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.

Symbol	Unicode	Name	Meaning
C	—	Counterpart	Cross-world relation
@	—	Actual	Current world
◇_c	—	Possibility	Via counterpart
w	—	World	Possible world

Metatheory. First-order formalization. No quantifying-in problems. Captures de re reading. Multiple systems.

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

Limitations. Counterpart selection. Inconstancy objections. Modal realism controversial. Counterintuitive for some.