One of the most interesting alternatives to conventional possible-world semantics is the "counterpart theory" developed by David Lewis in his 1968 essay, "Counterpart Theory and Quantified Modal Logic." What distinguishes this theoretical framework from that of conventional modal logic is counterpart theory's thesis that "worlds don't share individuals. Worlds don't overlap. Each individual is ine xactly one world"(Steinhart, p. 102). For example, whereas in conventional possible-world semantics, Bob might be happy in one world but sad in another. In David Lewis' counterpart theory, however, Bob does not even exist in more than one world. He cannot be happy in one world and sad in another because he does not even exist in more than one.

As Steinhart notes, counterpart theory postulates a reality with the modal structure (V, W, I, δ, C, f). V is "an ordered tuple of vocabulary item (words)"(Steinhart, p. 102). In other words, it is a list of words. W is the set of possible worlds; something with which possible-world semantics is familiar. The same is true of I; it is the set of individuals. Where counterpart theory becomes very interesting is its contribution of δ and C to the notation. "Item δ is a function that associates every world with the set of individuals in that world. The item C is the counterpart relation"(Steinhart, p. 102). In other words, keeping in mind that counterpart theory differs from conventional modal logic and possible world semantics insofar as it argues that each object exists in only one world, the notation δ is used to specify the objects which exist exclusively within that world. Likewise, C is intended to link that object with another object in another world that, while not identical, is "maximally similar" to it(Steinhart, p. 102).

"We consult δ to determine whether an individual is in a world. That is, x is in world w iff x is in δ(w). The item δ can be used to define a worldmate relation on individuals. We say x is a worldmade of y iff there is some world w such that x is in δ (w) and y is in δ(w). According to counterpart theory, worlds are non-empty; they do not overlap (they share no individuals); and they exhaust the set of possible individuals. Hence the worldmate relation is an equivalence relation that partitions the set of individauls into euivalence classes. Each equivalence class belongs to a world. It is all the things in that world and no other. The function δ maps each world onto its eequivalence class of worldmates.

The counterpart relation associates an individual with its counterparts. You are represeted at other worlds by your counterparts. The counterpart relation is a relation of similarity. Roughly, your counterpart at some world is the thing in that world that is maximally similar to you. On this view, your counterpart in your world is you. You are maximally similar to yourself. Since each thing is a counterpart of itself, the counterpart relation is reflexive. But what about symmetry? It might seem obvious that if x is a counterpart of y, then y is a counterpart of x. But we need not require that. And we need not require transivity. Further, an individual at one world can have many counterparts at another world"(Steinhart, p. 102).

For the counterpart theorist, a proper noun refers not to possible entities which may be in different situations or possess different properties across different worlds. Instead, the proper name "refers to a single thing that exists in exactly one world"(Steinhart, p. 102).The predicate, however, while referring to only a single extension, is mapped onto individuals which span multiple worlds. For example, "dog" refers to the set of both actual and non-actual canines.

Let's look at a simple example of how this might work. The numbers associated with each world w below are subscripts in Steinhart's original text. Suppose we have a set of worlds {w1, w2, w3, w4} and a set of individuals {A1, B1, C1, D1, A2, B2, C2, D2, A3, B3, C3, D3, A4, B4, C4, D4}. Now, let us define δ as "an inclusion function...The inclusion function associates each world with thet set of individuals in that world"(Steinhart, p. 103). Thus:

δ (w1) = {A1, B1, C1, D1}

δ (w2) = {A2, B2, C2, D2}

δ (w3) = {A3, B3, C3, D3}

δ (w4) = {A4, B4, C4, D4}

Each counterpart is an individual with the same letter but a different number. For example, B1 is the counterpart of B2, B3, and B4, and each one is a counterpart of all the others.

To get a more precise sense of how this all works, let's go to the original source: Davis Lewis' 1968 essay, "Counterpart Theory and Quantified Modal Logic." It ought to be noted that for Lewis, counterpart theory is not itself a form of modal logic. "Counterpart theory and quantified modal logic seem to have the same subject matter; seem to provide two rival ways of formalizing our modal discourse"(Lewis, 1968). Rather, his possible-world semantics provides an alternative to modal logic, furnishing us with the tools to speak of possible worlds without the notation distinctive of modal logic.

