On Russell’s First Paradox and The

Excluded-Middle Logical Reasoning




Doron Shadmi





Abstract


For more than 100 years the first paradox of Russell is considered as a problem in the foundations of what is called Naïve Set-Theory.

In this short paper we show that this paradox is based on elements that have no unique self-identity, and we can conclude that Russell's paradox cannot be more then a false statement in the framework of excluded-middle logical reasoning.

We also show that excluded-middle logical reasoning framework is a limited logical system.




Keywords: Unique self identity, Excluded-middle logical reasoning,

False statement, Limited logical system.




Russell's first paradox by standard logical reasoning:

( http://www.wikipedia.org/wiki/Russell%27s_paradox )

Consider the set M to be "The set of all sets that do not contain themselves as members". Formally: A is an element of M if and only if A is not an element of A. In the sense of Cantor, M is a well-defined set. Does it contain itself? If we assume that it does, it is not a member of M according to the definition. On the other hand, if we assume that M does not contain itself, than it has to be a member of M, again according to the very definition of M. Therefore, the statements "M is a member of M" and "M is not a member of M" both lead to a contradiction. So this must be a contradiction in the underlying theory.


A new point of view on Russell's first paradox:

In excluded-middle reasoning, each element must have a one and only one unique identity.

An element without a unique identity cannot be a participator in the excluded-middle "game".

Russell's paradox arises because we let to an element, which has no unique identity, to be a participator in our "game".


For example:

The identity of the barber of Seville cannot be defined because it is based on self contradiction which is:

1) He is from Seville.

2) He is a man.

3) He shaves all of the men in Seville (which means: he is included)

4) Only if they do not shave themselves.


By this last condition he contradicts its own identity because:

To shave all (which means: he is included) of the men in Seville only if they do not shave themselves, means that all is not_all (or a = not_a ).



The same contradiction of self identity, can be shown in the set that includes all of the sets only if they do not include themselves as their own members.

To include all (which means: it is included) of the sets only if they do not include themselves, means that all is not_all (or a = not_a ).

An element which has no self and unique identity cannot be a legitimate participator in an excluded-middle logical reasoning.

Also please be aware to the fact that the set that includes all of the elements that do not have well-defined and unique identity, has a unique self identity, and we can conclude that no one of the existing members of this set can be a legitimate participator in an excluded-middle logical reasoning system (we also can conclude that these existing members are beyond the domain of an excluded-middle logical reasoning system, which means that excluded-middle logical reasoning system is a limited logical system).

In short, no false result can be used as a logical basis to produce a paradox in an excluded-middle logical reasoning system, or in other worlds: no false statement can be considered as a paradox in an excluded-middle logical reasoning system.


a is a if and only if it is not_a (it means that a contradicts its own self identity) is not a paradox but a false statement, exactly as a is not_a is nothing but a false statement.


In other words: (a is a if and only if it is not_a) is (a is not_a).


M is M if and only if it is not_M is nothing but a false statement.


Therefore Russell's paradox is not defined within excluded-middle reasoning.

I'm sure I'm not as well versed in this as you are, but if M is a well-defined set, and yet both conditions (being a member and not being a member) lead to a contradiction, it smells to me of something along the lines of Godel's Incompleteness Theorem. In very general terms, we can define the set M, yet we cannot prove it's existence or nonexistence. To me it sounds like an unprovable truth.

Hi Fallen Angel,

An element in an excluded-middle logical reasoning cannot have more then one unique identity, and if we examine M we cannot define its unique identity in the framework of excluded-middle logical system.

So M is not well-defined in this framework, and we need to go beyond the excluded-middle reasoning, if we want to find if M is well-defined.

Our need to go beyond the limitations of some system, to get an answer about some examined element, is the basic idea of Godell's Incompleteness Theorem.

In short, any consistent system is also limited, and we need to go beyond it to increase our understanding about some fallen shadow of something that exist beyond the limitation of our current logical system.

