According to David Lewis, there is no inter-world identity. That is, there is no identity-relation between objects of two possible worlds. There may be a resemblance-relation in the sense that the object o1 of the possible world w1 may be have the same properties as the object o2 of the possible world w2. But if that were the case, then no modalities de re (of a thing) would be possible. That is, we could not assess the truth of, for instance, the following statement: "I could have been a doctor". This would be translated to modal logic in the following way: "There is a possible world where I am a doctor". But if there is no cross-worlds identity, then there is no sensible way of understanding the "I" in the statement.

But as it happens, Lewis does not abandon modalities de re. Instead, he has a theory he names Counterpart-theory. According to that theory, the statement "x could have had the property A" must be translated as "there is a world w where x's counterpart x' has the property A". Or, perhaps "there is a world w' where x' has a property A and x' is the counterpart of x of the world w". Alright, but what is this counterpart-relation? That is, what criteria do we have for determining whether x' truly is the counterpart of x?

According to Lewis, the matter is complicated. There is no straightforward way of determining it, but he agrees that it depends on the properties and relations x' has, weighed in some cosmic way. For instance, the counterpart might be determined by similar origins - I assume it would be like me being born as I was to the family I was in reality born, but then diverging (as Lewis puts it) to be a doctor instead of a student in philosophy. By divergence Lewis means that there are two worlds, w' and w'' that are identical (in relation to properties) to each other to the moment t, and different from each other after the moment t - so the world diverge at that point (yet they are not the same world splitting, but two different worlds from the start). Sometimes, on the other hand, it is not the origins that matter - for instance when I say that "I could have been born an American". In this case something else must determine my counterpart I' that was born in America. Lewis does not pretend to know all the relevant factors and weighings.

Fine, now I will show how Lewis's Counterpart Theory can be brought down. First I will define a term Lewis-vague. It is a humoristically tinged property that objects have that make them counterparts of each other. Because Lewis says that he does not know the relevant factors, but adds (with a fine show of optimistic faith) that there are those factors and that they are not arbitrary, I will name those relevant factors together as Lewis-vague. They exist, they are not arbitrary, but we do not know what they are. This way whatever position Lewis holds, I will hold the same for the relevant parts.

Then let us define three worlds. (Don't worry, there is a non-formal explanation below these :) ).

First the actual world w*. In the actual world there is an object x that has the Lewis-vague properties A including a measurable property (for instance height) a. (The object itself may, for instance, be me).

Second we define a world w'. In w' there is an object x' that has the Lewis-vague properties A' that resemble A apart from one difference: instead of a it has a property a', where a' > a. Thus in w' x' is the counterpart of x.

Third we define a world w''. In w'' there is an object x'' that has the Lewis-vague properties A'' that resemble A' (and A) apart from one difference: instead of a and a' it has a property a'', where a'' > a' > a. Also, in w'' there is the an object that is Lewis-vaguely identical to x'. This means that in w'' there is x' and x'' both, but otherwise w'' is Lewis-vaguely identical to both w' and w*.

Now let us look at the following modal statement: "It is possible that x would be x''" (where x'' is the x'' defined above, therefore the world it is in is the world w''). This must be translated as: "There is a world w'' where x'' is the counterpart of x". But because of the definitions given above, this is impossible! Why? Because in the world w'' there is w' that is because of the Lewis-vaguety the counterpart of x. Whatever the Lewis-vague relevant properties are that determine the counterpart, the counterpart must by definition be x'. So it is actually impossible that the modal statement expressed before would be true. But it is very clearly a possibility!

I assume that the formalism above might have thrown some people off. The situation is actually this: we have for instance me. Then we have someone who is like me in every (relevant) respect, but is 5cm taller than me. Then we also have, in the same world, someone who is like me in the same respect, but instead of being 5cm taller, he is 10cm taller. So because the 5cm taller person must be my counterpart, the 10cm taller cannot be one. So in Lewis's modal theory, the statement: "it is possible that I would be 10cm taller than I am in a world where there is someone (perhaps you?) exactly like me, but 5cm taller" would be false.

Now Lewis could wiggle out in many different ways (and, in fact, after reading one of his books where he replies to objections, I have no doubt that he would come up with some way of wiggling out!). One would be to say that it simply is impossible, although it looks possible. But in this case I would have to note that Lewis's theory is a theory about possibilities, and that there is no reason to take his formulation as the primary one. His whole assumption that there are such worlds is based on a metaphysical interpretation of the modal theory of possibilities, and because there is no logical contradiction in the proposition given above (and no conceivable contradiction of any kind, I think), it should be possible. Yet Lewis's theory does not allow it. Cart before the horse, David my dear.

Another way to wiggle out would be to say that there can be two or more counterparts in the world. Even though I do not think I understand how this would exactly be possible, I am willing to accept this. That is because it has no impact at all to the argument: I can change it to disclude that there are two counterparts: "It is possible that I would be less alike me in Lewis-vague properties than someone else who is not me". It does not make the statement in the least bit less plausible, but deals away with Lewis's (imagined) argument.

I contend that Lewis's counterpart theory does not work. Even if he made it work, he would have to give up on the non-arbitrariness of the Lewis-vague properties. He would have to agree that there is no way we can determine the counterpart. In fact, he would have to come up with a sort of counterpart-property that simply some objects have and some not. He would have to explain what this property is like. I do not think he can do that. And I do not even think he wants to do that.

Nonetheless, for all that it is worth, I think that there is no other theory about modalities de re that fits the bill. But that is more disastrous to Lewis, who postulates concerete possible worlds. He simply must account for the de re -part in some way, because the worlds are real. Unlike his colleague, who takes the possible worlds simply as linguistic apparati. The latter modal logician can simply say that "x could be y" means that "there is a possible world where x is y" without in any way explaining how x is y - it just is a manner of expression and that is all.

So, even though this argument is not against Lewis's modal theory as such, it destroys his attempt at explaining de re -modalities, and if he cannot come up with a way of making it work, I think his whole modal theory crumbles: it cannot explain how particulars could be different. It would prove the uneasy feeling many philosophers share that Lewis's theory, even if true, has nothing to do with possibilities. (Such concerns have been expressed by, for instance, Plantinga).