Saul Kripke, answered from the texts and cited to the page.
The informal analysis is simple enough to state: a proposition is necessary if and only if it is true in all possible worlds.1 That is the core idea. The formal apparatus of model theory is then built around it — a formula is valid in a domain if it is valid in every model of that formula in that domain, and the machinery of complete assignments for formulas in a domain gives us the extensional semantic counterpart to modal interpretation.2
But I want to be careful about what that apparatus is and is not doing. Possible worlds semantics, as I developed it and as others have used it, is not a reductive analysis of modal notions in any philosophically significant sense.3 It does not uncover the ultimate nature of necessity or possibility from the ground up. The notion of a possible world has its roots in ordinary ideas about ways the world might have been, but it comes at a much greater and subsequent level of abstraction than our direct modal judgments do — judgments like "it might have been the case that." Anyone who cannot understand the idea of possibility is not going to understand the idea of a possible world either.3
The main motivation for the possible worlds analysis was that it enabled modal logic to be treated by the same set-theoretic model-theoretic techniques that had proved so successful in extensional logic.3 Now, a very common misunderstanding — one I have been at some pains to correct — is that possible worlds are something like distant countries or foreign planets that we discover by some kind of powerful telescope.4
On that picture, the problem of transworld identity becomes pressing: here is Nixon in the actual world, and over there is some other world full of objects with various properties, and we need a criterion to determine which of those objects, if any, is Nixon.5 Quine pressed something like this, and David Lewis made it explicit with his counterpart theory — on Lewis's view, no object literally exists in more than one world, so we speak only of counterparts of Nixon, chosen by resemblance according to our favorite criteria.5
I think this takes the metaphor of possible worlds far too seriously.5 A possible world is not given to us by discovery; it is stipulated.4 When I say "in some other possible world I would not have given this lecture today," I am just imagining the situation in which I didn't decide to give it, or decided to give it on another day.4 We don't need to imagine everything true or false in such a situation — only those things relevant to the question at hand — but in principle a total description of the world is what we are gesturing at.4
And crucially: why can't it simply be part of the description of a possible world that it contains Nixon, and that in that world Nixon lost the election?4 Once we see that such a situation is possible, we are given that the man who lost the election in that world is Nixon — because that is part of the stipulation.4 There is no prior problem of identification to solve.
We are not coming across Nixon; we are talking about him. There is also a further complication worth flagging, which concerns the domain of objects across worlds. Variable domain semantics is required — some things that actually exist might not have existed, and there might have been things other than what actually exist.6 The same applies to all possible worlds considered generally.
This matters for predicates as well as names: when I turn to cases like "unicorn," the question of what it would take for there to have been unicorns is not answered simply by pointing to a possible world containing horse-like animals with horns.6 On my view, that does not yet describe a definite possibility in the relevant sense — but that is a further argument, and it depends on the same underlying point about how possible worlds are individuated and what it takes for a genuine possibility to be in view.
The basis of the informal analysis which motivated these definitions is that a proposition is necessary if and only if it is true in all 'possible worlds.'A Completeness Theorem in Modal Logic, p. 4
A is said to be valid in D if and only if A is valid in every model of A in D... an adequate semantic counterpart to this interpretation is given by the concept of a complete assignment for A in a domain D.A Completeness Theorem in Modal Logic, p. 4
I do not think of 'possible worlds' as providing a reductive analysis in any philosophically significant sense... In practice, no one who cannot understand the idea of possibility is likely to understand that of a 'possible world' either. The main and the original motivation for the 'possible worlds analysis'—and the way it clarified modal logic—was that it enabled modal logic to be treated by the same set theoretic techniques of model theory that proved so successful when applied to extensional logic.Naming and Necessity, pp. 27–28
A possible world is given by the descriptive conditions we associate with it... 'Possible worlds' are stipulated, not discovered by powerful telescopes. There is no reason why we cannot stipulate that, in talking about what would have happened to Nixon in a certain counterfactual situation, we are talking about what would have happened to him.Naming and Necessity, pp. 51–52
Some people in other possible worlds have dogs whom they call 'Checkers'... none of them can really be said to be Nixon; they are only counterparts of Nixon... All of this talk seems to me to have taken the metaphor of possible worlds much too seriously in some way. It is as if a 'possible world' were like a foreign country, or distant planet way out there.1. Identity and Necessity, pp. 10–11
What he (I) wanted to say was simply that there might have been things other than what actually exist. Also, some of the things that actually exist might not have. And the same thing applies to all possible worlds. So a 'variable domain' semantics for modal logic is required.Reference and Existence, p. 43