Modal logic was
axiomatized by
Rudolf Carnap and
Ruth Marcus in 1946. The principal modern work putting forth a
metaphysics exploring model logic is the 1970 book
Naming and Necessity by
Saul Kripke, a transcription of a series of lectures.
He denies Russell's thesis that an ordinary name is an implicit definite description, and says rather that a certain set of more or less conventional historical events lead to a definite thing being given a definite name. Statements about that thing in all possible worlds then refer to that thing, not to something more or less similar to it that happens to exist in another possible world.
For example, Russell might say that when we talk about Scott we mean the man who wrote Marmion and who was born in such a place on such a date...; that is we are saying "there exists someone who did all these things, and precisely one person fits these descriptions...". We can then ask whether he wrote Waverley, if that was not an original part of the defining description. The problem with this is if historical documents turn up that show that Scott's barber, not Scott, wrote Marmion: the reference for the description unravels. We don't know clearly who we're talking about when we say "Scott": do we still mean "the author of Marmion"? After all, that might be the only thing we know about "Scott".
Kripke says that somewhere in the process of birth and childhood Scott got the name Scott, was identified as Scott, and this identification has been transmitted to us. We still mean him even if all the facts we thought we knew about him turn out to be false (he wasn't born in Scotland, it was his barber who wrote the poetry and novels, and so on).
Moreover, this naming is rigid, that is it is fixed on this person across all possible worlds. In a world where a barber wrote Waverley and Redgauntlet and all the rest, that barber still wasn't Scott, because 'Scott' always means the person we're talking about in this (real) world.
This might sound a bit trivial, but it isn't: it's possibly the biggest revolution in logic since the 1950s.