Russell's Paradox and the Excluded-Middle reasoning

[Russell E. Rierson]

This is not "self-identity"; this is the "fallacy of composition". In general, the whole does not have the properties of its parts.

"self-identity" says that the set of all sets that contain themselves must be the set of all sets that contain themselves.

The correct logic is:

If A then B

A

therefore B



Yet the fallacy of composition is equivalent to:

If A then B

B

therefore A

which is incorrect logic.

The set of natural numbers has the identity "natural number" that distributes over all members of the "set"[the whole distributes over the parts].

The most fundamental identity distributes over all elements of the "Universal Set". True, one specific aspect is not a universal property but the universal property can be the first step in the logical deduction that eventually leads to the specific aspect.

U[X[Y[Z...{ }]]]


Russell's paradox is a form of the liars paradox:


This statement is false

Which leads to Goedel's incompleteness theorem.

 

[Lama]

No set can be its opposite ("contain" , "does_not_contain") without first losing its own existence.

Therefore the circular state of Russell's Paradox does not exist.

Now, Let us go deeper then that:

No set can be its opposite ("contain" , "does_not_contain") without first losing its own identity.

Therefore the circular state of Russell's Paradox cannot be found.

There is a very deep idea here that can be used as the basis of what I call "A non-naive Mathematics".


By a non-naive Mathematics the existence of an element does not depend on its name.

For example, let us take two different points.

The existence of the points is not depending on their names.

It means that the two points can have any pair of different names.

Now, let us say that names are what we call numbers.

So each number, when mapped with some point, give it its unique identity.

We get here two basic systems:

The absolute system:

Made of infinitely many points, which their existence does not depend on their identity (which is some unique name that can be mapped to each one of them).

The relative system:

Made of infinitely many possible unique names that when mapped with some absolute point, they determinate its identity.

It means that the identity of any absolute point relatively can be changed by the current name that we give it (after two arbitrary and unique names are given, the rest of points/names mapping is well-defined, relatively to an arbitrary name, which is used as a global name of the entire points/names mapping).

In the case of numbers, the global name is actually a unique scale factor over
the entire real-line (for more detailes about the real line, please look at https://www.physicsforums.com/showthread.php?t=30254)

This interaction between absolute/relative concepts, is maybe the deepest foundation of the language of Mathematics and can be used a solid basis to define its organic dynamical structure.

 

[Hurkyl]

"There does not exist an x such that x is not equal to x" is a perfectly correct statement. This does not mean that we aren't allowed to write "x != x"; it simply means that this statement is false.


And Russel, the fallacy of composition is not equivalent to what you wrote.

 

[Lama]

Hi Hurkyl,

Please read carefully post #37 (all of it, including the link) thank you.

 

[master_coda]

Hi Hurkyl,

Please read carefully post #37 (all of it, including the link) thank you.

Perhaps you should start reading our posts.

 

[Lama]

I read yours, and this is the reason why I came up with a new theory of a non-naive-mathematics.

 

[Lama]

No, it is simpler than that.

By a non-naive Mathematics the existence of an element does not depend on its name.

 

[master_coda]

No, it is simpler than that.

By a non-naive Mathematics the existence of an element does not depend on its name.

Apparently you don't realize that in actual math, the existence of something does not depend on its name either. Naming has absolutely nothing to do with Russel's paradox, although prehaps your lack of understanding has led you to believe that it does.

 

[TeV]

http://plato.stanford.edu/entries/russell-paradox/

Accordingly the solution to the paradox can be found in the key word "naive" set theory.Obviously, some classes are not well defined sets that obey logical operations.In other words,they are not correctly defined.
Axiomatic set theories are required to prevent paradoxes.

 

[Lama]

Apparently you don't realize that in actual math, the existence of something does not depend on its name either. Naming has absolutely nothing to do with Russel's paradox, although prehaps your lack of understanding has led you to believe that it does.

No, master_coda you are the one how misunderstand the meaning of identity.

If A is "The set of all_sets_that_do_not_contain_themselves" then its own self identity is not to contain itself.

If B is "The set of all_sets_that_do_contain_themselves" then its own self identity is to contain itself.

Now, A self identity cannot be related to any Blue(=B) property.

Also, B self identity cannot be related to any Red(=A) property.


Russell's paradox is based on this state:

                                 ,---> self identity [COLOR=Blue][B]B[/B] [/COLOR]---,                                                        |                           | 
Self identity [COLOR=Red][B]A[/B][/COLOR] is observed as   |                        |
                                 |                        |
                                 '--- self identity [COLOR=Red][B]A[/B][/COLOR] <---'

Now, we have to understand that there is no way to observe identity A as if it has also properties of identity B and also to keep its A identity on the same time.

