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Liar Liar Pants on Fire

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Stephanos
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0 posted 01-03-2013 06:26 PM       View Profile for Stephanos   Email Stephanos   Edit/Delete Message      Find Poems   Click to visit Stephanos's Home Page   View IP for Stephanos

Read this on Wikipedia.  I've heard that the Liar's Paradox has never been adequately solved.  My question for you arm-chair logicians is why wouldn't this be a solution?  

quote:
Arthur Prior asserts that there is nothing paradoxical about the liar paradox. His claim (which he attributes to Charles Sanders Peirce and John Buridan) is that every statement includes an implicit assertion of its own truth. Thus, for example, the statement, "It is true that two plus two equals four", contains no more information than the statement "two plus two equals four", because the phrase "it is true that..." is always implicitly there. And in the self-referential spirit of the Liar Paradox, the phrase "it is true that..." is equivalent to "this whole statement is true and ...".

Thus the following two statements are equivalent:
This statement is false.
This statement is true and this statement is false.

The latter is a simple contradiction of the form "A and not A", and hence is false. There is therefore no paradox because the claim that this two-conjunct Liar is false does not lead to a contradiction. Eugene Mills and Neil Lefebvre and Melissa Schelein present similar answers.


Ron, anyone?
Grinch
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1 posted 01-06-2013 08:10 PM       View Profile for Grinch   Email Grinch   Edit/Delete Message      Find Poems  View IP for Grinch


quote:
why wouldn't this be a solution?


Because, in the self-referential spirit of the Liar Paradox, you can't know for sure the truth value of "it is true that..."

Ron
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2 posted 01-06-2013 09:48 PM       View Profile for Ron   Email Ron   Edit/Delete Message      Find Poems   Click to visit Ron's Home Page   View IP for Ron

LOL @ Grinch. That's exactly why the Liar Paradox has persisted to fascinate for these few thousand years; it calls into question the very existence of Truth or, at least, our ability to perceive it.

Stephen, I think Prior's proposed solution has a few problems, starting with it's initial assertion. "Every statement includes an implicit assertion of its own truth" can be disproved in several ways, though of course only one is necessary to invalidate the assertion. Remember the roomful of monkeys typing away for a thousand years and producing the entire collected work of Shakespeare? Or imagine twirling your spoon in a bowl of alphabet soup, then looking down to discover the letters had spelled out "bad soup." I don't think anyone would argue that such random statements assert their own truth. Don't want random? The common phrase "willing suspension of disbelief," I think, lends lie to Prior's assertion. The artist, and especially the performing artist, tell his audience he is going to lie and then he lies with absolutely no assertion of truth (except perhaps below the surface and rarely in direct relation to the lie being told).

Want more?

Bertrand Russell formulated the Liar Paradox in terms of set there, correctly recognizing that self-reference lies at the heart of the paradox, and indeed, at the heart of most logical paradoxes. In joining Prior's implied statement with the direct statement, the referents are subtly changed. "This statement is true and this statement is false" are no longer clearly self-referencing. "This statement" can't unambiguously be applied to two statements. The appearance of non-contradiction is directly the result of removing (or at least greatly diluting) the self-referential characteristic of the paradox.

Finally, Prior's solution doesn't apply to versions of the Liar Paradox that are not directly self-referential (it can't since it relies on removing the self-reference), such as multi-sentence versions.

The next sentence is false.

The preceding sentence is true.

Change those sentence per Prior's version ...

This sentence is true and the next sentence is false.

This sentence is true and the preceding sentence is true.

... and you find yourself stuck right back in the paradoxical loop. Indeed, I think one of the characteristics of the Liar Paradox is that any time a solution looks like it might work, a simple rephrasing of the Paradox leads us right back into the same conundrum.


Essorant
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3 posted 01-07-2013 02:49 AM       View Profile for Essorant   Email Essorant   Edit/Delete Message      Find Poems   Click to visit Essorant's Home Page   View IP for Essorant

I think a "solution" to the "paradox" is understanding that the adjective "false" is contradicted and therefore can't be given the value of "true" or "false".  But there is a "true" value from the other words because the statement is indeed "this (statement)..." "(a) statement" and "is (something)"
   

           true    false
this         1       0
statement    1       0
is           1       0
false        0       0
           ______________          
             3       0


That makes three points as true, and no points as false.   Since nothing adds up to a point on the "false" side then the points on the "true" side get the victory.  The statement is true.    
 
Stephanos
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4 posted 01-16-2013 09:59 PM       View Profile for Stephanos   Email Stephanos   Edit/Delete Message      Find Poems   Click to visit Stephanos's Home Page   View IP for Stephanos

Ron:
quote:
The artist, and especially the performing artist, tell his audience he is going to lie and then he lies with absolutely no assertion of truth (except perhaps below the surface and rarely in direct relation to the lie being told).

Want more?


Wait a minute before you give me more.  I don't think random produces much of anything, and even when it does, we adopt it into our own rational framework.