If you ask my opinion, then this is the basis of any evolution in any system and the Language of Mathematics is not beyond paradigm shifts.

what do you mean by "unique identity"? you're not referring to bivalence are you?

what do you mean by "unique identity"? you're not referring to bivalence are you?

In an excluded-middle reasoning an element (set, number, ...) can have simultaneously a one and only one unique name (identity).

And I do not mean to some variable symbols like 'a', 'A', 'b' , 'B' ... and so on.

The identity of an element is its literal name like: a number named 'pi', a number named 'e', a number named '1', a number named '0', a set named 'not_all_sets_that_do_not_contain_themselves' ... and so on.

Strictly speaking, a well-defined element in an excluded-middle logical reasoning system, cannot be but an element that has a one and only one unique literal name.

Now, the set that includes 'all of the elements that do not have well-defined and unique identity', has a unique self identity.

Therefore it is a well-defined set in the framework of excluded-middle logical reasoning, but no one of its members can be considered as a well-defined element within the framework of excluded-middle reasoning (the best that can be done is to say that the members of this well defined set are false and true , neither true nor false, contingently true or false etc.)

There is here a positive approach of Godel's incompleteness theorem, which says: Within any consistent system, there can be found at least one well-defined set, which its content cannot be well-defined within the framework of the current logical system.

In short, in any consistent system we can find pointers, which lead us beyond the domain of the current system, or in another words:

Each consistent system includes within it the seeds of its paradigm shift, and in my opinion this is the essence of the Language of Mathematics.

Each consistent system includes within it the seeds of its paradigm shift, and in my opinion this is the essence of the Language of Mathematics.

Ahhh, I like the way you put that.

I agree with your reasoning about a set that includes all of the elements that do not have well-defined and unique identity having a unique self identity, and how from there you go on to explain that the members of this set cannot be well-defined within excluded-middle reasoning system.

What I don't see is the connection you make between the following. It seems to me that the set that contains all sets that do not contain themselves is a set with a unique identity of "the set that contains all sets that do not contain themselves." The statements "M is a member of M," and "M is not a member of M" are statements about the uniquely identifiable set "that contains all sets..." I can see where you conclude that this leads to a situation where paradigm shift is necessary to resolve the contradiction, however, I do not see your reasoning as to how the contradiction of the two statements renders M as not uniquely identifiable. More in a bit....

The same contradiction of self identity, can be shown in (M=) the set that includes all of the sets only if they do not include themselves as their own members.

To include all (which means: it is included) of the sets only if they do not include themselves, means that all is not_all (or a = not_a ).

In short, no false result can be used as a logical basis to produce a paradox in an excluded-middle logical reasoning system, or in other worlds: no false statement can be considered as a paradox in an excluded-middle logical reasoning system.


a is a if and only if it is not_a (it means that a contradicts its own self identity) is not a paradox but a false statement, exactly as a is not_a is nothing but a false statement.


In other words: (a is a if and only if it is not_a) is (a is not_a).


M is M if and only if it is not_M is nothing but a false statement.


Therefore Russell's paradox is not defined within excluded-middle reasoning.

ok, here is how i understand the problem

M is a set, A is a set, < means "is included in", /< means "is not included in"

we define M as:

M = { A | A /< A}

the two propositions "M is a member of M" and "M is not a member of M" result in the following:

(M < M) -> (M /< M) and eq. 1
(M /< M) -> (M < M) where "->" means "implies" eq. 2

now from the "implies" truth table:

0->0 is true
0->1 is true
1->0 is false
1->1 is true

therefore, if one of those statements ( M < M ) or ( M / < M ) is false AND the other is true, there is a contradiction. however if both are false or both are true, then we have two true statements. if bivalence is not a requirement, it is possible for both ( ( M < M ) and ( M / < M ) ) be true or both be false, hence eq. 1 and 2 are both true. if we don't hold on to bivalence, there is no conflict above, hence i don't see how eq. 1 and 2 demonstrate that M = { A | A /< A} does not have a unique identity.

the reason i say that we should not hold on to bivalence in this case is because we both mentioned that proof of the truth of ( M < M ) or ( M /< M ) is beyond the scope of the system and requires meta-reasoning.

or is that just another way of showing that the paradox requires meta-reasoning and hence cannot be decided by excluded-middle logical system? :D

Trust language to get in the way of reasoning.

or is that just another way of showing that the paradox requires meta-reasoning and hence cannot be decided by excluded-middle logical system?

