You might be interested in Capgras Delusion. A = A but ~A = A.

I confess that the law seems tautological, trite, useless — Art48

Actually it is quite problematic.
Some years ago I think I discovered a paradox I called Angelo Cannata’s paradox, since I haven’t found it anywhere else. it is exactly about the law of identity.

Another problem arises if we consider the subjectivity involved in whatever we think of.

The law of identity is expressed in various ways: A is A. Whatever is, is. A man is a man. Whatsoever is white is white. — Art48

No. This is a tautology.

A non-tautological identity is A is B.

That's not a paradox, that's confusing basic algebraic functions with a tautological axiom.

Can you better explain, please?

A = A is simply the most basic form of saying that ~A = A is false. It is the axiom that tells us that contradictions are always false.

Martin Heidegger . . . links the law of identity "A=A" to the Parmenides' fragment (....for the same thing can be thought and can exist). Heidegger thus understands identity starting from the relationship of Thinking and Being, and from the belonging-together of Thinking and Being. — Art48

How does this assist us?

It becomes a non-trivial question when it is difficult to determine identity. This article has a good example from the world of physics: https://nautil.us/quantum-mechanics-is-putting-human-identity-on-trial-3977/

Identity also becomes tricky in metaphysics. For example, many forms of nominalism embrace trope theories, which is the idea that objects that share traits (e.g. triangularity) do so by sharing tropes. Tropes are not universals, that is abstract objects that exist outside of their instantiations.Tropes exist only as properties that objects have. For example, there are no triangles outside actual triangular objects in trope theory.

Within these trope theories, many theories posit that objects are identified by their tropes. Sum up all the things that can be said about an object and that is the identity of the object.

But this view has problems. Imagine two identical red balls, ball A and ball B. We'd like to say that these two balls are two different things, that we have two balls if we pick both up. We probably do not want to say "ball A and ball B are actually the same ball in two different places at once." However, if we claim that things are the sum of their traits, we end up in a pickle, because now the red balls DO seem to be identical and thus they are one ball in two places.

Saying they are two different balls because they are in two different locations is not that helpful either. Relative location is a derived trait, one that changes with context. If such derived traits are part of identity then you would be a different person when you're north of your house than you are when you're south of it.

Going back to the link above, this has some real world implications, because small objects like atoms are very much like our balls A and B. But, to make things extra tricky, they are not even definitely in one place or another at any given time, and when we measure the location of such a particle in a system it is impossible for us to know "which one" we have measured with certainty.

This brings up a larger philosophical question: if the things we are composed of seem to lack a unique identity, or rather they share an identity with many of their kind, all of whom exist in different locations at once, how do we get our unique identity?

Look at how the subject of identity is handled in formal logic, philosophical logic, mathematical logic, set theory, and mathematics. Those clear up a lot of questions (though there are still some philosophical questions that arise).

A is A is kind of boring, but it gets a little more interesting when we think about such things as

Water is H2O
George Washington is (fill in your description)
9 is 4 + 5
Hesperus is Phosphorus

I've studied the law of identity for quite some time. It is rather perplexing, and requires a good deal of thought to properly understand. It is stated as 'a thing is the same as itself'. Despite the appearance of meaninglessness, there are a number of facets to this law, which we can consider.

The first thing is that this law puts the identity of the thing within the thing itself. This is important to understand, because it means that a thing's true identity is not what we, as human beings say it is, it's true identity is what it is, itself. So if we say "that person is a man", "man" is not that thing's true identity. And even if we say "that person is Art48", "Art48" is not that thing's true identity, because the thing's true identity is the thing itself. The thing itself is the thing's identity

The next thing is that the law of identity allows that a thing might be continuously changing, yet maintain its status as the same thing. This is very difficult to conceive of, because normally when a thing changes, we would say that it ceases to be what it was, and it becomes something different. But since we've allowed that the thing's true identity is within the the thing itself, then what we say about the thing is irrelevant to the thing's identity. So despite the fact that we might say that the thing has changed from being this, to being that, within itself the thing has maintained its identity, as itself, and it may continue to be the same thing which it was.

The law of identity may be denied, as I believe Hegel did. But doing this renders the other laws of logic, noncontradiction, and excluded middle, as useless. This is because these laws put restrictions on what we can truthfully say about a thing, by determining what is impossible for us to truthfully say about a thing. However, if a thing has no identity within itself, then there is no such thing as a thing, and it would be meaningless to talk about what we can truthfully say about a thing.

