Are paradoxes only created by self-reference? Do non-self-referencing paradoxes exist?

Are paradoxes only created by self-reference? Do non-self-referencing paradoxes exist?

Other urls found in this thread:

en.wikipedia.org/wiki/Unexpected_hanging_paradox
en.wikipedia.org/wiki/List_of_paradoxes
en.wikipedia.org/wiki/Tarski's_undefinability_theorem).
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Aren't all paradoxes just wordplay caused by the vagueness of expression? Do any paradoxes exist that can not be settled by defining them properly?

do Kant's Antinomies count?

Looks like wordplay to me. Does he even provide one single paradox, those are just pairs of contradictory statements, nowhere does he imply that any of them are true, only that both can't be true, and even then exact definitions of the used words will reveal loopholes.

paradox is self-contradictory statement, so by definition, yes.

Technically Yablo's paradox isn't self-referential.
Depending on your definition, Richard's paradox may also qualify.

This one I guess:

What is the likelihood to solve this question by random choice?
a) 25%
b) 25%
c) 50%
d) 0%

What about this one OP?
en.wikipedia.org/wiki/Unexpected_hanging_paradox

Yablo: The definitions are meaningless. Nonsensical things can be defined.
1. All amounts over 2 is 1 is false.
2. All amounts over 3 is 1 is false.
3. All amounts over 4 is 1 is false.
Statements 2 and 3 are not false because 1 is true. If the statement was, "all and only all", but is it just a subset of all values producing a false result. The only thing it proves, is that language is not precise.
Richard: The only thing he demonstrates is that there are different orders of infinite. Just because the first set of definitions for some (infinite) subset of the real numbers is infinite, does this not mean these infinites have the same cardinality. The set of sentences that describe all real numbers consisting of only the number 4 is limitless. "four" "four and four".. It is infinite. The set of all integers is infinite. But these are two different infinites, so there is no contradiction.

The percentages. Since a and b are the same answer, they should be considered as one when answering. There are only 3 possibilities. 0% 25% 50% The likelihood to choose the correct RESULT by chance is 1/3 if only the possible results are considered. If one chooses a box to tick from the 4 possibilities, then there is a 1/2 chance to answer 25% and a 1/4 chance to answer 0% and 1/4 for 50%. In this case there is no answer that fulfills the conditions. The question is meaningless. It is not real.
The hanging is just a misunderstanding. Timing is a surprise, so it can not be last minute, since it wont be a surprise the moment before last minute. Fair enough. But any time before last minute will be a surprise. It could be now (thursday) or last minute (friday) so if we should take the judges words literally, anything from monday to friday is ok. But I suspect the meaning of the judge was not that. It was simply that we won't tell you, not that we guarantee you can not know, so in reality friday might have been an option, since arguably the surprise only needs to hold at the time of the statement, not on thursday evening.

Most paradoxes are not self-referencing and most paradoxes are not caused by wordplay or vagueness.

Show me one that is not resolvable by demonstrating that there may be more than one interpretation, i.e. it there is wordplay at play.

en.wikipedia.org/wiki/List_of_paradoxes

Take your pick.

You are the one making the claim here, so show the evidence. If I pick one to pick apart, you will just tell me that maybe that one is resolvable, but those other ones are not. You pick one that you think is bulletproof and I'll start shooting.

>they should be considered as one when answering
Ok you are arguing semantics here. Let's phrase it like this: If you pick a letter a to d, how likely is it to pick the letter belonging to the likelihood to answer this question right.
Also: Lets assume there where only 3 possibilities.
> In this case there is no answer that fulfills the conditions.
Correct, and therefore the probability to pick the right one should be 0 shouldn't it? And therefore the chance to pick the correct answer would be 1/3 leaving you with a contradiction.

>It is not real.
>How can this question be real if your eyes ain't real???!!??!?!!
No, seriously tho: Define real.

It is impossible to pick the correct answer because the option to do so is not presented. This is not a paradox. This is a use of language that is not in concert with reality. It is nonsense.

If the probability to pick 1/3 is zero and that is the "answer" then the question itself is again not in line with reality, with what can be. The option of zero when chosen yielding 1/4 is nonsense. It has as much substance as
pick one:
a) it is impossible to pick a

Those are both instances of things being defined that are not real, that is not possible to exist and/or not possible be defined. Real is that which can be. That which is not contradictory. That which is precise. Some thing can be included in a contradictory statement and still be real, but the entire contradictory statement as a package is not real.

