Actually, you could set up an aircraft such that the "elevator/tail assembly" always was providing an upwards force. It would look a bit strange, but it can be done. Usually there's other design constraints - tail lift direction isn't an overriding concern.
The terms we use - "wing", "tail", "canard" - all refer to lifting surfaces mounted at different places on the fuselage. If we were to call the one at the front the "wing" and the one at the back the "tail", then any canard-configured aircraft has a "tail" which is much larger than its "wing", and the "tail" always provides up force in normal trimmed flight. Aerodynamically, the names we give to the surfaces is irrelevant; the air doesn't care what we call the parts of the aircraft.