Both Java and C# share the same model of pure interface
types. You declare a pure interface by going
Mailable, then you can declare you implement it with
class Customer implements Mailable (in Java). A class
may implement any number of pure interfaces. One of the things this
model ignores is that you have implicit interfaces whenever
you have a class.
The public implicit interface
Customer is all the public members declared on
Customer. (This implicit interface is present in every OO language
I've seen.) One thing that neither Java nor C# allow you to do is
to implement an implicit interface - that is you cannot write
ValuedCustomer implements Customer
What would it mean to implement an implicit interface? Essentially it would tell the type system that the ValuedCustomer class implements all the methods declared in the public interface of Customer but does not take any of its implementation, that is its public method bodies, and non public methods or data. In other words we have interface-inheritance but not implementation-inheritance.
It would be equivalent to changing Customer into an interface that contains all the public methods of customer, and then having a CustomerImpl class that implements this interface.
Why might this be useful? One case I remember from the past was in the old days of Java, before the current collections framework. We wanted to replace the Vector class with an implementation of our own, but couldn't because Vector was a class and we could only subclass it. From time to time you run into cases like this when libraries don't provide interfaces to allow free substitution, without this feature we're stuck.
This particularly comes up these days with testing. There are lots of times when you want to stub out stuff, but it's difficult or impossible unless you have an interface. It also leads to defining pure interface types when the only reason to do it is to support substitution for testing. While using an InterfaceImplementationPair is a common approach it's one that many of us don't favor. Implicit Interface Implementation would be a much cleaner approach.
So why don't languages allow this? I don't really know - but then I'm not a language designer. I once had the chance to ask Anders Heljsberg about this and his reply was much along the same lines as his preference to only have overriding if you explicitly declare members as virtual. Essentially it's a concern about subclasses (or implementers in this case) breaking the superclass, something which touches on a much broader topic how to use subclassing. However this was only a brief conversation over dinner so I'm not convinced we really fleshed out the discussion.
After writing this my old colleague Mike Rettig pointed out that one of the problems with this is that classes actually define multiple implicit interfaces. In Java, for example, the customer class actually defines four implicit interfaces: public, protected, package, and private. If an object collaborates with a Customer it might use any of these interfaces (another instance of customer can use private features.) If we want to implement an implicit interface we either have to implement everything, or we to define how far we are going. I don't know how hard it would be for the type system to keep track of that.
Of course the examples I've given mostly work fine with just public implicit interfaces, so in practice that might be enough.
Ian Griffiths pointed out that the issue is possibly about mixing classes and interfaces. Microsoft's COM technology actually had a sharp separation between the two: "if you want to use an object in COM you have to do so via an interface. So you're always able to make your own implementation." This was made pretty transparent in VB 6 where COM interfaces could be generated behind the scenes.
This issue doesn't come up with dynamically typed languages. If you want to implement another class's interface all you need to do is implement the same methods and just use the object where you need it. You don't need to implement every method, just the ones that are being used in the particular interaction you're focused on. It's a scheme that works very nicely for testing - Smalltalkers referred to it for a while as the Imposter pattern. It's also quite common to use dynamic proxies to do this kind of thing in Java, although I feel that implicit interface implementation would be more communicative.
Does any of this matter? Mostly it seems to be an issue for testing, it's much harder to insert TestDoubles if you can't use a reimplementation - and often subclassing won't do the trick if the superclass wants a real connection to the database. It may well only be an issue for testing - Robert Conley told me that he uses VB6's ability to do reimplementation a lot for testing, but never found a need for it in production code.