Most sentences contain presuppositions. For instance:
(1) a. Tom's sister left.
Depending on the situation, this sentence can be true or false. But regardless of whether it is true or false, it presupposes that Tom has a sister. Presupposition is identified using negation. If a sentence is negated, the presupposition nevertheless remains intact, e.g.
(1) b. Tom's sister did not leave.
For both (1a) and (1b), the presupposition is present that Tom has a sister. Negating the sentence did not remove the presupposition.
One way to understand presuppositions is to consider the predicates that sentences contain. Sentence (1a) contains two predicates, each associated with a separate proposition:
1st proposition: 'Tom's sister left.'
2nd proposition: 'Tom has a sister.'
Each of these propositions contains a main predicate. Predicates are thus capable of constituting propositions.
The key observation about presupposition is that one of the predicates is primary and any other predicate(s) that are present are secondary. In this case, the predicate 'leave' is the primary predicate and the predicate 'have' is a secondary predicate. Presupposition always resides with secondary predicates. From a syntactic point of view, the primary predicate is usually associated with the main verb of the main clause, whereas secondary predicates are expressed elsewhere in the sentence, for instance in a subordinate clause or in a noun phrase.
In the example sentence from the question, i.e. Alex stopped playing the piano, two separate predicates are clearly present, namely 'stop' and 'play'. Of these two, 'stop' is the primary predicate and 'play' is the secondary predicate. When the primary predicate is negated, it impacts the truth associated with the primary predicate only; the negation has no impact on the truth of the secondary predicate:
(2) a. Alex stopped playing the piano.
b. Alex did not stop playing the piano.
Both of these sentences presuppose the playing, i.e. Alex was definitely playing the piano, regardless of whether he did or did not stop. While the absence/presence of the negation reverses the truth value of the proposition expressed by the predicate 'stop', it has no impact on the truth value of the proposition expressed by the predicate 'play'.
To sum up so far, presupposition is closely associated with the predicates that are expressed in sentences. Primary predicates in sentences do not carry presupposition, whereas secondary predicates can carry presupposition.
Interestingly, however, the nature of the primary predicate can impact the ability of a secondary predicate to bear presupposition. Consider the predicate 'say' in this regard:
(3) a. Tom said he was hungry.
b. Tom did not say he was hungry.
These sentences contain two predicates each time, 'say' and 'hungry'. The primary predicate is 'say' and the secondary predicate is 'hungry'. The primary predicate 'say' is such that it blocks its secondary predicate from bearing presupposition. Whether or not Tom really was hungry or not is unknown, but rather what is known is merely that Tom did or did not claim he was hungry. The absence/presence of the negation has no impact on this state of affairs.
Other predicates, in contrast, allow their secondary predicates to bear presupposition, e.g.
(4) a. Tom knows that Susan is hungry.
b. Tom does not know that Susan is hungry.
In both cases, the speaker is saying that Susan is definitely hungry, regardless of whether Tom does or does not know it. Verbs that allow the secondary predicate to bear presupposition like this are known as factive. Verbs like 'say' above that block their secondary predicate from bearing presupposition can be designated as non-factive.
To sum up, the presence of presupposition is closely associated with the nature of predicates. Some predicates allow presupposition, whereas others do not. There is no easy path to understanding in such cases. One has explore which primary predicates allow presupposition, and which ones do not, and which types of secondary predicates always bear presupposition, and which ones are influenced by the primary predicate.