Counterpart theory

How can You be one year older? One way is to say that there is a possible world where You exist, someone identical to You, and have the property of being one year older then You are in the actual world. But how is that possible? If something is identical to something, then they have the same properties. But one of the things - is one year older then the other thing. Another way is for You to have a Counterpart, in that possible world, that has the property of being one year older then You. This article discusses the theory behind that.

Counterpart theory (CT) is a theoretical framework that uses the counterpart relation (hereafter C-relation) as a replacement for the identity relation between objects in different possible world/times/spaces. Identity is a reflexive, symmetric and transitive relation. The counterpart relation is only a similarity relation, it doesn’t have to be transitive or symmetric.

The C-relation is also known as genidentity (Carnarp 1967), I-relation (Lewis 1983) and the unity relation (Perry 1975). To assume transworld identity between objects in different possible worlds is to assume that the same object can exist in different possible worlds. CT reject that claim and instead think of object as located in only one possible world, they are, so called, worldbound individuals.

CT can also be about identity between objects in different times. The view that the same object can exist in different times is often called endurantism, and it claims that objects are ‘wholly present’ at different moments. That view should be separated from the view that objects have temporal parts or are perduring. A C-relation between objects at different times is called the stage view or exdurantism.

The theory, in a way, overlaps both semantics and metaphysics. That is because it can be looked both as a theory about how ordinary names or indexicals refer, in different context, or as a theory about what makes de re modal or temporal statements true.

The basics or how Lewis’s concrete possible worlds starts to delivers possibilities
David Lewis view on possible worlds are sometimes called possibilism or modal realism.

The possibilities that CT is suppose 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 says that “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.

An important part of the way Lewis’s worlds deliver possibilities is the use of the parthood relation, because then we can have in our hands some neat formal machinery, mereology. An axiomatic system that describes the formal relationship between parts and wholes, and the parts to the parts within the wholes. Especially important, and most reasonable, according to Lewis, is the strongest form that accepts the existence of, so called, mereological sums or the thesis unrestricted mereological composition (Lewis 1986:211-213).

The formal theory
As a formal theory counterpart theory can be used to translate sentences in 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)

Then 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-A8 make the standard counterpart system.

Comments on the axioms
A1 exclude 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 exclude so called trans-world individuals, 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-A4 make counterparts worldbound, excluding an individual that has a non-worldbound counterpart.

A5-A6 restricts the use of the CT-relation so that it only can be used across different possible worlds.

A7-A8 make only 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)

Motivations for Counterpart theory
As was mentioned earlier, CT can be applied to both possible worlds and between times. Depending on what we are talking about there are different reasons for assuming CT to be the correct relation between different entities.

In possible worlds
David Lewis defended possibilism. The view that a possible world is a concrete, maximal connected spatio-temporal region. The actual world is one of the possible worlds, it also concrete. Because a single concrete object demands spatio-temporal connectedness a possible concrete object can only exist in one possible world. Still we say true things like: It is possible that Hubert Humphrey won the 1968 US President Election. How is it true? Humphrey has a counterpart in another possible world that wins the 1968 election in that world.

But Lewis also argues against three other alternatives that might be compatible with possibilism: overlapping individuals, transworld individuals and haecceity.

Some philosophers, like for instance Peter van Inwagen (1985), see no problem with the identity and then also no problem with so called trans-world identity. Lewis seems to share this attitude. He says:


 * "… like the Holy Roman Empire, it is badly named. […] In the first place we should bear in mind that Trans-World Airlines is an intercontinental, but not as yet an interplanetary carrier. More important, we should not suppose that we have here any problem with identity.


 * We never have. Identity is utterly simple and unproblematic. Everything is identical to itself; nothing is ever identical to anything else except itself. There is never any problem about what makes something identical to itself; nothing can ever fail to be. And there is never any problem about what makes two things identical; two things never can be identical.


 * There might be a problem about how to define identity to someone sufficiently lacking in conceptual resources – we note that it won't suffice to teach him certain rules of inference – but since such unfortunates are rare, even among philosophers, we needn't worry much if their condition is incurable.


 * We do state plenty of genuine problems in terms of identity. But we needn't state them so.” (Lewis 1986:192-193.)

So Lewis point is to look at other places for reasons for and against other ways for objects to be trans-world identical.

Overlapping individuals
An overlapping individual, has a part in the actual world and a part in another world. Because identity is not problematic, we get overlapping individuals by having overlapping worlds. Two worlds overlap, if they share a common part. But some properties of overlapping objects are for Lewis troublesome (Lewis 1986:199-210).

