While in lecture, Prof. Potts argued "no, it is still a presupposition," but that doesn't seem to line up with my intuition.
Consider the statement: Sam stopped smoking.
Obviously, this presupposes that Sam smoked at one point. It's impossible to get around this because the definition of "stopped" entails that he was smoking at some point.
But now consider the negation: Sam didn't stop smoking.
I agree that we would intuitively understand this as implying that Sam smoked in the past, but this meaning seems like an implicature to me, not a presupposition. "Sam didn't stop smoking" doesn't semantically necessitate that he ever smoked in the same way that "Sam stopped smoking" did.
"Sam didn't stop smoking" opens up the two possibilities of either (1) Sam never smoked (so it would be impossible to stop smoking) or (2) Sam was a smoker and still is. We collapse on the latter meaning due to the counterfactual reasoning that whoever said "Sam didn't stop smoking" would have been a noncooperative speaker and violated Grices maxims of non-ambiguity, relevance, quantity, and brevity if Sam was never a smoker.
What makes something a presupposition is not that its entailed, but rather that it is being assumed to be part of the common knowledge by an utterance. This assumption can be entailed either by the original statement or by the meaning of an implicature of the utterance.