In possible-world semantics, what do nested modal quantifiers mean?

Question

I'm trying to learn modal logic and I'm having trouble translating sentences like [][]A and []<>A into natural language using possible-world semantics. The first statement seems to read, "In every possible world (in every possible world A)", and the second, "In every possible world (in some world A)".

But it's not clear to me how we could have possible worlds inside of possible worlds. I want to say that worlds don't stack in that way; a possible world inside another possible world is just not a world at all.

Have I translated the sentences wrong, or am I wrong to object to nested possible worlds?

Answer 1

You're not wrong, but rather than thinking of possible worlds as being "inside" other possible worlds, it's better to think of some worlds as being possible relative to others.

For example, let's say it's possible to fly to the moon. That means that there is a world - call it v - which is possible relative to our world, in which we do fly to the moon.

But there might be another possible world - call it w - where we have used up all the fuel, and so it is not possible to fly to the moon. In this case, v is not possible relative to w.

According to possible worlds semantics, []A is true at a possible world w if and only if A is true at every world v which is possible relative to w.

So [][]A is true at w if and only if for every world v possible relative to w, and every world u possible relative to v, A is true at u.

Likewise <>A is true at w if and only if at some world v possible relative to w, A is true at v. So []<>A is true at w if and only if at every world v possible relative to w, there is some world u such that A is true at u.

One reason for your confusion might be that in one of the simplest modal logics, S5, every world is considered possible relative to every other.

In this case if []A or <>A are true at any world, they are true at every world, and so [][]A and []<>A end up saying much the same thing.

But in other modal logics in which not every world is considered to be possible relative to others, more interesting things can happen.

Answer 2

A modal sentence such as []p has its meaning inside a world. It does not tell you that p holds for every world but p holds in the related worlds.

For example, suppose that we have only one world W in which p holds and R is empty relation. Even if there is no world ~p holds, but we still have []~p holds in W since W is isolated.

If you think possible worlds as people and <> as a belief of someone, it will be less confusing. If I know a person, say John, who believes that everyone he knows believes p, then, for me, <>[]p holds. In this case, John does not need to believe that everyone in the earth believes p. Furthermore, if I don't know John and everyone I know does not believe []p, then for me, <>[]p does not hold even if John is alive.

Answer 3

You may want to back off to the original meaning in Aristotelian wording, and then rephrase it in possible world semantics. [][]A or <>[]A or []<>A in a more direct interpretation means 'It is necessary that A is necessary' or 'It is possible that A is necessary' or 'It is necessary that A is possible'.

But 'For all x in X, for all y in X' just means 'For all x and y in X' whether X is the class of possible worlds, or just a mundane and ordinary set. So directly stacking modes is stacking up pointless quantifiers. All the 'Necessaries' or 'Possibles' collapse, and any 'Possible' eats any 'Necessary' inside it, just the way 'For all' and 'There exist' work when they are not quantifying over worlds.