The problem is with an objects accidental intrinsic properties, like shape or weight, that supervene on its parts. Humphrey could have the property of having six fingers on his left hand. How does he do that? It can’t be true that Humphrey has both the property of having six fingers and five fingers on his left hand. What we might say is that he has five fingers at this world and six fingers at that world. But how should these modifiers be understood?

But the devil is in the details. Maybe there is a way to spell these things out (See McDaniel 2004). If Lewis is right – the defender of overlapping individuals has to accept genuine contradictions or defend the view that every object has all their properties essentially.

Trans-world individuals
Take Humphrey, if he is a trans-world individual he is the mereological sum of all of the possible Humphreys in the different worlds. He is like a road that goes through different regions. There are parts that overlap, but we can also say that there is a northern part that is connected to the southern part and that the road is the mereological sum of them. The same thing with Humphrey. One part of him is in one world, another part in another world.

Lewis can't say no to mereological sums, they are unrestricted – in that way there is transworld individuals - but we don't speak of them. In a sense they cannot possibly exist (Lewis 1986:211). Some kind of restricted quantification is happening here. Here is his speech:


 * ”It is possible for something to exist iff it is possible for the whole to exist. That is, iff there is a world at which the whole of it exists. That is iff there is a world such that quantifying only over parts of that world, the whole of it exists. That is iff the whole of it is among the parts of some world. That is, iff it is part of some world – and hence not a trans-world individual. Parts of worlds are possible individuals; trans-world individuals are therefore impossible individuals.

A theory of Trans-world individuals agree on what exists with CT, but they disagree about what names or kinds of things are about. There might be a similar disagreement between perdurantism and the stage view. See further below - exdurantism.

Haecceity
A haecceity or individual essences is a property that only a single object instantiates. Ordinary properties, if one accepts the existence of universals, can be exemplified by more than one object at a time. Another way to explain a haecceity is to distinguish between suchness and thisness, where thisness have a more demonstrative character.

David Lewis give the following definition of a haecceitistic difference: “two worlds differ in what they represent de re concerning some individual, but do not differ qualitatively in any way.” (Lewis 1986:221.)

Lets take two examples. It is possible that I would be one of two identical twins. According to haecceitism there is a non-qualitive property, namely being me, that one of them has, in one possible world or that the other one has in another qualitatively identical world. But CT don't need that, distinct possibilities don't need distinct worlds – “a single world may provide many possibilities, since many possible individuals inhabit it” (Lewis 1986:230). CT can satisfy the twin intuition because it allows for multiple counterparts in one possible world.

The second case is the possibility that I might have been someone else, say David Lewis. What I am contemplating is a world identical to this world but where I am David Lewis. According to haecceitism there is a qualitatively identical world where the non-qualitative properties are shifted so I am David Lewis. But according to CT we can understand it very broad so that every counterpart of me just has the property of being a person, thus it is possible for me, in the actual world, of being David Lewis (Lewis 1986:231-232).

Between Times
Ted Sider believes that perdurantism and the stage view about how ordinary material objects persist through time. Perdurantism is the view that material objects are not wholly present at each single instant of time, instead objects have temporal parts. Sometimes, especially in the theory of relativity as it is expressed by Minikowski a perduring object is called a spatiotemporal worm, therefore perdurantism is sometimes called wormtheory.

Temporal parts can explain how an object instantiates temporary intrinsic properties (compare with the discussion of overlapping individuals and counterpars).

But perdurantism is not the only view that accepts the existence of temporal parts. Ted Siders defend a view called exdurantism or the stage view. The stage view uses a C-relation between the temporal parts of perduring objects. (Another defence of exdurantism is in Hawley 2001)

The point in saying that objects exdure instead of perdure is, first perdurantism is a good solution to the, so called, paradox of material constitution or the paradox of material coincidence. Basically most material objects go through a change of material parts. Take me, for instance, yesterday a certain atom on my left hand was part of me, today it isn't. Lets say that I am made up of all the atoms that compose me. But how can I have the two incompatible properties of both having an atom on my left hand as a part of me and not having it as a part of me? According to perdurantism different temporal parts are composed of different atoms.

But according to Sider, perdurantism can't handle all what is puzzling with the paradox, so secondly, to have the best solution to the paradox, we have to defend exdurantism.

First Sider defends a revisionary way of counting. Instead of counting individual objects we could count by parts of an object - Sider discuss the example of counting road segments instead of roads simpliciter. That way we count not only by identity (Sider 2001:188-192). (Compare with Lewis 1993.)