If we need 3-d to define something in 2-d, then this thing is not well-defined in 2-d framework.

If we examine any version of the paradox we always get the proposition: all = not_all.

Anything which is based on all=not_all cannot be but a false statement within excluded-middle framework, and we do not need any truth tables for this.

Therefore Russell’s paradox is no more than a false statement in excluded-middle reasoning.

there is no conflict above, hence i don't see how eq. 1 and 2 demonstrate that M = { A | A /< A} does not have a unique identity.


Unique identity in an excluded-middle reasoning cannot be based on self identity contradiction, or on more than a one self identity.

In an excluded-middle reasoning any self identity which needs its nagation to be well-defined, is not well-defined.

No proposition can make a statement about itself...

If we look at this propositoin, we can say that within an excluded-middle reasoning, if a self reference of a proposition changes the propositon, then and only then it cannot be refered to itsef, because in an excluded-middle reasoning, each element has exactly one and only one uniqe identity.

By tautology x = x means: x is itself, otherwise we cannot talk about x.

Now we can ask if a teotology is also recursive, for example: x = x = x = ...

If we do not get any new information by this recursion, then x = x is enough, which is like a one_step_recursion.

So, Russel's paradox is like if by teotology we examine if x is not_x or x = not_x , which is no more then a false statement from an exluded-middle point of view.

In an excluded-middle reasoning no false statement is a paradox.

For example:

If the Barber of Seville does not shave himself, then he does not fit to his own self identity, which is:

To shave all of the people in Seville, only if they do not shave themselves, and in this case we can conclude that all = less_than_all or in other words: all = not_all



If the Barber of Seville shaves himself, then he does not fit to his own self identity, which is:

To shave all of the people in Seville, only if they do not shave themselves, and in this case we can conclude that all = more_than_all or in other words: all = not_all



Some conclusions:

a) The self identity of the Barber of Seville is based on the false statement all = not_all.

b) Self identity, which is based on a false statement, is no more then a false statement.

c) No false statement is a paradox in excluded-middle reasoning.

d) Therefore Russell's paradox is not defined in excluded-middle reasoning.



In general we can conclude the above about any self-referenced definition, which includes in it all condition.

If an all condition is omitted form a self-referenced definition, then the possibility of self identity as a false statement, is avoided in an excluded-middle reasoning.

If the statement contradiction = not-contradiction is a contradiction is false, the statement contradiction = not-contradiction is not a contradiction is true?

In logic we can say that our true result is a false statemant.

This is the reason why some false reuslt can be found in our logical system.

Only the true stands behind any result.

Also in excluded-middle reasoning any examined concept cannot have more than one unique identity,
so a = not_a cannot be but a false statemant (which is the true reuslt) in this case.

Ok, now let's say, Person A lied his Whold life, except for one instance,
where it is not determined if he lied, or not. At that instance he said, "I
have lied my whole life, and will do so until I die". is this a false
statement or not?

Ok, now let's say, Person A lied his Whold life, except for one instance,
where it is not determined if he lied, or not. At that instance he said, "I
have lied my whole life, and will do so until I die". is this a false
statement or not?


A liar (in an excluded middle system) cannot say that he is a liar, because then he says the true.

except for one instance,
where it is not determined if he lied, or not.

If x = liar, then x cannot be but a liar, and no undereminate state can refer to him in excluded-middle reasoning if x is a liar.

