alt Hacker NewsI genuinely do not see it. If you have a collection of all sufficient properties for Q, you can construct Q's necessary property by just rolling them all up into a big or-statement. "You are high status if you get into Yale or have a lot of money or are really funny or ...", like that.
Again my specific claim here is merely that such a statement exists, nontrivially, for this kind of problem. Not even that we can write it down in full or whatever. I don't see why that's illegal.
For the specific question, I gave some examples upthread: for example, there isn't something that "it takes" to marry a spouse who confers high status on you, except for luck later in life, which Yale's admissions office can't predict and thus can't use as an admission criterion. There are persistent attributes that improve your chances of marrying well and staying married—pre-existing high status, conventional attractiveness, health, intelligence, sanity, etc.—but none of them are either necessary or sufficient.
If we're talking about the abstract problem in classical propositional logic, there's no requirement in classical logic that a proposition Q be true for some external reason. It can just be true. Causality is outside the scope of propositional logic.
It's true that, in classical propositional logic, there is necessarily a proposition that is necessary for Q's truth, that is, a proposition that Q implies. There are infinitely many of them, in fact. Q implies Q, for example. It also implies Q or not Q, because in classical propositional logic, any proposition implies all tautologies. You can't list them all, and it wouldn't help.
If you have some finite list of atomic propositions to try to compose your set S from, there is no guarantee that listing all of the sufficient conditions from that list (other than Q itself) will give you an S whose disjunction is a necessary condition. A simple model of this is P = no, Q = yes, S = {P}.