The Law of Identity

[Speakpigeon]

This thread is motivated by a dispute about the Law of Identity in the Logic & Philosophy of mathematics forum. However, this thread is about the Law of Identity. It isn't a thread about logic itself. So, please, remember we don't care about logic per se here.
Some more context...
The Law of Identity has been assumed as an axiom of logic since Aristotle some 2,400 years ago, but a few people apparently choose to deny the Law of Identity. This is their constitutional right, of course, but some of them, possibly all of them even, may not really understand much what the Law of Identity means.
So, here is your chance to articulate eloquently what you think the Law of Identity really means for all of us.
I'm not interested in quirky theories about something that would not be the Law of Identity. I'm interested in what you think the Law of Identity means to most people, even those who have never thought about it, and including what it meant to people like Aristotle who are long dead now.
Still, whatever can be found in encyclopedias about the subject is open to debate and we can perhaps improve our understanding of it by sharing our most intimate intuitions about the Law of Identity.
So, please, don't ramble. Keep to the point and leave the question of the logic of it to the other forum.
Thanks.
EB

[henry quirk]

The thing is the thing.

As I say elsewhere: my coffee cup is my coffee cup and no other. I am me and no other.

That's how I see it.

[Scott Mayers]

"The law of Identity" is a logical apriori assumption or agreement to what one means minimally about what is considered logical. For any given concept, symbol, or referent, if given some concept referenced by a label, say X, it is related to the concept referred by it uniformly by the label, X, wherever it is used.

I prefer to think of it in terms of Newton's first law of Inertia: given any state of being or constant change, that state remains as it was unless or until some external factor causes it to change. For logic, it means that if you assign some label as a referent, don't change the meaning of the label in the context of an argument.

As to the expressions of the form,

xRy

The R represents the relationship of the symbol x on the left, to the symbol y on the right. If we want to define some R as something such that some meaning of x is assigned to y, such that the meaning of y identifies with x as being uniquely referring to the same meaning by either force or comparison, then we call that R an Identity relationship of y to x.

There are many kinds of identities. To logic, this defines that if given any particular single assumption, it remains unchanged as a conclusion. This is an "implication". Another type is a binary implication, or "equality".

A definition that uses a symbol on one side, usually the left, IDENTIFIES the meaning of something more explanatory on the other.

"Cat" is "an animal" [meaning, a cat is identified (at least) as an animal]; but "animal" is-not "a cat"

If the symbols on both sides imply each other identically, a reflexive type, then we might have:

"a cat" is "any cat" is identical to "any cat" is "a cat"

Symmetric, Reflexive, and Associative properties are types of Identities.

So are the rules,

"1 or x = 1" , "1 and x = x" are both Identity claims that define what the meaning of "1" is.

The "law of Identity" then is the general meaning that defines "TRUTH" relationships of the system of reasoning that are 'acceptable' or 'fit'.

It is an agreement of those opting to play a game to follow the exact same rules with the same meanings shared rather than to arbitrarily confuse or change the meaning of the words or symbols used arbitrarily. In politics, they call this the "rule of law" to mean, let us agree that any laws we create or change becomes the Ruler of the people, unless or until we have some new reason to change it officially. For physics, it is Newtons' "Law of Inertia".

Why require this when it seems it already applies? It doesn't apply if people do not DEFINE their TERMS in a philosophical debate and is often the reason FOR disagreement. But if we are at a comedy club, we actually expect the opposite: to show how the same meanings are actually differently interpreted and makes us laugh for not realizing:

"How did they find out that the girl who died in the movie, Jaws, had dandruff?"

"They found her Head and Shoulders™ on the beach."

Humorous contrary misidentifying a definition:

"to find one's head and shoulders on a beach" =?= "to find one's Head & Shoulders™ on a beach."

Since we assume the girl died in the movie, we associate finding one's head and shoulder's on the beach to be unrelated to one who had dandruff puzzling.

[Logik]

The law of identity is the law of identity.
It is the LAW of language AND the LAW of thought! Because there is no difference between language and thought.

To learn to speak is to learn to think and vice versa.

The LAW of identity speaks to symbol-uniqueness.

DO NOT USE TWO SYMBOLS TO MEAN THE SAME THING AT THE SAME TIME!

