Simple Proof of the Law of Identity

[yahooyoda]

If one argues that A = A is false due to the perception that the two A's are different because they are written down in physical 3D space and because the two A's are located at different spatial coordinates...

then the equation: A ≠ not-A [A is not equal to not-A] is true.


If A ≠ not-A is true, it is an equaivalent statement of the law of identity: that a thing is itself.

[Nick]

If one argues that A = A is false due to the perception that the two A's are different because they are written down in physical 3D space and because the two A's are located at different spatial coordinates...

If someone argues A=A on that basis, then they aren't really arguing against A=A because they are actually arguing against A = (some other object which they are also calling A). They are imagining an entity, that the person who initially stated A=A, was never even talking about. They are missing the point that when one points out that A=A, they are necessarily saying that A, although written twice in the equation, represents the same entity.

[yahooyoda]

Bilocation of a singlie entity, for example, how an electron can be at two places at once but it still has a single identity.

[Jehu]

The Law of Identity is not properly rendered “A = A’, but “A is A”, for to be identical is not the same as to be equivalent. For example, four quarter are equal to a dollar bill, but they are not one and the same thing. Likewise, when two people say that they have bought the identical car, they do not mean that they have bought one and the same car, but two similar cars of the same make, model, year, colour, etc. The Law of Identity – originally rendered “Everything is different from another, but the same with itself.” [Plato – Theaetetus], is a linguistic principle which prohibits two or more definitions be associated with the same term – except where the appropriate definition may be easily inferred from the context in which the term is used.

Without such a governing principle, rational discourse would not be possible, for we would not know which definition (concept) a speaker intended us to call to mind when we hear a given term. For this reason, one can think of the formulation “A is A” as a proposition wherein the first “A” represents the subject (term), and the second “A”, the predicate (associated definition). Another way of expressing it is that “A thing is one and the same with its essential nature.”, for the ancient essentialists held that a definition was merely an expression of the essence of a thing – “essence” here meaning that set of characteristics which were essential to a thing’s being the sort of thing that it is”.

[Nick]

Bilocation of a singlie entity, for example, how an electron can be at two places at once but it still has a single identity.

That example doesn't even come close to touching A=A.

If we are defining a particle as having the attribute of bi-location, then that is exactly what it is (a particle which bi-location), i.e. A=A.

[Trevor Salyzyn]

Bilocation of a singlie entity, for example, how an electron can be at two places at once but it still has a single identity.

That's way too complicated. The Law of Identity is simple. It's so simple that there's no point in talking to someone who doesn't believe it, if such a person even exists.

[David Quinn]

If one argues that A = A is false due to the perception that the two A's are different because they are written down in physical 3D space and because the two A's are located at different spatial coordinates...

then the equation: A ≠ not-A [A is not equal to not-A] is true.


If A ≠ not-A is true, it is an equaivalent statement of the law of identity: that a thing is itself.

Well thought-out and expressed.

-

[divine focus]

If one argues that A = A is false due to the perception that the two A's are different because they are written down in physical 3D space and because the two A's are located at different spatial coordinates...

If this were the case, you can superimpose an A and an equal sign and say "This A is this A."

[yahooyoda]

Can truth be defined from within the system? ...what is the matrix?


http://en.wikipedia.org/wiki/Tarski%27s ... ty_theorem

Tarski's undefinability theorem, stated and proved by Alfred Tarski in 1936, is an important limitative result in mathematical logic, the foundations of mathematics, and in formal semantics. Informally, the theorem states that arithmetical truth cannot be defined in arithmetic.

The theorem applies more generally to any sufficiently strong formal system, showing that truth in the standard model of the system cannot be defined within the system.

[...]


The resulting theorem applies to any formal languages with negation and sufficient capability for self-reference that Gödel's Diagonal Lemma holds. First-order arithmetic satisfies these preconditions, of course.

