The "Problem of Transworld Identity", like so-called "Problems of Identity through Time" for persons or material objects, is a misnamed problem. There are no philosophical problems about identity. The problem should be called "The problem of de re, or non-qualitative, possibilities." Intuitively, we wish to say not only that it is possible that there be somebody very like I actually am, born in Memphis on Aug. 6, 1970, raised in Moscow, TN, 6' 5'' tall, male, dark hair, named "Brock Sides" and married to a woman named "Janet", etc., but who is a Republican; we want to say that is possible that I be a Republican. (Even though I am not.)
Those who believe that our sentential operators "It is possible that" and "It is necessary that" are quantifiers over possible worlds (however they conceive of things) must say that in some way these possible worlds represent, or misrepresent, this universe we live in. It is possible that there be a talking horse: those who believe in possible worlds will analyze this as "There is a possible world according to which there is a talking horse", i.e. a possible world that misrepresents this universe we live in as containing a talking horse.
Different accounts of possible worlds differ in their account of how this representation takes place. Modal realists, such as David Lewis, hold that a possible world represents the universe (for Lewis, "the actual world") as containing a talking horse by having a literal talking horse as a part. Ersatzers, who think it utter extravagance to believe in talking horses at all, however unactual, may hold several different views. "Linguistic ersatzers", for example, who believe that possible worlds are something like consistent stories that don't leave out any details, hold that a possible world misrepresents the universe as containing a talking horse by containing (as a part, or as a member) a sentence that means "There is a talking horse."
But this possibility, that there be a talking donkey, is a "de dicto" or merely qualitative possibility. It doesn't represent any particular donkey as talking. (The only thing it represents is the universe as a whole.) But as I noted above, we should like to say not only that it is possible that there be a talking horse, but also it is possible that Seattle Slew be a talking horse; or, as we may put it (to make the label "de re" truly deserved), it is possible of Seattle Slew that she be a talking horse. A possible world according to which Seattle Slew is a talking horse must not only represent the universe as containing a talking horse, but it must represent Seattle Slew as a talking horse.
It might be thought that the ability of possible worlds to represent de re is supervenient on their ability to represent the universe in a purely qualitatively way. Certain worlds represent the universe as being very similar to the way the world actually is, but with slight differences. Some world represents the universe as containing a dark-haired, 6' 5'' tall, male, born on Aug. 6, 1970, called "Brock Sides", and Republican. In virtue of this purely qualitative, or de dicto, representation, it represents me as being a Republican. But several thought experiments seem to show that de re representation of possibilities doesn't supervene on their purely qualitative representation. Here's an example due to Lewis: suppose that we in fact live in a universe of one-way eternal recurrence, in which there is a first epoch but no last epoch, and that I in fact live in the seventeenth epoch. (Never mind if we don't actually live in such a world; surely this is possible, and our theory of modality should be able to handle such cases.) Some would like to say that it is possible that the universe be qualitatively exactly like it actually is, but that I live in the third epoch instead. One who believes in possible worlds must say (if she agrees that this is a possibility), that there is a possible world that represents the universe exactly like it actually is, but that represents me as living in the third epoch. This possible world agrees with the actual world (the world that doesn't misrepresent the universe in any way) in all its purely qualitative, de dicto representation, but it disagrees with the actual world in its de re representation. Thus de re representation of possibilities does not supervene on de dicto representation of possibilities.
Another example that seems to show the need for de re representation independent of purely qualitative representation is an example of Chisholm's. Chisholm assumes that it is possible that Adam be a little more like Noah than he actually is, i.e. that there is a world that represents Adam as being a little more like Noah than he actually is, and vice-versa for Noah. If that world had been actual, it still would have been possible that Adam be a little more like Noah, and Noah a little more like Adam. So there is another world that represents Adam as even more like Noah, and Noah even more like Adam. By continual iteration, we eventually get to a world that represents the universe as qualitatively like it actually is, but which represents Noah as occupying the Adam role (being the first man, being married to Eve, getting kicked out of Eden for eating the apple, etc.) and Adam as occupying the Noah role (building the ark, etc.). This world doesn't differ from the actual world (the world that doesn't misrepresent the universe in any way) in its qualitative representation, but does differ from the actual world in its de re representation of Adam and Noah. Again, de re representation of possibilities doesn't supervene on de re representation of possibilities.
How might a modal realist account for this representation de re independent of qualitative representation? Here's one way a modal realist might try to account for this: since a possible world represents the universe as containing a talking donkey by having a talking donkey as part, a possible world represents me as being a Republican be having me as a part: I am a part of that world, and in that world I am a Republican.