It is good advice. Advice geared towards avoiding equivocation.

And then guess what Aristotle goes and does? He equivocates identity and equality.
He used "=" to mean TWO DIFFERENT THINGS.

These are different PROPOSITIONS.

IDENTITY: for all x: identity(x) = identity(x) => TRUE
EQUALITY: for all x: x = x => UNDEFINED

A proposition's possible results are: True, False, Undecided or Undecidable.

Demonstration: https://repl.it/@LogikLogicus/IdentityvsEquality

Code: Select all

TYPE: int, Value: 1
IDENTITY: A = A: True
EQUALITY: A = A: True
TYPE: Human, VALUE: <__main__.Human object at 0x7f41accc21d0>
IDENTITY: B = B: True
EQUALITY: B = B: False

That is Aristotle's error. Equivocation.

He didn't practice what he preached.

[Scott Mayers]

logik

See my post before yours above. Aristotle only used the "implication" verb of "is", he did not expand onto other forms of identity. George Boole, in "the Laws of Thought" begun the more formal expansion of this in the 1800s.

The Identity Law is about an agreement to make any system of 'logic' CONSISTENT to its rules. It just means you can't borrow some meaning from another system of reasoning without explicitly agreeing to it.

[Logik]

logik

See my post before yours above. Aristotle only used the "implication" verb of "is", he did not expand onto other forms of identity. George Boole, in "the Laws of Thought" begun the more formal expansion of this in the 1800s.

The Identity Law is about an agreement to make any system of 'logic' CONSISTENT to its rules. It just means you can't borrow some meaning from another system of reasoning without explicitly agreeing to it.

He fucked up either way. To say "x is x" (x = x) is NOT a proposition, but everything else is....That's just special pleading.

This is a PROPOSITION: for all x: x = x => TRUE
This is a DIFFERENT PROPOSITION: for all x: id(x) = id(x) => TRUE
This is a YET ANOTHER: for all x: x != x => False

The above is consistent but incomplete. There are many more propositions that one can make!

https://en.wikipedia.org/wiki/Decidability_(logic) is key to all human thought AND language.

[Scott Mayers]

In set theory, they define their first law, often called "extension", to define what it means when one uses the symbol "=" EXTENDED from first order logic's use of the implication and biconditional symbols, → and/or ↔:

Law of Extension
For any classes A and B,
(A = B) means (A → B) and (B → A)

This is set theory's logical identity law regarding the system itself. It tells you what "=" means as a rule to agree to remain consistent throughout as a minimum.

[Logik]

In set theory, they define their first law, often called "extension", to define what it means when one uses the symbol "=" EXTENDED from first order logic's use of the implication and biconditional symbols, → and/or ↔:

Law of Extension
For any classes A and B,
(A = B) means (A → B) and (B → A)

This is set theory's logical identity law regarding the system itself. It tells you what "=" means as a rule to agree to remain consistent throughout as a minimum.

I reject set theory. I accept Lambda calculus.
I reject MUH ( https://en.wikipedia.org/wiki/Mathemati ... hypothesis )
I accept Ultraintuitionism ( https://en.wikipedia.org/wiki/Ultrafinitism )

Symbols are primitives, not integers. Symbol-manipulation and information-processing is thought. Computation!

A set is a TYPE of thing.
A digit is a TYPE of thing.
An integer is a TYPE of thing.

IF (A = B) means (A → B) and (B → A) it implies that A ⇔ B

[Scott Mayers]

logik

See my post before yours above. Aristotle only used the "implication" verb of "is", he did not expand onto other forms of identity. George Boole, in "the Laws of Thought" begun the more formal expansion of this in the 1800s.

The Identity Law is about an agreement to make any system of 'logic' CONSISTENT to its rules. It just means you can't borrow some meaning from another system of reasoning without explicitly agreeing to it.

He fucked up either way. To say "x is x" (x = x) is NOT a proposition, but everything else is....That's just special pleading.

This is a PROPOSITION: for all x: x = x => TRUE
This is a DIFFERENT PROPOSITION: for all x: id(x) = id(x) => TRUE
This is a YET ANOTHER: for all x: x != x => False

The above is consistent but incomplete. There are many more propositions that one can make!