"We have an alternative. Instead of formalizing our modal discourse by means of modal operators, we could follow our usual practice. We could stick to our standard logic (quantification theory with identity and without ineliminable singular terms) and provide it with predicates and a domain of quantification suited to the topic of modality. That done, certain expressions are available which take the place of modal logic. The new predicates required, together with the postulates on them, constitute the system I call

Counterpart Theory"(Lewis, 1968).

Let's look at the primitive predicates which Lewis introduces in order to form his counterpart theory:

"*Wx *(*x *is a possible weorld

*Ixy *(*x *is in possible world *y*)

*Ax *(*x *is actual)

*Cxy *(*x *is a counterpart of *y*)"(Lewis, 1968)

We see here exactly what we discussed before. Rather than distinctive notation for the discussion of modal concepts, Lewis has introduced distinct predicates, which he believes will allow us to discuss the sort of modal concepts which modal logicians want to investigate without the contribution of new notation.

Lewis actually introduces 8 postulates for his modal logic. He summarizes them thus: "The domain of quantification is to contain every possible world and everything in every world. The primitives are to be understood according to their English readings and the following postulates:

P1: ∀x∀y (Ixy ⊃ Wy)

(Nothing is in anything except a world)

P2: ∀x∀y∀z(Ixy & Ixz .⊃ y = z)

(Nothing is in two worlds)

P3: ∀x∀y(Cxy ⊃ ∃zIxz)

(Whatever is a counterpart is in a world)

P4: ∀x∀y(Cxy ⊃ ∃zIxz)

(Whatever has a counterpart is in a world)

P5: ∀x∀y∀z(Ixy & Cxz .⊃ x = z)

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

P6: ∀x∀y(Ixy ⊃ Cxx)

(Anything in a world is a counterpart of itself)

P7: ∃x(Wx & ∀y(Iyx = Ay))

(Some world containsa ll and only actual things)

P8: ∃x∀x

(Something is actual)"(Lewis, 1968)

P2 is particularly important for Lewis' unique understanding of the identity relation. For Lewis, things are identical to themselves, but they are never identical to their counterpartts in opposite worlds. Objects exist only in one world, rather than existing in many worlds but having different properties, as in other forms of possible-world semantics.

"...identity literally understood is no problem for us. Within any one world, things of every category are individuated just as they are in the actual world; things in different worlds are

neveridentical, by P2. The counterpart relation is our substitute for identity between things in different worlds. Where some would say that youa re in several worlds, inw hich you have somewhat different properties and somewhat different things happen to you, I prefer to say that you are in the actual world and no other, but you have counterparts in several other worlds. Your counterparts resemble you closely in content and context in important respects. They resemble you more closely than do other things in their worlds. But they are not really you. For each of them is in his own world, and only you are here in the actual world. Indeed we might say, speaking casually, that your counterparts are you in other worlds, that they and you are the same; but this sameness is no more a literal identity than the sameness between you today and you tomorrow. It wtould be better to say that your counterparts are men youwould have been,had the world been otherwise"(Lewis, 1968).

Thus, for Lewis, possible-world semantics functions rather as a kind of counterfactual semantics, or a counterfactual view of modality. Your counterparts in other possible worlds are what could have been, or what would have been with respect to your counterpart, had the world been different. Indeed, Lewis sharply differentiates his counterpart theory, and its semantics, from the modal logic of thinkers like Kripke, Montague, Kanger, Hintikka and Carnap,

"on which one thing is allowed to be in several worlds. A reader of this persuasion might suspect that he and I differ only verbally: that what I call a thing in a world is just what he would call a {thing, world} pair, and that what he calls the same thing in several worlds is just what I would call a class of mutual counterparts. but beware. Our difference is not just verbal, for I enjoy a generality he cannot match. The counterpart relation will not, in general, be an equivalence relation. So it will not hold just between those of his {thing, world} pairs with the same first term, no matter how he may choose to identify things between worlds"(Lewis, 1968)