Sider writes: "even if we knew that some material object would go through some fission, and split into two it "we would not say" there are two objects located at the same spacetime region.

The consequence is that exdurantism accept the same ontology as perdurantism, quantifying over spatiotemporal worms, but when we ordinary talk (not only in English) our predicates don't quantify over them. We, most of the time, talk with restricted quantifiers (Sider 2001:192). Further, ordinary sortals like person, lamp or tree designat momenarily temporal parts.

But here an objection creeps up: How can we predicate temporal properties of these momentary temporal parts?

It is here that the C-relation comes in play. Take a sentence like "Ted was once a boy", the truth condition is that there exist some person stage x such that prior to the utterance of the sentence such that x is a boy and x bears the temporal counterpart relation to Ted (Sider 2001:193).

It can be interesting to compare Siders lose descprition of exdurantims with Yuri Balashov attempt and defining exdurantism (Balashov 2007).

Counterpart theory and the necessity of identity
Kripke three lectures on proper names and identity, (1980), raised the issues of how we should interpret statements about identity. Take the statement that the Evening star is identical to the Morning star, aka. Venus, this seems to be an [a posteriori] identity statement. We discover that the names designate the same thing. The traditional view, since Kant, has been that statements or propositions that are necessary true are a prior. But in the end of the sixties Saul Kripke and Ruth Barcan Marcus offered proof for the necessary truth of identity statements. Here is the Kripkes version (Kripke 1971):


 * (1) ∀x $$\Box$$(x = x)               [Necessity of self-identity]
 * (2) ∀x∀y [x = y → ∀P(Px → Py)]                [Leibniz law]
 * (3) ∀x∀y [x = y → ($$\Box$$(x = x) → $$\Box$$(x = y))]    [From (1) and (2)]
 * (4) ∀x∀y [x = y → $$\Box$$(x = y)]   [From the following principle A → B → C ⇒ A → C and (3)]

If the proof is correct or identity statements are necessarily true anyway, the distinction between the a prior/a posteriori and necessary/contingent becomes less clear. (For some interesting comments on the proof see Lowe 2002.) The statement that for instance “Water is identical to H2O” is (then) a statement that is necessarily true but a posteriori. If CT is the correct account of modal properties we still can keep the intuition that identity statements are contingent and a priori because counterpart theory understands the modal operator differentially than standard modal logic.

The relationship between CT and essentialism is of interest. (Essentialism, the necessity of identity and rigid designators form an important troika of mutual interdependence.) According to David Lewis claims about an objects essential properties can be true or false depending on context (Chapter 4.5 in 1986 he calls against constancy, because an absolut conception of essences is constant over the logical space of possibilities.). He writes:


 * But if I ask how things would be if Saul Kripke had come from no sperm and egg but had been brought by a stork, that makes equally good sense. I create a context that make my question make sense, and to do so it has to be a context that makes origins not be essential. (Lewis 1986:252.)

Counterpart theory and rigid designators
In Kripke (1980) he interpreted proper names as rigid designators where a rigid designator pick out the same object in every possible world. For someone that accepts contingent identity statements the following semantic problem occurs (semantic because we deal with de dicto necessity) (Rea 1997:xxxvii):

Take a scenario that is mentioned in the paradox of coincidence, a statue, call it “Statue” fashioned by melding two pieces of clay together, those two pieces are called “Clay”. Statue and Clay seem to be identical, they exist at the same time, and we could incinerate them at the same time. The following seem true


 * (7) Necessarily, if Statue exist then Statue is identical to Statue.

But,


 * (8) Necessarily, if Statue exist then Statue is identical to Clay

is false.

Counterpart theory, qua-identity and individual concepts can offer solutions to this problem.

Arguments for inconstancy
Ted Sider gives roughly the following argument (Sider 2001:223): inconstancy is had if a proposition about essence of an object is true in one context and false in another. C-relation is a similarity relation. What is similar in one dimension is not similar regarding another dimension. Therefore the C-relation can have the same difference and express inconstant judgements about essences.

David Lewis offers another argument. The paradox of coincidence can be solved if we accept inconstancy. We can then say that it is possible for a dishpan and a piece of plastic to coincide, in some context. That context can then be described using CT.

Other motivations for Counterpart theory
Ted Sider makes the important point that David Lewis considered his defence of CT as a forced move, from his defence of modal realism or possibilism. Sider needs CT to have an optimal solution to the paradox of material coincidence, a solution that in no way is forced upon him, therefore it is interesting to mention other motivations for CT.