The following is a proof of Tarski's undefinability theorem in its most general form by reductio ad absurdum. Suppose that an L- formula True(x) defines T*. In particular, if A is a sentence of arithmetic then True("A") is true in N if and only if A is true in N. Hence for all A, the Tarski T-sentence True("A") ↔ A is true in N. But Gödel's diagonal lemma yields a counterexample to this equivalence: the "Liar" sentence S such that S ↔ ¬True("S") holds. Thus no L-formula True(x) can define T*. QED.

[Philosophaster]

The Law of Identity is not properly rendered “A = A’, but “A is A”, for to be identical is not the same as to be equivalent.

In formal logic, the "=" sign is used to express an identity relation. The logical equivalence sign, on the other hand, looks like an "=" with an extra horizontal line.

[Kunga]

This A=A formula is the basis of logic ?
If so...i can see that clearly.

[Gurrb]

Could be A=A just as existence=existence. 3x=6 if x is 2, but in physical space they are obviously different. They are rendered equal because they exist occupying the same 'space' as the other, but indeed of different location. Nothing can truly be equivalent to another, for its 'form' can infinitely be divided into different parts, just as a golf ball is 'equal' to Earth as they are composed of the same amount of parts- infinity.

[Jehu]

Could be A=A just as existence=existence. 3x=6 if x is 2, but in physical space they are obviously different. They are rendered equal because they exist occupying the same 'space' as the other, but indeed of different location.

The one, (3x=6), is an assertion of ‘equivalence’, while the second, (existence = existence), is an assertion of ‘identity’.

Nothing can truly be equivalent to another, for its 'form' can infinitely be divided into different parts, just as a golf ball is 'equal' to Earth as they are composed of the same amount of parts- infinity.

It is true that nothing can truly be equivalent to another, however, not for the reason you assert; but simply because to be ‘another’ is to be ‘not identical’. As to the notion that there is anything that entails an infinite number of parts, this is simply not logically tenable. The term “infinite” applies only to processes which can never be completed, and does not signify a quantity– no matter how large. Consider, for example, the ‘set of natural numbers’ which is said to be a infinite set, because it contains no final term. What is meant by, ‘infinite set’, is that there is no logical end to the quantity of natural numbers that may be created by simply adding 1 more; but one should not infer from this that such a set exists in its entirety – for then it would necessarily have a final term.

[jufa]

Does 1 + 1 equal 2? And is 2

an equaivalent statement of the law of identity: that a thing is itself?

And should 2 represent

that a thing is itself.

Does it eliminate the identity of 1 which is the true identity expressing itself as the 1 # 2?

Or does 1 multiply itself to express 2 making itself bilocational, yet not a thing of itself because 2 could not exist if 1 + itself was not still 1?

Never give power to anything a person believes is their source of strength - jufa

[Gurrb]

for 2 to truly be '2', both of the 1s must in turn be exactly identical. could 2 possibly exist by this truth? if the 1s are no identical, then 2 is simply half and half; comprised of two different parts. just as water and oil are two liquids, which when mixed together make another liquid, but clearly of two different components. i think numbers should be loosely upheld as metaphor, as they are governed by a strict set of rules; whereas we think objects of existence are, but can never be certain. flawed in 1 cannot equal 2, but two of 1 can?

[jufa]

Two is 1 number, which means regardless of the reasoning applied, the substance and essence and identity of all numbers is 1. Your testimony to this truth is found in your own words.

for 2 to truly be '2', both of the 1s must in turn be exactly identical.

Never give power to anything a person believes is their source of strength - jufa

[Pam Seeback]

jufa it is true that all numbers are the number 1 in substance-essence, only appearing to be different to the five physical senses. This statement about the nature of numbers can be applied to the nature of consciousness/awareness itself, that is, that there is one consciousness/awareness of one substance-essence, only appearing to be different to the five physical senses. This is the appearance of a self, two selves, many selves.

What can be concluded from this insight is that 1 is the living metaphor of the permanent nature of reality. What can be concluded further is that all numbers after 1 are but duplicates, rendering multiplicity an illusion (a lie).

This means that the true nature of reality is of One Self, One Consciousness, One Awareness, One Thinker, One Thought.