How to define non-trivial necessary and sufficient conditions?

Question

Mathematicians often speak of finding necessary and sufficient conditions for some property $P$. But $P$ is a necessary and sufficient condition for $P$. So, how do we determine what a non-trivial necessary and sufficient condition is? Is there a formal and rigorous definition of such a thing?

Answer 1

No. Why should there be? $Q$ is a necessary and sufficient condition for $P$ iff $P$ and $Q$ are logically equivalent. It is a value judgement whether the logical equivalence is non-trivial.