Necessary conditionhood is a notion I employ all over the place, and I've only just noticed that I'm not really sure what it amounts to.
Suppose P is a necessary condition for Q. Here are some of the things I've probably taken that to mean on various occasions:
1. Q materially implies P
2. Q strictly implies P
3. ~P materially implies ~Q
4. ~P strictly implies ~Q
5. If Q then P
6. If ~P then ~Q
7. Necessarily, if Q then P
8. Necessarily, if ~P then ~Q
9. If Q were the case then P would be the case
10. If ~P were the case then ~Q would be the case
On other occasions I've taken necessary conditionhood to involve some more substantial kind of dependence, so that if P is a necessary condition for Q, then Q's obtaining depends causally, explanatorily, or in some other interesting way, on P's obtaining. So for instance another thing I've taken it to mean is:
11. If ~P were the case then ~Q would be the case *because* of ~P
All this goes for sufficient conditions too, mutatis mutandis.
Some questions this raises: is one of these understandings right and the others wrong? Or do we in fact have a wide range of notions of necessary conditionhood? If the latter, is that a good state of affairs? Should we take more care to spell out what we mean each time we use the phrase 'necessary condition'?