C# Multiple Interface Inheritance does not allow public access modifier with same name

Question

So this has me perplexed.
Suppose two interfaces.

public interface a
{
    void foo();
}

public interface b
{
    void foo();
}

Both of those interfaces have a function foo, I have a class that provides explicit implementation:

public class alpha : a, b
{
    // why can't I put an access modifier here?
    // How would you be able to hide this from a derived class
    void a.foo()
    {
        Console.WriteLine("a");
    }

   void b.foo()
    {
        Console.WriteLine("b");
    }
}

And a Class that is derived from alpha

public class beta : alpha
{
}

How do you make foo private or protected since alpha doesn't allow access modifiers on explicit imlementation, what can stop someone from calling:

var be = new beta();
(be as b).foo();

EDIT

How come when I don't explicitly provide implementation I can provide an access modifier?

public class alpha : a, b
{
   //why this compile? 
   public void foo()
    {
        Console.WriteLine("both");
    }

}

Answer 1

Since interface a is public, any class that implements a must make the methods of a publicly accessible, either implicitly (through public methods) or explicitly. Explicit implementations are "sort-of" private since they can only be accessed through the interface.

In short, there is no way to completely "hide" foo - your class implements both a and b so those methods must me made accessible through some means.

This would be true even if you only had one interface - having multiple interfaces with a method name collision just forces you to make the implementations explicit. If you had one interface, foo would either have to be public or explicit.