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Transworld identity concerns what makes a particular object in one possible world the same object in another possible world. The question is which counterpart of an object in world w2 is genuinely that object versus merely similar. This requires criteria for tracking identity across modal space, involving understanding rigidity of designation and essential properties.
From your study of possible worlds semantics, you can state what it means to say "Aristotle might have been a farmer": there is a possible world in which Aristotle is a farmer. But this raises a question your semantics didn't quite answer: what makes the farmer in that world *Aristotle*, rather than just someone who resembles him? To say there's a world where Aristotle farms is to presuppose that we can track Aristotle across worlds — that there is some fact of the matter about which individual in world w₂ is the same person as the philosopher in the actual world. This is the problem of transworld identity, and it turns out to be deeply contested.
One major position is David Lewis's counterpart theory. Lewis held that individuals exist only in one possible world — the actual world contains the actual Aristotle, other worlds contain other, numerically distinct individuals. When we say "Aristotle might have been a farmer," we mean that some individual in another world who is sufficiently similar to Aristotle in the right respects — Aristotle's counterpart — is a farmer. Identity across worlds is replaced by counterpart relations: qualitative similarity of the right kind. The advantage is ontological clarity (no individual literally inhabits multiple worlds), but the cost is intuitive: it seems to make modal claims about Aristotle *about someone else*. Kripke pressed this with the Humphrey objection: if Hubert Humphrey might have won the 1968 election, he cares about that possibility as a possibility for *him*, not for some qualitatively similar but numerically distinct person. Counterpart theory, Kripke argued, misidentifies what's at stake in modal statements.
The alternative is genuine transworld identity: the very same individual — numerically one and the same — exists (or at least has properties) in multiple possible worlds. This view, associated with Kripke and others, raises a different problem: what determines which properties an individual must have in every world (its essential properties) versus which it could lack (its accidental properties)? Aristotle is necessarily human, plausibly; but might he have been born a day later, in a different city, to different parents? The essentialist tradition holds that some properties are constitutive of an individual's identity while others are incidental. But there is no consensus on where the line falls, and different principles (origin essentialism, kind essentialism, bare particular theories) draw it differently.
Your soft prerequisite — rigid designators — connects directly here. Kripke argued that proper names rigidly designate the same individual across all possible worlds. This presupposes that there is something — the individual itself — to be designated across worlds. The rigid designator framework thus tacitly commits to genuine transworld identity. Counterpart theory, by contrast, requires that apparent rigid designation is really a kind of shorthand for counterpart relations. Choosing between these frameworks isn't merely terminological; it shapes what we can coherently say about personal identity, modal claims about individuals, and the metaphysics of essence.
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