    truth-condition operation on conditions:
    ------------------------------------------------------------------------------
    truth-condition on false-condition implies {} = set's content ~exists = 0

    truth-condition on truth-condition implies ~{} = set's content exists = 1
    ------------------------------------------------------------------------------


    The paradox arises because of the mixing between conditions and sentences.


    A sentence is a statement, not a condition.


    The one who can be operated (on itself and/or on a sentence), is the boolean-condition.


    So, we must not mix between an operator and anything that is not an operator (like some statement).


    Let X = "This sentence is a false" = {sentence} = ~{} = content exists

    X is not a boolean-condition but it tells us what boolean-condition
    should be operated on it:

    false-condition on X = set's content ~exist, therefore there is no paradox.


    ------------------------------------------------------------------------------
    To be {truth sentence} or to be {false sentence}, that is not the question.


    To be(=~{}), or not to be(={}), that is the question.

    ------------------------------------------------------------------------------
     
    Last edited by a moderator: Aug 23, 2003
  13. Aug 23, 2003 #12

    Tom Mattson

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    You are simply begging my question here. I asked you on what basis you associate "falsity" with "nonexistence". All you did here is re-assert the relationship.

    The way I see it, you are making 2 mistakes here.

    1.) First, you are associating the number 0 with the empty set. There is no justification for that.

    2.) Second, you seem to be confusing identificaiton with predication. That is, you are taking the mistake in 1.) and compounding it by identifying "this sentence" with "falsity", rather than taking "the sentence" as the subject of the predicate "false".

    Just think about it. The sentence "this sentence is false" must exist, because the idea of it can be formed in your mind. That alone is sufficient to qualify as existence for an abstract object of logic.
     
    Last edited: Aug 23, 2003
  14. Aug 23, 2003 #13
    The boolean system is based on X^2 = X

    Therefore X=1 or X=0.


    X = 1 = The universal set = {{}} = set's content exists = truth-condition

    X = 0 = The empty set = {} = set's content does not exist = false-condition


    So, in a boolean system there are only two states:

    {} = no members = 0 = false-condition

    ~{} = all members = 1 = truth-condition





    Sorry Tom,


    But you did not read the second part of my answer.

    Please go back and raed the second part of it.

    After you read it, then and only then you will understand my answer.


    Yours,

    Doron
     
    Last edited by a moderator: Aug 23, 2003
  15. Aug 23, 2003 #14

    Tom Mattson

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    Yes, I know.

    Once again, you are begging the question as to why truth value should be associated with existence.

    I did read it, and I stand by my objections.
     
    Last edited: Aug 23, 2003
  16. Aug 24, 2003 #15
    Hi Tom,

    In a boolean system any element logically belongs to {}(=0) XOR ~{}(=1)


    For example:

    Let X = "2 is in {3,4}" is false, therefore X logically belongs to {} = emply set = 0.

    Let X = "2 is in {2,7}" is truth, therefore X logically belongs to ~{} = {2,7} = non-empty set = 1.


    As you can see, in both cases X is some existing sentence.


    But we are not talking about the existence of a sentence as some thought or as some string of notations.

    We are talking about the logical existence of some element in a boolean logical system.

    Therefore, any element that refers to ITSELF (directly or indirectly) as logically false ("This X is false"), logically belongs to {} = empty set.

    The paradox is examined in the boolean logical space.

    Because its generator logically send itself to {}, there is no logical base for the paradox.

    Therefore, there is no paradox in the logical boolean space, but in some nonlogical imagination of the thinker.
     
    Last edited by a moderator: Aug 24, 2003
  17. Aug 24, 2003 #16

    Hurkyl

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    The defining property of the empty set is that nothing is an element of the empty set...
     
  18. Aug 24, 2003 #17
    Hi Hurkyl,

    Thank you, I mean logically belong ...


    I'll fix it.
     
  19. Aug 24, 2003 #18

    Hurkyl

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    Well, it doesn't really logically belong to the empty set either (whatever that means)... logic doesn't know what sets or numbers are; those are additional structures imposed by mathematical theories.


    In mathematical logic, "true" and "false" are not fundamental ideas; deducibility is the fundamental idea. However, one can define things called a truth assignments which is a logical function from well-formed sentences to either true or false.


    So, in set theory, for any truth assignment T, we have:

    T( T(2 is in {3, 4})=true )=false

    (I don't recall with 100% confidence, but I think truth assignments are fair game to be used like this)


    But the point is that truth assignments map things to the objects "true" and "false", and logic does not state that these two objects have any connection with 0, 1, {}, {2, 7}, or whatever ~{} is supposed to mean.
     
  20. Aug 24, 2003 #19
    ~ = not

    In a logical boolean space any state holds(=~{}} or does not hold(={}}
     
  21. Aug 24, 2003 #20

    Hurkyl

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    The problem is that {} is not a truth value, only "true" and "false" are truth values (at least in conventional mathematical logic). You can't negate it, xor it with anything, or say that sentences have some fundamental relationship with it.
     
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