«لا بالضرورة»
Not necessarily: the relation is not necessary in virtue of the subject itself.
This is not the same as declaring the first judgment false. A particular person may actually be writing, but writing is not essential to that person's existence; the person can remain human without writing.
Bi-l-fiʿl concerns actuality, whereas lā bi-l-ḍarūrah denies essential necessity. What happens and what must happen are logically distinct.