The suspension of disbelief is a way to enjoy fiction knowing it isn't really happening, or propositional.  I'm familiar with it when I consider monkeys typing Shakespeare or watching "The Dark Knight Rises".  But if you want to say the Liar's paradox is a kind of fiction, I'll buy that.  I think it's fascinating, but I think the mystery is dramatized.

quote:
"This statement" can't unambiguously be applied to two statements.


Why not?  Why does a statement have to be limited to one adjective rather than two?


quote:
Finally, Prior's solution doesn't apply to versions of the Liar Paradox that are not directly self-referential (it can't since it relies on removing the self-reference), such as multi-sentence versions.

The next sentence is false.

The preceding sentence is true.

Change those sentence per Prior's version ...

This sentence is true and the next sentence is false.


But I think to correctly reflect Prior's solution, and thinking, you'd change them to:

The next sentence is false and true.

The preceding sentence is true and false.


Stephen  


Stephanos
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5 posted 01-16-2013 10:00 PM       View Profile for Stephanos   Email Stephanos   Edit/Delete Message      Find Poems   Click to visit Stephanos's Home Page   View IP for Stephanos

Essorant, you solved the liar's paradox by philosophically dissolving "false", and saying it doesn't exist?  wow!  Where did you put your magic wand?  ;-)
Stephanos
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6 posted 01-16-2013 10:14 PM       View Profile for Stephanos   Email Stephanos   Edit/Delete Message      Find Poems   Click to visit Stephanos's Home Page   View IP for Stephanos

Ron, you're into magic.  Don't you see the Liar's Paradox as first and foremost a bit of sleight of hand ... or tongue?  lol.
Ron
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7 posted 01-16-2013 11:29 PM       View Profile for Ron   Email Ron   Edit/Delete Message      Find Poems   Click to visit Ron's Home Page   View IP for Ron

On the contrary, Stephen, I think the Liar Paradox, and others of similar ilk, are a direct result of Godel's Incompleteness Theorem (also detailed on the same Wiki page you referenced in your first post). While he addressed mathematics, many would argue that linguistics necessarily follows the same path. In any non-trivial system, including Language, there will always be propositions that can't be proven either true or false. It's the nature of the beast.

The Liar Paradox essentially shows we are non-trivial.

I'll even crawl out here at the very tip of this tree limb and go a bit further. I believe logical paradoxes, the Heisenberg Uncertainty principle, and division by zero are very subtle clues to the nature of our reality.
Essorant
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8 posted 01-16-2013 11:53 PM       View Profile for Essorant   Email Essorant   Edit/Delete Message      Find Poems   Click to visit Essorant's Home Page   View IP for Essorant

Hi Stephanos,

Only in a similar sense to saying 2 is "dissolved" in "2 - 2".  

The meaning "false" (as well "true") is contradicted in the statement by the word "false", therefore it can't have a fixed "false" or "true" value.   Therefore, the only value you are left with is the true value of the other words, true because they are backed up by the evidence/existence of the statement itself.  Obviously the words "this statement is..." are true because the statement exists.  It can be proven.  While the word "false" in this case merely contradicts itself and therefore doesn't retain any "true" or "false" meaning.

[This message has been edited by Essorant (01-17-2013 11:37 AM).]

Stephanos
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9 posted 01-17-2013 01:47 AM       View Profile for Stephanos   Email Stephanos   Edit/Delete Message      Find Poems   Click to visit Stephanos's Home Page   View IP for Stephanos

Ron, fascinating.  Could you comment on what you mean by nontrivial ... and on how you think it describes our nature?  

Thanks.
Stephanos
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10 posted 05-11-2014 01:55 AM       View Profile for Stephanos   Email Stephanos   Edit/Delete Message      Find Poems   Click to visit Stephanos's Home Page   View IP for Stephanos

Ron:  
quote:
Prior's solution doesn't apply to versions of the Liar Paradox that are not directly self-referential (it can't since it relies on removing the self-reference), such as multi-sentence versions

The next sentence is false.

The preceding sentence is true.

Change those sentence per Prior's version ...

This sentence is true and the next sentence is false.

This sentence is true and the preceding sentence is true.



Just wondering why the following would not more closely represent Prior's thinking:  

This sentence is true therefore the next sentence is true and false.  

This sentence is true, therefore the preceding sentence is true and false.  

I know I've taken liberty with the sentences.  But my point is that whenever logical syllogism is applied to the liar's paradox, whether in the form of one sentence or two, the contradiction (rather than paradox) becomes apparent.

It's late ... I could be way off base here.  
Stephanos
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11 posted 05-11-2014 02:02 AM       View Profile for Stephanos   Email Stephanos   Edit/Delete Message      Find Poems   Click to visit Stephanos's Home Page   View IP for Stephanos

Also Ron ... To answer your example of fiction ... The context of fiction supplies a preceding statement of its own.  "The following motion picture ..."  The very smell of over priced buttered popcorn tells me:  "I am pretending when I say that Batman watches over Gotham City."  In fact when ever these implicit or explicit statements are missing we say folks are liars or crazy.  
 
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