Ever since the toilet was invented and the opposite sexes shared it, the argument of whether the seat should go up or down has raged.
The discourse is easily settled. Both the seat and lid should go down. That lid is there for a reason. The reason? A plume of human excrement mist erupts when you flush. You don't want to breathe that in. If you do you might want to chase it with a few squirts of febreze.