We can all accept that “there are many ways things could have been besides the way they actually are”. There's no problem with that. However, David Lewis believed that this sentence involves an “existential quantification”. Why does he believe that? Surely you can only existentially quantify over that which exists. That’s why it’s existential and not modal. Ways things that could have been don't actually exist. There could be three-headed snakes; though there aren’t. Therefore we can’t quantify over three-headed snakes. As Quine put it: “To be is to be the value of a variable.” Things that could be can’t be the values of variables.
However, Lewis qualified his argument by saying that people who believe in possibilities “believe in the existence of entities”. Why does Lewis’s argument follow? Believing in ways things could be doesn't entail (or imply) a belief in their existence. So if these “ways things could have been” are possible worlds, then possible worlds don't exist.
Again, why does belief in possibilities entail (or imply) existential quantification? Again, David Lewis might have been a bricklayer; though David Lewis the bricklayer didn't and still doesn’t exist. That is, we can’t quantify over a bricklaying David Lewis - unless it’s just someone with the same name. (Not that Quinian quantification applies to single persons qua single persons.)
Lewis, however, preempts my problem and asks:
“If our modal idioms are not quantifiers over possible worlds, then what else are they?”
Quine (in his ‘What There Is’) has provided us with strong arguments against such extravagant Meinongism. (Though was Lewis an extravagant Meinongian?) This problem itself brings in a whole host of accompanying problems about the references of words and the names of entities that seemingly don’t actually exist.
Though I can quite happily talk about a round square: does this somehow bring about the round square’s existence? I can talk about a possible Murphy who has three thousand girlfriends. Does my talk alone bring into existence this possible Murphy? However, it's still a fair question. What is it we're talking about when we talk about “way things could have been”? Indeed who is this God who “doesn’t exist”?
The same is true, according to Lewis, when he talked about some x being necessary (rather than possible). What are we talking about when we talk about this or that being necessary? What are we referring to? What makes this or that necessary? Necessity can’t be seen in one world – in our world. Therefore it must be seen at every possible world.
When we say that
"2 + 2 equals 4" is necessarily true.
what are we saying? We're saying that this equation is true at every possible world - even in a world made of alcohol seas or one without our physics. We can only make sense of necessity - in this instance and in all instances - by believing in possible worlds that make our statements of necessity true. Without possible worlds, what is it that makes "2 + 2 equals 4" necessarily true? After all, it may be true in our world; though how do we know that it's true at all other possible worlds? We know by imagining other possible worlds of all shapes and forms and then we quickly realise that 2 + 2 equals 4 must be necessarily true at these worlds too. If 2 + 2 equals 4 were true only in our world, then it wouldn’t be necessarily true.
Let’s be clear what Lewis believed about possible worlds.
Are they simply theoretical constructs to him? Or are they convenient posits which somehow solve a whole host of problematic modal issues? Are they fictions-for-a-purpose? Or, in Lewis’s own words, are they “linguistic entities”? The answer in all cases is: absolutely not! Lewis is a (concrete) realist when it comes to possible worlds. That’s what he’s famous for. He wants to “be taken literally”.
Though what should we take literally?
Well, for a start, possible worlds are like our world, according to Lewis. They are, in fact, very similar to our world. So what’s different about them? Different things go on in them than go on in our world. We could say that - departing a little from Lewis - possible worlds have exactly the same constituents as our world; though they're differently configured. So possible worlds have legs, buses, atoms, trees, tables and so on. They also have David Lewises, Houses of Parliaments and so on. However, at one possible world David Lewis is a bus conductor. At another world David Lewis is Prime Minister. And perhaps at others there are different configurations of atoms and molecules. (Perhaps there's no H2O molecules - i.e., water - at certain possible worlds.)
This is where things get complicated.
Lewis says that all these other possible worlds exist; though they aren't “actual”. What does that mean? Well, for a start, the word “actual” is indexical (like “here”, “there” and “now”). That is, what is and what isn’t actual is dependent upon (or contextual to) the circumstances of utterance. That is, our world is actual to us; and other possible worlds are merely, well, possible. However, in W it's the case that W is actual; and our world, to them, is merely possible. So every possible world is actual according to itself; though only possible according to every other possible world.
Can we make sense of this distinction between actual and possible/existent? At a prima facie level, actual and existent seem to be virtual synonyms. However, as I’ve said, actual and possible don’t seem to be synonyms in Lewis’s scheme.
Strangely enough, Lewis actually says that the "unactualised inhabitants [of possible worlds] do not actually exist". That is according to us, not them. Again, actuality is indexical. Can we make sense of this strange ontology? Lewis is explicit:
"To actually exist is to exist and to be located at our actual world…"
Here Lewis seems to be conflating existence and actuality. That is, other-worldly persons don't exist and they aren't actual (because they don't exist?). What sort of existence or being do they have? If they "don't exist according to our world”, then what kind of existence do they have? There seems to be a logical contradiction looming here. Other-worldly persons both exist and don't exist. They don't exist according to us; though they do exist according to their own worlds. What's going on here? To some, the conclusion can only be that other possible worlds don't exist. In fact, in Lewis's own words, it "does not follow that realism about possible worlds is false". So what did Lewis say to this? He came out with this Zen-like statement: "…there are more things than actually exist." So some things that don't exist do actually…well, what, have being? Exist? Again, the word 'unactualised' seems to be synonymous with 'non-existence'; whereas earlier Lewis offered a distinction. However, Lewis - in this paper a least - doesn't offer us a precise account of his ontological position; which would, hopefully, clear away some of problems. Again, non-actuals have some kind of, well, existence. If not existence, then what - being?
The extent of Lewis's realism about possible worlds can be seen in the following passage. In it he says: "…there is much about them [possible worlds] that I do not know…" So they certainly aren't imaginative creations. If they were, then Lewis, presumably, would have known everything about them. Possible worlds are therefore like unknown planets. There are unknown planets out there; though we know precisely nothing about them. (Or we can see them vaguely through telescopes; though we still don't know much about them.)
Lewis didn't just believe in possible worlds because he thinks that they exist. He thinks that their existence solves various philosophical problems. So his interest or belief in possible worlds wasn't, as it were, entirely objective (if that's the right word). So what did possible worlds do for Lewis and other possible-worldists? They "systematize [our] pre-existing modal opinions". That is, they serve a philosophical purpose over and above the fact of their existence.
What are these other worlds like, according to Lewis?
Although earlier in this paper Lewis wrote that possible worlds are very much like our world, only reconfigured. However, he did say that the physics of these possible worlds may be different to our own. Indeed it follows from a belief in possible worlds that certain possible worlds must have alternative physics.
However, Lewis didn’t accept that possible worlds have alternative logics or alternative mathematics. And isn't that the primary point (if they need a point) of possible worlds? Of course Lewis wasn't talking about specific or generally-accepted logical or mathematical systems; only that logical and mathematical truths and realities will be true and real regardless of our efforts to codify them. Thus Lewis was also a realist (rather than an anti-realist) about logic and mathematics. That is, there may be some logical or mathematical truths or realities than we human beings can never - or will never - know or be able to formulate (e.g., Goldbach's theorem).
And why are logical or mathematical truths necessarily true? Because they're true at all possible worlds. Their necessity comes from their being true at all possible worlds. Thus, in order to guarantee or insure necessity and possibility, we need possible worlds.