What is the logic behind the "using" keyword in C++?

It is used in different situations and I am trying to find if all those have something in common and there is a reason why the "using" keyword is used as such.

using namespace std; // to import namespace in the current namespace
using T = int; // type alias
using SuperClass::X; // using super class methods in derived class

In C++11, the using keyword when used for type alias is identical to typedef.

7.1.3.2

A typedef-name can also be introduced by an alias-declaration. The identifier following the using keyword becomes a typedef-name and the optional attribute-specifier-seq following the identifier appertains to that typedef-name. It has the same semantics as if it were introduced by the typedef specifier. In particular, it does not define a new type and it shall not appear in the type-id.

Bjarne Stroustrup provides a practical example:

typedef void (*PFD)(double);    // C style typedef to make `PFD` a pointer to a function returning void and accepting double
using PF = void (*)(double);    // `using`-based equivalent of the typedef above
using P = [](double)->void; // using plus suffix return type, syntax error
using P = auto(double)->void // Fixed thanks to DyP

Pre-C++11, the using keyword can bring member functions into scope. In C++11, you can now do this for constructors (another Bjarne Stroustrup example):

class Derived : public Base { 
public: 
    using Base::f;    // lift Base's f into Derived's scope -- works in C++98
    void f(char);     // provide a new f 
    void f(int);      // prefer this f to Base::f(int) 

    using Base::Base; // lift Base constructors Derived's scope -- C++11 only
    Derived(char);    // provide a new constructor 
    Derived(int);     // prefer this constructor to Base::Base(int) 
    // ...
};

Ben Voight provides a pretty good reason behind the rationale of not introducing a new keyword or new syntax. The standard wants to avoid breaking old code as much as possible. This is why in proposal documents you will see sections like Impact on the Standard, Design decisions, and how they might affect older code. There are situations when a proposal seems like a really good idea but might not have traction because it would be too difficult to implement, too confusing, or would contradict old code.

Here is an old paper from 2003 n1449. The rationale seems to be related to templates. Warning: there may be typos due to copying over from PDF.

First let’s consider a toy example:
template <typename T>
class MyAlloc {/*...*/};

template <typename T, class A>
class MyVector {/*...*/};

template <typename T>

struct Vec {
typedef MyVector<T, MyAlloc<T> > type;
};
Vec<int>::type p; // sample usage
The fundamental problem with this idiom, and the main motivating fact for this proposal, is that the idiom causes the template parameters to appear in non-deducible context. That is, it will not be possible to call the function foo below without explicitly specifying template arguments.
template <typename T> void foo (Vec<T>::type&);
So, the syntax is somewhat ugly. We would rather avoid the nested ::type We’d prefer something like the following:
template <typename T>
using Vec = MyVector<T, MyAlloc<T> >; //defined in section 2 below
Vec<int> p; // sample usage
Note that we specifically avoid the term “typedef template” and introduce the new syntax involving the pair “using” and “=” to help avoid confusion: we are not defining any types here, we are introducing a synonym (i.e. alias) for an abstraction of a type-id (i.e. type expression) involving template parameters. If the template parameters are used in deducible contexts in the type expression then whenever the template alias is used to form a template-id, the values of the corresponding template parameters can be deduced – more on this will follow. In any case, it is now possible to write generic functions which operate on Vec<T> in deducible context, and the syntax is improved as well. For example we could rewrite foo as:
template <typename T> void foo (Vec<T>&);
We underscore here that one of the primary reasons for proposing template aliases was so that argument deduction and the call to foo(p) will succeed.

The follow-up paper n1489 explains why using instead of using typedef:

It has been suggested to (re)use the keyword typedef — as done in the paper [4] — to introduce template aliases:
template<class T> 
    typedef std::vector<T, MyAllocator<T> > Vec;
That notation has the advantage of using a keyword already known to introduce a type alias. However, it also displays several disavantages among which the confusion of using a keyword known to introduce an alias for a type-name in a context where the alias does not designate a type, but a template; Vec is not an alias for a type, and should not be taken for a typedef-name. The name Vec is a name for the family std::vector< [bullet] , MyAllocator< [bullet] > > – where the bullet is a placeholder for a type-name. Consequently we do not propose the “typedef” syntax. On the other hand the sentence
template<class T>
    using Vec = std::vector<T, MyAllocator<T> >;
can be read/interpreted as: from now on, I’ll be using Vec<T> as a synonym for std::vector<T, MyAllocator<T> >. With that reading, the new syntax for aliasing seems reasonably logical.

I think the important distinction is made here, aliases instead of types. Another quote from the same document:

An alias-declaration is a declaration, and not a definition. An alias- declaration introduces a name into a declarative region as an alias for the type designated by the right-hand-side of the declaration. The core of this proposal concerns itself with type name aliases, but the notation can obviously be generalized to provide alternate spellings of namespace-aliasing or naming set of overloaded functions (see ✁ 2.3 for further discussion). [My note: That section discusses what that syntax can look like and reasons why it isn't part of the proposal.] It may be noted that the grammar production alias-declaration is acceptable anywhere a typedef declaration or a namespace-alias-definition is acceptable.

Summary, for the role of using:

template aliases (or template typedefs, the former is preferred namewise)
namespace aliases (i.e., namespace PO = boost::program_options and using PO = ... equivalent)
the document says A typedef declaration can be viewed as a special case of non-template alias-declaration. It's an aesthetic change, and is considered identical in this case.
bringing something into scope (for example, namespace std into the global scope), member functions, inheriting constructors

It cannot be used for:

int i;
using r = i; // compile-error

Instead do:

using r = decltype(i);

Naming a set of overloads.

// bring cos into scope
using std::cos;

// invalid syntax
using std::cos(double);

// not allowed, instead use Bjarne Stroustrup function pointer alias example
using test = std::cos(double);

Another difference between typedef and using: you may do this:

    using vector3d_t = double[3];
    vector3d_t v = {1,2,3};
    v[1] = 4;

It is not possible with typedef:

    typedef double[3] vector3d_t; // Compilation error

What is the logic behind the "using" keyword in C++?

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