In philosophy, specifically in the area of metaphysics, counterpart theory (CT) is an alternative to standard (Kripkean) possible-worlds semantics for interpreting quantified modal logic. Counterpart theory still presupposes possible worlds, but differs in certain important respects from the Kripkean view. The form of the theory most commonly cited was developed by David Lewis, first in a paper and later in his book On the Plurality of Worlds.
Differences from the Kripkean view Counterpart theory (CT), as formulated by Lewis, requires that individuals exist in only one world. According to the standard account of possible worlds, a modal statement about an individual (e.g., "it is possible that x is y") says 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. By contrast, according to counterpart theory, this statement says 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 that 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 thought would require a single individual to both have and simultaneously not have particular properties.Lewis (1986), pp. 199-202.
The 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), p. 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.
Add also the following “principle of recombination,” which Lewis describes this way: “patching together parts of different possible worlds yields another possible world […]. can coexist with anything else, […] provided they occupy distinct spatiotemporal positions.”Lewis (1986), pp. 87-88. But these possibilities should be restricted by CT.
The counterpart relation
The counterpart relation (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), pp. 252-254. I-relation,Lewis (1983). and the unity relation.Perry (1975), p. 9.
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), pp. 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:
( is a possible world) ( is in possible world ) ( is actual) ( is a counterpart of )
Lewis laid out the following axioms:Lewis (1968), p. 114.
A1. (Nothing is in anything except a world)
A2. (Nothing is in two worlds)
A3. (Whatever is a counterpart is in a world)
A4. (Whatever has a counterpart is in a world)
A5. (Nothing is a counterpart of anything else in its world)
A6. (Anything in a world is a counterpart of itself)
A7. (Some world contains all and only actual things)
A8. (Something is actual)
It is an uncontroversial assumption to assume that the primitives and the axioms A1 through A8 make the standard counterpart system.
Comments on the axioms
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. (Symmetry of the counterpart relation)
R2. (Transitivity of the counterpart relation)
R3. (Nothing in any world has more than one counterpart in any other world)
R4. (No two things in any world have a common counterpart in any other world)
R5. (For any two worlds, anything in one is a counterpart of something in the other)
R6. (For any two worlds, anything in one has some counterpart in the other)
Motivations for counterpart theory CT can be applied to the relationship between identical objects in different worlds or at different times. Depending on the subject, there are different reasons for accepting CT as a description of the relation between different entities.
In possible worlds
David Lewis defended modal realism. This is the view that a possible world is a concrete, maximal connected spatio-temporal region. The actual world is one of the possible worlds; it is 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 presidential election. How is it true? Humphrey has a counterpart in another possible world that wins the 1968 election in that world.
Lewis also argues against three other alternatives that might be compatible with possibilism: overlapping individuals, trans-world individuals, and haecceity.
Some philosophers, such as Peter van Inwagen,Van Inwagen (1985). see no problem with identity within a world . 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), pp. 192-193.
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), pp. 199-210.
The problem is with an object's accidental intrinsic properties, such as shape and weight, which 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?
According to McDaniel, if Lewis is right, the defender of overlapping individuals has to accept genuine contradictions or defend the view that every object has all its properties essentially.McDaniel (2004), pp. 147-148.
How can you be one year older than you are? One way is to say that there is a possible world where you exist. Another way is for you to have a counterpart in that possible world, who has the property of being one year older than you.
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 these parts. The same thing with Humphrey. One part of him is in one world, another part in another world.
"It is possible for something to exist if it is possible for the whole to exist. That is, if there is a world at which the whole of it exists. That is, if there is a world such that quantifying only over parts of that world, the whole of it exists. That is, if the whole of it is among the parts of some world. That is, if 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."
Haecceity
A haecceity or individual essence 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 has a more demonstrative character.
David Lewis gives 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), p. 221.