The end of philosophy-(it was fun while it lasted)

[Chikoka]

All logical systems contain as an axiom the axiom of identity. "A=A" and A!=!A".

Without it , no other axiom can work. What if i told you that this axiom violates itself?

Let us begin.

Equality "=" is a relation, which by definition is an ordered pair. If the two "A's" in "A=A" differ in any respect (like order) then they cannot be the same.

There we have it folks. The end of philosophy!

Theism anyone?

[Togo1]

They don't differ in order.

[Chikoka]

Are you aware of the theory of relations?

[Togo1]

Are you aware of the theory of relations?

No, please do expand.

Are you aware of the difference between equality and identity?

[Chikoka]

Any expression to mean anything must have parts.That is why those two "A's" HAVE to be different. If they are not different then the expression has no parts and so is devoid of information.

The theory of relations needs the elements "related" to ...well....stand in relation to each other. This stand must of necessity involve order. Identity is expressed using a relation. A type of relation called an equivalence relation.

[Togo1]

Any expression to mean anything must have parts.That is why those two "A's" HAVE to be different. If they are not different then the expression has no parts and so is devoid of information.

A=A is devoid of information, and can be expressed with no parts.

The theory of relations needs the elements "related" to ...well....stand in relation to each other. This stand must of necessity involve order. Identity is expressed using a relation. A type of relation called an equivalence relation.

But the order is in the expression of the elements, not the elements themselves. Putting a series of fruits in alphabetical order does not confer applies with the attribute of firstness. The order of the fruit is entirely a function of the order in which you have placed them, and is not true of the fruit themselves.

Ok, so expanding your demonstration into plain language, you're arguing that because the same item can be expressed in an equation more than once, each expression must refer to a different item.

Can you see how the contradiction is included in your choice of expression?

[Daviddunn]

If the two "A's" in "A=A" differ in any respect (like order) then they cannot be the same.

Well, if they are the same (I.e. A), then they cannot differ in order. If you think this can be different, how would you express this difference? Because for me, the statement 'A=A', is the same as the statement 'A=A'! This might seem tautological but consider the following. It is not like 'A=B', which can differ in order, (I.e. B=A). Isn't this called the property of commutativity of equality in mathematics?

Anyway, I think this type of problem was addressed by Frege in his distinction between sense and reference, in answer to Moore's problem if I am not mistaken. So, it goes like A=B, where 'A' and 'B' are names with different senses, but the same reference. An example we can take is 'Hillary Clinton' and 'the wife of Bill Clintion'. These two names have different senses (which I hope everybody understands), but they have the same reference, and this is what is meant by saying 'Hillary Clinton = the wife of Bill Clinton'.

So, if a name like 'A' is thought to have a reference, then that thing is identical to itself necessarily. In 'A=A', there might indeed be two "A's" as you say (I.e. the two 'A' one writes on each side of the identity symbol), but these are just names which refers to the same thing, and it is that which makes the statement meaningful.

[Chikoka]

If the two "A's" in "A=A" differ in any respect (like order) then they cannot be the same.

Well, if they are the same (I.e. A), then they cannot differ in order. If you think this can be different, how would you express this difference? Because for me, the statement 'A=A', is the same as the statement 'A=A'! This might seem tautological but consider the following. It is not like 'A=B', which can differ in order, (I.e. B=A). Isn't this called the property of commutativity of equality in mathematics?

Anyway, I think this type of problem was addressed by Frege in his distinction between sense and reference, in answer to Moore's problem if I am not mistaken. So, it goes like A=B, where 'A' and 'B' are names with different senses, but the same reference. An example we can take is 'Hillary Clinton' and 'the wife of Bill Clintion'. These two names have different senses (which I hope everybody understands), but they have the same reference, and this is what is meant by saying 'Hillary Clinton = the wife of Bill Clinton'.

So, if a name like 'A' is thought to have a reference, then that thing is identical to itself necessarily. In 'A=A', there might indeed be two "A's" as you say (I.e. the two 'A' one writes on each side of the identity symbol), but these are just names which refers to the same thing, and it is that which makes the statement meaningful.

If the two "A's" reffer to the same thing. that is the the same as saying (A=A) = A. Now there are three A's. which are all not identical.

ENTER "the Infinite Regress"

-- Updated Thu Aug 25, 2016 11:33 am to add the following --

Any expression to mean anything must have parts.That is why those two "A's" HAVE to be different. If they are not different then the expression has no parts and so is devoid of information.

A=A is devoid of information, and can be expressed with no parts.

The theory of relations needs the elements "related" to ...well....stand in relation to each other. This stand must of necessity involve order. Identity is expressed using a relation. A type of relation called an equivalence relation.

