Re: Liar paradox, one more time

From: Daryl McCullough (daryl_at_atc-nycorp.com)
Date: 07/30/04 

Date: 30 Jul 2004 07:55:38 -0700

Poker Joker says...

>Let's just say that a sentence is semantically acceptable if its meaning
>is well understood; Almost everyone who would be expected to receive
>the message would gather the same meaning. With regards to the
>following sentence: "This sentence is not true." What meaning do you
>gather? Do you think it is the same meaning I gather?

Yes, you're right. But what that shows is that there is something
funny about the properties "is true" and "is not true". If you replace
these properties by any other well-defined property, then there is no
problem. For example: "appears on page 32 of my copy of 'The Hobbit'"
or "is 40 characters in length". The corresponding self-referential
sentences are perfectly meaningful:

     This sentence appears on page 32 of my copy of 'The Hobbit'
     (Meaningful, but false)
     This sentence is 40 characters in length
     (Meaningful, and true if I counted correctly)

So, there is something special about "is true" (and other semantic
properties such as "is meaningful"). While the claim that a string
is 40 characters in length is always meaningful, the claim that a
string is true (or not true) is not always meaningful.

Here's a way of looking at the Liar paradox that perhaps sheds some
light on it.

Suppose that we enumerate all grammatically correct English sentences
of fewer than 1000 words in length. We put each sentence onto a separate
index card, and we give the cards to God (or any other omniscient
being). God sorts the cards into three piles, depending on the sentence
written on them. Pile A contains those cards that are true. Pile B
contains those cards are false. Pile C contains those cards that are
meaningless, or whose meaning is too ambiguous to be considered definitely
true or definitely false.

At some point, God comes upon a card upon which is written "This card
will not be placed into Pile A". This sentence is perfectly meaningful.
It's a prediction about what God is going to do to a particular card,
and it's perfectly unambiguous which card is being referred to (the
card on which the sentence is written). However, God is stuck:

     If He puts the card in Pile A, then the sentence is false,
     and so Pile A has a false sentence in it.

     If He doesn't put the card in Pile A, then the sentence is
     true, and so there is at least one true sentence that is not
     put into Pile A.

The conclusion is this: Not even God can sort all sentences into
piles according to whether they are true or not. 

--
Daryl McCullough
Ithaca, NY

Relevant Pages

  • Re: Liar paradox, one more time
    ... God sorts the cards into three piles, ... Pile A contains those cards that are true. ... > meaningless, or whose meaning is too ambiguous to be considered definitely ... God comes upon a card upon which is written "This card ...
    (sci.logic)
  • Re: Liar paradox, one more time
    ... God sorts the cards into three piles, ... Pile A contains those cards that are true. ... >> It's a prediction about what God is going to do to a particular card, ... If He puts it in pile D, then He *doesn't* put it in pile A, ...
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  • Re: Liar paradox, one more time
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  • Re: Liar paradox, one more time
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  • Re: OOA?
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