# Counterpart theory

Term used in connection with the modal realis tanalysis of necessity, possibility, and counterfactual conditional statements (those where the antecedent is presented as being false).

Consider ‘If Hitler had invaded England he would have won.’ Assuming his invasion was a possibility, there will be possible worlds in which he does, and in some of these he wins and in some he loses (assuming neither of these is impossible).

‘If Hitler…’ will be true if (ignoring a minor subtlety) the nearest world in which he invaded is also one in which he won. One problem (apart from providing criteria of ‘nearness’) is of that transworld identity: in remote worlds, is Hitler still really Hitler?

To answer this and associated problems he is given a counterpart in certain of the possible worlds. Counterpart theory then studies the conditions under which he has such a counterpart, and what is involved in his having one.

Source:
D K Lewis, On the Plurality of Worlds (1986)

## Differences from the Kripkean view

Counterpart theory (hereafter “CT”), as formulated by Lewis, requires that individuals exist in only one world. The standard account of possible worlds assumes that a modal statement about an individual (e.g., “it is possible that x is y”) means that there is a possible world, W, where the individual x has the property y; in this case there is only one individual, x, at issue. On the contrary, counterpart theory supposes that this statement is really saying that there is a possible world, W, wherein exists an individual that is not x itself, but rather a distinct individual ‘x’ different from but nonetheless similar to x. So, when I state that I might have been a banker (rather than a philosopher) according to counterpart theory I am saying not that I exist in another possible world where I am a banker, but rather my counterpart does. Nevertheless, this statement about my counterpart is still held to ground the truth of the statement that I might have been a banker. The requirement that any individual exist in only one world is to avoid what Lewis termed the “problem of accidental intrinsics” which (he held) would require a single individual to both have and simultaneously not have particular properties.

In its formalization, counterpart theoretic formalization of modal discourse also departs from the standard formulation by eschewing use of modality operators (Necessarily, Possibly) in favor of quantifiers that range over worlds and ‘counterparts’ of individuals in those worlds. Lewis put forth a set of primitive predicates and a number of axioms governing CT and a scheme for translating standard modal claims in the language of quantified modal logic into his CT.

In addition to interpreting modal claims about objects and possible worlds, CT can also be applied to the identity of a single object at different points in time. The view that an object can retain its identity over time is often called endurantism, and it claims that objects are ‘wholly present’ at different moments (see the counterpart relation, below). An opposing view is that any object in time is made up of temporal parts or is perduring.

Lewis’ view on possible worlds is sometimes called modal realism.

### The basics

The possibilities that CT is supposed to describe are “ways a world might be” (Lewis 1986:86) or more exactly:

(1) absolutely every way that a world could possibly be is a way that some world is, and
(2) absolutely every way that a part of a world could possibly be is a way that some part of some world is. (Lewis 1986:86.)

Add also the following “principle of recombination,” which Lewis describes this way: “patching together parts of different possible worlds yields another possible world […]. [A]nything can coexist with anything else, […] provided they occupy distinct spatiotemporal positions.” (Lewis 1986:87-88). But these possibilities should be restricted by CT.

## The counterpart relation

The counterpart relation (hereafter C-relation) differs from the notion of identity. Identity is a reflexive, symmetric, and transitive relation. The counterpart relation is only a similarity relation; it needn’t be transitive or symmetric. The C-relation is also known as genidentity (Carnap 1967), I-relation (Lewis 1983), and the unity relation (Perry 1975).

If identity is shared between objects in different possible worlds then the same object can be said to exist in different possible worlds (a trans-world object, that is, a series of objects sharing a single identity).

### Parthood relation

An important part of the way Lewis’s worlds deliver possibilities is the use of the parthood relation. This gives some neat formal machinery, mereology. This is an axiomatic system that uses formal logic to describe the relationship between parts and wholes, and between parts within a whole. Especially important, and most reasonable, according to Lewis, is the strongest form that accepts the existence of mereological sums or the thesis of unrestricted mereological composition (Lewis 1986:211-213).

## 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.

• A1 excludes individuals that exist in no world at all. The way an individual is in a world is by being a part of that world, so the basic relation is mereological.
• A2 excludes individuals that exist in more than one possible world. But because David Lewis accepts the existence of arbitrary mereological sums there are individuals that exist in several possible worlds, but they are not possible individuals because none of them have the property of being actual. And that is because it is not possible for such a whole to be actual.
• A3 and A4 make counterparts worldbound, excluding an individual that has a non-worldbound counterpart.
• A5 and A6 restrict the use of the CT-relation so that it is used within a possible world when and only when it is stood in by an entity to itself.
• A7 and A8 make one possible world the unique actual world.

### Principles that are not accepted in normal CT

R1 Cxy → Cyx
(Symmetry of the counterpart relation)
R2 Cxy ∧ Cyz → Cxz
(Transitivity of the counterpart relation)
R3 Cy1x ∧ Cy2x ∧ Iy1w1 ∧ Iy2w2 ∧ y1≠y2 → w1≠w2
(Nothing in any world has more than one counterpart in any other world)
R4 Cyx1 ∧ Cyx2 ∧ Ix1w1 ∧ Ix2w2 ∧ x1≠x2 → w1≠w2
(No two things in any world have a common counterpart in any other world)
R5 Ww1 ∧ Ww2 ∧ Ixw1 → ∃y (Iyw2 ∧ Cxy)
(For any two worlds, anything in one is a counterpart of something in the other)
R6 Ww1 ∧ Ww2 ∧ Ixw1 → ∃y (Iyw2 ∧ Cyx)
(For any two worlds, anything in one has some counterpart in the other)

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