Any "paradoxical" statement (or system of statements) for which neither True or False can be applied, has a complementary statement which can take either value.

For example,
(non-paradoxical)
>The sky is blue
{T}
>The sky is not blue
{F}

(paradoxical)
>This statement is true
{T,F}
>This statement is false
{}

Both True and False are contained within any matched pair, they're just not necessarily distributed evenly.

What does this actually prove? That it is possible to define things that have no connection to reality? What is the value of presenting this self-evident fact in such a convoluted form?

"True" and "False" don't exist in reality.

They're platonic ideals, like circles.

(The sky is not always blue.)

>That which is not contradictory.
But thats the definition of a paradox my mane.
You are simply saying "Nuhh uhh it isn't a paradox because it contradicts itself!!!"

Within a certain frame of reference it is true. When a person states the color of the sky, they usually mean earths atmosphere in daylight with no clouds or dust. Blue is correct (TRUE) within the frame of reference that is implied. If the frame of reference is not defined precisely it is not surprising that people make the wrong assumptions, especially when the subject is something more abstract than the color of the sky. Even if you only accept that true and false only exist as ideas, then you should allow them to be the result of systems consisting of pure ideas, logic. But I don't think we need to go this far. True and false can be successfully applied to statements in natural languages about the real world, if the scope, the frame of reference is defined precisely.

I'm saying that no paradox has any instances in the real world. The meaningfulness and usefulness of a paradox is to find problems in reality. If it is just wordplay it is meaningless. If it is shown that a defined paradox does not exist in reality then it has no bearing on the world. What value has it then? What is it even?

By definition, a paradox is a sentence (or set of sentences) that are syntactically well-formed but cannot be assigned a consistent truth-value under Tarskian semantics.

Saying that paradoxes are "meaningless" is tautological because meaningless is the defining property of paradoxes, and saying that they do not "exist reality" betrays your ignorance of incompleteness (en.wikipedia.org/wiki/Tarski's_undefinability_theorem).

What you are saying is that paradoxes are sentences that make sense in one sense, but makes no sense in another sense, and this exact property is what makes them paradoxes.

I still don't see the point in them. Is it just a special subset of all meaningless statements, and that rule of "syntactically well formed" and "no truth value assignable" is the qualifying test? What use do we have of them? What can they tell us about reality, or even language, or even meaninglessness? What useful information can be extracted from the fact that some meaningless statements are paradoxes?

...

it's to separate the men from the boys

So the fact that my take on the argument is often proven to be true and usually instinctively known to be true, is evidence of its falsehood?

if your argument is proven to be true, then of course it cannot be evidence of its falsehood

Show me one paradox that is real. Just one.

>usually mean
look how this rationalist think he is an empiricist.

what do you mean?

A paradox that shows an actual logical disconnect in nature. Something that is precisely defined and still result in a logical trap. If that exists you prove that nature is broken. If all your paradoxes are just erroneous use of reasoning logic or language then they have not connection to reality and can not tell us anything about anything. They are just poetry.

>actual logical disconnect in nature
what is that
>Something that is precisely defined and still result in a logical trap
{x : x not in x }
>If that exists you prove that nature is broken
how does a 'broken nature' follow? what even is that

In all seriousness, paradoxes in all shapes and sizes can show us the limitations of the reasoning system in which the paradox is formulated. E.g. Russell's paradox actually inspired improvements in logic, which arguably made the field robust enough to serve as a foundation for mathematics and the technology we all love and use today

>Aren't all paradoxes just wordplay caused by the vagueness of expression?
only most.

The rest involve fantasy like time travel and philosophical questions that aren't interesting.

>What is the likelihood to solve this question by random choice?
>a) 25%
>b) 25%
>c) 50%
>d) 0%
That is not a paradox, that is being asinine.

With "actual logical disconnect in nature" I mean an instance when something REAL is simultaneously true and false.

Using a formal notation does not guarantee a precise definition. That definition is clearly nonsense, so how can it be precise?

By "broken nature" I mean an instance where a REAL thing is simultaneously true and false. A paradox come to life.

So you are saying that defining nonsensical objects in a formal notation shows us that the formal notation has no built in mechanism to guarantee that all statements in that notation are valid in the real world. That is something. And finding these loopholes made it possible to define more rigid notations to avoid these problems. Well. Good.

Not a paradox.

It is simply a false statement.

Plus the barber could keep a beard and therefore never be shaven.

