Does the elevator/stabilator generate upward force?

Ok, this is a followup to the previous discussion about forward control
pressure during the landing rollout. Can someone tell me if the
elevator/stabilator can actually generate an _upward_ force, or does it
simply generate a downward force for climbing and less downward force
for a decent?

Does the nose fall due to the CG being forward of the center of lift,
or does the elevator actually push the tail up?

We are talking about tricycle gear planes, not taildraggers... The PA28
series specifically.

--Dan