# Models for Counterparts

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DOI: 10.1007/s10516-010-9120-1

- Cite this article as:
- Torza, A. Axiomathes (2011) 21: 553. doi:10.1007/s10516-010-9120-1

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## Abstract

Lewis proposed to test the validity of a modal thesis by checking whether its possible-world translation is a theorem of counterpart theory. However, that criterion fails to validate many standard modal laws, thus raising doubts about the logical adequacy of the Lewisian framework. The present paper considers systems of counterpart theory of increasing strength and shows how each can be motivated by exhibiting a suitable intended model. In particular, *perfect counterpart theory* validates all the desired modal laws and therefore provides a way out of the logical objection. Finally, a weakening of perfect counterpart theory is put forward as a response to some metaphysical objections.

### Keywords

Counterpart theoryModal logicIdentityIntended model## 1 Lewisian Counterpart Theory

^{1}Nonetheless, counterpart theory has drawn a considerable amount of interest in its own right.

^{2}One of the features making it so attractive is the fact that it provides a straightforward criterion to determine the validity of a given modal thesis:

The basic counterpart-theoretic language \(\mathcal{L}_{{\bf C}}\) includes three constants:A modal formula is valid iff its translation is a theorem of counterpart theory.

^{3}

*W*(*x*):*x is a world**I*(*x*,*y*):*x is at world y**C*(*x*,*y*):*x is a counterpart of y*.

*w*is defined by recursion:

^{4}

- (T1)
[ϕ]

^{w}is ϕ, if ϕ is atomic - (T2)
\([\neg\phi]^{w}\) is \(\neg[\phi]^{w}\)

- (T3)
\([\phi\wedge\psi]^{w}\) is \([\phi]^{w}\wedge[\psi]^{w}\)

- (T4)
\([\phi\vee\psi]^{w}\) is \([\phi]^{w}\vee[\psi]^{w}\)

- (T5)
\([\phi\rightarrow\psi]^{w}\) is \([\phi]^{w} \rightarrow [\psi]^{w}\)

- (T6)
[(∃

*x*)ϕ]^{w}is \((\exists x)(I(x,w)\wedge[\phi]^{w})\) - (T7)
\([(\forall x)\phi]^{w}\) is \((\forall x)(I(x,w)\rightarrow[\phi]^{w})\)

- (T8)
\([\diamondsuit\phi(\bar{x})]^{w}\) is \((\exists z)(\exists\bar{y})(W(z)\wedge I(\bar{y},z)\wedge C(\bar{y},\bar{x})\wedge[\phi(\bar{y})]^{z})\)

- (T9)
\([\square\phi(\bar{x})]^{w}\) is \((\forall z)(\forall\bar{y})(W(z)\wedge I(\bar{y},z)\wedge C(\bar{y},\bar{x}) \rightarrow[\phi(\bar{y})]^{z})\).

**CT**is defined by the following axioms:

- (P1)
\((\forall x)(\forall y)(I(x,y)\rightarrow W(y))\)

- (P2)
\((\forall x)(\forall y)(\forall z)(I(x,y)\wedge I(x,z)\rightarrow z=y)\)

- (P3)
\((\forall x)(\forall y)(C(x,y)\rightarrow(\exists z)(I(x,z)))\)

- (P4)
\((\forall x)(\forall y)(C(x,y)\rightarrow(\exists z)(I(y,z)))\).

**CT**is the lack of specific conditions imposed on the counterpart relation

*C*: all we know is that it is a binary relation defined on the class of world-bound individuals. Whether

*C*has further properties depends on the nature of the

*intended model*of counterpart theory, which is the model we take to be materially adequate.

^{5}Comparative similarity relations being reflexive, Lewisian counterpart theory must include the axiom

- (P5)
\((\forall x)(\forall y)(I(x,y)\rightarrow C(x,x))\).

*Feys’ principle*:

- (T)
\(\phi\rightarrow\diamondsuit\phi\).

^{6}Since the counterpart relation in Lewis’ intended model is defined by comparative similarity, it satisfies the

*Maximality Principle*(MP):

This principle is inconsistent with the symmetry conditionif

xatwis a counterpart ofy, no individual atwis more similar toythanx.

- (P6)
\((\forall x)(\forall y)(C(x,y)\rightarrow C(y,x))\).

*Brouwer’s principle*:

- (B)
\(\phi\rightarrow\square\diamondsuit\phi\)

*w*, but my brother is strictly more similar to him than I am. So I am not a counterpart of my counterpart. It follows that, even though I live in Massachusetts, it is not necessarily possible that I live in Massachusetts.

*Minimality Principle*(mP):

where the relevant respects and the similarity threshold are determined contextually. This principle is inconsistent with the transitivity axiomif

xis a counterpart ofy,xandyare sufficiently similar in relevant respects

- (P7)
\((\forall x)(\forall y)(\forall z)(C(x,y)\wedge C(y,z)\rightarrow C(x,z))\).

*Becker’s principle*:

- (4)
\(\square\phi\rightarrow\square\square\phi\)

Consider for instance a counterpart relation stressing mereological similarity: the counterpart of an object *x* is composed of (duplicates of) at least 50% of the parts of *x*. So, although my bike couldn’t have had less than a half of the parts it actually has, it is possible that it could have had less than a half of the parts it actually has.

- (5)
\(\diamondsuit\phi\rightarrow\square\diamondsuit\phi\).

*Principle of Qualitative Supervenience*(Q):

This condition prevents a counterpart relation from being either locally functional or locally 1–1. It isif

xis qualitatively indistinguishable fromx′ andyis qualitatively indistinguishable fromy′, thenxis a counterpart ofyiffx′ is a counterpart ofy′.

*locally functional*when an individual cannot have multiple counterparts at the same world:

- (P8)
\((\forall w)(\forall x)(\forall y)(\forall z)(C(x,z)\wedge C(y,z)\wedge I(x,w)\wedge I(y,w)\rightarrow x=y)\).

*locally 1–1*if multiple individuals from the same world cannot share a counterpart:

- (P9)
\((\forall w)(\forall x)(\forall y)(\forall z)(C(z,x)\wedge C(z,y)\wedge I(x,w)\wedge I(y,w)\rightarrow x=y)\).

*Necessity of Identity*(NI) nor the

*Necessity of Distinctness*(ND) come out valid:

- (NI)
\(x=y \rightarrow\square x=y\)

- (ND)
\(\neg x =y \rightarrow\square \neg x= y\).

*Leibniz’ Law*:

- (LL)
\(x=y \rightarrow(\phi(x,x)\rightarrow\phi(x,y))\).

\({\bf CT}^{\bf T}:= {\bf CT}\cup\{\hbox{P}5\}\)

\({\bf CT}^{\bf TB}:= {\bf CT}^{\bf T}\cup\{\hbox{P}6\}\)

\({\bf CT}^{\bf TB4}:= {\bf CT}^{\bf TB}\cup\{\hbox{P}7\}\)

\({\bf CT}^{\bf TB4N}:= {\bf CT}^{\bf TB4}\cup\{\hbox{P}8, \hbox{P}9\}\).

On the other hand, since counterpart theory is meant to provide an interpretation of metaphysical (or broadly logical) modalities, accepting the Lewisian system is tantamount to accepting a weak modal logic. This fact may easily be turned into an argument against counterpart theory. For all of the aforementioned modal theses—namely (T), (B), (4), (5), (LL), (NI) and (ND)—are usually regarded as logical truths. Indeed, such principles are routinely used to distinguish metaphysical possibility from other kinds of possibility (epistemic, doxastic, temporal, physical etc).^{7}

Call that the *logical objection* to counterpart theory. In the rest of the paper I submit a way to block the objection by defending *perfect counterpart theory*, which corresponds to the system \({\bf CT}^{\bf TB4N}\). The task will be carried out by developing a method to construct an intended model of perfect counterpart theory. Since both Lewisian and perfect counterpart theory have an intended interpretation, but only the latter validates all the desired modal laws, I will conclude that the stronger system should be preferred from a purely logical point of view.

In the concluding Sect. 1 consider a critical objection to transitive counterpart relations and, a fortiori, to perfect counterpart theory. Even so, the present approach provides a quite natural fallback position. Insofar as the counterpart relation can be tweaked by adding or subtracting properties as desired, it is possible to construct intended models for systems intermediate between Lewisian and perfect counterpart theory. In particular, we can construct a model \({\mathcal{M}}\) having all the properties of an intended model of perfect counterpart theory except for the transitivity of the counterpart relation. I will conclude that such a model \({\mathcal{M}}\) yields the best combination of logical strength and material adequacy.

