How is identity defined in the law of identity A = A?
Is identity the values and properties we assign to the object, or is it a circular definition where A is itself, or is it like a unique Id assigned to the object?
Does the law of identity allow us to assign itself as the value of something with itself being a vague circular notion without any specific definition, which can change depending on what we want to do?
The law of identity just says that a thing is identical to itself.
So the Eiffel Tower is identical to itself; the number 2 is identical to itself; the universe is identical to itself; time is identical to itself; the portrait of Descartes is identical to itself; God is identical to itself; the spin of a boson is identical to itself, etc.
Identity in this context is a relation between two things. Identity is reflexive, symmetric, and transitive. So, precisely, whatever x, x is identical to x.
As such, it goes, or should go, without saying.
However, the law of identity is usually understood as implying that all occurrences of a name used in a logical argument should refer to the same thing. Thus, in the modus ponens (Fx → Gy) ∧ Fx ⊢ Gy, for example, the first and second occurrences of the letter F should be understood as referring to the same thing. Similarly, the first and second occurrences of the letter x should be understood as referring to the same thing. And likewise for G and y.
I don't see anything else to say on the subject.
