Counterpart Theory

Origin. Lewis (1968). No transworld identity. Counterparts represent de re. Modal realism. Foundation for modal metaphysics.

Models. Individuals world-bound. Counterpart relation C. De re modality via counterparts. No same object across worlds.

Formalism.

Counterpart relation: C(x, y, w₁, w₂): y in w₂ is counterpart of x in w₁. Similarity-based. Not transitive, not symmetric.

Possibility de re: ◇Fx (object a satisfies): ∃w∃y. w accessible ∧ C(a, y, @, w) ∧ Fy at w. Counterpart has F.

Necessity de re: □Fx (object a satisfies): ∀w∀y. (accessible(w) ∧ C(a, y, @, w)) → Fy at w. All counterparts have F.

No transworld identity: a exists only at actual world. "Same" across worlds = counterpart. Identity is world-bound.

Multiple counterparts: a may have many counterparts in w. Or none. Context determines which.

Inconstancy: Same sentence, different counterpart relations. "Humphrey might have won" vs "The loser might have won." Context-dependence.

Modal realism (Lewis): Other possible worlds exist. Concrete, real. Actuality is indexical.

Fictionalist variant: Worlds as fictions. Counterparts in fiction. Avoid ontological commitment.

Symbols.

Metatheory. Lewis (1968) presented counterpart theory as a first-order theory with four primitives — being a world, being in a world, being actual, and being a counterpart of — a set of postulates governing them, and a translation of quantified modal formulas into that language; the modal apparatus is eliminated rather than interpreted, so the theory inherits the completeness and compactness of classical first-order logic together with its undecidability. Because the translation quantifies over all worlds without restriction, the de dicto propositional logic is S5. The de re logic is weaker and matches no standard system: Lewis's postulates make each thing its own counterpart, so C is reflexive and the de re T schema holds, but C is neither symmetric nor transitive, so the de re 4 and B schemas fail. The necessity of identity fails too, since one object may have two distinct counterparts in a single world. Hazen (1979) showed the translation is not faithful to any normal quantified modal logic and mishandles iterated modalities, and Kracht and Kutz (2002) developed counterpart frames into a general semantics that is complete for quantified modal logics Kripke semantics leaves incomplete.

Applies to. Modal metaphysics. Personal identity. Essentialism. Transworld identity.

Limitations. The translation quantifies over all worlds without restriction, so the de dicto logic is fixed at S5 and no weaker alethic notion can be modelled without revising the postulates. The counterpart relation is an unconstrained primitive: the postulates make it reflexive and impose no further conditions, so the theory does not settle the truth value of a de re modal claim, and Lewis's inconstancy thesis makes that indeterminacy deliberate rather than a gap to be closed. Non-transitivity blocks iteration — □Fa does not yield □□Fa — so iterated essence is inexpressible, and the failures of the necessity of identity and of distinctness follow directly from C's not being one-to-one. Fara and Williamson (2005) argued that counterpart-theoretic semantics cannot express certain modal claims combining actuality with iterated modality. Kripke's objection remains a representational one: a claim about a counterpart of Humphrey is a claim about someone other than Humphrey.

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