## 2 Symmetric Counterpart Theory

The counterpart relation in Lewis’ intended model satisfies the Maximality Principle (MP). As I pointed out, the resulting theory features an asymmetric counterpart relation. The present section argues that (MP) should be dropped and counterparts defined via simple similarity.^{8} This revision yields an intended model of *symmetric counterpart theory* which corresponds to the system \({\bf CT}^{\bf TB}\).

I now aim to show that de re modal statements get the wrong truth-conditions if the counterpart relation satisfies (MP).^{9} Consider the statement: “Julius Caesar could have failed to be a dictator”. What conditions must be realized in order for it to be true in an intended \({\bf CT}^{\bf T}\)-model? Let *x* be the actual Julius Caesar and *w* be a world which is just like the actual world @ except that the duplicate *y* of the actual Julius Caesar has a twin brother *y*′. The two twins are virtually indistinguishable and live very similar lives. The only relevant aspect that tells them apart is that *y*′, unlike *y*, fails to become a dictator of (the counterpart of) Rome. This fact makes the actual Julius Caesar strictly more similar to *y* than *y*′, so by (MP) *y*′ is not a counterpart of *x*. It follows that the sentence “Julius Caesar could have failed to be a dictator” cannot be true in virtue of the fact that *y*′ fails to be a dictator.

Now take a world *w*′ which is just like *w* except that there is only one duplicate *y*′′ of Julius Caesar, namely one that is not a dictator. Since no individual at *w*′ is more similar to *x* than *y*′′ is, then *y*′′ is a counterpart of *x*. Thus, it is true in virtue of *y*′′ that the actual Julius Caesar could have failed to be a dictator.

The different modal role played here by *y*′ and *y*′′ hints at a problem. If we agree that *x* and *y*′′ are similar enough, so that what is the case for *y*′′ is possible for *x*, there seems to be no reason to deny that what is the case for *y*′ is also possible for *x*. For the only relevant difference between the two cases is that *y*′ has a twin brother who is a dictator while *y*′′ does not, and it is this fact alone that prevents *y*′ from being a Lewisian counterpart of *x*. On the other hand, it is precisely because Caesar’s duplicate *y*′ has a brother dictator that *y*′ is not a dictator and, therefore, the actual Caesar may have failed to be a dictator. One can indeed point at a situation like the one depicted by *w* and say: “Look, there might have been two identical twins instead of our Julius Caesar. Only one of them would eventually become a dictator. So, it could have gone either way: Julius Caesar may or may not have been a dictator”. In brief, one way Caesar might have failed to be a dictator is that he could have been trumped by a possible twin. To make sense of this possibility, *y*′ must then count as a counterpart of Julius Caesar, too.^{10}

As anticipated, one way to accommodate such cases is to drop (MP). The obvious fallback interpretation is to let the intended counterpart relation be defined by similarity, rather than comparative similarity. Since similarity is both reflexive and symmetric, we are now allowed to assume axiom P6 which yields the system of symmetric counterpart theory \({\bf CT}^{\bf TB}\). Under Lewis’ translation scheme, this theory validates the modal principle (B).

## 3 Approximate Counterpart Theory

The aim of this section is to motivate *approximate counterpart theory*, which corresponds to the system \({\bf CT}^{\bf TB4}\), by constructing an intended model for it. The task will carried out by adding transitivity to the counterpart relation of an intended \({\bf CT}^{\bf TB}\)-model.

One way to define an equivalence counterpart relation is to conjoin a theory of possible worlds and a theory of kinds (or sortal essences): *x* is a counterpart of *y* iff *x* and *y* belong to the same kind.^{11} Thus, I could take my counterparts to be all possible rational animals (to borrow Aristotle’s definition); or, on a more fine-grained scale, all organisms sharing my genome; or, to frame it in Lewis’ terms, all things with my perfectly natural properties; etc. The problem with this approach is that it makes counterpart theory, as well as the resulting modal logic, contingent upon one specific theory of properties. In doing so, it also exposes the modal framework to whatever objections are leveled against the underlying metaphysics. Finally, even if sortal essentialism could be worked out for some special cases, such as mathematical entities, physical and chemical elements and organisms, it is hard to implement in the case of artifacts and other objects.^{12} I will then set aside sortal essentialism and pursue the task from a completely different angle.

*D*of organisms. The goal is to define identity conditions for

*kinds*of organisms in

*D*. One approach is to look for a relation

*R*that partitions the elements of

*D*into equivalence classes. Then we could say that two kinds are identical if they are the same equivalence class

*modulo*

*R*. However, we cannot realistically assume the existence of such an equivalence relation: all we have is a similarity relation

*S*on

*D*. Thus, the goal is to define an equivalence relation which is as close as possible to the given similarity relation

*S*. This is in essence the problem which Carnap grappled with in the

*Aufbau*, when he attempted to define identity conditions for qualities. Williamson (1986, p. 381) suggested two methods to obtain the desired approximation: “Beneath the philosophical task of understanding the identity conditions of certain entities lies the logical task of defining a clear sense in which an equivalence relation

*R*can approximate a relation

*S*, which is not itself an equivalence relation. We cannot insist that

*S*be both necessary and sufficient for

*R*, but we do have an obvious pair of fall-back positions: either that

*S*be sufficient for

*R*but not necessary, or that it be necessary but not sufficient. […] In either case, we should expect to be able to find a closer approximation to

*S*.” The available similarity relation

*S*can then be approximated by a strictly weaker or strictly stronger equivalence relation

*R*. However, the two approaches are not symmetric:

The approach from above always yields a unique outcome becauseWilliamson considers two ways of constructing an adequate substitute relation. One way is to search for a smallest equivalence relation

R^{+}such that \(R\subseteq R^{+}\). Let us call this theapproach from above. Such anR^{+}always exists and is always unique. Another way is to look for a largest equivalence relationR^{−}such that \(R^{-}\subseteq R\). Such anR^{−}always exists (on the assumption that the Axiom of Choice holds) but is not typically unique. We will call this theapproach from below.^{13}

*R*

^{+}is the

*transitive closure*of

*R*. Applied to the counterpart relation

*C*of a \({\bf CT}^{\bf TB}\)-model, a nontrivial approximation from above will force some individuals

*x*and

*y*that are not

*C*-counterparts to be

*C*

^{+}-counterparts. On the other hand, for any nontrivial approximation from below

*C*

^{−}there will be

*x*and

*y*that are

*C*-counterparts without being

*C*

^{−}-counterparts. Consequently,

*C*makes less facts possible than

*C*

^{+}and more facts possible than

*C*

^{−}.

In general, there may well be cases in which approximations from below are preferable to ones from above. When it comes to choosing how to approximate a counterpart relation, however, the uniqueness condition is essential. For if we select the approach from below, the class of de re truths would be contingent upon which approximate counterpart relation has been picked by a particular application of the axiom of choice. Due to the nonconstructive nature of this axiom, there would be no way to identify the approximation from below which has been produced. Consequently, the semantic value of de re statements would be epistemically inaccessible. For that reason we ought to choose the approach from above. Namely, the best approximation *C*^{+} of the counterpart relation *C* in a \({\bf CT}^{\bf TB}\)-model is the transitive closure of *C*.

The intended model of \({\bf CT}^{\bf TB4}\) is obtained from the intended model of \({\bf CT}^{\bf TB}\) by approximating from above the counterpart relation. I will then refer to \({\bf CT}^{\bf TB4}\) as *approximate counterpart theory*. Insofar as it features an equivalence counterpart relation, this system also validates the modal principles (4) and (5). In an intended \({\bf CT}^{\bf TB4}\)-model \({\mathcal{M}}\), de re statements are only approximately true (false) compared to the intended \({\bf CT}^{\bf TB}\)-model from which \({\mathcal{M}}\) is derived. We have traded off logical strength for semantic precision.

Now, a cautionary note. In Sect. 5 I present some metaphysical objections to transitive counterpart relations, whose flaws far outweigh the advantage of having a modal logic which validates (4) and (5). To cope with this issue, I will eventually backtrack and show how transitivity can be “subtracted” from an intended model of perfect counterpart theory.

## 4 Perfect Counterpart Theory

Our third and last task is to defend *perfect counterpart theory*. To achieve the goal I will show how an intended \({\bf CT}^{\bf TB4N}\)-model can be obtained from an intended \({\bf CT}^{\bf TB4}\)-model. The resulting system will validate the main modal laws of identity, namely the Necessity of Identity and Distinctness, as well as Leibniz’ Law.

