Consider someone in this world and someone is another possible world. Does it make sense to say that they are the same height (or that they aren’t the same height)? Or, more formally, if there are objects x and y, each of which belongs to a different world, are there predicates that can reasonably said operate over both x and y?
The answer is more complicated than it might seem. First we must investigate what it means to say that two individuals in this world are the same height. There are two ways of doing this. One is the direct way, which is to say that two individuals are the same height if and only if when put back to back neither would overshadow the other (or that a line drawn between the top of their heads would be parallel to the ground, ect). The other is the indirect way, which is to say that two individuals are the same height if and only if when measured the result is the same for each. In essence this is to break down the same-height relation, S, over a and b (Sab), into a comparison between two properties (Ha = Hb). But we must take an additional step to ensure that the indirect method is valid, which is to ensure than when we measure the height of each individual we do so in the same way. We wouldn’t, for example, consider one individual, whose height was measured by the number of yards-sticks, to be the same height as an individual whose height was measured in meters, even if the numerical quantities were the same. In essence we then need a direct way to compare the methods by which we measure the two individuals. For example, we might insist that the methods of measurement agree when measuring the same object.
So in both cases we are forced to define our relation, in some way, through a direct interaction. In the direct way we say that the relation holds only if some result would follow were the two objects in a certain situation, and in the indirect way we say that the relation holds if they both have some property, but to make sure the property is the same one for both objects we must stipulate that, in principle, whenever one property holds the other can be determined to hold as well. And so in both cases the possibility of inter-world relations is ruled out.* Consider, for example that there are some possible worlds identical to ours in which everything is, say 20% larger (from the size of the universe to the smallest constituent of matter, with the physical laws appropriately adjusted as well), and there are others were everything is 20% smaller. Now, given the complete collection of possible worlds how could you tell if an individual in this world was the same height as an individual in some other possible world? You can’t, because there is no way to tell if you are dealing with an individual in one of these “re-sized” worlds, since their meter sticks will have been altered as well.**
The above holds for what I call “real” properties and relations. “Real” properties, and relations, are the ones that are based in the physical facts, and thus in some way reveal information about the world to us. And usually these are the properties that concern both philosophers and scientists. But there is another class of properties and relations, the constructed properties. A constructed property, or relation, is not one that holds based on some fact about the world, but rather by definition. For example, I could define the property Q to hold for me, you, and the cat, and no one else, and it would be a viable constructed property. And because of this there is no reason a constructed relation couldn’t hold between the individuals of different possible worlds. However, such properties, being constructed, cannot be legitimately said to be based on any facts, and thus cannot reflect a “real” relation between worlds.
Now whether this matters depends on how you view possible worlds. If you view them as sort of really “out there” then you are concerned with real properties, and thus must maintain that there are no inter-world relations. On the other hand, if you view possible worlds as a mathematical construction that provides a model for certain systems of logic, then you are interested in constructed properties. Even if you view possible worlds as possible rearrangements of real facts of this world then you are interested in constructed properties, because in such a case you are treating the world as a collection of facts and constructing another arrangement of those facts for some purpose. Now admittedly, for simplicity’s sake, when we talk about constructed possible worlds we use the names of real properties, like height. But this is simply convenience, and not a reflection that we really believe that people in our constructed world really have height in the same way we do.
So why does this matter? Well in terms of the possible worlds dealt with in mathematics it doesn’t. But recently (in the history of philosophy) Kripke, and others, have adopted possible worlds for use in the philosophy of language. And, on the basis of inter-world relations, insist that we can or can’t reference objects in other possible worlds under certain circumstances. I won’t go into the exact nature of the claims here, it should suffice to point out that if reference is something real, based on the physical facts, then talk about these possible worlds can’t have a bearing upon it, because basing their arguments, as they do, upon various inter-world relations (such as being the same thing, having the same properties, ect), their possible worlds must be of the constructed variety. But if they are the constructed variety it can mean little to say that we can or can’t make reference to things in them, since they don’t really exist, and thus is as nonsensical as arguing about whether we can or can’t make reference to unicorns (or at least our conclusions as to whether we can or can’t meaningfully reference unicorns shouldn’t impact the validity of the theory). They don’t exist, so there are no facts of the matter to worry about.
* Nor can you construct inter-world relations based on counting, because counting requires an agreement on what you are counting, which, again, is not something you can legitimately establish between possible worlds.
** There is one exception to this rule, which is the relation not-in-the-same-world can legitimately be said to hold for individuals of different worlds, because it is based on the one fact that is “outside” of their possible world, namely which possible world it is.