But the order is in the expression of the elements, not the elements themselves. Putting a series of fruits in alphabetical order does not confer applies with the attribute of firstness. The order of the fruit is entirely a function of the order in which you have placed them, and is not true of the fruit themselves.

Ok, so expanding your demonstration into plain language, you're arguing that because the same item can be expressed in an equation more than once, each expression must refer to a different item.

Can you see how the contradiction is included in your choice of expression?

Then you agree it is meaningless and hence lacks the ability to give meaning to the other axioms in any system?

If they are the same thing there cannot be more than one

Any way we would like to express the Axiom we express an order. Even all the talk of sameness in normal english uses it as i have shown above. The problem is that we cannot think of such n axiom without using order.that means it is meaningless to us and means the same as this: hjfukfu . As they are both meaningless and hence reffere to the same thing.

[Daviddunn]

If the two "A's" reffer to the same thing. that is the the same as saying (A=A) = A. Now there are three A's. which are all not identical.

ENTER "the Infinite Regress"

There is no infinite regress here. This is a case where Wittgenstein would say that we are trying to say that which can only be shown! Anyway let us try to work this out, and attempt if not to say that which can only be shown but at least point to that which is shown! By the way, this is known as the ostentive reference. A technical term which you should get yourself familiar with as it contains the key to the problem that you present in the OP. It is an interesting subject by the way.

Let me take an example first. The tennis player Roger Federer, he is a Swiss and he has won 17 Grand Slam single titles (a record). He is a father etc... Some people have seen him played and met him face to face. There is only one Roger Federer who fits this description. That would be the reference.

Now the name 'Roger Federer', is a sequence of characters, namely {'R', 'o', 'g', 'e', 'r', ' ', 'F', 'e', 'd', 'e', 'r', 'e', 'r'}. This name (I.e. the sequence of characters) can be written many times either physically (on paper) or digitially (on the screen of your computer). Each time the name 'Roger Federer' is written, each name occupies different spaces ( either on paper or on a screen). So, with respect to the space which the characters occupy, they are different. However, with respect to the form/shape of the characters, the names are similar. The latter is recognised.

Now, the important part is here. Each time the name 'Roger Federer' is mentioned or written, it is not the case that the man who bears this name, gets a copy of himself made, such that each name refers to one of such copy! Otherwise, the world would be crowded with Roger Federers, and this would be known by now! So even though, one duplicates the name, the man who bears this name does not get duplicated. And also, this man cannot become characters which one can write and get displayed on a screen or on paper.

The problem with the statement from you that I quoted above, is that you are confusing the name with the reference. The name you can write but the reference can only be shown! So, in your statement "(A=A)=A", the As are just names which you have repeated three times. You can repeat them as many times as you want but never would you be able to juxtapose a name and its reference in this way. You will end up merely juxtaposing names, but the reference is only shown by the ostentive reference.

Think about this for sometime if you did not get it right away. It is a tricky problem, so it is alright if you do not get it now.

[Nicolas]

You state: "All logical systems contain as an axiom the axiom of identity. "

Then we have to do deal with the problem of identity vs essence I think.

It's rather the beginning of philosophy and not it's end!

while identity is made up within relationships, ( social relationships, states of being and systems, equations etc. )
essence stands on it's own ( outside of any system or equation )

Theism any-ONE? Let's give it a try:

In Essence A = A even without the equation. But to identify, one must name/ analyze things.

In theological terms:
Essence: G_d = G_d
Identity: G_d = JC : a specific ( empirical and legendary biological person who walked the earth 2000 years ago etc, etc,

Logical it appears self evident that A = A or G_d = G_d
but it takes scientific or theological inquiry or experience to claim that A = B or G_d = JC

Oneness or essence always transcends identity. One Essence can have multiple identities or names for that matter.
( for the christian thinker the concept of the trinity is therefore no problem)

"You can steal identity but never essence. "

What is "=" ? That is the mystic question.

[A_Seagull]

All logical systems contain as an axiom the axiom of identity. "A=A" and A!=!A".

Without it , no other axiom can work. What if i told you that this axiom violates itself?

Let us begin.

Equality "=" is a relation, which by definition is an ordered pair. If the two "A's" in "A=A" differ in any respect (like order) then they cannot be the same.

There we have it folks. The end of philosophy!

Its not so much the end of philosophy as it is the beginning of philosophy!

Ditch formal logic, it has never achieved anything worthwhile anyway!

[Chikoka]

The problem is symbols.If you were to (try) showing me you would only manage to "say" it in a different langauge/manner. In any case this platform would be totally inappropriate for showing anything (whatever that could mean) since you can only use words ...i.e. say_ings.