- 1.
\(\square x=x\)

- 2.
\(x=y\rightarrow(\square x=x\rightarrow\square x=y)\).

^{14}

However, Kripke’s argument is question-begging. For, although identity is indeed defined via (LL), it is a matter of controversy whether this law is restricted to extensional languages, or whether a complex predicate ϕ can also express modal properties such as ‘being identical to *x*′.^{15} So, Kripke’s line of reasoning remains inconclusive unless it is proven that (LL) holds unrestrictedly.

Gibbard (1975) questioned Kripke’s background assumptions by offering an argument in support of contingent identity. If someone makes a clay statue of Goliath, goes the argument, the statue *G* will be identical to the particular lump of clay *L* constituting it.^{16} Things could have gone otherwise, though. Consider a possible course of events in which the sculptor uses marble instead, while the lump of clay remains untouched and is never turned into a statue. In this world both *G* and *L* exist and are not identical, therefore (NI) fails.

Lewis’ own account of Gibbard’s case appeals to the inconstancy of the counterpart relation, i.e. the fact that different contexts induce different counterpart relations.^{17} If we consider the statue of Goliath *qua* statue, its counterparts will be statues having similar size and shape. If we consider the statue of Goliath *qua* lump of clay, its counterparts will be lumps of clay closely resembling the clay that constitutes the statue. There is not a single counterpart relation, so there is not a single kind of de re possibility. It is now easy to reduce Gibbard’s scenario to a case of inconstant counterpart relation. According to a relation *R*_{1} that favors such properties as shape and size, the clay statue of Goliath has a counterpart *G*′ at some world *w* which is a marble statue of Goliath. According to another relation *R*_{2} stressing similarity of material constitution, at the same world *w* the statue has a distinct counterpart *L*′ which is an amorphous lump of clay.

- (a)
*G*is*L*and*G*may not have been*L*

- (b)
*G*is*L*and there is a world*w*with counterparts*G*′ of*G*and*L*′ of*L*such that*G*′ is not*L*′.

*G*′ and

*L*′ represent the clay statue at the same world

*w*with respect to

*one*counterpart relation. In other words,

*G*and

*L*are possibly distinct iff they are distinct individuals of a world according to a

*single*way of representing them. On the other hand, Lewis’ solution to Gibbard’s puzzle involves

*two*counterpart relations, each inducing a distinct way of representing the actual clay statue:

- (c)
*G*is*L*and there is a world*w*containing an*R*_{1}-counterpart*G*′ of*G*and an*R*_{2}-counterpart*L*′ of*L*such that*G*′ is not*L*′.

*G*′ and

*L*′ are picked by different counterpart relations

*R*

_{1}and

*R*

_{2}, therefore their distinctness does not fall in the scope of a single modal operator. On Lewis’ reading, Gibbard’s case is expressible in counterpart theory but not in the ordinary modal language, so it is not an instance of contingent identity. Since scenarios like (c) involve inconstant counterpart relations, I will use the expression

*inconstant identity*to distinguish them from genuine cases of contingent identity such as (b).

^{18}

As it turned out, the most prominent argument in support of contingent identity can be explained away in other terms. Lewis’ solution to Gibbard’s puzzle in fact does not undermine (NI), it only shows that counterpart theory has more expressive power than quantified modal logic, since the former but not the latter can describe cases of inconstant identity. Nonetheless, we know that (NI) fails in counterpart theory: “I distinguish two different ways that something might have multiple counterparts in another world. The first way is that there are different counterpart relations, differing in the comparative weights or priorities they give to different respects of comparison, which favour different candidates. This is the inconstancy of counterparts that we have been considering. The second way is that there might be a single counterpart relation, given by a single system of weights or priorities, which on occasion is one-many: it delivers multiple counterparts because of ties—say, the tie between a pair of twins. It is not that one system of weights and priorities favours one twin, and another favours the other.”^{19}

In the case of Gibbard’s puzzle, for example, we can define a weaker counterpart relation *R*_{3} as the union of *R*_{1} and *R*_{2}. *R*_{3} codifies a way of representing de re which accounts for similarity by either shape and size or material constitution, where neither kind of properties outweighs the other. When interpreted via *R*_{3}, the contingent identity statement (a) comes out true.

- (d)
David may have been the first or second born of two identical twins.

^{20}

- (e)
There is a world

*w*with two identical twins whose first born represents David, and there is a world*w*′ with two identical twins whose second born represents David. - (f)
There is a world

*w*with two identical twins, each representing David.

I claim that (e) and not (f) provides the correct truth condition of (d).

Notice first of all that sentence (f) is strictly stronger than (e), and that (f) violates (NI) whereas (e) does not. Now, the truth condition of (d) must be at least as strong as (e). For (d) entails that it is a possibility for David to be the first twin of a pair, and that it is also a possibility for him to be the second twin of a pair. Rephrased in possible-world language, this indeed amounts to (e). What (d) does not entail is that these two possibilities are realized in the same world. Hence, (d) is weaker than (f). Insofar as (e) and (f) were taken to be the only viable candidates, it follows that (e) provides a necessary and sufficient condition for (d), whereas (f) provides only a sufficient condition.

Lewis’ claim that the case of identical twins yields an instance of multiple counterparts must therefore be qualified: modal sentences like (d) may be but *ought not* to be interpreted via instances of multiple counterparts. Once again, there is no specific modal reason to embrace contingent identity.

- (g)
David may not have been David

- (h)
David may not have been Lynch

is satisfiable.

I suspect Lewis would try to make sense of (h) by saying that different ways of naming David Lynch evoke multiple ways of representing him. This solution is not viable, though. Different ways of representing individuals are associated to different counterpart relations. I just argued that the inconstancy of the counterpart relation leads to inconstant identity and is irrelevant to such cases of contingent identity as (h). Moreover, if proper names are to be interpreted de re they should not evoke any representational content, on pain of conflating de re and de dicto reading.

*dictum de omni*: what is true of all things is true of each thing.

^{21}For on the one hand the necessity of self-identity is valid already in minimal counterpart theory. However, the formula

- (i)
\(\forall y \neg\square x=y\).

- (j)
\(\square x=x\wedge \forall y \neg\square x=y\)

- (k)
David is necessarily self-identical, and nothing is necessarily identical to David.

As pointed out in Sect. 1, the reason why Lewis’ counterpart relation is not locally functional and 1–1 is that he assumed the principle of qualitative supervenience (Q). Thus, in order to restore the laws of identity (NI), (ND) and (LL), an intended model must be exhibited in which the counterpart relation is not purely qualitative. This is accomplished now in the construction of the intended \({\bf CT}^{\bf TB4N}\)-model.

*choice counterpart relation*. Given a \({\bf CT}^{\bf TB4}\)-model \({\mathcal{M}}=\langle{\bf M},I,W,C,\{R^{j}\}_{j\in J} \rangle\), where \(\{R^{j}\}_{j\in J}\) is a set of nonlogical predicate symbols, consider the set of world-bound individuals

*w*and world-bound individual

*x*, let

*K*

^{x}(

*w*) be the set of

*w*-counterparts of

*x*:

*K*

^{x}is the collection of sets of the counterparts of

*x*at each world, that is

*a partial choice set*\({\mathcal{K}}^{x}\) for

*K*

^{x}. The reason why I am imposing that the choice sets be partial will be clear soon. Now, let \(\{{\mathcal{K}}^{x}\}_{x\in{\bf B}}\) be a collection containing, for each world-bound individual

*x*, a partial choice set \({\mathcal{K}}^{x}\). Such a collection induces a

*a choice counterpart relation*\(\widehat{C}\)

*derived from*

*C*:

*choice refinement of*

*C*. A model \(\widehat{\mathcal{M}}\) obtained by replacing

*C*in \(\mathcal{M}\) with a choice refinement \(\widehat{C}\) of

*C*is said to be a

*choice model derived from*\(\mathcal{M}\).

It is noteworthy that a choice model derived from a \({\bf CT}^{\bf TB4}\)-model is in general not a \({\bf CT}^{\bf TB4}\)-model, for example if the choice counterpart relation \(\widehat{C}\) fails to be reflexive. On the other hand, the following fact holds:

**Theorem 1**

*For every*\({\bf CT}^{\bf TB4}\)*-model *\({\mathcal{M}}\)*there is a choice*\({\bf CT}^{\bf TB4}\)*-model*\(\widehat{\mathcal{M}}\)*derived from*\({\mathcal{M}}\).

