Suppose we have two propositions p and q, and an implication of this form:
p → □q
I can think of two potential interpretations of the meaning of this in terms of possible worlds:
If p is true in this world, q must be true in all possible worlds.
In any world where p is true, it necessarily follows q is true in that possible world as well. But this should not be taken to rule out that there may be worlds where neither is true. For example, it is necessarily true that if Steve is a bachelor than Steve does not have a living spouse, but there can still be worlds where Steve is not a bachelor and does have a living spouse.
Which of these, if either, is the generally accepted meaning? If anyone can point to a reference by a professional that spells this out that'd be ideal, but if not I'll accept an answer based on personal expertise if there seems to be agreement among those answering.