Necessary but not sufficient conditions are more important than you think

Most of the time when I was reading propositions/theorems/corollaries stating necessary-but-not-sufficient conditions I used to feel almost defrauded: you’re selling me something that sounds useful but actually tells me nothing about the statement I’m trying to prove, since on its own it isn’t enough to prove it.

What I’d been missing, though, were several use-cases where knowing a necessary-but-not-sufficient condition can save you:

• As a circuit breaker in your proof:
  • if “N is necessary for P (but may not be sufficient)”, then the moment you realize in your proof that N==false, you can safely exit early and conclude P==false.
  • if instead N==true, you still can’t conclude anything about P (hence my original frustration).
  • Under this use case, the necessary-but-not-sufficient condition is extremely useful, paradoxically, when you can prove that it is not satisfied and terribly useless when it is verified
• For piggy-backing additional properties or statements:
  • Suppose that, whether from your proof’s initial assumptions or from something derived along the way, you observe P==true: trivially, every condition required for P to exist must hold, otherwise P couldn’t exist in the first place.
  • Consequently, if any prop/theorem/proof states “N is a necessary condition for P (but may not be sufficient)”, then P==true hands you N==true for free.
  • This is also what the notation P => N really means: if N is a necessary (but perhaps not sufficient) condition for P, then P is a sufficient (but perhaps not necessary) condition for N.
  • However, if your aim is to get N for free but P==false, then you still can’t say anything about N.
  • In contrast to the previous case, here the necessary-but-not-sufficient condition is incredibly useful (and generous) whenever it applies because of the existence itself of P, and completely useless when P==false