To get deeper....if you were to hold up an apple (perhaps that's what you mean), it exists as an appearance in my mind ...a similar appearance to what appears when you show someone else the "same" apple.
Though similar , those appearances are separated by subjective interpretation (minds) and cannot be the same.Again i challenge that you have simply shown me an apple. you have shown me what appears (to you firstly through your senses) to be an apple. If all our appearances are different THAT is the violation of A=A. This is not a new argument. A refers to 'A' just as the appearance refers to the apple. Different minds have different copies.This doesn't necessarily mean that there is no objective reality, just that you cant reveal/show me enough of it to demonstrate the axiom.

[A_Seagull]

So0me comments:
First off : "All logical systems contain as an axiom the axiom of identity. "A=A" .Without it , no other axiom can work"
This is not true at all. A logical system does not require an axiom of identity. If you doubt me, then prove it, logically!

Second : as stated 'Roger Federer' can be considered to be a string of symbols. That string can also be recognised as a label for a person.

Third the symbol "=" can also be considered to be an abstract symbol that can be manipulated by a logical system and has no meaning until it is recognised as a label for an identity.

[Consul]

All logical systems contain as an axiom the axiom of identity. "A=A" and A!=!A".

Not all do, because it's not an axiom of negative free logic, in which "A = A" is false if A doesn't exist. In this logical system it is false that everything is self-identical, but it is true that everything existent is self-identical.

If the two "A's" in "A=A" differ in any respect (like order) then they cannot be the same.
There we have it folks. The end of philosophy!

You are wrong, and it has already been explained to you why.
In "A = A" the left token and the right token of the letter type <A> are certainly different from one another, but this formal sentence is not about letter tokens or letter types: it says neither that the left token of the letter type <A> = the right token of the letter type <A> (if it did it would be false), nor that the letter type <A> = the letter type <A> (if it did it would be true). For what it says is that the thing referred to by "A" = the thing referred to by "A", with "A" being a placeholder for a proper name.

-- Updated August 26th, 2016, 6:33 pm to add the following --

So0me comments:
First off : "All logical systems contain as an axiom the axiom of identity. "A=A" .Without it , no other axiom can work"
This is not true at all. A logical system does not require an axiom of identity. If you doubt me, then prove it, logically!
Second : as stated 'Roger Federer' can be considered to be a string of symbols. That string can also be recognised as a label for a person.
Third the symbol "=" can also be considered to be an abstract symbol that can be manipulated by a logical system and has no meaning until it is recognised as a label for an identity.

"[M]ost texts distinguish first-order logic without identity from first-order logic with identity. In the latter, but not the former, the sign '=' for identity is logical. In the model theory for first-order logic with identity, it is stipulated that a sentence of the form 'a = b' is true in a given interpretation just in case the denotation of 'a' is identical to the denotation of 'b'. In the deductive system, there are specific axioms or rules for this term (e.g. identity-introduction and identity-elimination rules). In first-order logic without identity, the '=' sign is treated as any other non-logical binary relation symbol."

(Shapiro, Stewart. Varieties of Logic. New York: Oxford University Press, 2014. p. 55)

That is, in first-order logic with identity, the identity sign "=" is a logical constant with a fixed semantics, and in first-order logic without identity it is not.

-- Updated August 26th, 2016, 6:37 pm to add the following --

Ditch formal logic, it has never achieved anything worthwhile anyway!

Yes, it has: clear rational thought and coherent argumentation!

-- Updated August 26th, 2016, 6:43 pm to add the following --

What is "=" ? That is the mystic question.

There's nothing mystical about identity.

"We never have [a problem about identity]. Identity is utterly simple and unproblematic. Everything is identical to itself; nothing is ever identical to anything else except itself. There is never any problem about what makes something identical to itself; nothing can ever fail to be. And there is never any problem about what makes two things identical; two things never can be identical. There might be a problem about how to define identity to someone lacking in conceptual resources—we note that it won't suffice to teach him certain rules of inference—but since such unfortunates are rare, even among philosophers, we needn't worry much if their condition is incurable."

(Lewis, David. On the Plurality of Worlds. Oxford: Blackwell, 1986. p. 192-3)

[Wirius]

You might misunderstand what the "=" means. In English replace "=" with "is" So "A is A" A=A

John is John. Sky is Sky. There is no ordering, there is merely the statement of equivalence. 5 is a grouping of 2 and 3. 5=2+3 There isn't a five on one side and a 2 and a 3 somewhere else. It is the observation that a grouping of 2 and 3 is the same as the grouping we call 5.