It's a little more than notation: for a long time informal set theory was thought to be rigorous enough. And when you think about it, why not? Until, that is, you encounter a paradox. Same thing with incompleteness of arithmetic.

Something REAL cannot be true or false, since it is not a proposition, not a truth baring statement.

I think your confusion lays in the fact that you conflate thought with existence. A paradox is something that lives on the thought-side, and only by reflection can it say something about our relation to the world. And indeed, it is this latter reflection that gives the paradox its worth.

Everything you said is dumb.

A paradox is a statement or set of statements that falsifies itself.

It has no relation to reality or nature in any way.

>3=4
LOL I CREATED A PARADOX LOOK HOW INCOMPLETE MATHMATICS AND LOGIC IS!

IT IS A PARADOX HOLY SHIT

that is you, you are that dumb.

I don't know man,
>3=4
seems like just a false statement to me

X is a barber with the property 3
X the barber also has the property 4

You see X = 3 and X = 4 because I told you so and both are true statements, but

X = 3 AND X = 4 cannot exist in the same Universe {} therefore it is a paradox! It is both true and false!

Thinking that any statement precisely defined within set theory is true is just arrogant. All statements must be evaluated on their own.

I'm not confused. I see no use for paradoxes. They are over-hyped quirks. Reflecting over a paradox can tell you nothing because the paradox itself is a mistake.

Since it has no relation to the real world it is useless. Just like poetry or religion.

Religion is a social technology that has the use of teaching and enforcing moral norms upon the members of the society.

You can replicate the benefits of religion in other ways, but you end up creating an atheistic state religion like communism has.

Without any religion people just do whatever and people lose a sense of community and don't trust each other and you end up with an ANCAP paradise/hell.

Banach Tarski paradox is not self referential

>no self reference
>solve this question

it is not a paradox, but a consequence of axioms. It is only called a paradox, because it is not possible in our physical world

>It is only called a paradox, because it is not possible in our physical world
It's only called a paradox because it contradicts our desire for isometries to preserve volume.

what are properties 3 and 4

>Thinking that any statement precisely defined within set theory is true is just arrogant
nobody thinks that

>I see no use for paradoxes.
they can show us the limitations of our reasoning, which we can then use to improve said reasoning.

Look, I'm not some paradox-enthusiast who thinks they are the pinnacle of human cognition, but to say they are over-hyped quirks just goes against 2000+ years of history, in which we have painstakingly built the logical structures that we now have and use in science and technology abundantly.

>en.wikipedia.org/wiki/Unexpected_hanging_paradox
In what way is that a paradox?

LOL YOU DIDN'T SURPRISE ME! YOU CAN'T HANG ME NOW!

The prisoner only knows it will be a weekday but not what day it will come. His surprise or non-surprise isn't real since the prisoner doesn't know. And it doesn't matter anyways, not like they're going to stop the execution.

It is being willfully retarded and making up rules that don't apply.

>The reason the Banach–Tarski theorem is called a paradox is that it contradicts basic geometric intuition. "Doubling the ball" by dividing it into parts and moving them around by rotations and translations, without any stretching, bending, or adding new points, seems to be impossible, since all these operations ought, intuitively speaking, to preserve the volume. The intuition that such operations preserve volumes is not mathematically absurd and it is even included in the formal definition of volumes. However, this is not applicable here, because in this case it is impossible to define the volumes of the considered subsets, as they are chosen with such a large porosity. Reassembling them reproduces a volume, which happens to be different from the volume at the start.

it is a paradox m8

To become will and to be, or to become will and not to be?

>Since a and b are the same answer, they should be considered as one when answering.
no, you haven't established that A could be correct and B could still be wrong. Also you haven't established any connection between the question and the answers, nor has any restriction that only one answer may be chosen.

Also if the question is to be answered randomly then it doesn't matter anyways.

The entire problem is an exercise in being willingly retarded.

You can't throw a shitload of A Priori rules and then claim the thing exists in a logical vacuum.

The properties 3 and 4 are mutually exclusive, therefore any object that has both properties is a paradox.

Why don't you try it and find out, idiot.

an object cannot be a paradox

Quotes from Wikipedia on Banach Tarski:
"decomposition of the ball into a finite number"
Since there is a finite number of pieces they all have a volume, a size.
"However, this is not applicable here, because in this case it is impossible to define the volumes of the considered subsets, as they are chosen with such a large porosity"
Now magically they lose their volume, and thus they are something akin to point particles. And a point need not be stretched to fill the void in scaling.
You can not have it both ways. Either they have volume or they don't. The definition is erroneous.

