The brown dog is out in the snow.
The brown dog is playing in the snow with another dog.
The dog still may not be playing in the snow, and there is no mention of another dog.
exists p1 s1 s2 x1 x2.(n_snow(x2) & n_dog(x1) & a_brown(s1) & Theme(s1,x1) & a_out(s2) & Manner(p1,s2) & in(x1,x2))
exists e1 s1 x1 x2 x3.(n_dog(x3) & n_snow(x2) & with(e1,x1) & n_dog(x1) & n_dog(x1) & -(x1 = x3) & in(e1,x2) & Actor(e1,x1) & v_play(e1) & a_brown(s1) & Theme(s1,x1))