Can two different objects be identical? In the book I am reading I have been given multiple definitions of the identity relation, they are,


*

*a=b if a and b are identical


*a=b if a and b refer to the same object

and for the negation of identity I was given,



*a=b is false if a and b refer to different objects.


I am confused because 1 and 2 seem like they mean different things. With definition 1 you could have two different objects, one named 'a' and one named 'b', but with the exact same properties so that they are identical and a=b would be true however a=b would also be false by definition 3 since they are two different objects. I think I might be misinterpreting the definition of 'identical' since no proper definition was given in the book.
Could you please tell me which definition is the correct one?
 A: One of the properties of $a$ is the object to which it refers.  That is, if both $a$ and $b$ have all the same properties, then they refer to the same object.
We can relax this and talk about equivalence classes of objects, but the logical identity is as strict as one can make.  If there is any property that discriminates between the two of them, they aren't identical.
A: "Can two different objects be identical?"
No.
Just no.
At least according to your book.

With definition 1 you could have two different objects, one named 'a' and one named 'b', but with the exact same properties so that they are identical

I don't blame you for thinking that is what the definition seems like.  But it clear from context that according to this book, that is simply not possible.  
No two different objects have the exact same properties.  The only way that the thing named 'a' and thing named 'b' have exactly the same properties is if 'a' and 'b' are just two names for the same thing.  If 'a' and 'b' were different things they'd have at least one different property.
At least according to this book.
And, as mentioned, in a comment.  If you have a specific object, say this coffee mug in my hand.  Then one of the properties it has is "is the specific thing that it is; this coffee mug in my hand".  Now you might have another coffee mug that is in every other aspect exactly like mine.  But when it comes the the property "is the specific thing that it is" yours has a different property; it is your mug.  Mine has the property; it is my mug.  So that is one property they do not have in common.
