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: http://www.xrefer.com/entry.jsp?xrefid=552831 http://www.xrefer.com/entry.jsp?xrefid=553229 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?