How can mainstream OO languages support strong binary operations by classes? Section 24.2  in Types and Programming Languages by Pierce compares ADT and existential objects,in terms of how well they support strong binary operations:


  Other binary operations cannot be implemented without concrete, privileged access to the representations of both abstract values. For example,
  suppose we are implementing an abstraction representing sets of numbers. After scouring several algorithms textbooks, we choose a concrete representation of sets as labeled trees obeying some particular complex
  invariant. An eﬃcient implementation of the union operation on two sets
  will need to view both of them concretely, as trees. However, we do not
  want to expose this concrete representation anywhere in the public interface to our set abstraction. So we will need to arrange for union to have
  privileged access to both of its arguments that is not available to ordinary client code—i.e., the union operation must live inside the abstraction
  boundary. We call such operations strong binary operations.


and says strong binary operations can't be expressed as methods of existential objects:


  Strong binary operations, on the other hand, cannot be expressed as
  methods of objects in our model. We can express their types just as we
  did for weak binary methods above:

NatSet = {∃X, {state:X, methods: {empty:X, singleton:Nat→X,
member:X→Nat→Bool,
union:X→NatSet→X}}}

  
  . But there is no satisfactory way to implement an object of this
  type: all we know about the second argument of the union operation is
  that it provides the operations of NatSet, but these do not give us
  any way to ﬁnd out what its elements are so that we can compute the
  union.


but then says that mainstream OO languages provide support for strong binary operations by classes (instances of which are supposedly existential objects):


  The
  classes in mainstream object-oriented languages like C++ and Java are designed to allow some forms of strong binary methods, and are actually best
  described as a kind of compromise between the pure objects and pure ADTs
  that we have seen in this chapter. In these languages, the type of an object
  is exactly the name of the class from which it was instantiated, and this type
  is considered distinct from the names of other classes, even if they provide
  exactly the same operations (cf. §19.3). That is, a given object type in these
  languages has a single implementation given by the corresponding class declaration. Moreover, subclasses in these languages can add instance variables
  only to those inherited from superclasses. These constraints mean that every
  object belonging to type C is guaranteed to have all the instance variables
  deﬁned by the (unique) declaration of class C (and possibly some more). It
  now makes sense for a method of such an object to take another C as an argument and concretely access its instance variables, as long as it uses only
  instance variables deﬁned by C. This permits strong binary operations such as
  set union to be deﬁned as methods. 


When a method in a class has an argument which is another object of the same class, how can the method "concretely access the instance variables" of the other object? 

Does the book assume that the class makes the instance variables only privately accessible, and doesn't provide public accessible methods to access the instance variables? (I guess yes, because it seems to me that the third quote has the same situation as   the second quote where it be impossible for union method on existential objects.)

Thanks.
I will attempt an answer, though, the discussion in comments with @yuval-filmus seems to be going in the right way.

Let's recap: The book discusses ADTs against objects in their strictest sense.

ADTs are entirely public about their unique representation. Belonging in the ADT means satisfying said representation, and so binary methods can rightfully assume that both operands have that exact representation.

Objects are entirely opaque about their non-unique representation. Belonging in the object type means satisfying its interface, with no guarantees as to how this is achieved internally, and so binary methods may only assume that the two objects can respond to the same set of messages, but are quite possibly implemented entirely differently.

Even if the interface has a getter getFoo, that is no guarantee that the object has a foo field that you could poke at directly. Maybe some mock object is faking it with a method that returns a constant Foo. All you know as a client is that it supports a getFoo call.



Where the class-based object approach differs from the pure object approach is that the knowledge of the type of an object (the class it belongs to) gives you some guarantees about the structure of the object, and so, allows you to partially break the abstraction barrier.  Because all objects of a class must have been built with, at least, a call to the constructor of that class, you can guarantee the presence of some fields (initialized during that constructor), and so the language lets you access those directly, foregoing the mandatory message-passing of the pure object style.

This is a strength and a weakness:


because you have access to the concrete representation, you might be able to implement operations more efficiently,
but because you ask for a member of the class, a "mock" implementation is no longer a valid object for this operation: if you want to allow such an object, you must define an interface, and you are effectively back to a pure-object approach.




As in my other answer to one of your question, this paper is extremely relevant:

http://www.cs.utexas.edu/~wcook/Drafts/2009/essay.pdf
"On Understanding Data Abstraction, Revisited", by William R. Cook

Of particular interest are section 3.3 on autognosis, which hints at why binary methods in the ADT style are strong, whereas in the object style are weak, and section 5 which explains a bit the difference between what is considered a pure object approach and the more general objects available in Java. (You can program in the pure object style in Java, but you have to follow a given discipline about not breaking certain abstraction boundaries, which relates to the discussion you've had in the comments)



I hope this makes sense, I'm sorry you had trouble with my other answer, this is a hard topic to get your head around, but once you have fixed your misunderstandings you will see it is not that complex!