So your proposition is
>"There exists an X such that X=3 and X=4."

FALSE

Truth value assigned.

Not a contradiction.

Take literally any Paradox

List the properties and assign the properties to the object

Then use a series of logical AND statements to test all of the properties together.

With paradoxes you will always get a false.

>With paradoxes you will always get a false.

"This statement is false"

FALSE

...wait that means the statement is TRUE! OMG!

Bob is a Barber

Bob shaves everyone who doesn't shave themselves.

Does Bob shave himself?

Property A: Bob shaves everyone who doesn't shaves himself.
Property B: Bob shaves himself
Property C: Bob does not shave himself

When A is true, B cannot be true, A AND B = FALSE
When A is true, C cannot be true, A AND C = FALSE

it is a false statement, all paradoxes are like this.

paradox doesn't mean contradiction
paradox means apparent contradiction which isn't really there

>This statement is false"

Property A: The statement has the property of being logically true.
Property B: The statement has the property of being logically false.

If A is true, B cannot be true.
If B is true, A cannot be true.

Therefore A AND B = false.

Therefore it is a false statement, a "paradox" if you were.

> really there
how can paradoxes be real if the truth isn't real?

I'm not a historian. I don't care about the difficulties faced when inventing the wheel. The wheel is a reality. I'm not saying that the ability to formulate nonsensical statements should not be known. It should be known for what it is. A paradox should never be presented as proof of a broken nature. It is not that.

but A AND B is true user...

but Bob either shaves himself or he doesn't, hence B OR C, hence (A & B) OR (A & C), but those were both false! huuuuuuh?

> is called a paradox is that it contradicts basic geometric intuition.
Just like the fact that hot water is better for extinguishing fires than cold water. Is that a paradox too?

only you presented paradoxes as proof of a 'broken nature', which is an absurd claim if it even means anything.

I'm not a historian either, but I don't have the hubris to think we can't learn from past mistakes.

You're a child, wondering in the world, who doesn't know where anything comes from, and only accepts the immediate tangible reality as a given. Luckily, some of us do care where wheels, planes and computers come from, so that we can make better versions of the products and of ourselves.

You can not win an argument with subtle insults. If the only value of a paradox is in showing the limits of language or formal notation, then that is fine. That is a good thing. Trying to learn anything beyond that from a paradox is not possible. Once you find a paradox, there is nothing you can do with it. It is beyond reason.

A and B are literally the same answer (25%). They are either both true or both false or the question in itself is meaningless.

Solve [math]x+1=2[/math]:
A: x=1
B: x=1
C: x=2
D: x=1
Pick one at random (uniformly distributed). Is the probability of getting a right answer 50%?

Pick one what? One RESULT or one OPTION.
There are 2 possible results.
There are 4 possible options.

If we consider the OPTIONS, the probability of getting the right answer is 3/4. But the probability that there is one and only one right answer is 0. This is different from the question discussed above where the answering itself affects the result.

The probability of choosing the right answer randomly if considering the possible RESULTS that are presented is 50%.

wheres the monty hall problem

Nice one. It is however not a paradox.

There is a 1/3 chance to choose correctly in the first round. There is a 1/2 chance to choose correctly in the second round.
Keep -> 1/6 chance to pick these choices (correct 50% or the time)
Change -> 1/6 chance to pick these choices (correct 50% or the time)


There is a 2/3 chance to choose wrong in the first round. There is a 1/2 chance to choose correct in the second round.
Keep -> 1/3 chances to pick these choices (correct 0% of the time)
Change 1/3 chances to pick these choices (correct 100% of the time)

This does not add up. The problem is that the Game show hosts choice is never independent of the contestants choice. If the Game show host only picks a door with a goat behind it, then you should switch. The game show host can choose a door with a goat from 2 doors if you have picked correctly 1/3. If you picked a bad one 2/3 then the game show host only has one choice to make. There is a 1/3 * 1/2 = 17% chance to win if you keep your choice, of the right door. If you keep your choice of the wrong door you are guaranteed to lose. if you switch your choice of the wrong door, you are guaranteed to win. 2/3 * 1 = 67% of the time.

You are never asked to pick one out of three.
You are asked to pick one out of three and then asked if you want to keep that choice. There are 6 possible choices.

