The Axiom Of Identity *Challenged*

[Scott Mayers]

helps us define classes of things or universals

Dog = 4 legged creature, with fur, a muzzle...

The definition;
is equivalent (means the same thing as)

Dog = Dog

By transitivity . ITs equivalent because its an equivalence relation.

This is not a logical definition and would be a 'tautology' [a normal logical equivalence]. This does not make your 'definition' necessarily 'wrong', but INCOMPLETE.

For instance, properly, your definition in logic would use the sign, "≡", in logic to emphasize equivalence or, in computers, "==". These aren't necessary to use these exact symbols but point to the kind of distinctions that is used. If I were to correct your definition to one of the assignment meaning, it could be:

A dog is a domesticated carnivorous mammal [Generic Class of Equivalence between members] THAT is capable of being domesticated for human uses or as a socially adaptable pet [qualifying additional explanation to explain how it differs from other "domesticated carnivorous mammals"].

This definition doesn't have to be exactly like this but note how it explains how it belongs to some large class (equal or 'same') and how it differs from others of the same class by some principle of difference more specifically. They call the general class description, the Genus, and the specifying difference between others of the same class as the Species or Differential to clarify the symbol or word being used to summarize it.

To help understand this, in computers, you might use the 'definition' form of using the relation of equality as follows:

Dog = Dog + Wolf

This kind of equality reassigns the meaning of Dog to include the meaning of both the previous definition of 'Dog' with the addition of the meaning of 'Wolf'. But regular Equivalence would be demonstrated as

Dog == Dog

[Scott Mayers]

No, a 'form' is simply an abstraction (relative to us independently)

Isnt that what the axiom of identity does? Tell us how a thing is related to other things?
I.e. what it is not.

Dog != cat, zebra, donkey....(rest of the unviverse)

other than itself

Dog = Dog.

The Axiom of identy is the one that forms the skin (boundary) of a form.

An Axiom of identity is meant to define the kind of symbol being used for this relationship whether real or not.
In algebra, the Axiom of Identity initially begins by first indicating the meaning of two ideas that 'mean' the same use the symbol, "=", and that the symbols being used are arbitrary to what they could mean when in use.

So A = A,

tells us that the '=' is the relation symbol being used to relate two things, ideas, real or unreal, as the SAME only. It is not a definition of 'A' as a definition is incomplete without illustrating what is also DIFFERENT. Can you see how the "=" sign here is the 'symbol' being defined where the 'A's being used are both the SAME in it kind of symbol, but DIFFERENT in that one 'A' is on the left and the other copy is on the right. Thus this is NOT a definition of 'A' but IS a definition of '='.

Can you see that you actually agree and is why you were concerned. Your reasoning IS what made them have to use different but related symbols to explain equivalence verses assignment. But we often get these confused often in arguments using language that we shift to both these meanings without realizing where or when we do.

I'm not sure if you are discussing some particular 'theory of relations' or to the defined term 'relation' with respect to logic or math. There a relation is any process, machine, or structure of argument, that takes members of one domain as 'inputs' and relates them to a range of other members called a 'range'. The domain members may require the range to be the same or different depending on what you define it as. In this way, the members of the range can be copies of the domain or some subset of them. A function is a specific type of relation that takes one (or more) input members from the Domain and outputs them to a Unique member from the Range. For most purposes, we use the Range to at least be some subset (including possibly the whole) of the Domain.

There is no problem in general with relations ..its only when it describes a things relationship to itself. It implies that a things domain is exactly the same as its range...no problem with most of that except exact is a meaningless concept.

I understand what you are thinking. This is part of what I just mentioned above and might help you see this. But note that the 'relation' symbol above is '=' and is defined only to state that the meaning of the components of both are identical in meaning. It is NOT and assigning relation but meant to relate the "meaning". It is confusing though when you then later learn that algebra often uses the '=' sign to also assign as in,

Let A = B + C.

[Moyo]

Scott

"=" is a verb while "==" is a noun. I mean that "==" is illogical, not "=".
But after your "=" happens, straight away it can be substituted with "==". So your attack does not hold.
I see you travel from theme to theme,
The title of this thread was to challenge the axiom of identity. The way we identify something. If you have any other way of identifying something then that should also be known as "an axiom of identity".

