Models for Counterparts

Lewis (1968) introduced counterpart theory as an ingredient in a more comprehensive metaphysical framework, namely modal realism. ^{Footnote 1} Nonetheless, counterpart theory has drawn a considerable amount of interest in its own right. ^{Footnote 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:

A modal formula is valid iff its translation is a theorem of counterpart theory. ^{Footnote 3}

The basic counterpart-theoretic language \(\mathcal{L}_{{\bf C}}\) includes three constants:

• W(x): x is a world

• I(x, y): x is at world y

• C(x, y): x is a counterpart of y.

Lewis provides a translation scheme from the language \({\mathcal{L}}_{{\bf QM}}\) of quantified modal logic to \({\mathcal{L}}_{{\bf C}}\). The paraphrase of a modal formula at world w is defined by recursion:^{Footnote 4}

1. (T1)

   [ϕ]^w is ϕ, if ϕ is atomic

2. (T2)

   \([\neg\phi]^{w}\) is \(\neg[\phi]^{w}\)

3. (T3)

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

4. (T4)

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

5. (T5)

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

6. (T6)

   [(∃x)ϕ]^w is \((\exists x)(I(x,w)\wedge[\phi]^{w})\)

7. (T7)

   \([(\forall x)\phi]^{w}\) is \((\forall x)(I(x,w)\rightarrow[\phi]^{w})\)

8. (T8)

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

9. (T9)

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

The minimal system of counterpart theory CT is defined by the following axioms:

1. (P1)

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

2. (P2)

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

3. (P3)

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

4. (P4)

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

The main fact to be noticed about 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.

In Lewis’ intended model, the counterpart relation is defined by comparative similarity: “Your counterparts resemble you closely in content and context in important respects. They resemble you more closely than do the other things in their worlds.”^{Footnote 5} Comparative similarity relations being reflexive, Lewisian counterpart theory must include the axiom

1. (P5)

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

Given the above translation scheme, the resulting system will validate Feys’ principle:

1. (T)

   \(\phi\rightarrow\diamondsuit\phi\).

Insofar as nothing is strictly more similar to me than myself, I am one of my counterparts. Therefore, what is the case for me is also a possibility for me.^{Footnote 6} Since the counterpart relation in Lewis’ intended model is defined by comparative similarity, it satisfies the Maximality Principle (MP):

if x at w is a counterpart of y, no individual at w is more similar to y than x.

This principle is inconsistent with the symmetry condition

1. (P6)

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

Since P6 cannot be added to Lewis’ theory, this does not validate Brouwer’s principle:

1. (B)

   \(\phi\rightarrow\square\diamondsuit\phi\)

For example, a duplicate of my brother is my counterpart at world 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.

A counterpart relation defined by comparative similarity meets the Minimality Principle (mP):

if x is a counterpart of y, x and y are sufficiently similar in relevant respects

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

1. (P7)

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

Since Lewis’ theory cannot include P7, it does not validate Becker’s principle:

1. (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.

Incidentally, if the counterpart relation fails to be both symmetric and transitive (in Lewis’ case it is neither), another modal principle fails to hold:

1. (5)

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

Lewis’ relation of comparative similarity is purely qualitative. Namely, it satisfies the following Principle of Qualitative Supervenience (Q):

if x is qualitatively indistinguishable from x′ and y is qualitatively indistinguishable from y′, then x is a counterpart of y iff x′ is a counterpart of y′.

This condition prevents a counterpart relation from being either locally functional or locally 1–1. It is locally functional when an individual cannot have multiple counterparts at the same world:

1. (P8)

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

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

1. (P9)

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

A counterexample to P8 is the scenario in which two duplicates of me, say two twins, exist at the same world. By (Q), if either is my counterpart (which would be unreasonable to deny), the other must also be my counterpart. P9 fails in the symmetrical case in which I am the counterpart of two twins. Since Lewis had to reject P8 and P9, neither the Necessity of Identity (NI) nor the Necessity of Distinctness (ND) come out valid:

1. (NI)

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

2. (ND)

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

Moreover, if (NI) fails, so does Leibniz’ Law:

1. (LL)

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

This concludes the analysis of Lewis’ intended model. The upshot is that Lewisian counterpart theory corresponds to the weakest system in the following list:

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

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

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

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

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).^{Footnote 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.

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.^{Footnote 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).^{Footnote 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.^{Footnote 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).

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.^{Footnote 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.^{Footnote 12} I will then set aside sortal essentialism and pursue the task from a completely different angle.

Consider the following example. We start with a set 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:

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

The approach from above always yields a unique outcome because 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.

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.

Is identity a necessary relation? As Kripke (1971) pointed out, we can easily derive (NI) from the conjunction of two facts, namely

1. 1.

   \(\square x=x\)

2. 2.

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

The first premise follows by necessitation from the reflexivity of identity (REF), whereas the second is an instance of Leibniz’ law (LL). But identity is defined by (REF) and (LL), hence any theory which fails to validate (NI) misinterprets identity.^{Footnote 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′.^{Footnote 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.^{Footnote 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.^{Footnote 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.

The crucial point here is that Lewis’ account of Gibbard’s alleged case of contingent identity is not a case of contingent identity. Indeed, the counterpart-theoretic translation of

1. (a)

   G is L and G may not have been L

is

1. (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′.