*Proof*

In every \({\bf CT}^{\bf TB4}\)-model \({\mathcal{M}}\) the identity relation on *M* is a reflexive, symmetric and transitive choice refinement of the counterpart relation. \(\square\)

### Observation:

The proof of Theorem 1 relies on the condition that, for every world-bound individual *x*, a choice set \({\mathcal{K}}^{x}\) is by definition *partial*. For suppose \({\mathcal{M}}\)*is an*\({\bf CT}^{\bf TB4}\)-model with world-bound individuals *x*, *y* and *z* such that *x* and *y* coexist at world *w*, whereas *z* exists at *w*′ and is the *unique**C*-counterpart of both *x* and *y* at *w*′. Let us assume there is an equivalence choice counterpart relation \(\widehat{C}\) derived from *C* defined as above, except that the choice sets \({\mathcal{K}}^{x}\) and \({\mathcal{K}}^{y}\) are total. It follows that both \(\widehat{C}(z,x)\) and \(\widehat{C}(z,y)\). By symmetry this entails \(\widehat{C}(x,z)\) and \(\widehat{C}(y,z)\), against the assumption that \(\widehat{C}\) is a choice counterpart relation and so that every individual has at most one counterpart at each world.

The following two facts are trivial.

**Lemma 1**

*A choice counterpart relation is locally functional.*

**Lemma 2**

*A symmetric choice counterpart relation is locally 1–1.*

Hence:

**Theorem 2**

*Every choice*\({\bf CT}^{\bf TB4}\)*-model is a*\({\bf CT}^{\bf TB4N}\)*-model.*

*Proof*

A straightforward consequence of Lemma 1 and Lemma 2. \(\square\)

This fact shows that, given an intended model of symmetric counterpart theory \({\mathcal{M}}\), we can always construct a model of perfect counterpart theory \(\widehat{\mathcal{M}}\) derived from it. It may be tempting to choose a choice model \(\widehat{\mathcal{M}}\) as our intended model of perfect counterpart theory. However, this proposal is inadequate in two respects, one epistemic and one semantic.

On the epistemic side, the problem traces back to the axiom of choice. Since a \({\bf CT}^{\bf TB4}\)-model is in general associated with multiple choice \({\bf CT}^{\bf TB4}\)-models, and choice models are generated in a typically nonconstructive fashion, it is impossible to know what particular choice counterpart relation \(\widehat{C}\) has replaced the original *C*. As a consequence, we cannot know what class of de re statements comes out true in the derived choice model. Notice that the problem is essentially the same as the one that troubled us in the construction of intended models of approximate counterpart theory. For in that case we could not rely on approximations from below due to their non-uniqueness.

The problem of non-uniqueness of choice models cannot be solved in the most obvious way, that is by imposing conditions that guarantee the existence of exactly one choice \({\bf CT}^{\bf TB4}\)-model derived from a given \({\bf CT}^{\bf TB4}\)-model. However, the lack of uniqueness can be made harmless in the following way: for any \({\bf CT}^{\bf TB4}\)-model, any two intended choice \({\bf CT}^{\bf TB4}\)-models based on it *ought to be satisfy the same class of modal formulae*. I call this the *epistemic desideratum*.

*w*, Big J.C. and Small J.C.: Big J.C. is 7 feet tall, while Small J.C. is only 5 feet tall. Provided that our own Julius Caesar goes by two names, “Gaius” and “Caesar”, the following sentences are true in \({\mathcal{M}}\) in virtue of Big J.C. and Small J.C.:

- 1.
it is possible for Gaius to be 5 feet tall;

- 2.
it is possible for Gaius to be 7 feet tall;

- 3.
it is possible for Caesar to be 2 feet taller than Gaius.

*w*.

In a choice model \(\widehat{{\mathcal{M}}}\) derived from \({\mathcal{M}}\), the same individuals Big J.C. and Small J.C. fail to make (3) true. So far, so good: choice models have been introduced precisely to filter out such unwanted modal facts. Indeed, the truth of sentence (3) presupposes the contingency of identity. On the other hand, \(\widehat{{\mathcal{M}}}\) cannot satisfy both (1) and (2), although neither sentence is true in \({\mathcal{M}}\) in virtue of Julius Caesar’s having multiple counterparts. This shows that replacing *C* with \(\widehat{C}\) may force individuals to lose possibilities which are consistent with the necessity of identity.

The Caesar example has a moral: in order to obtain an intended \({\bf CT}^{\bf TB4N}\) model based on the intended \({\bf CT}^{\bf TB4}\)-model, *we want to eliminate only those possibilities that involve splitting or merging counterpart relation*, hence preserving such de re truths as (1) and (2). I call this the *semantic desideratum*.

^{22}on \({\mathcal{M}}\) that satisfies the

*Perfection Principle*(Π):

Before moving on to the systematic construction of choice models that meet (Π), I want to discuss the underlying idea by applying it to the Julius Caesar scenario, here represented in Fig. 1.for every

n-tuple \(\bar{x}\) of individuals at worldwin \({\mathcal{M}}\), the modal properties of \(\bar{x}\) in \(\widehat{{\mathcal{M}}}\) are exactly the modal properties of \(\bar{x}\) in \({\mathcal{M}}\) that involve no splitting or merging counterpart relation.

The goal, in this case, is to obtain a model derived from \({\mathcal{M}}\) where both (1) and (2) are true, while (3) is false. Intuitively, \({\mathcal{M}}\) is a collection of worlds with a counterpart relation *C* defined on the set of the inhabitants of those worlds. The first key move is to expand the ontology in such a way that, for every world *w*, an infinite number of copies of *w* is added to \({\mathcal{M}}\).^{23} At this point we need to extend *C* to the new individuals. This will be done in such a way that the extended counterpart relation *C*^{*} mirrors the behavior of *C*: *x* is a *C*^{*}-counterpart of *y* in the expanded structure iff the original structure contains duplicates *x*′ of *x* and *y*′ of *y* such that *x*′ is a *C*-counterpart of *y*′.

*w*containing Big J.C. an Small J.C. will have two copies

*w*′ and

*w*′′, each containing a duplicate of Big J.C. and a duplicate of Small J.C.—call them Big (Small) J.C. 1 and Big (Small) J.C. 2. Since the extended relation

*C*

^{*}simulates the behavior of

*C*, the duplicates of Big J.C. and Small J.C. will be

*C*

^{*}-counterparts of the actual Julius Caesar. The situation is represented in Fig. 2 (although only two of the infinitely many duplicates of

*w*are shown there).

*C*

^{*}that satisfies the Perfection Principle (Π). With respect to worlds

*w*′ and

*w*′′, the resulting choice relation \(\widehat{C}^{*}\) can pick counterparts for Caesar in one of these two ways:

- a.
\(\langle\hbox {Big J.C. 1},\hbox {Caesar}\rangle,\langle\hbox {Small J.C. 2},\hbox {Caesar}\rangle\)

- b.
\(\langle\hbox {Small J.C. 1},\hbox {Caesar}\rangle,\langle\hbox {Big J.C. 2},\hbox {Caesar}\rangle\).

*both*“it is possible for Julius to be 5 feet tall” and “it is possible for Caesar to be 7 feet tall”, but not “it is possible for Caesar to be 2 feet taller than Julius”. The same will be the case if \(\widehat{C}^{*}\) picks counterpart as in (b). This shows that (limited to the de re properties of Caesar with respects to worlds

*w*,

*w*′ and

*w*′′) either model satisfies (Π) and can be chosen as an intended model of perfect counterpart theory. In other words, the choice between the two models is immaterial when it comes to the modal properties of Julius Caesar.

The construction must be generalized so as to cover all *possibilia*. Moreover, since some de re properties involve nested modalities, the newly introduced worlds will have to be duplicated as well, and so on. This fact introduces an extra dimension to the picture. The task can be pursued by a recursive definition of the extended model. Let us then turn to the formal construction.

