Self-reference paradoxes

[lpetrich]

The simplest of these is the  liar paradox:

This statement is false.

A version of it is in Kurt Gödel's famous incompleteness theorem. In any formal system that includes Peano's axioms of arithmetic, an axiomatic formulation of the natural numbers, one can construct a theorem G that states "G is not a theorem".

Another one is the  barber paradox suggested by Bertrand Russell in connection with his set-theory paradox.

A barber in a certain town is a man who shaves every man who does not shave himself (assume that all the men can grow facial hair). Who shaves the barber?
A hairdresser in that town is a woman who styles the hair of every woman who does not style her own hair (assume that all the women can grow scalp hair). Who styles the hairdresser's hair?

That set-theory paradox is  Russell's paradox. Some sets are members of themselves, like the set of all abstract ideas. Others are not, like the set of all physical objects. A set that is not a member of itself we can call a "normal set". Is the set of all normal sets itself a normal set?

Any other such paradoxes?

 

[Don2 (Don1 Revised)]

 

[Bomb#20]

A version of it is in Kurt Gödel's famous incompleteness theorem. In any formal system that includes Peano's axioms of arithmetic, an axiomatic formulation of the natural numbers, one can construct a theorem G that states "G is not a theorem".

Well, one can construct a true statement G that states "G is not a theorem". But since you can't construct a proof of G from Peano's axioms, G isn't a theorem.

Another one is the  barber paradox suggested by Bertrand Russell in connection with his set-theory paradox.

A barber in a certain town is a man who shaves every man who does not shave himself (assume that all the men can grow facial hair). Who shaves the barber?

Not really a paradox, just a false statement, sort of like "A certain triangle has four sides".

Any other such paradoxes?

"Yields falsehood when appended to its own quotation" yields falsehood when appended to its own quotation.

Suppose not all natural numbers are interesting. Then there must be a smallest uninteresting number. What an interesting property! Therefore every natural number is interesting. But for what property is some arbitrary number N proven to be interesting, when we've just ruled out the existence of the very property we used in the proof?

 

[Don2 (Don1 Revised)]

Titus 1:12.

One of themselves, even a prophet of their own, said, The Cretians are always liars, evil beasts, slow bellies.

 

[Juma]

"Yields falsehood when appended to its own quotation" yields falsehood when appended to its own quotation.

But that is just wrong, not false.

 

[Bomb#20]

"Yields falsehood when appended to its own quotation" yields falsehood when appended to its own quotation.

But that is just wrong, not false.

Well, I'm not sure what distinction you're making between "wrong" and "false", and I don't really want to refight the whole of last year's "Logic and self referential statements" thread, so let's just skip to the end of it. Whatever your objection to

"Yields falsehood when appended to its own quotation" yields falsehood when appended to its own quotation.​

is, do you have the same objection to

The result of inserting a copy of itself in quotation marks into the sentence "The result of inserting a copy of itself in quotation marks into the sentence between the fourteenth and fifteenth words is a sentence that isn't true." between the fourteenth and fifteenth words is a sentence that isn't true.​

?

 

[Juma]

But that is just wrong, not false.

Well, I'm not sure what distinction you're making between "wrong" and "false", and I don't really want to refight the whole of last year's "Logic and self referential statements" thread, so let's just skip to the end of it. Whatever your objection to

"Yields falsehood when appended to its own quotation" yields falsehood when appended to its own quotation.​

is, do you have the same objection to

The result of inserting a copy of itself in quotation marks into the sentence "The result of inserting a copy of itself in quotation marks into the sentence between the fourteenth and fifteenth words is a sentence that isn't true." between the fourteenth and fifteenth words is a sentence that isn't true.​

?

"False" and "true" are two completary values in the room V that you assign to a proposition according some rule R.

What R are you using in your example?

 

[Bomb#20]

Well, I'm not sure what distinction you're making between "wrong" and "false", and I don't really want to refight the whole of last year's "Logic and self referential statements" thread, so let's just skip to the end of it. Whatever your objection to

"Yields falsehood when appended to its own quotation" yields falsehood when appended to its own quotation.​

is, do you have the same objection to

The result of inserting a copy of itself in quotation marks into the sentence "The result of inserting a copy of itself in quotation marks into the sentence between the fourteenth and fifteenth words is a sentence that isn't true." between the fourteenth and fifteenth words is a sentence that isn't true.​

?

"False" and "true" are two completary values in the room V that you assign to a proposition according some rule R.

What R are you using in your example?

I don't see how that makes a difference as long as the rule R isn't self-contradictory. If you think it matters, for the sake of discussion we can make R as ridiculous and unbelievably stupid as you please. Let R call a proposition "true" if and only if the proposition matches the version of the Lucas-verse where Greedo shot first.