The "paradox" arises because we rape by force identity A to keep its own identity and also to say that it has a B property.


Conclusion:

Russell's Paradox is nothing but a brutal action of a rough mind.

 

[matt grime]

you'd do a lot better i f you didn't use such silly terms as 'rape' and 'brutal' and 'ruff' (which is a frilly neck garment by the way) and actually learned about the stuff you're pronouncing on. Russell's paradox simply arises from the naive attempt to say that a set is a collection of objects with a rule for belonging to that set.

 

[Russell E. Rierson]

And Russel, the fallacy of composition is not equivalent to what you wrote.

Wake up and smell the logic Hurkyl. It is a MAD world!

The composition fallacy is a form of "modus ponens" error:

http://www.illc.uva.nl/j50/contribs/eemeren/eemeren.pdf

The fallacies of composition and division

Frans H. van Eemeren, University of Amsterdam and New York University
Rob Grootendorst, University of Amsterdam

1. Introduction
In the pragma-dialectical conception of argumentation fallacies are defined as violations of rules that further the resolution of differences of opinion. Viewed within this perspective, they are wrong moves in a discussion. Such moves can occur in every stage of the resolution process and they can be made by both parties. Among the wrong moves that can be made in the argumentation stage are the fallacies of composition and division. They are violations of the rule for reasonable discussions that any argument
used in the argumentation should be valid or capable of being validated by making explicit one or more unexpressed premises. In this paper the fallacies of composition and division are analyzed in such a way that it becomes clear that the problem at stake here is in fact a specific problem of language use.

2. Properties of wholes and the constituent parts
There are several ways of violating the dialectical rule that the reasoning that is used in argumentation should be valid or capable of being validated by making explicit one or more unexpressed premises. To make this clear, first, the argument has to be reconstructed that is used in the argumentation. Next, an intersubjective reasoning procedure has to be gone through to establish whether the argument is indeed valid (van
Eemeren and Grootendorst 1984: 169).

A well-known violation of the validity rule consists of confusing necessary and
sufficient conditions in reasoning with an 'If ... then' proposition as a premise.

There are two variants. The first is the fallacy of affirming the consequens, in which, by way of a 'reversal' of the valid argument form of modus ponens, from the affirmation of the consequens (by another premise) is derived that the antecedens may be considered confirmed. The second is the fallacy of denying the antecedens, in which by way of a similar reversal of the valid argument form of modus tollens the denial of the consequence is derived from the denial (by another premise) of the antecedens.

There are also other violations of the validity rule. A violation that often occurs is unjustifiably assigning a property of a whole to the constituent parts. Or the other way around: unjustifiably assigning a property of the constituent parts to the whole. The properties of wholes and of parts are not always just like that transferable to each other. Sometimes the transfer leads to invalid reasoning:

a This chair is heavy

b Therefore: The lining of this chair is heavy

 

[moshek]

Lama :

You have a nice name in Hebrew mean "Way".

Well I see that you treat symbol as mathematical object and by these Russell paradox have a new meaning. Please tell me and how is all that relate if at all to the Epilog of the book "Nature's number" by Ian Stewart and his interesting new idea about Morfomatica?


Thank you
Moshek
:shy:

 

[Lama]

You have a nice name in Hebrew mean "Way".

"Lama" in Hebrew is "Why?" and not "Way".

 

[Lama]

you'd do a lot better i f you didn't use such silly terms as 'rape' and 'brutal' and 'ruff' (which is a frilly neck garment by the way) and actually learned about the stuff you're pronouncing on. Russell's paradox simply arises from the naive attempt to say that a set is a collection of objects with a rule for belonging to that set.

Thank you, I corrected "ruff" to "rough".

I used 'rape' and 'brutal' and 'rough' not as mathematical terms but to clearly show how some fundamental parts of Modern Mathematics do not hold water.

 

[Hurkyl]

No, master_coda you are the one how misunderstand the meaning of identity.

No, I must insist that you are the one misunderstanding identity.

Self-identity says "A thing is equal to itself", which is something vastly different than the fallacy of composition, which says "A thing satisfies the properties of its parts".


Some other examples:

The "set of all individual numbers" is clearly not an individual number.
The "set of everything worth less than a dollar" would obviously be worth more than a dollar.
This "set of all blue objects" is clearly a red object.

