.. 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 concludeP==false . - if instead N==true , you still can’t conclude anything aboutP (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
- if “N is necessary for P (but may not be sufficient)”, then the moment you realize in your proof that
- 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 forP to exist must hold, otherwiseP 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 youN==true
for free. - This is also what the notation
P => N
really means: ifN is a necessary (but perhaps not sufficient) condition forP , thenP is a sufficient (but perhaps not necessary) condition forN . - However, if your aim is to get N for free butP==false , then you still can’t say anything aboutN . - 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 whenP==false
- Suppose that, whether from your proof’s initial assumptions or from something derived along the way, you observe