Every \({\bf CT}^{\bf TB4}\)-model can be decomposed into a set of disjoint submodels, each defined by a world jointly with the individuals inhabiting it. Formally, let \({\mathcal{M}}=\langle{\bf M},I,W,C,\{R^{j}\}_{j\in J} \rangle\) be an \({\bf CT}^{\bf TB4}\)-model and \(w\in{\bf M}\) a world. The *world-model in*\({\mathcal{M}}\)*induced by**w* is the submodel of \({\mathcal{M}}\backslash\{C\}\) restricted to the domain \(\{z|z=w\vee I(z,w)\}\). I will refer to that world-model as \({\mathcal{M}}[w]\). Also, I will use the notation \({\mathcal{M}}[w]\prec {\mathcal{M}}\), that is \({\mathcal{M}}[w]\) is *a submodel of*\({\mathcal{M}}\), to indicate that \({\mathcal{M}}[w]\) is a world-model in \({\mathcal{M}}\). The Greek letters σ, τ, … are used to denote isomorphisms of world-models. Hence, \(\sigma({\mathcal{M}}[w])\) is an isomorphic copy of \({\mathcal{M}}[w]\); \({\mathcal{M}}[w]\simeq_{\sigma}{\mathcal{M}}[w']\) means \(\sigma({\mathcal{M}}[w])={\mathcal{M}}[w']\). Also, I make the assumption that whenever σ(*x*) = *y*, then there are world-models \({\mathcal{M}}[w]\), \({\mathcal{M}}[w']\) such that \(x\in{\mathcal{M}}[w]\), \(y\in{\mathcal{M}}[w']\) and \({\mathcal{M}}[w]\simeq_{\sigma}{\mathcal{M}}[w']\).

*C*to the pair of world-models \(\langle{\mathcal{M}}[w],{\mathcal{M}}[w']\rangle\), that is

*w*,

*w*′ in \({\mathcal{M}}\):

We can now proceed to the recursive construction of expanded models.

*an expansion*\({\mathcal{M}}^{*}\) of \({\mathcal{M}}\) is defined thus:

- (0)
\({\mathcal{M}}^{0}={\mathcal{M}}\).

- (1) \({\mathcal{M}}^{1}\) is a model of \({\mathcal{L}}_{{\bf C}}\) such that
- a.
\({\mathcal{M}}^{0}\) is a submodel of \({\mathcal{M}}^{1}\);

- b.
let \(\kappa_{0}=card(\{z|W(z)\hbox { and } z\in{\bf M}^{0}\})\). For each world \(w\in{\bf M}^{0}\), \({\mathcal{M}}^{1}\backslash {\mathcal{M}}^{0}\) contains \(\mu\cdot\kappa_{0}\) disjoint copies of \({\mathcal{M}}^{0}[w]\). The collection of newly introduced world-models in \({\mathcal{M}}^{1}\) isomorphic to a given \({\mathcal{M}}^{0}[w]\) is represented by the set \(\{\sigma_{i,j}({\mathcal{M}}^{0}[w])\}_{i\leqslant\mu,j\leqslant\kappa_{0}}\).

- c.
for every \(x,y\in {\bf M}^{\bf 1}\), \({\mathcal{M}}^{1}\models C_{1}(x,y)\) iff there are \(x',y'\in {\bf M}^{\bf 0}\) and isomorphisms σ, τ s.t. σ(

*x*′) =*x*, τ(*y*′) =*y*and \({\mathcal{M}}^{0}\models C(x',y')\). Also, for every*j*∈*J*, \({\mathcal{M}}^{1}\models R^{j}_{1}(x_{1},\ldots, x_{n})\) iff there are \(x'_{1},\ldots, x'_{n}\in {\bf M}^{\bf 0}\) and isomorphisms σ_{1}, …, σ_{n}s.t. \(\sigma_{1}(x'_{1})=x_{1}\), \(\sigma_{n}(x'_{n})=x_{n}\) and \({\mathcal{M}}^{0}\models R^{j}(x'_{1},\ldots, x'_{n})\); - d.
there is no model of \({\mathcal{L}}_{{\bf C}}\) that satisfies a-c and is a proper submodel of \({\mathcal{M}}^{1}\);

- a.
- n+2) \({\mathcal{M}}^{n+2}\) is a model of \({\mathcal{L}}_{{\bf C}}\) such that
- a.
\({\mathcal{M}}^{n+1}\) is a submodel of \({\mathcal{M}}^{n+2}\);

- b.
let \(\kappa_{n+1}=card(\{z|W(z)\hbox { and } z\in{\bf M}^{n+1}\backslash {\bf M}^{n}\})\). For each world \(w\in{\bf M}^{n+1}\backslash {\bf M}^{n}\), \({\mathcal{M}}^{n+2}\backslash {\mathcal{M}}^{n+1}\) contains μ·κ

_{n+1}disjoint copies of \({\mathcal{M}}^{n+1}[w]\). The collection of newly introduced world-models in \({\mathcal{M}}^{n+2}\) isomorphic to \({\mathcal{M}}^{n+1}[w]\) is represented by the set \(\{\tau_{i,j}({\mathcal{M}}^{n+1}[w])\}_{i\leqslant\mu,j\leqslant\kappa_{n+1}}\). - c.
for every \(x,y\in {\bf M}^{\bf n+2}\), \({\mathcal{M}}^{n+2}\models C_{n+2}(x,y)\) iff there are \(x',y'\in {\bf M}^{\bf n+1}\) and isomorphisms σ, τ s.t. σ(

*x*′) =*x*, τ(*y*′) =*y*and \({\mathcal{M}}^{n+1}\models C_{n+1}(x',y')\). Also, for every*j*∈*J*, \({\mathcal{M}}^{n+2}\models R^{j}_{n+2}(x_{1},\ldots, x_{n})\) iff there are \(x'_{1},\ldots, x'_{n}\in {\bf M}^{\bf n+1}\) and isomorphisms σ_{1}, …, σ_{n}s.t. \(\sigma_{1}(x'_{1})=x_{1}\), \(\sigma_{n}(x'_{n})=x_{n}\) and \({\mathcal{M}}^{n+1}\models R_{n+1}^{j}(x'_{1},\ldots, x'_{n})\); - d.
there is no model of \({\mathcal{L}}_{{\bf C}}\) that satisfies a-c and is a proper submodel of \({\mathcal{M}}^{n+2}\).

- a.
- ω)
\({\mathcal{M}}^{*}=\bigcup_{n<\omega}{\mathcal{M}}^{n}\).

*C*

_{1}and

*C*

_{n+2}be well-defined in point c (steps 1 and

*n*+ 2). I will show this for step 1; the other case is alike. Given \(x,y\in {\bf M}^{\bf 1}\) let \({\mathcal{M}}^{1}\models C_{1}(x,y)\). Now assume there are \(x',y',x'',y''\in {\bf M}^{\bf 0}\) and isomorphisms σ, τ, σ′, τ′ s.t.

σ(

*x*′) =*x*and τ(*y*′) =*y*;σ′(

*x*′′) =*x*and τ′(*y*′′) =*y*.

*C*in \({\mathcal{M}}^{0}\) is the approximation from above of some reflexive and symmetric counterpart relation

*C*

^{−}. If it is already the case that

*C*

^{−}(

*x*′,

*y*′) then it must be that

*C*

^{−}(

*x*′′,

*y*′′), as

*C*

^{−}is a similarity relation (hence, it meets (Q)),

*x*′ and

*y*′ are qualitatively indiscernible and so are

*x*′′ and

*y*′′. It follows trivially that

*C*(

*x*′′,

*y*′′). If it is not the case that

*C*

^{−}(

*x*′,

*y*′), there is a \(z\in{\bf M}^{\bf 0}\) s.t.

*C*

^{−}(

*x*′,

*z*) and

*C*

^{−}(

*z*,

*y*′). Since

*C*

^{−}meets (Q), it must be that

*C*

^{−}(

*x*′′,

*z*) and

*C*

^{−}(

*z*,

*y*′′). But

*C*is the transitive closure of

*C*

^{−}, therefore

*C*(

*x*′′,

*y*′′).

The same line of reasoning will show that the interpretation of the predicate symbols \(R^{j}_{1}\) and \(R^{j}_{n+2}\) is well-defined in point c (steps 1 and *n* + 2).

The next theorem shows that the class of \({\bf CT}^{\bf TB4}\)-models is closed under expansion.

**Theorem 3**

*Every expanded model*\({\mathcal{M}}^{*}\)*is a *\({\bf CT}^{\bf TB4}\)*-model.*

*Proof*

- (0)
\({\mathcal{M}}^{0}={\mathcal{M}}\) is a \({\bf CT}^{\bf TB4}\)-model.