Tell me how you identify what a is , other than using the axiom of identity bearing in mind that the axiom of identity both uses identifyand starts of with a is ...

[Moyo]

Tell me how you identify what a is , other than using the axiom of identity bearing in mind that the axiom of identity both uses identify and starts of with a is ...

If you stil dont get how obvious this is and insist on assigning form to be what identifys something...then i am not attacking the form way of saying what something is but the axiom of identity way..and it clearly does so...this would be in line with the title of this thread.

[Scott Mayers]

Tell me how you identify what a is , other than using the axiom of identity bearing in mind that the axiom of identity both uses identify and starts of with a is ...

If you stil dont get how obvious this is and insist on assigning form to be what identifys something...then i am not attacking the form way of saying what something is but the axiom of identity way..and it clearly does so...this would be in line with the title of this thread.

Okay, I understand you so far.

I'm wondering now what particular "axiom of identity" you are thinking of now? Some systems define them differently and why I'm asking. If you have one in mind can you define it and the system you are thinking of so that I can relate specifically?

For logic (as by the 'classical laws' from the Greeks), the Law of Identity is the following:

In logic, the law of identity is the first of the three classical laws of thought. It states that “each thing is the same with itself and different from another”. By this it is meant that each thing (be it a universal or a particular) is composed of its own unique set of characteristic qualities or features, which the ancient Greeks called its essence. Consequently, things that have the same essence are the same thing, while things that have different essences are different things.[1]

In its symbolic representation, “A is A”, the first element of the proposition represents the subject (thing) and the second element, the predicate (its essence), with the copula “is” signifying the relation of “identity”.[2] Further, since a definition is an expression of the essence of that thing with which the linguistic term is associated, it follows that it is through its definition that the identity of a thing is established.[3] For example, in the definitive proposition:[4]"A lawyer is a person qualified and authorized to practice law", the subject (lawyer) and the predicate (person qualified and authorized to practice law) are declared to be one and the same thing (identical). Consequently, the Law of Identity prohibits us from rightfully calling anything other than "a person qualified and authorized to practice law" a "lawyer".

Note here that the word, "is" is what they call the copula where we were using '=' and/or '==' depending on interpretation. Notice how in the second paragraph they mention the point of introducing a concept using a "definition" to which is the initial assignment. There you can see what I was thinking in that it defines the left side as the "subject" and the right (with the copula) as the "predicate". The left says above that the subject is the "thing" and the object of the predicate (the second element after the copula) is the 'essence'. But recognize that this "thing" is a symbol or object or even, action, that represents the actual essence (being) of it. That is, I could use an idol to represent some idea, not just a word; I can also use an action, like pointing to something specifically, to indicate the essence of what I'm pointing to (denoted object or action, etc.)

So IF this is the Axiom you are referring to, let me know if this either already agrees with you or if you still were thinking something different.

[Moyo]

n its symbolic representation, “A is A”, the first element of the proposition represents the subject (thing) and the second element, the predicate (its essence), with the copula “is” signifying the relation of “identity”

Subject and predicateare not identical otherwise they wouldnt have made that distinction of use. So how can it signify Identity

Yes thats what i mean

[Scott Mayers]

In Euclidean geometry, they use the word, "congruence" to mean that something is similar or has proportional mapping to the original object. So if something is the same in most ways but different only in one other, it acts as a type of equality akin to the assigning but a 'static' type. For example all squares are congruent means that even though the particular essence of each square you could draw could be made up of an infinite set of sides of different measures, each square is congruent but not NECESSARILY equal. But all "equal" squares are considered those that are both congruent AND have a measure of the same size.

I also mentioned that the ancients often more particularly defined the first such experience of a thing as the 'unit' to which we might use the word, "unique" and while any repeated uses of it elsewhere could 'appear' as the same, they would treat these too as "congruent" only like what I think you mean. The Pythagoreans used this way of approaching it and I think it can be wise so be very, very specific.