 

[Hurkyl]

Wake up and smell the logic Hurkyl. It is a MAD world!

The composition fallacy is a form of "modus ponens" error:

Your quote from that link seems to say exactly the opposite...

 

[Lama]

A thing satisfies the properties of its parts

In this case A cannot be in a state that do not satisfies the properties of its parts, and this it exactly what I say.


The set of all dogs is not a dog itself but it has a common property with each dog that included in it, which we can call it "Dogness".

So the most basic identity of each dog and the most basic identity of the set of all dogs is "Dogness" (in this case this basic identity is also like a one_step_recursion, which is equivalence to the tautology x=x).

Now in the case of Russell's Paradox, the most basic identity of each member "not_to_contain_itself" (which is like the "Dogness" example) and the most basic identity of the set of "all_members_that_do_not_contain_themselves" is "not_to_contain_itself" (in this case this basic identity is also like a one_step_recursion, which is equivalent to the tautology x=x).

Strictly speaking, the "Dogness" identity example is equivalent to the "not_to_contain_itself" identity case.

The "set of everything worth less than a dollar" would obviously be worth more than a dollar.

In this case it cannot be a member of itself because the most basic identity here is "worth less than a dollar".

IN EACH CASE WE HAVE TO DEFINE THE MOST BASIC PROPERTY, AND ONLY THEN WE CAN CONCLUDE IF THIS PROPERTY MEANS THAT WE HAVE TO INCDLUDE THE SET IN ITSELF.

FOR EXAMPLE: THE SET OF ALL_MEMBERS_THAT_CONTAIN_THEMSELVES MUST CONTAIN ITSELF AS A MEMBER OF ITSELF, BECAUSE “TO_CONTAIN_YOURSELF” IS THE MOST BASIC IDENTITY IN THIS CASE.

IN SHORT, RUSSELL'S PARADOX DOES NOT HOLD WATER JUST BECAUSE OF THE REASON THAT THERE IS NO LOGIC STATE HERE THAT FORCE US TO INCLUDE THE SET IN ITSELF.

If A is "The set of all_sets_that_do_not_contain_themselves" then its own self identity is not to contain itself.

If B is "The set of all_sets_that_do_contain_themselves" then its own self identity is to contain itself.

Now, A self identity cannot be related to any Blue(=B) property.

Also, B self identity cannot be related to any Red(=A) property.


Russell's paradox is based on this state:

                                 ,---> self identity [COLOR=Blue][B]B[/B] [/COLOR]---,                                                        |                           | 
Self identity [COLOR=Red][B]A[/B][/COLOR] is observed as   |                        |
                                 |                        |
                                 '--- self identity [COLOR=Red][B]A[/B][/COLOR] <---'

Now, we have to understand that there is no way to observe identity A as if it has also properties of identity B and also to keep its A identity on the same time.

The "paradox" arises because we force identity A to keep its own identity and also to say that it has a B property.

 

[matt grime]

you should stop using your 'real life' intuition in mathematics, Doron, in particular your notion of 'sharing' some element of 'dogness' which is a spurious example to do with your subjective notion of degree.

 

[Lama]

Hi Matt,

Please refreash your screen and read all of my previous post, thank you.

 

[matt grime]

but the set of dogs displays no aspect of 'dogness' ie being a dog. its elements do. learn, please, before spouting asinine garbage.

 

[Lama]

The set of dogs cannot be called "the set of dogs" if it has nothing to to with dogs.

"Dogness" is the abstract proprty of anything which is related to dogs, and this abstract property is the self identity of "the set of all dogs".

"DOGNESS" IS NOT ANY PARTICULAR DOG.

 

[Lama]

By the way, to say that a set is not its members is a non-abstract and trivial thing to say.

If you build your abstarct logical reasoning on this non-abstract trivial thing, then you can't go far.

 

[master_coda]

By the way, to say that a set is not its members is a non-abstract and trivial thing to say.

If you build your abstarct logical reasoning on this non-abstract trivial thing, then you can't go far.

Of course, your idea of basing reasoning on abstract, contradictory things is far better.


Understand this simple fact - you cannot assume that a set must share any properties with the things it contains. If you do make this assumption, one of two things will happen:

1) your system of logic will be inconsistent

2) your system of logic will be unable to make any deductions about sets that do not share any properties with their elements, and will therefore be much, much weaker than existing logical systems

 

[matt grime]

The set of dogs cannot be called "the set of dogs" if it has nothing to to with dogs.

