The dogs are running in the snow.
The dogs are chasing each other.
Just because the dogs are running doesn't mean they are chasing each other.
exists e1 x1 x2.(n_snow(x2) & n_dog(x1) & in(e1,x2) & Actor(e1,x1) & v_run(e1))
exists x1.(n_dog(x1) & all x2.(n_other(x2) -> exists e1.(Theme(e1,x2) & Actor(e1,x1) & v_chase(e1))))