What would happen if the elevators were set to opposite angles?
This would be an elevon application (ailerons + elevator).  Found on flying wings and deltas.  Theoretically possible on a standard tail.

With elevons, one up and one down will roll the aircraft with out pitching it.  Interesting that this can be done from the front, back, or center of the aircraft. 

Someone may wish to correct me, I believe Burt Rutan put both elevator and aileron function in the canard of one of his planes to simplify the control linkages.

So, as a designer, yes this is possible.