Why do we actually need Private or Protected inheritance in C++?

In C++, I can't think of a case in which I would like to inherit private/protected from a base class:

class Base;
class Derived1 : private Base;
class Derived2 : protected Base;

Is it really useful?

It is useful when you want to have access to some members of the base class, but without exposing them in your class interface. Private inheritance can also be seen as some kind of composition: the C++ faq-lite gives the following example to illustrate this statement

class Engine {
 public:
   Engine(int numCylinders);
   void start();                 // Starts this Engine
};

class Car {
  public:
    Car() : e_(8) { }             // Initializes this Car with 8 cylinders
    void start() { e_.start(); }  // Start this Car by starting its Engine
  private:
    Engine e_;                    // Car has-a Engine
};

To obtain the same semantic, you could also write the car Class as follow:

class Car : private Engine {    // Car has-a Engine
 public:
   Car() : Engine(8) { }         // Initializes this Car with 8 cylinders
   using Engine::start;          // Start this Car by starting its Engine
};

However, this way of doing has several disadvantages:

your intent is much less clear
it can lead to abusive multiple inheritance
it breaks the encapsulation of the Engine class since you can access its protected members
you're allowed to override Engine virtual methods, which is something you don't want if your aim is a simple composition

Private can be useful in quite a few circumstances. Just one of them are policies:

Is partial class template specialization the answer to this design problem?.

Another occasion where it is useful is to forbid copying and assigning:

struct noncopyable {
    private:
    noncopyable(noncopyable const&);
    noncopyable & operator=(noncopyable const&);
};

class my_noncopyable_type : noncopyable {
    // ...
};

Because we don't want that the user has a pointer of type noncopyable* to our object, we derive privately. That counts not only for noncopyable, but many other such classes too (policies being the most common).

Public inheritance models IS-A.
Non-public inheritance models IS-IMPLEMENTED-IN-TERMS-OF.
Containment models HAS-A, which is equivalent to IS-IMPLEMENTED-IN-TERMS-OF.

Sutter on the topic. He explains when you'd choose non-public inheritance over containment for implementation details.

For instance, when you want to reuse the implementation, but not the interface of a class AND override its virtual functions.

Why do we actually need Private or Protected inheritance in C++?

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