Counterpart theory compared to qua-theory and individual concepts
Assuming contingent identity it is informative to compare CT with other theories about how to handle de re representations.

Qua-theory

Kit Fine (1982) and Alan Gibbard (1975) (according to Rea 1997) are defences of qua-theory. According to qua-theory we can make talk about some of an objects modal properties. The theory is handy if we don't think it is possible for Socrates to be identical with a piece of bread or a stone. Socrates qua person is essentially a person.

Individual concepts

According to Rudolph Carnap in modal contexts variables refer to individual concepts instead of individuals. An individual concept is then defined as a function of individuals in different possible worlds. Basically individual concepts deliver semantic objects or abstract functions instead of real concrete entities as in CT.

Counterpart theory and epistemic possibility
Kripke accepts the necessity of identity but agrees with the feeling that it still seems that it is possible that Phospherus is not identical to Hespherus. For all we know it could be that they are different, he says:


 * What, then, does the intuition that the table might have turned out to have been made of ice or of anything else, that it might even have turned out not to be made of molecules, amount to? I think that it means simply that there might have been a table looking and feeling just like this one and placed in this very position in the room, which was in fact made of ice, In other words, I (or some conscious being) could have been qualitatively in the same epistemic situation that in fact obtains, I could have the same sensory evidence that I in fact have, about a table which was made of ice. The situation is thus akin to the one which inspired the counterpart theorists; when I speak of the possibility of the table turning out to be made of various things, I am speaking loosely. This table itself could not have had an origin different form the one it in fact had, but in a situation qualitatively identical to this one with respect to all evidence I had in advance, the room could have contained a table made of ice in place of this one. Something like counterpart theory is thus applicable to the situation, but it applies only because we are not interested in what might not be true of a table given certain evidence. It is precisely because it is not true that this table might have been made of ice from the Thames that we must turn here to qualitative descriptions and counterparts. To apply these notions to genuine de re modalities, is from the present standpoint, perverse. (Kripke 1980:142.)

So to explain how the illusion of necessity is possible, according to Kripke, CT is an alternative. Therefore CT forms an important part of our theory about the knowledge of modal intuitions. (For doubt about this strategy, see Della Roca, 2002. And for more about the knowledge of modal statements, see Gendler and Hawthorne, 2002.)

Arguments against Counterpart theory
Let’s start with the most famous one.

The Humphrey Objection
Because a counterpart is never identical to something in another possible world Kripke raised the following objection against CT:


 * "Thus if we say "Humphrey might have won the election (if only he had done such-and-such), we are not talking about something that might have happened to Humphrey but to someone else, a "counterpart"." Probably, however, Humphrey could not care less whether someone else, no matter how much resembling him, would have been victorious in another possible world. Thus, Lewis's view seems to me even more bizarre than the usual notions of transworld identification that it replaces. (Kripke 1980:45 note 13.)

One way to spell out the meaning of Kripkes claim is by the following imagitive dialogue: (Based on Sider MS)


 * Mr Against: Kripke mean that Humphrey himself don’t have the property of possibly winning the election, because it is only the counterparts that wins.


 * Miss For: The property of possible winning the election is the counterpart winning the election.


 * Mr Against: But they can't be the same property because Humphrey have different attitudes to them: he cares about him having the property of possibly winning the election, not for the property of having a counterpart wins the election.


 * Miss For: But properties don't work the same way as objects, our attitudes towards them can be different, because we have different descriptions – they are still the same properties. That lesson is taught by the Paradox of Analysis.

(These arguments, if they are correct, also apply to exudrantism, see for instance Sider 2001:194-195.)

Not all modal sentences or propositions can be translated to CT
CT is inadequate if it can't translate all our modal sentences or intuitions. Fred Feldman mention two sentences (Feldman 1971):


 * (1) I could have been quite unlike what I in fact am.


 * (2) I could have been more like what you in fact are than like what I in fact am, and at the same time, you could have been more like what I in fact am than what you in fact are.

Argument specifically against exdurantism
One argument against the stage view is that it is hard to understand how it differ from perdurantism and endurantism. Balashov (2007) is an attempt to more formally spell out exdurantism.

Another argument against the stage view is  Peter Ungers problem the problem of the many. Because the stage view tryied to answer a semantic objects connected to how many persons are located at a place in space and time by saying there is one person there, but the problem of the many says it is not neither definitely one or not definitely one. (Sider 2001:192 and compare with Lewis 1993)