If «فُلَانٌ يُعْطِي» occurs in a context praising generosity, it asserts giving as a characteristic. A negative expression, «فُلَانٌ لَا يُعْطِي», can similarly characterize a person as not giving without invoking one specifically recoverable object.
The scope of affirmation and negation must be read in context. A transitive verb without an overt object does not automatically mean “gives this particular thing” or “does not give that particular thing.”
The first question is whether the utterance concerns the existence or absence of the action, or instead what happens to a specified object.