Two men barbecuing at a beach.
Two men are barbecuing while on the beach.
At a beach implies location whereas on the beach describes the relationship between the people and the beach. While at the beach, they can do things on the beach, not in it.
exists e1 x1 x2 x3.(a_topic(x1) & at(e1,x3) & n_beach(x3) & Actor(e1,x2) & v_barbecue(e1) & (x1 = x2) & n_man(x1) & card_eq_2(x1))
exists e1 x1 x2.(n_beach(x2) & on(e1,x2) & r_while(e1) & Actor(e1,x1) & v_barbecue(e1) & n_man(x1) & card_eq_2(x1))