Some stuff from another forum on this subject:


No proposition can make a statement about itself...

If we look at this propositoin, we can say that within an excluded-middle reasoning, if a self reference of a proposition changes the propositon, then and only then it cannot be refered to itsef, because in an excluded-middle reasoning, each element has exactly one and only one uniqe identity.

By tautology x = x means: x is itself, otherwise we cannot talk about x.

Now we can ask if a teotology is also recursive, for example: x = x = x = ...

If we do not get any new information by this recursion, then x = x is enough, which is like a one_step_recursion.

So, Russel's paradox is like if by teotology we examine if x is not_x or x = not_x , which is no more then a false statement from an exluded-middle point of view.

In an excluded-middle reasoning no false statement is a paradox.

Again:

The element x_AND_not_x cannot be defined in excluded-middle reasoning, because any examined concept cannot have more than a one unique identity.

Therefore Russell's Antinomy is nothing but a false statemant and not a paradox in excluded-middle framework.


note the correct use of iff, sometimes denoted <=>, and not =, since 'equals' is not an operator in boolean logic

'=' is used here for the tautology of a = a.

a = not_a is no more than a false statment in excluded-middle reasoning.


A and not_A

A and not_A cannot be defined in excluded-middle reasoning, because any examined concept cannot have more than a one unique identity.


Discussing this kind of stuff as if you're working on or attempting to resolve a 100 year old problem is a joke.

Our true result in this case is no more then a false statement, and all the big affords that professional mathematicians put in their theories to avoid this "paradox", are no more than a full gas in neutral.

contiune...

On Godel's incompleteness theorem:

In an excluded-middle reasoning an element (set, number, ...) can have simultaneously a one and only one unique name (identity).

And we do not mean to some variable symbols like 'a', 'A', 'b' , 'B' ... and so on.

The identity of an element is its literal name like: a number named 'pi', a number named 'e', a number named '1', a number named '0', a set named 'not_all_sets_that_do_not_contain_themselves' ... and so on.

Strictly speaking, a well-defined element in an excluded-middle logical reasoning system, cannot be but an element that has a one and only one unique literal name.

The set that includes 'all of the elements that do not have well-defined and unique identity' has a unique self identity.

Therefore it is a well-defined set in the framework of excluded-middle logical reasoning, but no one of its members can be considered as a well-defined element within the framework of excluded-middle reasoning (the best that can be done is to say that the members of this well-defined set are false and true, neither true nor false, contingently true or false etc.)

This is a positive approach of Godel's incompleteness theorem, which says:

Within any consistent system, there is at least one well-defined set, which its content cannot be well-defined within the framework of the current logical system.

In short, in any consistent system we can find pointers, which lead us beyond the domain of the current system, or in another words:

Each consistent system includes within it the seeds of its paradigm shift, and in my opinion, this is the essence of the Language of Mathematics.

sorry for no input for quite a while, been off the net, and i think i understand where you're coming from where you describe the recursive equals. a = not a = a = not a...

you might be onto something with recursive tautology, but it seems outside a logic system.

and i've read your arguments, and i think i understand what you are arguing about no unique identity, based on a = ~a being a contradiction. however, i just don't believe that contradiction cannot define the unique identity of a set (i may be wrong in this, haven't had extensive training in logic, tho i've done some set theory, but on its own bases i personally don't see a problem defining it as a contradiction, despite the arguments you presented). i'll agree with you that the system requries meta-reasoning and thus cannot be decided by excluded-middle reasoning, but i disagree that M does not have unique identity.

Good luck with the idea and look forward to more posts coming from you.

If a_AND_not_a (=M) is considered as a well-defined element in excluded-middle reasoning, then and only then it has a unique literal name (=identity) within excluded-middle framework.

