According to possible world semantics, "it's necessary that P" means that
P is true in all worlds accessible from this one. Different modal logics
correspond to different restrictions on the accessibility relation. Before
the invention of possible world semantics, people argued about which modal
logic is the correct one, but now philosophers realize that different
notions of accessibility (and the corresponding notions of modality) are
useful at different times, so there is no single correct modal logic.
That's my one paragraph summary of possible world semantics. Please
correct me if I'm wrong, or read these articles if you're not familiar
with this topic:
My questions is, why not just quantify over the possible worlds and refer
to the accessibility relation directly? This way you can talk about
multiple accessibility relations simultaneously, and you don't have to
introduce new logical symbols (i.e. the box and the diamond). Is
modality just a syntactic shorthand now?