- n+1)Assume \({\mathcal{M}}^{n}\) is a \({\bf CT}^{\bf TB4}\)-model.
- [P1]
If \({\mathcal{M}}^{n+1}\models I(x,y)\), by points b and d of the construction (steps 1 and n+2) there are \(x',y'\in{\bf M}^{\bf n}\) and σ s.t. σ(

*x*′) =*x*, σ(*y*′) =*y*and \({\mathcal{M}}^{n}\models I(x',y')\). By inductive hypothesis \({\mathcal{M}}^{n}\models W(y')\) and, by the isomorphism σ, \({\mathcal{M}}^{n+1}\models W(y)\). - [P2]
Let \({\mathcal{M}}^{n+1}\models I(x,y)\wedge I(x,z)\). Were

*y*and*z*distinct, the world-models \({\mathcal{M}}^{n+1}[y]\) and \({\mathcal{M}}^{n+1}[z]\) would overlap, against the construction. - [P3]
If \({\mathcal{M}}^{n+1}\models C(x,y)\), by point c there are \(x',y'\in{\bf M}^{\bf n}\) and σ, τ s.t. σ(

*x*′) =*x*, τ(*y*′) =*y*and \({\mathcal{M}}^{n}\models C(x',y')\). Hence, \({\mathcal{M}}^{n}\models I(x',z')\) for some \(z'\in{\bf M}^{\bf n}\). But \({\mathcal{M}}^{n}\models I(x',z')\) iff \({\mathcal{M}}^{n+1}\models I(\sigma(x'),\sigma(z'))\) iff \({\mathcal{M}}^{n+1}\models I(x,\sigma(z'))\). So, there exists a \(z\in{\bf M}^{\bf n+1}\) s.t. \({\mathcal{M}}^{n+1}\models I(x,z)\). - [P4]
Likewise. \(\square\)

- [P1]

The following fact shows that an intended \({\bf CT}^{\bf TB4}\)-model \({\mathcal{M}}\) and its expansion \({\mathcal{M}}^{*}\) satisfy a *mirror property*: indiscernible individuals in the two models satisfy the same modal conditions.

**Theorem 4**

*Given an expansion*\({\mathcal{M}}^{*}\)

*of the intended*\({\bf CT}^{\bf TB4}\)

*-model*\({\mathcal{M}}\)

*, let*\({\mathcal{M}}[w]\simeq_{\sigma}{\mathcal{M}}^{*}[w^{*}]\)

*. Take a formula*\(\phi(\bar{x})\in{\mathcal{L}}_{{\bf QM}}\)

*and variable assignments*\(\xi:Free(\phi)\rightarrow {\mathcal{M}}[w]\)

*and*\(\xi^{*}:Free(\phi)\rightarrow {\mathcal{M}}^{*}[w^{*}]\)

*such that*\(\xi^{*}(x)=\sigma\cdot\xi(x)\)

*. Then,*

*Proof*

*x*

_{i}in the tuple \(\bar{x}\), \(\tau^{-1}\cdot\lambda^{*}(x'_{i})\) is a counterpart of ξ(

*x*

_{i}). But this follows from the construction of \({\mathcal{M}}^{*}\). Hence we obtain the desired fact:

Now that we know what an expanded model looks like, we can proceed to show how to derive from it a choice \({\bf CT}^{\bf TB4}\)-model satisfying (Π).

*complete*if for all worlds \(w,w'\in{\bf M}\) and every \(X\in\overline{C}_{\langle w,w'\rangle}\) there is a world

*w*′′ = σ(

*w*) s.t., if

*I*(

*x*,

*w*) and

*I*(

*y*,

*w*′) then

To prove this claim it needs to be shown that, for every intended \({\bf CT}^{\bf TB4}\)-model \({\mathcal{M}}\):if \({\mathcal{M}}\) is an intended \({\bf CT}^{\bf TB4}\)-model,

an intended\({\bf CT}^{\bf TB4N}\)-model is a complete choice\({\bf CT}^{\bf TB4}\)-model derived from an expansion of\({\mathcal{M}}\).

(\(\varepsilon1\)) there exists a complete choice \({\bf CT}^{\bf TB4}\)-model derived from an expansion of \({\mathcal{M}}\);

(\(\varepsilon2\)) any complete choice \({\bf CT}^{\bf TB4}\)-model derived from an expansion of \({\mathcal{M}}\) satisfies (Π).

**Theorem 5**

*Let*\({\mathcal{M}}\)*be an expanded model. Then there is a complete choice*\({\bf CT}^{\bf TB4}\)*-model derived from*\({\mathcal{M}}\).

*Proof*

*C*is the counterpart relation in \({\mathcal{M}}\), it suffices to show that for every \(n\,\geqslant\, 0\) there is an equivalence choice counterpart relation \(\widehat{C}_{n}\) derived from

*C*that is perfect w.r.t. every \(\overline{C}_{\langle w,w'\rangle}\), where \(w'\in{\mathcal{M}}^{n}\).

- (0)
Due to the construction of \({\mathcal{M}}\), every \({\mathcal{M}}[w']\prec{\mathcal{M}}\) is isomorphic to some \({\mathcal{M}}[w]\prec{\mathcal{M}}^{0}\), so let

*w*= π(*w*′). Moreover, for every world \(w\in{\mathcal{M}}^{0}\) there are μ·κ_{0}copies of \({\mathcal{M}}[w]\) in \({\mathcal{M}}^{1}\backslash{\mathcal{M}}^{0}\), represented by the collection \(\{\sigma_{i,j}({\mathcal{M}}[w])\}_{i\leqslant\mu,j\leqslant\kappa_{0}}\). Hence, there is a choice refinement \(\widehat{C}'_{0}\) of*C*s.t.[α_{0}] for every \(w'\in{\mathcal{M}}\) and \(w_{r}\in{\mathcal{M}}^{0}\) (\(r\,\leqslant\,k_{0}\)), every \(X_{i\leqslant\mu}\in\overline{C}_{\langle w',w_{r} \rangle}\) and every*x*,*y*s.t.*I*(*x*,*w*′) and*I*(*y*,*w*_{r}):- i.
*X*_{i}(*x*,*y*) iff \(\widehat{C}'_{0}(\sigma_{i,r}\cdot\pi(x),y)\).

Let \(\widehat{C}'_{0}\) be a minimal choice counterpart relation derived from

*C*that satisfies α_{0}. Being minimal, it picks*only*the individuals in \({\mathcal{M}}^{1}\backslash{\mathcal{M}}^{0}\) specified by α_{0}. Let \(\widehat{C}_{0}\) be the smallest equivalence relation including \(\widehat{C}'_{0}\). Thus, \(\widehat{C}_{0}\) is an equivalence choice counterpart relation derived from*C*that satisfies α_{0}. - i.

- n+1)
Due to the construction of \({\mathcal{M}}\), every \({\mathcal{M}}[w']\prec{\mathcal{M}}\) is isomorphic to some \({\mathcal{M}}[w]\prec{\mathcal{M}}^{n+1}\backslash{\mathcal{M}}^{n}\), so let

*w*= τ(*w*′). Also, for every world \(w\in{\mathcal{M}}^{n+1}\backslash{\mathcal{M}}^{n}\) there are \(\mu\cdot\kappa_{n+1}\) copies of \({\mathcal{M}}[w]\) in \({\mathcal{M}}^{n+2}\backslash{\mathcal{M}}^{n+1}\), represented by the set \(\{\rho_{i,j}({\mathcal{M}}[w])\}_{i\leqslant\mu,j\leqslant\kappa_{n+1}}\). Hence, there is a choice refinement \(\widehat{C}'_{n+1}\) of*C*s.t.[α_{n+1}] for every \(w'\in{\mathcal{M}}\) and \(w_{s}\in{\mathcal{M}}^{n+1}\backslash{\mathcal{M}}^{n}\) (\(s\,\leqslant\,k_{n+1}\)), every \(X_{i\leqslant\mu}\in\overline{C}_{\langle{w',w_{s}}\rangle}\) and every*x*,*y*s.t.*I*(*x*,*w*′) and*I*(*y*,*w*_{s}):- i.
*X*_{i}(*x*,*y*) iff \(\widehat{C}'_{n+1}(\rho_{i,s}\cdot\tau(x),y)\) - ii.
\(\widehat{C}_{n}\subset\widehat{C}'_{n+1}\).

Let \(\widehat{C}'_{n+1}\) be a minimal choice counterpart relation derived from

*C*that satisfies α_{n+1}. Being minimal, it extends \(\widehat{C}_{n}\) only to the individuals in \({\mathcal{M}}^{n+2}\backslash{\mathcal{M}}^{n+1}\) selected by (i) in α_{n+1}. Let now \(\widehat{C}_{n+1}\) be the smallest equivalence relation including \(\widehat{C}'_{n+1}\). Thus, \(\widehat{C}_{n+1}\) is an equivalence choice counterpart relation derived from*C*that satisfies \(\alpha_{n+1}\). \(\square\) - i.