2 of them produce the right result all the time:
Wrong initial choice + switch door
2 produce the wrong result all the time:
Wrong initial choice + keep door
1 have a probability of 1/6 to be correct
Right initial choice + keep door
1 produce the wrong result all the time
Right initial choice + switch door

of the 6 choices 2 and 1/6 are correct.
there is a 1/3 chance to win.

You do not know if you chose right in the first round. After the game show host makes his pick, you are no more certain of your choice.

Not him but Russlle's Paradox should count. There's no vagueness or wordplay there.

This is an interesting one.
What exactly is part of this set. What is a set that is not a member of itself? Isn't a set just a collection of things, and this collection has to include itself? What is the collection if not itself?

From the assumption that all sets must contain themselves, there is no sets that do not contain themselves, and since the set of all things not containing themselves is empty, it does not even contain itself but it should? At this point we should introduce an axiom forbidding empty sets.
If empty sets are allowed, can the inclusion of them even be defined?

All sets either contain themselves or none do. If all do, it implies that the members of a set is known and countable. If it does not contain itself it can never be precisely constructed, only defined, we keep adding an endless amount of itself to itself.

>What is a set that is not a member of itself?

A set.

>Isn't a set just a collection of things, and this collection has to include itself?

A set is a collection of elements. Sets aren't required to include themselves, e.g. the set of {1,2,3} is not 1 or 2 or 3 and the set of blue cars is not a blue car.

>What is the collection if not itself?

For any set U, U has an equivalence relation with U. Sets are always themselves.

>From the assumption that all sets must contain themselves, there is no sets that do not contain themselves, and since the set of all things not containing themselves is empty, it does not even contain itself but it should?

That would be a paradox.

>At this point we should introduce an axiom forbidding empty sets.

You shouldn't require all sets to be elements of themselves. The empty set is central to axiomatic set theory.

>If empty sets are allowed, can the inclusion of them even be defined?

Yes.

>All sets either contain themselves or none do.

Not for ZFC.

> If all do, it implies that the members of a set is known and countable.

It doesn't.

>If it does not contain itself it can never be precisely constructed, only defined, we keep adding an endless amount of itself to itself.


There are an infinite number of sets that do not contain themselves that are constructed and well defined in axiomatic set theory, e.g. every natural number (without "adding and endless amount of itself to itself").

Yes and they're tedious

So, if the set { 1, 2, 3 } contains itself, its contents is { 1, 2, 3, { 1, 2, 3} } ?
But this changes its contents, and thus, it changes what we need to include into itself, ad infinitum.
If this is not the contents of a set containing itself, then what is?

I've never understood this attitude of people who clearly don't understand X but argue X is overrated/useless/trivial...

That is how you understand. Make your arguments, then see if they hold. If you don't understand something do not accept it, rather challenge it.

say you have 3 doors (1/3 chance to win for each doors right?)
Now the host open one door and reveal a goat
If you keep your door you will still have 1/3 chance to win
However if you change to the other door you now have 1/2 chance to win

The set of {1,2,3} does't contain itself. An example of a set that contains itself is the set containing all sets.

Also misspoke earlier, sets that contain themselves and Russel's paradox is from naive set theory (and others), not ZFC. ZFC's axiom of regularity implies sets can not contain themselves.

ZFC does have sets with infinite elements, though. The axiom of infinity says that there exists a set I, that for every element x in I, the union of x and {x} is an element of I. I'm too tired right now, but I'm curious what the proof is that there exists a x in I such that the intersection of x and I is the empty set.

What would a set that contain itself look like? Is it something more than a set B containing all elements of set A in addition to the entire set A as a whole as one element?

So the set containing (as elements) all sets, has one element that contains itself? Is that the self it is before adding itself, or the self it is after adding it?

1. ƎA : A ∈ A

2. B would contain A and the elements of A. B would not contain itself.

3/4. The set V containing all sets, has infinite elements including but not limited to: V, {V}, {{V}}, ...

Interesting. Does a set containing itself always have an infinite amount of elements?

Going back to reconsider Russell's paradox, in light of this better understanding, the problem seems to be that we assume that the specification of the set is exhaustive, before we consider the ramifications of them.
Why would we in the first place assume, that these formal specifications are infallible? Just because they seem to be true, or have been demonstrated to be true in some cases?
Isn't this an example of vague expression, not as defined by the rules of the notation, but as defined by the results?

There's no proof, at least not the informal notion of 'proof'.
That's why it's an axiom.

>Do non-self-referencing paradoxes exist?

Yes and no.

If he has a beard he's clearly not shaving himself.

unexpected hanging