In C++ Anything Is Possible. The only limitation is you!

enum_bitwise_overload.cpp

/* Demonstrate operator|() overloading for enum type.
   If you want to hide the fact that your enum's numerical encoding is not actually bit flaggable.
   Please don't hurt me, I promise you will never see this in production code from me.
   
   ~~aed.20160429
   
   Runnable version:
   http://coliru.stacked-crooked.com/a/bfedb189f1929447
*/

#include <stdio.h>
#include <algorithm>

enum E {
  E1 = 1,
  E2 = 2,
  E3 = 3,
  E4 = 4
};

E operator|(const E & e1, const E & e2)
{
  E emin = std::min(e1,e2);
  E emax = std::max(e1,e2);
  if (emin == E1 && emax == E2)
  {
    return E3;
  }
  return emax;
}

int main()
{
  E en = E1;
  printf ("1: %d\n", en);      // prove it's printable.
  printf ("2: %d\n", en | E1); // basically no-op.
  printf ("3: %d\n", en | E4); // pick highest.
  printf ("4: %d\n", en | E2); // special case.
  printf ("5: %d\n", (int)en | (int)E4); // force a different operator. cf. #3 above.

  return 0;
}