*explicit translation scheme*from \({\mathcal{L}}_{{\bf QM}}\) to \({\mathcal{L}}_{{\bf C}}\) is just like Lewis’ translation scheme, except for formulae governed by modal operators. The new clauses for the explicit translation \([\phi]^{\ddot{w}}\) of ϕ at world

*w*are as follows:

- (T8′)
\([\diamondsuit\phi(\bar{x})]^{\ddot{w}}\) is \((\exists z)(\exists\bar{y})(W(z)\wedge I(\bar{y},z))\wedge C(\bar{y},\bar{x})\wedge ID(\bar{x},\bar{y})\wedge[\psi(\bar{y})]^{\ddot{z}})\)

- (T9′)
\([\square\phi(\bar{x})]^{\ddot{w}}\) is \((\forall z)(\forall\bar{y})(W(z)\wedge I(\bar{y},z))\wedge C(\bar{y},\bar{x})\wedge ID(\bar{x},\bar{y})\rightarrow[\psi(\bar{y})]^{\ddot{z}})\)

The next fact shows that the explicit-translation modal facts holding in an expanded model coincide with the Lewis-translation modal facts holding in a complete choice \({\bf CT}^{\bf TB4}\)-model derived from it.

**Lemma 3**

*Let*\(\widehat{{\mathcal{M}}}\)

*be a complete choice*\({\bf CT}^{\bf TB4}\)

*-model derived from the expanded model*\({\mathcal{M}}\)

*. Given a world*\(w\in{\mathcal{M}}\)

*, a formula*\(\phi(\bar{x})\in{\mathcal{L}}_{{\bf QM}}\)

*and a variable assignment*\(\xi:Free(\phi)\rightarrow {\mathcal{M}}[w]\)

*(where*\({\mathcal{M}}[w]=\widehat{{\mathcal{M}}}[w]\)

*), it follows that*

*Proof*

*m*be the smallest integer s.t. \(w\in{\mathcal{M}}^{m}\), and therefore \(w\in\widehat{{\mathcal{M}}}^{m}\). By Theorem 5, there is a world \(v'\in\widehat{{\mathcal{M}}}^{m+1}\backslash\widehat{{\mathcal{M}}}^{m}\) s.t. π(

*v*) =

*v*′ and for every \(i\,\leqslant\,n\):

The following corollary shows that Theorem 4 remains true under the explicit translation scheme.

**Corollary 1**

*Given an expansion*\({\mathcal{M}}^{*}\)

*of the intended*\({\bf CT}^{\bf TB4}\)

*-model*\({\mathcal{M}}\)

*, let*\({\mathcal{M}}[w]\simeq_{\sigma}{\mathcal{M}}^{*}[w^{*}]\)

*. Take a formula*\(\phi(\bar{x})\in{\mathcal{L}}_{{\bf QM}}\)

*and variable assignments*\(\xi:Free(\phi)\rightarrow {\mathcal{M}}[w]\)

*and*\(\xi^{*}:Free(\phi)\rightarrow {\mathcal{M}}^{*}[w^{*}]\)

*such that*\(\xi^{*}(x)=\sigma\cdot\xi(x)\)

*. Then,*

*Proof*

It is a straightforward variation of Theorem 4. \(\square\)

Finally, the next fact guarantees that any complete choice \({\bf CT}^{\bf TB4}\)-model derived from an expanded model satisfies \(\epsilon2\).

**Corollary 2**

*Given*

*a formula*\(\phi\in{\mathcal{L}}_{{\bf QM}};\)*an intended*\({\bf CT}^{\bf TB4}\)*-model*\({\mathcal{M}};\)*a complete choice*\({\bf CT}^{\bf TB4}\)*-model*\(\widehat{{\mathcal{M}}}\)*derived from an expansion of*\({\mathcal{M}};\)*worlds*\(w\in{\mathcal{M}}\)*and*\(w'\in\widehat{{\mathcal{M}}}\)*s.t.*\({\mathcal{M}}[w]\simeq_{\sigma}\widehat{{\mathcal{M}}}[w'];\)*variable assignments*\(\xi:Free(\phi)\rightarrow{\mathcal{M}}[w]\)*and*\(\xi':Free(\phi)\rightarrow\widehat{{\mathcal{M}}}[w'],\)*where ξ′(x) = σ·ξ(x),*

*then*

*Proof*

By Corollary 1 and Lemma 3.\(\square\)

The last result concludes the construction of an intended model of perfect counterpart theory.

Let’s recap. We start with our intended \({\bf CT}^{\bf TB4}\)-model \({\mathcal{M}}\). The goal is to construct an intended \({\bf CT}^{\bf TB4N}\)-model based on \({\mathcal{M}}\) (that is, an intended model of perfect counterpart theory). Using the Caesar case, I argued that an intended \({\bf CT}^{\bf TB4N}\)-model is a choice \({\bf CT}^{\bf TB4}\)-model satisfying an epistemic and a semantic desideratum. Both desiderata are codified by the Perfection Principle (Π). The construction of a choice \({\bf CT}^{\bf TB4}\)-model which meets (Π) involved two steps. First, I defined an expansion \(\mathcal{M}^{*}\) of \({\mathcal{M}}\). Second, I have shown that: (\(\varepsilon1\)) from each expansion \(\mathcal{M}^{*}\) of \({\mathcal{M}}\) we can derive a complete choice \({\bf CT}^{\bf TB4}\)-model \(\widehat{\mathcal{M}}^{*}\); and (\(\varepsilon2\)) any complete choice \({\bf CT}^{\bf TB4}\)-model thus obtained meets (Π). We can conclude that an intended \({\bf CT}^{\bf TB4N}\)-model exists. This fact suffices to justify perfect counterpart theory.

*x*refers to Al, the formal regimentation ϕ of this sentence is:

*B*and

*C*at some world

*v*such that

*C*is a counterpart of Al and

*D*and

*E*at some world

*w*such that

*E*is a counterpart of

*C*,

*D*is a counterpart of

*B*and

^{u}is true in every intended model of perfect counterpart theory based on \({\mathcal{M}}\). Namely, \([\phi]^{u}\) will be true of Al in virtue of suitable duplicate worlds

*v*′ and

*w*′′, as pictured in Fig. 4 (note how the choice counterpart relation

*C*

^{*}selects duplicates of the individuals picked by

*C*).

*x*and

*y*refer to

*A*= Cicero and

*B*= Catiline respectively, the paraphrase ψ is:

*w*-counterparts,

*C*and

*D*respectively, who are mutual friends; and the actual Catiline

*B*is a counterpart of

*C*. Figure 5 shows the relevant pattern of

*C*-counterparts. By Corollary 2, if \([\psi]^{\ddot{v}}\) is true, then every intended \({\bf CT}^{\bf TB4N}\)-model makes \([\psi]^{v}\) true. This fact is exemplified in Fig. 5, given a choice refinement

*C*

^{*}.

^{v}is false in any intended model of perfect counterpart theory.

## 5 Conclusion

I provided a defense of perfect counterpart theory by constructing an intended \({\bf CT}^{\bf TB4N}\)-model. To be precise, it has been shown how to derive an intended choice \({\bf CT}^{\bf TB4}\)-model from an intended \({\bf CT}^{\bf TB4}\)-model (which was independently motivated). Since counterparts are partially specified via the axiom of choice, the above construction constitutes a challenge to Lewis’ claim that a non-qualitative counterpart relation is a “contradiction in terms”. We can make sense of non-qualitative counterpart relations, provided that the application of the axiom of choice is properly regimented in an expanded model.

The existence of an intended model of counterpart theory provides a case for the system \({\bf CT}^{\bf TB4N}\). Since this system validates all the modal principles listed in Sect. 1, the counterpart theorist has a way out of the logical objection.

Moreover, the counterpart-theoretic framework enjoys a generality that Kripke semantics lacks. Since de re statements are not interpreted via transworld identity, we have the option to weaken or strengthen the modal logic by tweaking the counterpart relation as desired. This feature of Lewis’ approach suggests a solution to one last objection I now wish to consider.