 

[Juma]

"False" and "true" are two completary values in the room V that you assign to a proposition according some rule R.

What R are you using in your example?

I don't see how that makes a difference as long as the rule R isn't self-contradictory. If you think it matters, for the sake of discussion we can make R as ridiculous and unbelievably stupid as you please. Let R call a proposition "true" if and only if the proposition matches the version of the Lucas-verse where Greedo shot first.

Ok. Then the text is "false", self referencing or not. Case closed.

 

[Bomb#20]

I don't see how that makes a difference as long as the rule R isn't self-contradictory. If you think it matters, for the sake of discussion we can make R as ridiculous and unbelievably stupid as you please. Let R call a proposition "true" if and only if the proposition matches the version of the Lucas-verse where Greedo shot first.

Ok. Then the text is "false", self referencing or not. Case closed.

Well, if the text is false, then that implies the text isn't true, in Greedo-shot-first-world. Do you agree?

 

[Juma]

Ok. Then the text is "false", self referencing or not. Case closed.

Well, if the text is false, then that implies the text isn't true, in Greedo-shot-first-world. Do you agree?

Absolutely.

 

[Bomb#20]

Well, if the text is false, then that implies the text isn't true, in Greedo-shot-first-world. Do you agree?

Absolutely.

But that's exactly what the text asserts. So it makes an assertion you absolutely agree with. What more does it take to match Greedo-shot-first world and thereby satisfy rule R's criterion for "true"?

 

[Juma]

Absolutely.

But that's exactly what the text asserts. So it makes an assertion you absolutely agree with. What more does it take to match Greedo-shot-first world and thereby satisfy rule R's criterion for "true"?

Where is the paradox?

 

[Gila Guerilla]

. . .
This statement is false
. . .

The statement above is neither true nor false.

Neither is it a paradox, (IMHO), as it is impossible to parse logically.

I'd call a paradox something where two truths contradict, (maybe only apparently).
In relativity, there is the twin paradox,  Twin Paradox, in which two people begin and end
at the same point in time and space, but age very differently. The paradox is explained by time dilation.

The statement given above has no identifiable truth or falsity value.

 

[Bomb#20]

But that's exactly what the text asserts. So it makes an assertion you absolutely agree with. What more does it take to match Greedo-shot-first world and thereby satisfy rule R's criterion for "true"?

Where is the paradox?

The text satisfies the criterion to be called "false", and it also satisfies the criterion to be called "true". What makes a paradox, if not this?

 

[Juma]

Absolutely.

But that's exactly what the text asserts. So it makes an assertion you absolutely agree with. What more does it take to match Greedo-shot-first world and thereby satisfy rule R's criterion for "true"?

Sorry, i answered this wrong... : i think i misunderstood your R.

Exactly how are you ment to evaluate R? I thought you ment that the text existed verbatim in the manuscript, but that is obviosly not what you had in mind.
It seems to me that you havent specified any rule R yet.

 

[Bomb#20]

But that's exactly what the text asserts. So it makes an assertion you absolutely agree with. What more does it take to match Greedo-shot-first world and thereby satisfy rule R's criterion for "true"?

Sorry, i answered this wrong... : i think i misunderstood your R.

Exactly how are you ment to evaluate R? I thought you ment that the text existed verbatim in the manuscript, but that is obviosly not what you had in mind.
It seems to me that you havent specified any rule R yet.

Sorry, I wasn't clear. I meant R calls something "true" if it's shown in the relevant versions of the films, or if it follows from what's shown by normal reasoning. They never mention this in the movies, but we can infer that "Jabba the Hutt was alive during the game between Chewie and R2." is true in that universe, since he was alive before it and he was alive after it. Conversely, the text appears verbatim in the manuscript that Luke's father had already been killed at that time, but we know from elsewhere in the movies that that was a lie.

If this is at all bogging us down, by all means we can switch to R', the rule which calls a proposition "true" if and only if the proposition matches the real world. I only picked R to argue that the paradox will show up no matter what rule you pick.

 

[Nexus]

The text satisfies the criterion to be called "false", and it also satisfies the criterion to be called "true". What makes a paradox, if not this?

I don't know whether Chomsky said this or someone else connected it to his work, but the text while semantically correct has no inherent meaning. A meaningless string of words is not true or false just like its neither an acid or base.

 

[Random Person]

Some things you know to be true.

Some things you know are not true.

A paradox is something that you know isn't true but still think it is.

All wisdom is paradoxical.

 

[Juma]

Some things you know to be true.

Some things you know are not true.

A paradox is something that you know isn't true but still think it is.

All wisdom is paradoxical.

No, a paradox is that something seems obviously false but in fact is really true.