"Dogness" is the abstract proprty of anything which is related to dogs, and this abstract property is the self identity of "the set of all dogs".

"DOGNESS" IS NOT ANY PARTICULAR DOG.

I realize this isn't being conducted in your first language, but at least take care to read what is written. you are saying that the set of dogs displays the properties of being 'doggy'', and that is certainly not true. i did not say the set of dogs has nothing to do with dogs. that would require some agreement on what we mean by 'has to do with'.

Consider the set which contains the empty set, that set is not empty...

 

[Russell E. Rierson]

The set of dogs cannot be called "the set of dogs" if it has nothing to to with dogs.

"Dogness" is the abstract proprty of anything which is related to dogs, and this abstract property is the self identity of "the set of all dogs".

"DOGNESS" IS NOT ANY PARTICULAR DOG.

Perhaps "dogness" can be viewed as a form of constraint forcing the members included in the set of dogs to this defining aspect.

The set of all dogs is a subset of the set of all mammals...

Eventually, the set that includes "everything" is reached by removing nested constraints.

 

[Russell E. Rierson]

Your quote from that link seems to say exactly the opposite...

Perhaps you can demonstrate[with logical symbolism] how the composition fallacy is not a form of modus ponens error?

 

[Lama]

Consider the set which contains the empty set, that set is not empty...

A set is only a framework where we can examine our ideas, and its own existence does not depend on the properties of its contents.

Only its name (identity) is denpend on the properties of its contents.

Again you use a non-abstract approech of the set concept.

As for "Dogness", I use this world as the most geneal concept of anything that is realed to dogs, but also does not have to be a dog at all.

If you have another word instead of "Dogness" to what I wrote above, then I'll be glad to get it from you.

 

[Lama]

Understand this simple fact - you cannot assume that a set must share any properties with the things it contains. If you do make this assumption, one of two things will happen:

1) your system of logic will be inconsistent

2) your system of logic will be unable to make any deductions about sets that do not share any properties with their elements, and will therefore be much, much weaker than existing logical systems

1) Please show us how a set can get its name (identity) without any relation to the properties of its contents.

2) Please show us some mathematics where sets with no names (identities) are involved.

 

[master_coda]

1) Please show us how a set can get its name (identity) without any relation to the properties of its contents.

2) Please show us some mathematic where sets with no names (identiteis) are involved.

What does that have to do with anything? Of course you need to know about the properties of the contents of the set. My point was that just because a property holds true for every element in a set, you cannot then conclude that the property also holds true for the set itself.

 

[matt grime]

perhaps, doron (i can't be bothered to work out which psuedonym you'll end up using in the next post), you should learn from this that attempting to apply mathematical ideas in nonmathematical ways is fraught with danger and doesn't tell you maths is flawed, which is you opinion apparently, but that your attempts to misuse it are. One need only view your opinion about dogness to see this.

 

[Lama]

you cannot then conclude that the property also holds true for the set itself.

Identity of a set has nothing to do with 'true' or 'false'.


The identity of a set is based on the most abstract basis of its contents, which gives it its name.

Therefore no set can contrarict its own identity (name).

In the case of Russell's paradox the identity of a set is forced to stay invariant, when its most abstract basis of its identity is chaneged.

This is exactly as if we say: Black is White or vise vera.

 

[Lama]

perhaps, doron (i can't be bothered to work out which psuedonym you'll end up using in the next post), you should learn from this that attempting to apply mathematical ideas in nonmathematical ways is fraught with danger and doesn't tell you maths is flawed, which is you opinion apparently, but that your attempts to misuse it are. One need only view your opinion about dogness to see this.

Please read post #63

 

[master_coda]

Identity of a set has nothing to do with 'true' or 'false'.


The identity of a set is based on the most abstract basis of its contents, which gives it its name.

Therefore no set can contrarict its own identity (name).

In the case of Russell's paradox the identity of a set is forced to stay invariant, when its most abstract basis of its identity is chaneged.

This is exactly as if we say: Black is White or vise vera.

"No set can contradict its own name" isn't just something you can just assert. It doesn't even make sense. Your "proof" is nothing more than you saying "I made up a rule about sets, and Russel's paradox violates it, so the paradox must be wrong".

What are you going to do next? Tell us that 0 = 1 and so obviously x/0 = x?

 

[Lama]

Can a set which its identity is: "all_members_thet_do_contain_themselves" can contradict itself identity and include itself as a member of itself as if its identity is not important (or in other words: without losing its own identity)?

Can you be master-coda which is not master-coda?