Although I have argued all along in favor of perfect counterpart theory, I do have qualms about transitive counterpart relations. Indeed, the transitivity of identity is often mentioned as one of the main flaws of Kripke semantics. Several philosophers^{24} have pointed out that transworld identity leads to modal paradoxes of the sorites variety. To block them, in a Kripke-style system we can either drop the transitivity of the accessibility relation or appeal to a theory of essences. The former solution seems ad hoc^{25} whereas the latter is hard to implement. The counterpart-theorist has a third option: let the counterpart relation be intransitive by basing it on similarity. This was Lewis’ solution to the paradoxes.

A further reason to drop transitivity is that the construction of intended \({\bf CT}^{\bf TB4}\)-models trivializes de re modality when combined with the *principle of plenitude*, the thesis that any way the world can be is a way a world is.^{26} For an intended \({\bf CT}^{\bf TB4}\)-model is obtained by approximating from above the counterpart relation *C* of an intended \({\bf CT}^{\bf TB}\)-model. It is easy to see that, by plenitude, for any two possibilia *x* and *z* there are possibilia *y*_{1}, …, *y*_{n} s.t. *x* is arbitrarily close in similarity to *y*_{1}, *y*_{1} is arbitrarily close in similarity to *y*_{2}, … and *y*_{n} is arbitrarily close in similarity to *z*. This implies that *x* is a *C*-counterpart of *y*_{1}, *y*_{1} is a *C*-counterpart of *y*_{2}, … and *y*_{n} is a *C*-counterpart of *z*. But the approximation from above *C*^{+} of *C* is simply the transitive closure of *C*. So, *x* will be a *C*^{+}-counterpart of *z*. We can conclude that, since plenitude allows there to be a chain of *C*-counterparts linking any two possible individuals, everything is a *C*^{+}-counterpart of everything in an intended model of approximate counterpart theory. Hence the trivialization of modal facts: every de re possibility statement that is not first-order inconsistent will be true and every de re necessity statement that is not first-order valid will be false. By Corollary 2, this problem is inherited by the intended model of perfect counterpart theory.

If the above remarks are correct, a materially adequate system of counterpart theory cannot include the transitivity axiom P7. This fact prompts an obvious revision of the construction carried out in Sect. 4. First of all, let \({\bf CT}^{\bf TBN}\) be the system defined as \({\bf CT}^{\bf TB}\cup\{P8,P9\}\). Instead of showing the existence of an intended \({\bf CT^{TB4N}}\)-model by constructing a suitable choice \({\bf CT}^{\bf TB4}\)-model, we now want to show the existence of an intended \({\bf CT}^{\bf TBN}\)-model by constructing a suitable choice \({\bf CT}^{\bf TB}\)-model. Because the construction is a slight variation on what is done in Sect. 4, I will skip the details. Roughly, we start with an intended model of symmetric counterpart theory \({\mathcal{M}}\) and produce an expansion of it. Then we show that there exists a complete choice \({\bf CT}^{\bf TB}\)-model based on the expansion of \({\mathcal{M}}\). The resulting model will indeed be a \({\bf CT}^{\bf TBN}\)-model satisfying (Π), i.e. an intended model of \({\bf CT}^{\bf TBN}\).

Since \({\bf CT}^{\bf TBN}\) is the strongest theory whose intended model does not trivialize de re modalities, we can conclude that this system provides the best combination of logical strength and material adequacy.^{27}

See Hazen (1979), Stalnaker (2003a, b), Ramachandran (1990), Forbes (1985), Sider (2002), Varzi (2001), Fara (2008), Corsi (2002), Kracht and Kutz (2007), Torza (forthcoming) to mention just a few.

Cf. Lewis (1968, p. 123): “Translation into counterpart theory can settle disputed questions in quantified modal logic. We can test a suggested modal principle by seeing whether its translation is a theorem of counterpart theory.”

Here, \(\bar{x}\) stands for \(x_{1},\ldots, x_{n}\). So, \(\phi(\bar{x})\) is short for \(\phi(x_{1},\ldots x_{n})\), \(I(\bar{y},z)\) for \(I(y_{1},z)\wedge\ldots\wedge I(y_{n},z)\), and \(C(\bar{y},\bar{x})\) for \(C(y_{1},x_{1})\wedge C(y_{2},x_{2})\wedge\ldots\wedge C(y_{n},x_{n})\).

In applying the translation scheme, I make a metatheoretic assumption throughout: if \(\phi(\bar{x})\) is interpreted at *w*, then \(I(\bar{x},w)\).

For an overview of systems of modal logic and their interpretations, see Burgess (2009), Goldblatt (2005) and Chap. 21 of Blackburn et al. (2006). Specifically on metaphysical modalities, see Forbes (1985). However, I will later express doubts about the tenability of the modal theses (4) and (5).

Hazen (1979) was the first to suggest an interpretation of counterparthood by means of simple similarity.

It is noteworthy that (MP) alone is insufficient to characterize a viable notion of counterparthood. For otherwise every individual *x* would have counterparts at all worlds, including those worlds containing only individuals that share few or no properties with *x*. To see this, let *x* be Julius Caesar and *w* a world where there are no organisms. Since *w* includes some counterpart of *x*, the sentence “Julius Caesar may have been an inorganic being” will be true. So, depending on the nature of *w*, we will attribute to Caesar all sorts of bizarre possibilities: being a pebble, being a three-wheeled cycle etc. For the same reason, arguably true necessity statements such as “Julius Caesar is necessarily an organism” will come out false. Roughly, if we embrace a sufficiently rich ontology, every de re possibility statement that is not first-order inconsistent will be true, and every de re necessity statement that is not first-order valid will be false. If (MP) is the only condition imposed on the counterpart relation, metaphysical possibility will collapse to logical possibility. For this reason Lewis imposed the minimality principle (mP) on the counterpart relation.

One might reply that Lewis’ intended model has the resources to handle the Caesar scenario and, therefore, a semantically adequate counterpart theory is consistent with (MP). For a relation of comparative similarity results from a “system of weights or priorities” in which some respects of similarity outweigh others. The relative weight of a particular similarity respect is a function of the context. Now let *C*_{t} be a counterpart relation determined by a context *t* which assigns no weight to similarities and differences in political role and power. Since *y* and *y*′ do not differ at all except for the political role they play, according to *t* they will be equally similar to the actual Caesar *x*. Consequently, if *y* is a *C*_{t}-counterpart of *x*, so is *y*′. It can be concluded that, given the appropriate context, the sentence “Caesar could have failed to be a dictator” is indeed true in virtue of *y*′. The problem with this reply (which could be leveled against other criticisms of (MP) such as the ones of Feldman (1971), Hazen (1979)) is that it simply assumes the existence of a comparative similarity relation which supervenes on similarities and differences in particular respects. However, this assumption is undermined by a recent impossibility theorem. Morreau (forthcoming) has shown that no comparative similarity relation can be defined which satisfies a few desirable conditions. In effect, this result provides a general refutation of any counterpart relation defined by comparative similarity.

De Clercq and Horsten (2005, p. 272). Incidentally, they propose a third approximation technique which, although superior to the two considered by Williamson, applies only to finite domains. This limitation makes it unsuitable for the present purpose.

Moreover, (ND) is derivable from (NI) via (B) and the distribution principle (K), both valid in every Kripke frame with a symmetric accessibility relation.

The case for their identity can be made stronger by assuming that the lump of clay and the statue come into existence at the same time, for example by sticking together the two halves of the statue in the last step of its creation.

Incidentally, the incapacity of an ordinary modal language to express inconstant identity cannot be obviated by simply adding a pair of modal operators \(\{\diamondsuit_{n},\square_{n}\}\) for each counterpart relation *R*_{n}.

I am ruling out a third candidate, namely: “There is a world *w* with two identical twins such that either the first or the second born represents David”. Although closer to the surface structure of (d), this cannot be the correct condition. For if it were, it would be sufficient for the truth of (d) that David may have been the first but not the second born. This is obviously not the sense in which a competent speaker understands (d).

Incidentally, Lewis was agnostic about the existence of indiscernible worlds. See Lewis (1986, p. 80).

As Lewis (1986) himself acknowledged, modal realism trivializes the principle of plenitude. For if a way the world can be is *identified* with a way a world is, then the principle of plenitude amounts to the claim that any way a world is is a way a world is. Thus, to guarantee that there are enough possibilities, Lewis appealed to a mereological principle of recombination. But the question whether recombination adequately expresses plenitude is a problem for the modal realist and not for the counterpart theorist, hence it lies outside the scope of the present paper.

Unlike perfect counterpart theory, \({\bf CT}^{\bf TBN}\) fails to validate the modal theses (4) and (5).