Let us look again at this example:

If the Barber of Seville does not shave himself, then he does not fit to his own self identity, which is:

To shave all of the people in Seville, only if they do not shave themselves, and in this case we can conclude that all = less_than_all or in other words: all = not_all



If the Barber of Seville shaves himself, then he does not fit to his own self identity, which is:

To shave all of the people in Seville, only if they do not shave themselves, and in this case we can conclude that all = more_than_all or in other words: all = not_all



Some conclusions:

a) The self identity of the Barber of Seville is based on the false statement all = not_all.

b) Self identity, which is based on a false statement, is no more then a false statement.

c) No false statement is a paradox in excluded-middle reasoning.

d) Therefore Russell's paradox is not defined in excluded-middle reasoning.



In general we can conclude that no unique identity can be defined without the existenece of a framework.

If the barber shaves those, and only those men who do not shave themselves, then does the barber shave himself?


If an assertion A, is true and its negation, ~A is also true, it becomes a form of the "liars paradox".

Suppose a person called X, stands up and says, "This assertion is false."

Let S denote the statement uttered; let p be the proposition the person makes by uttering S. Then the utterance of the phrase "This assertion" refers to the claim p. It follows that, in uttering the words "This assertion is false," X is making the claim "p is false". Thus , p and "p is false" are one and the same:

p = [p is false]

By making the claim, X is implicitly referring to the context in which the claim is stated. Let c symbolically represent the context for which the sentence refers.

X's uttering of the words "This assertion" refers to the context, c, which entails p.

[c entails p]


That is to say, p must be the same as [c entails p] due to the fact that X is referring to both p and [c entails p] via the utterance of the phrase "This assertion."

If X's assertion is true then [c entails p] is true

p = [p is false]

[c entails p is false] is true


This creates a contradiction, ergo X's claim that [p is false] is false.

[c entails p is false] is false


This appears to be the same contradictory state of affairs as in the previous cases of the Liars Paradox.

Conclusion?:

c cannot be the appropriate context.

Consequently, the paradox becomes a theorem/demonstration.
When X utters the Liar sentence, X is uttering a falsehood, and the context in which the claim of falsehood is made cannot be the same as the context in which the Liar sentence S, was uttered...



Thus context c, becomes a subjective/qualia operator.

Suppose a person called X, stands up and says, "This assertion is false."


No well-defined element in an excluded-middle reasoning can refer directly or indirectly a false state to itself.

For example:


X = Liar

Y = Honest


X cannot have any property of Y, and Y cannot have any property of X, In an excluded-middle reasoning or in other words:

A true Liar cannot say directly (by using the word "I") or indirectly (by using the word "ALL" or "THIS" that points back to itself) that he is a Liar, because he can’t say the truth.

A true Honest cannot say directly (by using the word "I") or indirectly (by using the word "ALL" or "THIS" that points back to itself) that he is a Liar, because he can’t lie.


X has a one and only one unique self-identity.

Y has a one and only one unique self-identity.

The logical condition between X and Y in an excluded-middle reasoning is : X_XOR_Y.

The paradox is based on X_AND_Y but because any well-defined element in an excluded-middle reasoning cannot have more than one unique self-identity, then no element, which its identity is based on X_AND_Y is well-defined in the domain of excluded-middle reasoning.

It means that the Liar's paradox (and also Russell's first paradox) is not well-defined concept within excluded-middle reasoning.

The set that includes 'all of the elements that do not have well-defined and unique identity' has a unique self identity.

Therefore it is a well-defined set in the framework of excluded-middle logical reasoning, but no one of its members can be considered as a well-defined element within the framework of excluded-middle reasoning (the best that can be done is to say that the members of this well-defined set are false and true, neither true nor false, contingently true or false etc.)

This is a positive approach of Godel's incompleteness theorem, which says:

Within any consistent system, there is at least one well-defined set, which its content cannot be well-defined within the framework of the current logical system.

In short, in any consistent system we can find pointers, which lead us beyond the domain of the current system, or in another words:

Each consistent system includes within it the seeds of its own paradigm shift, and in my opinion, this is the essence of the Language of Mathematics.