Water is H2O
George Washington is (fill in your description)
9 is 4 + 5
Hesperus is Phosphorus — Richard B

Interesting.

Water is H20 is Identity - same thing, different names

George Washington is - insert the value statement of your choice - not about identity

9 = 2 + 5 - the law of identify allows us to say that 5 + 4 equals 9.

Hesperus is Phosphorus - is a property of a thing - not identity

The fact that one can refer to properties of a thing does not make the property identical to the thing.


Identity - A is A - A thing is what it is

Non-contradiction - A is not not A - A thing isn't what it isn't

Excluded middle - A or not A - No thing is neither or both.

These axioms seem to allow us to have maths and reason. The questions people seem to ask - are these structures in the human mind, or are they are like Platonism?

What can be its significance? — Art48

The 'law of identity' is the most truthful, possibly the only completely truthful, instance of 'is' or 'equals'. If you trace the western philosophical tradition back to Parmenides, as that quote you mention suggests, the idea is that only 'what is' truly is, and cannot be otherwise. The reason this perplexed that ancients, is that all of the objects of perception - the denizens of the empirical domain - constantly change, decay and turn into something else - so what are they really? Is there anything about them which is real or are they merely illusory?

Those kinds of questions are a long way removed from the attitude of modernity. I'm no expert in any of it, but am trying to re-trace the steps, so to speak, to understand the issues. It should be said that Parmenides and his contemporaries are representative of what is called the 'axial age' philosophies, which also include the Buddha and Lao Tzu. 'The term 'Axial Age,' coined by German philosopher Karl Jaspers (1883-1969), refers to the period between 900 and 300 BCE, when the intellectual, philosophical, and religious systems that came to shape subsequent human society and culture emerged.'

There's an interesting and little-known philosopher with the unusual name of Afrikan Spir, a German-speaking Russian (actually Ukrainian) neo-kantian. I notice the Wikipedia entry on him summarised his ontology thus:

For Spir the principle of identity is not only the fundamental law of knowledge, it is also an ontological principle, expression of the unconditioned essence of reality (Realität=Identität mit sich), which is opposed to the empirical reality (Wirklichkeit), which in turn is evolution (Geschehen).[26] The principle of identity displays the essence of reality: only that which is identical to itself is real, the empirical world is ever-changing, therefore it is not real. Thus the empirical world has an illusory character, because phenomena are ever-changing, and empirical reality is unknowable.

I've actually found a very good PDF of Afrikan Spir's book wherein this is developed, although I haven't tackled it yet.

~A = A contradiction. (A = A tautology.)
~A v A excluded middle.
A = B identity.

You say the sweetest things...

:smirk:

If ~(A = A) then, mathematically, A > A. This implies A + x = A where x > 0. However, from A + x = A, x = 0. This is contradiction [x > 0 & x = 0]! Hence, A = A. Did I just use the law of noncontradiction to prove the law of identity à la @180 Proof?

The law of identity just states that whatever exists is self-identical, like an ontological assertion.
(I guess, in terms of propositions, whatever proposition implies itself.)

When we start talking about Hamlet or the Moon, then we've already presupposed identity.
Otherwise, what would we be talking about? Meaning would be lost; meaning presupposes identity.

Not that the world has to oblige, though.

Going the other way around, we could say that tautologies are true by definition.
Well, actually, we do.
Within some (logical) system, a proof of a proposition could be showing that the proposition is related to a tautology via bi-implications.

The fallacy of equivocation, a consequence of polysemy, will wreak havoc if not for the law of identity.

1. This pen is good.
Ergo,
2. This pen will go to heaven.

Is the law of identity about consistency in the meaning of symbols or does it also extend over metaphysical identity? In other words, does the law of identity state something about things themselves and not just the words used to refer to them?

:chin:

Look at how the subject of identity is handled in formal logic, philosophical logic, mathematical logic, set theory, and mathematics. Those clear up a lot of questions (though there are still some philosophical questions that arise). — TonesInDeepFreeze

Spot on. One of the best examples of how logic can make our everyday language clear.

The law of identity: A = A

It feels odd that, unlike the other two laws of thought, there's an awkwardness to constructing a truth table for the law of identity.

A = A probably means A $\leftrightarrow$ A but then now we're talking about propositions and logic is about propositions. So, the law should be the law of identity (of propositions).