So the very first television, that you may have ever seen as a child may be considered your relative "unique" idea and every other television you see afterwards is merely a 'copy' of the original (with respect to you experiencing it). In math, Pythagoras would use the first use of '1' as the absolute unique unit and any other's as copies of it. I'm not sure if they would have added some notification to represent the copies or not if written. But it might be something like this:

First idea: 1
Copies of the first idea: 1c

So had they wanted to actually use a symbol of comparison, they would use something like:

1 ≅ 1c

So is this what you were seeking to show?

[Scott Mayers]

n its symbolic representation, “A is A”, the first element of the proposition represents the subject (thing) and the second element, the predicate (its essence), with the copula “is” signifying the relation of “identity”

Subject and predicateare not identical otherwise they wouldnt have made that distinction of use. So how can it signify Identity

Yes thats what i mean

The word "identity" as a verb or noun means to present the original as is or simply, that which is "the same". Notice that it doesn't qualify whether the 'same' is due to congruence or perfect in every way or the EXACT same thing. This is why you have to spell this out for each system of logic/math. But our normal uses of speaking lose the specific degree to which we use words and so you are correct to address this concern for being more precise.

[Arising_uk]

This means that in some way A is different from A since subject != essence, otherwise therd be no distinction.

Or you could argue like I think Kant did(maybe someone else?) that 'is' or 'being' is not a predicate?

Or maybe just define identity this way( P-> Q ^ Q -> P) or (¬P v Q ^ Q V ¬P)?

That there might be an issue in set theory or relational theory does not mean there is a problem per se as in propositional logic there appears not to be? If so then the issue appears to be with the application of the relational theory to monadic objects and not with existence, as it appears that a thing is that thing no matter how you cut it?

[Moyo]

First idea: 1
Copies of the first idea: 1c

So had they wanted to actually use a symbol of comparison, they would use something like:

1 ≅ 1c

So is this what you were seeking to show?

Yes

1 ≅ 1c

And even that first 1
.
is not ≅ 1 or itself. because its relationship with itself is also

1 = 1c

and even itself is not....

[Moyo]

Everyone :

Are we all idealist yet? I need to move on to juicyer material.

[Moyo]

This means that in some way A is different from A since subject != essence, otherwise therd be no distinction.

Or you could argue like I think Kant did(maybe someone else?) that 'is' or 'being' is not a predicate?

Or maybe just define identity this way

( P-> Q ^ Q -> P

[/i][/b]) or (¬P v Q ^ Q V ¬P)?

That there might be an issue in set theory or relational theory does not mean there is a problem per se as in propositional logic there appears not to be? If so then the issue appears to be with the application of the relational theory to monadic objects and not with existence, as it appears that a thing is that thing no matter how you cut it?

Propositions are english scentences that we sometimes represent symbolicaly. If you were to translate that underlined you have to use the word "is'" then you will be in the teritory "a is..."

[Moyo]

This means that in some way A is different from A since subject != essence, otherwise therd be no distinction.

Or you could argue like I think Kant did(maybe someone else?) that 'is' or 'being' is not a predicate?

Or maybe just define identity this way( P-> Q ^ Q -> P) or (¬P v Q ^ Q V ¬P)?

That there might be an issue in set theory or relational theory does not mean there is a problem per se as in propositional logic there appears not to be? If so then the issue appears to be with the application of the relational theory to monadic objects and not with existence, as it appears that a thing is that thing no matter how you cut it?

[Arising_uk]

Everyone :

Are we all idealist yet? ...

No, at least not in the sense that a 'God' is needed to support those ideas that are not sourced from us, i,e, perceptions.

I need to move on to juicyer material.

Who's stopping you?

[Arising_uk]

Propositions are english scentences that we sometimes represent symbolicaly. ...

Not quite, pretty much every language can be represented by logical symbols, propositions are essentially declarative sentences.

If you were to translate that underlined you have to use the word "is'" then you will be in the teritory "a is..."

Or I could say, if that is a brick then that is a brick or that is a brick or that is not a brick but it cannot both be a brick and not a brick.