David Lewis, answered from the texts and cited to the page.
Counterpart theory is my alternative to the standard treatment of de re modality through transworld identity — the view that when we say Humphrey might have won the election, we are committed to Humphrey himself existing in some other possible world and winning there. My view is different. In each possible world there exist things wholly confined to that world; no ordinary individual is literally a part of more than one world.1
What makes it true that Humphrey might have won is not that Humphrey himself shows up elsewhere, but that some other individual in another world — his counterpart — wins, and that counterpart resembles Humphrey closely enough in the relevant respects to play that role.2 The counterpart relation is a relation of similarity: X's counterparts in other worlds are all and only those things which resemble X closely enough in important respects, and more closely than do the other things in their worlds.3
This has several consequences worth noting. The counterpart relation is neither transitive nor symmetric in general, and it depends on which respects of similarity we weight as important — both features that distinguish it sharply from identity, which is transitive, symmetric, and context-independent.4 It also means that a single individual can have more than one counterpart in a given world: if two twins in world w₄ both resemble you closely and equally, both are your counterparts.5
Conversely, you may have no counterpart in some world if nothing there resembles you closely enough.6 Kripke's famous objection — that counterpart theory fails Humphrey because it is someone else who wins, not Humphrey himself — misses the point of the competition.7 Both counterpart theorists and ersatzers agree that Humphrey himself is not literally part of the other worlds; the other world represents him as winning without containing him.
Counterpart theory is simply explicit about what enters into that representation — a similar individual in the other world — whereas ersatzism invokes some abstract whatnot to do the same job, and is no better off for having done so quietly.8 There is also a reformulation available in terms of trans-world individuals — mereological sums of stages drawn from different worlds.
A \-possible individual* is one whose stages across worlds are all counterparts of one another, and maximal under that condition.9 This is definitional machinery, not a change of doctrine; ordinary individuals remain wholly parts of single worlds, and the counterpart relation does the same modal work as before, just expressed in a different idiom.
The advantages of the framework over quantified modal logic are real. Counterpart theory is a first-order theory, not a special-purpose intensional logic, and whatever obscurity it carries can be traced to two independent sources — uncertainty about which descriptions describe possible worlds, and uncertainty about the relative importance of different respects of similarity — rather than being irreducibly opaque in the way that quantified modal logic tends to be.10
an ordinary individual will exist according to other worlds, thanks to its other-worldly counterparts. Still, it is a part of one world only, and neither the whole nor any (particular) part of it is part of any other world.On the Plurality of Worlds, pp. 221–222
if we say 'Humphrey might have won the election (if only he had done such-and-such),' we are not talking about something that might have happened to Humphrey, but to someone else, a 'counterpart'.On the Plurality of Worlds, p. 204
The counterpart relation is a relation of similarity. X's counterparts in other worlds are all and only those things which resemble X closely enough in important respects, and more closely than do the other things in their worlds.Part 1: Ontology, pp. 49–50
the counterpart relation is not transitive or symmetric, as identity is; and second, because the counterpart relation depends on the relative importances we attach to various different respects of similarity and dissimilarity, as identity does not.Part 1: Ontology, pp. 49–50
Suppose x4a and x4b in world w4 are twins; both resemble you closely; both resemble you far more closely than anything else in w4 does; both resemble you equally. If so, both are your counterparts.Part 1: Ontology, p. 29
Suppose whatever thing x6 in world w6 it is that resembles you more closely than anything else in w6 is nevertheless quite unlike you; nothing in w6 resembles you at all closely. If so, you have no counterpart in w6.Part 1: Ontology, p. 29
Humphrey could not care less whether someone else, no matter how much resembling him, would have been victorious in another possible world.On the Plurality of Worlds, p. 204
I think counterpart theorists and ersatzers are in perfect agreement that there are other worlds (genuine or ersatz) according to which Humphrey - he himself! - wins the election. And we are in equal agreement that Humphrey - he himself — is not part of these other worlds.On the Plurality of Worlds, p. 204
If all stages of a trans-world individual Y are counterparts of one another, let us call Y counterpart-interrelated. If Y is counterpart-interrelated, and not a proper part of any other counterpart-interrelated trans-world individual, then let us call Y a *-possible individual.On the Plurality of Worlds, pp. 221–222
Counterpart theory has at least three advantages over quantified modal logic as a vehicle for formalized discourse about modality. (1) Counterpart theory is a theory, not a special-purpose intensional logic. (2) Whereas the obscurity of quantified modal logic has proved intractable, that of counterpart theory is at least divided, if not conquered. We can trace it to its two independent sources.Part 1: Ontology, p. 29