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In logic, the 'law of identity' classically says: 'Every thing that exists' has a specific nature. More abstractly, For all x, x=x. In a mathematical perspective, "For all x" is sensible to me when x is contained in a universal set U which is already been defined in a 'context'.

What does it mean by "For all x" or 'every thing that exists' or 'every being that exists' mathematically or logically? Is it a primitive notion? Also, what about 'existence'?, Some body argue that mathematically 'existence' is purely contextual.

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    Correct; the universal quantifier means "for all". Thus, when we interpret an universally quantified formula we set an interpretation with a specific domain (a "context") and thus "for all" means: "all the objects of the domain". "For all x (x >0)" when interpreted in the domain of naturals, reads as the statement: "Every natural number is greater than zero". Similar for "For all x (x is Philosopher)" when interpreted in the domain of human beings.
    – Mauro ALLEGRANZA
    Mar 9, 2022 at 7:58
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    It's not obvious that the law of identity does attribute a specific nature to things. It can just as well be understood as a defining characteristic of the relation of equality.
    – David Gudeman
    Mar 9, 2022 at 8:16
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    Just think that the second part of x=1 and x+1=2 is contingent, not necessary, without the law of identity.
    – RodolfoAP
    Mar 9, 2022 at 11:23
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    You are reading it wrong. The law of Identity was a philosophical concept not a Mathematical logic issue. Notice there are different types of LOGIC. The law of identity also has a semantic meaning. There are propositions that Express the same idea with different words (aka different sentences). The law of identity is about ideas referring to the same object or the same properties. So the idea "Tibby is a cat" is a sentence in English but the same idea is not the same written sentence in Mandarin. The proposition in each case is identical but the sentences visually look different.
    – Logikal
    Mar 9, 2022 at 20:26
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    To add further context about the law of identity in a non mathematics application the law of identity helps philosophers analyze proposition where the truth value is unknown in reality. For instance, "All black holes can emit information of any radioactive object that falls into it" is either true or false. We are unaware in reality which one it is. However if you contrapose the original proposition you still will not be aware of the true or falsity. I am unaware if the proposition is true or false but I know with certainty the contrapositive must have the same truth value as the original.
    – Logikal
    Mar 9, 2022 at 21:17

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In logic, the 'law of identity' classically says (...) More abstractly, For all x, x=x.

The law of identity which is assumed in formal logic is not x = x but "x is identical to x". Applied to numerical values, equality is fine but misleading outside them. The Eiffel Tower is identical, not equal, to the Eiffel Tower.

In logic, the 'law of identity' classically says: 'Every thing that exists' has a specific nature.

No. Saying that everything that exists has a nature has nothing to do with the law of identity.

To say that something has a nature is to say that some aspect of it remains constant over time, and the law of identity does not say that. Things that change over time are still identical to themselves because identity is identity between the thing and itself now. A variable may take different values as long as the same value is given at the same time to all its occurrences.

What does it mean by "For all x" or 'every thing that exists' or 'every being that exists' mathematically or logically?

Mathematicians are not interested in everything. They are only interested in a particular subset of everything, namely, mathematical concepts, so they always assume some domain.

Formal logic is obviously more general than mathematics and so does not require assuming any domain. As long as you don't assume that a variable is somehow restricted, it is not, and so by default it is thought of as potentially identical to any value whatsoever.

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X is X works just fine as an abstract logic claim. The problem comes when one tries to identify an X with an object in our world. The Sorites Paradox, the Ship of Theseus, and never stepping into the same river twice are classical demonstrations that all objects in our world are bundle objects, where X is X does not apply to them.

Essentialism is an effort to declare objects to have some hidden property that allows logic to be valid when applied to them. Most philosophers have rejected essentialism as very clearly falsified.

See this answer for a more extensive discussion of essentialism: How do essentialists deal with fuzzy essential properties?

As implied in that answer, many of the objects in our world are stable enough, that we can approximate them as X is X, and usefully apply logic to them, even if that logic is not valid per its own criteria. This is the pragmatic rationale for using logic in our world, even when it is formally an invalid process.

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