Lewis' counterpart relation is neither symmetric nor transitive. First, let's look at how it lacks the property of transitivity:

"It would not have been plausible to postulate that the counterpart relation was transitive. Suppose x1 in world w1 resembles you closely in many respects, far more closely than anything else in w1 does. And suppose x2 in world w2 resembles x1 closely, far more closely than anything else in w2 dodes. So x2 is a counterpart of your counterpart x1. yet x2 might not resemble you very closely, and something else in w2 might resemble you more closely. If so, x2 is not your counterpart"(Lewis, 1968).

In other words, the counterpart of your counterpart is not itself necessarily your counterpart. The following transitive relation does not obtain:

x1 is the counterpart of x. x2 is the counterpart of x1. Therefore, x2 is the counterpart of x.

Likewise, the counterpart relation lacks symmetry. In other words, it does not follow that if x bears some relation to y, then y bears some relation to x. Lewis explains:

"It would not have been plausible to postulate that the counterpart relation was symmetric. Suppose x3 in world w3 is a sort of blend of you and your brother; x3 resembles both of you closely, far more closely than anything else in w3 resembles either one of you. So x3 is your counterpart. But suppose also that the resemblance between x3 and your brother is far closer than that between x3 and you. if so, you are not a counterpart of x3"(Lewis, 1968).

Put simply, x3 might be your counterpart, but it does not necessarily follow that you are the counterpart of x3.

Furthermore, it follows that an object in one world can have more than one counterpart in another. Lewis explains:

"It would not have been plausible to postulate that nothing in any world had more than one counterpart in any other world. Suppose x4a and x4b in world w4 are twins; both resemble you closely; both resemble you far more closely than anything else in w4 does; both resemble you equally. If so, both are your counterparts"(Lewis, 1968).

It is also possible, Lewis argues, that two things in one world can have a common counterpart in another. Lewis explains:

"It would not have been plausible to postulate that no two things in any world had a common counterpart in any other world. Suppose you resemble both twins x4a and x4b far more closely than anything else in the actual world does. If so, you are a counterpart of both"(Lewis, 1968).

Lewis likewise insists that it is possible for something in a world to simply not be the counterpart of any other:

"It would not have been plausible to postulate that, for any two worlds, anything in one was a counterpart of something in the other. Suppose there is something x5 in world w5 - say, Batman - which does not much resemble anything actual. If so, x5 is not a counterpart of anything in the actual world"(Lewis, 1968).

Just as it would not be reasonable to insist that something in a world must *be *the counterpart of another, it would be just as unreasonable to suggest that every object in a world must *have *a counterpart in another. Lewis explains:

"It would not have been plausible to postulate that, for any two worlds, anything in one had some counterpart in the other. Suppose whatever thing x6 in world w6 it is that resembles you more closely than anything else in w6 is nevertheless quite unlike you; nothing in w6 resembles you clocsely at all. If so, you have no counterpart in w6"(Lewis, 1968).

Keeping in mind that counterpart theory is intended as an alternative to conventional modal logic, Lewis provides us with a means of translating sentences from one into the other. Let's look at how we might translate sentences originally articulated in modal logic into the language of counterpart theory.

Suppose we have ⋄a and □a. These symbols are the modal operators for possibility and necessity, respectively, applied to a variable. How do we translate this into the language of counterpart theory, which lacks such modal notation?

∀x (Wx ⊃ a^x)

In this case, instead of merely saying "possibly a," we could use the language of counterpart theory to say that the variable a "holds in any possible world x." a^x itself means "a holds in world x." We could also write ∃x (Wx & a^x), which would read "a holds in some possible world x."

Steinhart, Eric. "More Precisely: The Mathematics You Need To Do Philosophy." Broadview Press, 2009.

Lewis, David. "Counterpart Theory and Quantified Modal Logic." The Journal of Philosophy. Volume LXV, NO. 5, March 7, 1968.