This attempt runs up against what David Lewis calls a "devastating and decisive" objection: the problem of accidental intrinsics. I am actually 6'5'' tall. This is an intrinsic property of me. It is possible, however, that I be 6'7'' tall. According to the version of modal realism under consideration, there is a possible world that has me as a part, and in which I am 6'7'' tall. But I'm be both 6'5'' tall and be 6'7'' tall. The proponent of the version of modal realism under consideration must say that being 6'5'' tall is not an intrinsic property after all, but relational property, which I have actually in virtue of bearing a certain relation to the actual world, and a certain relation to that other possible world. (Being 6'5'', on this story, is like being a father: one is a father in virtue of bearing the "is a father of" relation to someone.) But it is absurd to say that being 6'5'' tall is a relational property. So this theory must be scrapped.
According to Lewis's version of modal realism, the possible worlds represent de re in virtue of having counterpart objects. A world represents me as being a Republican in virtue of having a Republican counterpart of me, a person similar to me in many respects, but not in political affiliation, as a part. This counterpart relation is a similarity relation that is vague and partly determined by context: it is not once and for all decided exactly which other-worldly objects are my counterparts.
According to Lewis's counterpart theory, our idioms of de re modal possibility make use not only of quantification over possible worlds, but also of an implicit counterpart relation: "I might have been a Republican", according to Lewis, means "There is a possible world W, and a counterpart of me that is a part of W, and who is a Republican." Lewis remains agnostic about the existence of qualitatively indiscernible worlds, but his counterpart theory can handle the possibility that the world be qualitatively just like it actually is, but differ in the facts about individuals. The world that represents the universe qualitatively just as it is, but with Adam occupying the Noah role and vice-versa may well be the actual world itself. Under a certain counterpart relation, Noah himself qualifies as a counterpart of Adam: thus there is a world (viz. the actual one) that is qualitatively just like the actual one, in which there is a counterpart of Adam (viz. Noah himself) who occupies the Noah role.
In Naming and Necessity, Kripke poses several objections to Lewis's counterpart theory of de re representation of possibilities. The most famous of these is probably the "Humphrey couldn't care less objection," concerning the possibility that Humphrey win the 1968 election, instead of Nixon: "Probably, however, Humphrey could not care less whether someone else, no matter how much resembling him, would have been [sic] victorious in another possible world." One thing to note: Lewis's theory does not yield the result that, if things had been different, someone very like Humphrey would have been victorious. If things had been different, Humphrey himself would have been victorious: and Humphrey himself has this property in virtue of having a counterpart, someone very like Humphrey, who is victorious in another possible world.
Lewis notes that this objection might have some force coming from an modal realist who advocated genuine overlap between worlds, i.e. that Humphrey himself is a part of another possible world. But this theory succumbs to the problem of accidental intrinsics. Kripke himself, however, is some sort of ersatzer. ("A possible world isn't a distant country we are coming across, or viewing through a telescope.") And if this objection has any force against Lewis's theory, it applies equally well to theories that make use of ersatz worlds. Linguistic ersatzers, for instance, hold that possible worlds are something like consistent stories that don't leave out any details: "Probably, however, Humphrey could not care less whether there is a consistent story that contains a sentence meaning 'Humphrey is victorious'." An ersatzer cannot get Humphrey himself contained in the possible world (except as an arbitrary name for himself in a "Lagadonian language") any more than can a modal realist who adopts counterpart theory.
Furthermore, it seems that the ersatzer will need to make use of counterpart theory as well in her theory of de re possibilities. This is illustrated by Chisholm's example of the possibility of Adam and Noah "swapping roles." A linguistic ersatzer can easily construct possible worlds according to which I and President Clinton swap roles: the consistent story says "Brock became governor of Arkansas, and then became president of the U.S. in 1992, etc." and "President Clinton was born in Memphis in 1970, and went to grad school in philosophy, etc." But "Brock" and "Clinton" refer to actual individuals, so the ersatzer can make use of names for us in his language, whereas "Adam" and "Noah" do not refer to anything actual: these expressions are only fictionally names. The actual Adam has no counterparts in any world, for there is no such person as Adam. The linguistic ersatzer needs to make some use of counterpart theory to describe the two possibilities of Chisholm's example: "There is a possible world according to which someone was the first man, etc., and according to which someone else built the ark; and there is a possible world according to which someone was the first man, etc., and someone else built the ark, etc.; and the open sentence in the first world 'x is the first man, etc.' is a counterpart sentence of the sentence 'x built the ark, etc.' in the second world, and vice-versa."
Kripke, Saul. Naming and Necessity. Harvard, 1980.
Lewis, David. On the Plurality of Worlds. Blackwell, 1986.
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