What is interesting is the conjunction of 1) the law of identity, 2)the identity of indiscernibles, and 3)the indiscernibility of identicals.

Identity of indiscernibles: entities x and y are identical if every property possessed by x is also possessed by y and vice versa.

indiscernibility of identicals: if two entities are identical with each other then they have the same properties.

Look at how the subject of identity is handled in formal logic, philosophical logic, mathematical logic, set theory, and mathematics. Those clear up a lot of questions (though there are still some philosophical questions that arise). — TonesInDeepFreeze

Precisely. In ordinary math we have an identity (valid for all or most values) and conditional equation (valid for a select set of values). 2(x+1)=2x+2 and 2(x+1)=3. Rough definitions.

Identity Equivalence of indiscernibles — Relativist

Words matter! Mass-Energy/Acceleration-Gravity identity equivalence!

:strong: :up:

It's a useful convention, allowing us to apply logic, make inferences, abstract and generalize etc etc... enabling us to built up knowledge.

It's important to keep in mind that the law of identity, and logic in general, is not about the world, but about language only.

We arbitrarily split classes of particular things off from the whole/the flux of existence by giving them labels, and decide that classes of things that are given the same labels are equal to themselves.... even though 'in reality' only particulars are equal to themselves, and only at the exact same time.

The fact that x is not exactly equal to x generally, doesn't matter all that much, because it still works for our intents and purposes. And we need this basic 'falsification', because without it we wouldn't be able to abstract from particulars to something more general... any kind of knowledge would be impossible.

Identity of indiscernibles
— Relativist

Words matter! Mass-Energy/Acceleration-Gravity identity equivalence! — Agent Smith

Yes, they matter, and I chose my words carefully. The identity of indiscernibles is a metaphysical principle of strict identity, not a physics statement.

Example: Joe describes object A to you. He gives you a complete list of of A's properties, including specifically where it was located at a specific point in time.

Separately, Tom described to you an object he calls B, describing a complete set of B's properties including precisely where and when.

Now, you compare A properties to B properties, confident both sets of properties are complete. You notice there's no discernible difference. Identity of indiscernibles implies A is B. The labels A and B refer to the same object. They have the same identity.

This is a good post overall, but the bit about Hegel needs an addendum.

The law of identity may be denied, as I believe Hegel did. But doing this renders the other laws of logic, noncontradiction, and excluded middle, as useless. This is because these laws put restrictions on what we can truthfully say about a thing, by determining what is impossible for us to truthfully say about a thing.

For Hegel, the real is the rational. He's all about trying to walk his readers off the ledge of radical skepticism. Part of the way forward from doubting everything is to accept that the world is rational, not illogical. Because if the world isn't rational, then nothing necessarily follows from anything else, and knowledge becomes impossible.

Hegel definitely wants to keep rationality at the center of his thought. In his early theological writings he sees Logos, unifying reason (identified with Christ in The Gospel of John), as the element in faith that can lift man up and give him freedom.

Hegel's problem with identity as it is commonly formulated comes from his attempt to rebuild logic. He wants to create a logic that is aware of the self as thinking subject, and of objects as existing for that subject. He wants to move past propositions such as, "the apple is red," that take the apple and its redness as existing outside of the perceiving mind. Identity has to be different because identity changes and grows more complete over time as our knowledge grows (as the dialectical progresses). And he doesn't want to look just at the apple as being a part of an individual subject's mind, since he is not a solipsist or subjective idealist, but how it is for all minds.

Thus, the apple's identity can unfold through the progression of history. The truth of the apple is the whole, and so covers the apple as well as the apple seed, the sprout from that seed, the apple sapling that comes from the sprout, the branch that comes off the sapling-turned-tree, the bud on that branch, and the apple that springs from the bud, complete with new seeds. A = A can't fit the truth. Is the seed the tree, the United States in 1795 the United States in 2025? Since the truth of a thing is the process of its becoming the law of identity as commonly used becomes defunct, but it isn't just tossed aside.

Yes, yes, and yes!

Hit the bullseye, you have! — Master Yoda

Space and time are properties. Missed that completely!

So, is for every predicate P if Px & also Py, x = y with especial care taken to include locus & tempus among the predicates.

What's your take on two cars of the same model? Would you still say identity of indiscernibles or would you switch to equivalence of indiscernibles?

Muchas gracias again for clearing up the matter for me! There's more to discuss but leave that for another day!