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Passions in Poetry

Self-referential Paradoxes

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Crazy Eddie
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since 09-14-2002
Posts 221


25 posted 10-12-2002 08:53 AM       View Profile for Crazy Eddie   Email Crazy Eddie   Edit/Delete Message      Find Poems  View IP for Crazy Eddie

Ron,

Having trouble making them drink?    

Using an example of a Paradox that allowed reference to grammatical mistakes in a forum full of poets and writers could be classed as an unfortunate accident waiting to happen.

This statement is true.

And

This statement is false.

Would have been safer examples.

I spent some time a couple of years ago trying to explain to my wife why the first was not a paradox and the second definitely was. That explanation may be of some use to clarify the term paradox and in particular the self-referencing aspect of the above examples.

The paradox is based upon a fundamental rule - that every proposition has a truth-value that is either true or false. It isn’t important whether the truth-value is known, only that it exists and is either true or false. In fact the truth-value can be both at different times and in different places and situations:

It is raining.

Could have a truth-value of true or false depending on whether it is in fact raining. It must be one or the other and cannot be both at the same time If we were to apply the same truth-value test to the first example - “This statement is true” – . We can only reference the statement itself to decided whether it is true or false, the proposition itself demands that the truth-value is in fact true but we can test that.

If we re-write the proposition to read:

This statement has a truth-value of true.

Then temporarily assign it a truth-value of true the logic of the proposition works – the truth-value of the statement is true and the proposition has a truth-value of true.

Now let’s try assigning a truth-value of false – the truth-value of the statement is false and the proposition has a truth-value of false, the logic works again. However because the statement is self-referencing, claiming that it is true, probability dictates that it is likely to have a truth-value of true but a truth-value of false is possible.

This statement is false.

This second example is more difficult (if not impossible) to assign a truth–value to, if we re-write the proposition we get this:

This statement has a truth-value of false.

Let’s temporarily assign a truth-value of false and check the logic – the truth-value of the statement is false and the proposition has a truth-value of false. This obviously doesn’t work, if the truth-value of the proposition is false the truth-value of the statement must be true.

Ok, let’s try temporarily assigning a truth-value of true– the truth-value of the statement is true and the truth-value of the proposition is true. This doesn’t work either, if the truth-value of the statement is true the truth-value of the proposition must be false.

The paradox is that it must be one or the other but it can’t be both or none, I think the common factor you’re looking for Ron could be classified as self-creating contradiction.

[This message has been edited by Crazy Eddie (10-12-2002 04:16 PM).]

Brad
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since 08-20-99
Posts 5896
Jejudo, South Korea


26 posted 10-12-2002 10:21 PM       View Profile for Brad   Email Brad   Edit/Delete Message      Find Poems  View IP for Brad

I'm not completely sure I'm applying this theory correctly but consider Russel's theory of types.  It seems clear that it is a paradox in first order logic but you can abstract it and say, "'Ther are three misteaks in this sentence'is true' is a paradox' is true. By abstracting into what's called second order logic, the statement itself is no longer paradoxical. Paradoxical statements in second order logic can be abstracted into third order logics and so on and so forth. If this is unclear, I'll try to explain the reasoning on set theory later. Now, as it turns out, this nice little trick has had some real world practical application. Still, I intuitively feel that were trading one logical problem, the paradox, for another, the infinite regression.

Or perhaps we should call it the infinite progression.

Brad
Ron
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27 posted 10-13-2002 01:04 AM       View Profile for Ron   Email Ron   Edit/Delete Message      Find Poems   Click to visit Ron's Home Page   View IP for Ron

Self-creating contradiction has some promise, Eddie. But I'm not sure it's a great deal more helpful than simply looking for the paradox. Let's look at a less obvious example.

Every rule has an exception.

If we define the statement as a rule, which isn't a stretch I think, then this is certainly a self-referencing statement. Is it true, false, or a paradox? If it's true, then there must be an exception to this rule, too, meaning it can't be true. It's a paradox, or as you say, a self-creating contradiction. But, if the statement is false? Then there's no contradiction and no paradox. Unfortunately, this falls in the class of "exhaustive" sets where it's impossible to prove it false. The best we can probably hope to do is find a rule with no "known" exception, and that's not good enough to prove that this maybe-a-paradox rule is false. In short, I don't really know if it's a paradox.

Any self-referencing statement, I think, is automatically suspect. But how suspect?

Brad, you pretty much got it, at least to my understanding. Russell's treatment was to allow anything within a given set to only reference things within lower order sets. With this method, he disallowed direct self-referencing (within the same set) and also prevented indirect self-referencing ("The following sentence is true. The preceding sentence is false."). Of course, it had this nasty little side-effect - I could no longer talk about me.

Gödel showed that the elimination of paradoxes necessarily introduced inconsistency. We "might" be able to get rid of paradoxes, but we really don't want to. That's cool. But I'd still like to understand a bit more about their nature.
Toerag
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since 07-29-99
Posts 5839
Ala bam a


28 posted 10-14-2002 05:34 PM       View Profile for Toerag   Email Toerag   Edit/Delete Message      Find Poems  View IP for Toerag

It all comes down to whether you are looking thru a windshield that is in fact a windshield as in (shielding one from the wind), or...if it's a window, a clear object intended for seeing through....I am quite sure the real mistake was in your spelling checker however

[This message has been edited by Toerag (10-15-2002 08:04 AM).]

Stephanos
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since 07-31-2000
Posts 3496
Statesboro, GA, USA


29 posted 10-15-2002 11:08 PM       View Profile for Stephanos   Email Stephanos   Edit/Delete Message      Find Poems   Click to visit Stephanos's Home Page   View IP for Stephanos

Maybe some of you could help me, as I am not familiar with the philosophical terminology related to this phenomenon under discussion...  Is there not among all the great minds who have written about self referential paradoxes, the notion or suggestion that in order for a sentence to be true or false, (logically) there is required some external standard or reference by which to judge it true or false.  


For when I read the sentence, "This sentence is false".  this is what happens in my brain ... (help me if I am hopelessly or philosophically or linguistically naive) ... I think to myself, "It is impossible to say meaningfully that this sentence is either false or true, because it refers to nothing distinct from itself which can be judged as true or false.  Any assignment of true or false would be completely arbitrary (for the sake of seeing where the sentence goes) which in my mind does not comport with the determination of any truth.  When I inquire why I should believe the sentence to be false, the only answer that can come from the sentence itself is "because it is false", begging the question.  

I am stuck in a position of wondering why the game should be played at all, because the whole structure of the sentence lends to an absurdity.  Paradoxes are often just truths in seeming juxtaposition, but paradoxes which cannot be explained may be absurdities, not real paradox at all.  The bar of logic seems to deny such a sentence a hearing at all, that's why it flounders outside of court in and endless regression or progression of stalemate.


Am I the only one who has seen something fundamentally wrong with the question itself?  Who would write such a sentence in real life, for the purpose of making a statement?  Is there nothing required of the intent of the stater?  Doesn't a statement necessitate a certain intrinsic integrity of purpose in order to be genuine?  ie  that it was really trying to state something, and not merely pose an intellectual or linguistic dilemma?

[This message has been edited by Stephanos (10-15-2002 11:11 PM).]

 
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