The "=>" you use is an implication form of identity that assigns the meaning of "x = x", as a whole term.
You are defining "x = x", a term about "=" as an identity of equality. So you have two forms within the same proposition.

The second assignment you have is to assign the value TRUE to the term, "id(x) = id(x)". Your "=>" is identical to "is" or implication in the same way Aristotle used it. But the "=" you use identifies equality of the subterms, "id(x)" which appears to be a FUNCTION, which is another kind of assignment. So there is a mixture of three IDENTITIES there. I'm guessing "id(x)" is Python's memory space IDENTIFIER?

For, "for all x: x != x => False", you are assigning the meaning of " x != x" to the value, FALSE, which is actually, 0, in the computer, and the "!=" by the meaning, "is not equal" as an equality identifier.

So you are using both two general types of identities AND assuming the meaning of 'FALSE' is zero, to the computer, the computer IDENTIFIES the term "x != x" to have a value of '0'. I'm not sure of the particular format of python, but the "x != x" looks like it is an ASCII STRING to the computer only. When you use x elsewhere, it has to replace the specific instances of 'x' in that string to the literal constant you put into some code.

[Scott Mayers]

In set theory, they define their first law, often called "extension", to define what it means when one uses the symbol "=" EXTENDED from first order logic's use of the implication and biconditional symbols, → and/or ↔:

Law of Extension
For any classes A and B,
(A = B) means (A → B) and (B → A)

This is set theory's logical identity law regarding the system itself. It tells you what "=" means as a rule to agree to remain consistent throughout as a minimum.

I reject set theory. I accept Lambda calculus.
I reject MUH ( https://en.wikipedia.org/wiki/Mathemati ... hypothesis )
I accept Ultraintuitionism ( https://en.wikipedia.org/wiki/Ultrafinitism )

Symbols are primitives, not integers. Symbol-manipulation and information-processing is thought. Computation!

A set is a TYPE of thing.
A digit is a TYPE of thing.
An integer is a TYPE of thing.

IF (A = B) means (A → B) and (B → A) it implies that A ⇔ B

You're missing the point. Lambda Calculus will simply have a distinct set of different rules but still requires a set of consistent axioms of which Identity is still at least required.

[I believe Lambda Calculus is just a metalanguage, used to discuss other languages, like a universal or general computer relative to programs. But if so, why mix python as a higher-order language to discuss a metalanguage? ]

[Logik]

The "=>" you use is an implication form of identity that assigns the meaning of "x = x", as a whole term.
You are defining "x = x", a term about "=" as an identity of equality. So you have two forms within the same proposition.

Yes,I am. I am indicating that there exists SOME MECHANISM by which the truth-value of "x = x" can be decided.
I am suggesting that the folowing: for ALL x: x = x => True can be re-written as follows:

for ALL x: f(x, x) = 1

I am suggesting that there exists a valid function which can DETERMINE the answer to f(x,x) to BE 1.

This is decidability.

The second assignment you have is to assign the value TRUE to the term, "id(x) = id(x)". Your "=>" is identical to "is" or implication in the same way Aristotle used it. But the "=" you use identifies equality of the subterms, "id(x)" which appears to be a FUNCTION, which is another kind of assignment. So there is a mixture of three IDENTITIES there. I'm guessing "id(x)" is Python's memory space IDENTIFIER?

Id(x) is memory-space identifier. Full stop. Semiotics. https://www.tutorialspoint.com/cprogram ... inters.htm

Identity refers to the signifier (in your head) NOT the signified.

For, "for all x: x != x => False", you are assigning the meaning of " x != x" to the value, FALSE, which is actually, 0, in the computer, and the "!=" by the meaning, "is not equal" as an equality identifier.

IF you insist on consistency. Which you don't have to. Paraconsistency is fine.

So you are using both two general types of identities AND assuming the meaning of 'FALSE' is zero, to the computer, the computer IDENTIFIES the term "x != x" to have a value of '0'. I'm not sure of the particular format of python, but the "x != x" looks like it is an ASCII STRING to the computer only. When you use x elsewhere, it has to replace the specific instances of 'x' in that string to the literal constant you put into some code.

