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From wikipedia I know that the principle (or law) of identity can be stated as : A is A.
This seems a pretty straightforward principle: something cannot be what it isn't. Why does this principle need to be stated? are there any logics which do not take as given this principle?
It isn't universal, though like you point out it seems it must be.
For example, we can say that given two objects, all of whose properties are the same, we can say that they are similar and not the same. In Leibnizs language they are indiscernible.
This point of view is used in modern algebra, where the technical notion is called isomorphism.
And it's taken as a basic principle in Category Theory, where one 'ought' not to say that two objects are the same, but isomorphic.
This might be seen as a hair-splitting distinction; and it is, but it turns out to be a useful one.
In mathematics, we need to explicitly state the reflexive rule for equality: For all x, x=x.
Consider, for example, the set S such that, for all x, x in S if and only if x = 1. Is 1 in S? From the definition of S, we have 1 in S if and only if 1=1. By reflexivity, 1=1. Therefore 1 is in S.
Well, there are Non-reflexive Logics which restrict identity to only some terms in the language, leaving it inapplicable to the other terms on the language. People like Newton da Costa and Décio Krause talk about. Also check out Quasi-set Theory. As for a straightforward example, isn't there a version of Classical Predicate Logic without identity? Just take away the equality symbol and bam, you've got FOPL sans-Law of Identity.
From what I understand, they discuss the applicability of such logics/set theories in the context of quantum mechanics (well outside of my wheelhouse). It has its roots in some statements by Schrödinger that it doesn't make sense to apply identity to quantum objects (hence why these logics are sometimes called Schrödinger Logics).
