An adumbration of the argument at III.11.1.359-60 of the Physics of the Shifa’ is as follows:
Given that possibility exists, it either exists as a substance or something that inheres in a substance.
It does not exist as a substance.
Therefore, it inheres in a substance.
Proof of the major: That possibility exists i.e., as a real feature of the world, is intuitively obvious. Ibn Sina draws attention to this simple fact of experience by noting that this is why we make statements of the sort “before the existence of any nonexistent thing, it was something whose existence was possible. So the possibility of its existence existed before [the things actual] existence.” Otherwise, he continues, “were [the possibility] not to exist, it would be nonexistent and there would be no possibility of its existence, in which case its existence would be impossible.” As for possibility existing as either a substance or something that inheres in a substance i.e., an accident, this follows from the Peripatetic division of all finite being into the ten Categories, the first of which is substance and the rest of which are accidents.
Proof of the minor: the Shaykh al-Ra’is writes:
If [possibility] were something subsisting in itself, which is neither in a substrate nor in a subject, then, as such, it would not be a correlative [or something related]; however, as possible existence, it is correlative to something and intelligible by the comparison. So it is not a substance subsisting essentially. Perhaps instead it is a certain relation or accident belonging to a substance. Now, possible existence cannot be a substance that has a relation, because that relation is associated with something assumed to be nonexistent. […]. Therefore, possibility itself is the relation, not some substance upon which a relation other than possibility follows, where the two, taken together, would be possibility.
Basically, the Shaykh’s intention is to deny two views about the status of possibility with a view to affirming a third. The views he denies are (1) that it is a substance and (2) that it is a substance-relation complex. The view he affirms is (3) that it exists in a substrate. He denies (1) because, unlike substances, possibilities are always correlative or relational. That is, they’re always related to something else and are hence only, as he says, “intelligible by the comparison” i.e., with the other things for which they are possibilities. A relation after all involves two terms. Substances though do not essentially involve relations; nothing else is needed for them to be completed or exist or be understood precisely as substances. Then he denies (2) because possibility or potentiality, being essentially for something, is also as a result privative; that is, it’s the (relative) absence of some kind of existence i.e., of the thing the possibility is a possibility for. This I think is the meaning of the Shaykh’s words when he says “possible existence cannot be a substance that has a relation, because that relation is associated with something assumed to be nonexistent [i.e., the thing to which the possibility is related].” In other words, the substance-relation composite, as something that actually exists, cannot as such be identified with possibility; for the latter, as a privation of sorts, lacks such actual existence which characterizes the former. And the conclusion follows.