As long as you maintain 1:1 relationship in your semantics, you are OK....

[Logik]

You're missing the point. Lambda Calculus will simply have a distinct set of different rules but still requires a set of consistent axioms of which Identity is still at least required.

No, I am not missing the point. I never rejected identity. I merely rejected YOUR understanding of identity in favor of mine.

Lambda calculus does NOT mandate your Alphabet. It mandates nothing whatsoever.

A Turing machine has infinite length tape. So you can have infinite-length alphabet.

If you can invent a million unique SYMBOLS and you assign unique SEMANTICS to all of them. then you are good to go.

The ONLY requirement is consistent semiotics. 1:1 relationship between SYMBOL and SEMANTICS.
Everything else is up for grabs.

[Scott Mayers]

You're missing the point. Lambda Calculus will simply have a distinct set of different rules but still requires a set of consistent axioms of which Identity is still at least required.

No, I am not missing the point. I never rejected identity. I merely rejected YOUR understanding of identity in favor of mine.

Lambda calculus does NOT mandate your Alphabet. It mandates nothing whatsoever.

A Turing machine has infinite length tape. So you can have infinite-length alphabet.

If you can invent a million unique SYMBOLS and you assign unique SEMANTICS to all of them. then you are good to go.

The ONLY requirement is consistent semiotics. 1:1 relationship between SYMBOL and SEMANTICS.
Everything else is up for grabs.

I'm not sure what you are thinking.(?) We of course can define the word, "True" to mean not true. This is just taking the string of "True" and turning it into the number based upon its ASCII code, so that whenever you use the words "True" and "(formula-string) in a program, it replaces it with addresses that links to some unique memory space. Then, if by 'not true' you mean the constant, '0', it puts that into the memory space assigned to the string, "True", and you assign the formula-string's address to the value you assigned to the word string, "True", the value '0' binary (for 'not true' in the metalanguage's meaning). So True is interpreted as the value most computers assign "False" to mean.

This is only a 'trick' if you think it is being anti-consistent. [and similar reason Einstein asserted to the Quantum Mechanical claims of ACTUAL indeterminacy to be true is false since "God does not play dice": the reality of the appearance still has to require that IDENTITY LAW]

Note: replace (formula-string) to be "x != x", for the instance you gave above. That you assign it to the word "True" when underneath the program, it assigned it to the '0', which is normally, the boolean, "False" meaning.

[Logik]

I'm not sure what you are thinking.(?) We of course can define the word, "True" to mean not true. This is just taking the string of "True" and turning it into the number based upon its ASCII code, so that whenever you use the words "True" and "(formula-string) in a program, it replaces it with addresses that links to some unique memory space. Then, if by 'not true' you mean the constant, '0', it puts that into the memory space assigned to the string, "True", and you assign the formula-string's address to the value you assigned to the word string, "True", the value '0' binary (for 'not true' in the metalanguage's meaning). So True is interpreted as the value most computers assign "False" to mean.

This is only a 'trick' if you think it is being anti-consistent. [and similar reason Einstein asserted to the Quantum Mechanical claims of ACTUAL indeterminacy to be true is false since "God does not play dice": the reality of the appearance still has to require that IDENTITY LAW]

Note: replace (formula-string) to be "x != x", for the instance you gave above. That you assign it to the word "True" when underneath the program, it assigned it to the '0', which is normally, the boolean, "False" meaning.

Grave misunderstanding. Concepts come before symbols.

All truth is conceptual first and foremost! The CONCEPT of "truth" comes before the SYMBOL "truth". How I express the CONCEPT of "truth" in language is immaterial.

I can use T. or 1 or ⊤ or ✔ or ䷼

It's just representation. The SYMBOL is what you put on the ticker-tape.

[Logik]

Note: replace (formula-string) to be "x != x", for the instance you gave above. That you assign it to the word "True" when underneath the program, it assigned it to the '0', which is normally, the boolean, "False" meaning.

When I state the PROPOSITION: "Scott Mayers is human => True".

I am stating that there exists a Turing machine which can determine the answer of the above proposition to be correct.

I am not changing the value of "True" I am using it like you would use it.

Is there a Turing machine that CAN decide the above? Yes. Any other human.