Counterpart Theory - The Formal Theory

The Formal Theory

As a formal theory, counterpart theory can be used to translate sentences into modal quantificational logic. Sentences that seem to be quantifying over possible individuals should be translated into CT. (Explicit primitives and axioms have not yet been stated for the temporal or spatial use of CT.) Let CT be stated in quantificational logic and contain the following primitives:

Wx (x is a possible world)

Ixy (x is in possible world y)

Ax (x is actual)

Cxy (x is a counterpart of y)

We have the following axioms (taken from Lewis 1968):

A1. Ixy → Wy

(Nothing is in anything except a world)

A2. Ixy ∧ Ixz → y=z

(Nothing is in two worlds)

A3. Cxy → ∃zIxz

(Whatever is a counterpart is in a world)

A4. Cxy → ∃zIyz

(Whatever has a counterpart is in a world)

A5. Ixy ∧ Izy ∧ Cxz → x=z

(Nothing is a counterpart of anything else in its world)

A6. Ixy → Cxx

(Anything in a world is a counterpart of itself)

A7. ∃x (Wx ∧ ∀y(Iyx ↔ Ay))

(Some world contains all and only actual things)

A8. ∃xAx

(Something is actual)

It is an uncontroversial assumption to assume that the primitives and the axioms A1 through A8 make the standard counterpart system.