Full Idea
'Strong necessity' requires the truth of 'necessarily φ' is all possible worlds. 'Weak necessity' merely requires that 'necessarily φ' be true in all worlds in which objects referred to within φ exist.
Gist of Idea
'Strong' necessity in all possible worlds; 'weak' necessity in the worlds where the relevant objects exist
Source
Theodore Sider (Logic for Philosophy [2010], 9.6.3)
Book Reference
Sider,Theodore: 'Logic for Philosophy' [OUP 2010], p.248
A Reaction
This seems to be a highly desirably distinction, given the problem of Idea 13719. It is weakly necessary that humans can't fly unaided, assuming we are referring the current feeble wingless species. That hardly seems to be strongly necessary.
Related Ideas
Idea 13719 Barcan Formula problem: there might have been a ghost, despite nothing existing which could be a ghost [Sider]
Idea 16466 Strong necessity is always true; weak necessity is cannot be false [Stalnaker]