Lewis’ translation scheme reduces an instance of contingent identity to a case of splitting counterpart relation: two individuals 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:

1. (c)

   G is L and there is a world w containing an R₁-counterpart G′ of G and an R₂-counterpart L′ of L such that G′ is not L′.

In this case, 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).^{Footnote 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.”^{Footnote 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.

The punch line is that Gibbard’s story may but ought not to be interpreted in counterpart theory as a case of contingent identity. The pressing question now is: Are there any modal scenarios that the counterpart theorist can only interpret in terms of contingent identity? If so then we do have a case against (NI); otherwise, the nonclassical behavior of intensional identity in Lewisian counterpart theory should be regarded as a byproduct of his intended model. Consider the sentence:

1. (d)

   David may have been the first or second born of two identical twins.

Lewis believes that modal truths involving “a tie between a pair of twins” are the typical case in which an individual can have multiple counterparts with respect to a single system of representation. How can we specify the truth condition of (d)? I see only two viable candidates:^{Footnote 20}

1. (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.

2. (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.

On the other hand, accepting the possibility of multiple counterparts yields bizarre modal phenomena. For the sentence

1. (g)

   David may not have been David

is certainly false, since the necessity of self-identity is a valid principle of counterpart theory. On the other hand, if our David is David Lynch, the rather odd statement

1. (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.

Another problem with contingent identity is that it leads to the failure of the logical principle dictum de omni: what is true of all things is true of each thing.^{Footnote 21} For on the one hand the necessity of self-identity is valid already in minimal counterpart theory. However, the formula

1. (i)

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

is \({\bf CT}^{\bf T}\)-satisfiable. Thus, the following formula can be true in Lewis’ framework:

1. (j)

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

which may be interpreted as

1. (k)

   David is necessarily self-identical, and nothing is necessarily identical to David.

This sounds a lot like a contradiction. The Lewisian must then face the fact that the cost of allowing multiple counterparts largely outweighs the benefits, if any. For on the one hand it makes identity contingent, leading to puzzling semantic and logical phenomena such as the satisfiability of (h) and (k). On the other, it is redundant when it comes to the explanation of substantive modal problems, such as Gibbard’s case or the possibility of having identical twins.

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.

We first need to refine the conceptual apparatus by introducing the notion of a 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

For every world w and world-bound individual x, let K ^x(w) be the set of w-counterparts of x:

If K ^x is the collection of sets of the counterparts of x at each world, that is

by an application of the axiom of choice to a subset of this collection we obtain 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:

I call \(\widehat{C}\) a 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.

There is also a semantic side to the issue. Assume that in our intended \({\bf CT}^{\bf TB4}\)-model \({\mathcal{M}}\) the actual Julius Caesar has two counterparts at some world 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. 1.

   it is possible for Gaius to be 5 feet tall;

2. 2.

   it is possible for Gaius to be 7 feet tall;

3. 3.

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

In particular, sentence (3) is true in \({\mathcal{M}}\) in virtue of Caesar’s multiple counterparts at 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.

To secure the existence of an intended \({\bf CT}^{\bf TB4N}\)-model meeting both the epistemic and the semantic desideratum it suffices to show that, given an intended \({\bf CT}^{\bf TB4}\)-model \({\mathcal{M}}\), there exists a choice \({\bf CT}^{\bf TB4}\)-model \(\widehat{{\mathcal{M}}}\) based^{Footnote 22} on \({\mathcal{M}}\) that satisfies the Perfection Principle (Π):

for every n-tuple \(\bar{x}\) of individuals at world w in \({\mathcal{M}}\), 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.

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.

Caesar’s C-counterparts

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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 \({\mathcal{M}}\).^{Footnote 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′.

For example, the world 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).

Caesar’s C ^*-counterparts

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The second key move is to select a choice refinement of 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:

1. a.

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

2. b.

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

If the choice counterpart relation \(\widehat{C}^{*}\) picks counterparts as in (a), the model will look like the one in Fig. 3. This structure verifies 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.

Caesar’s \(\widehat{C}^{*}\)-counterparts

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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 \({\mathcal{M}}[w]\simeq_{\sigma}{\mathcal{M}}[w']\).

Given a \({\bf CT}^{\bf TB4}\)-model \({\mathcal{M}}\), let \(C_{\langle w,w'\rangle}\) be the restriction of C to the pair of world-models \(\langle{\mathcal{M}}[w],{\mathcal{M}}[w']\rangle\), that is

Now, given the set

let μ be the cardinality of the largest \(\overline{C}_{\langle w,w'\rangle}\), for all w, w′ in \({\mathcal{M}}\):

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

Given an intended \({\bf CT}^{\bf TB4}\)-model \({\mathcal{M}}=\langle{\bf M},I,W,C,\{R^{j}\}_{j\in J} \rangle\), an expansion \({\mathcal{M}}^{*}\) of \({\mathcal{M}}\) is defined thus:

1. (0)

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

2. (1)

   \({\mathcal{M}}^{1}\) is a model of \({\mathcal{L}}_{{\bf C}}\) such that

   1. a.

      \({\mathcal{M}}^{0}\) is a submodel of \({\mathcal{M}}^{1}\);

   2. 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 \(\{\sigma_{i,j}({\mathcal{M}}^{0}[w])\}_{i\leqslant\mu,j\leqslant\kappa_{0}}\).

   3. 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 σ₁, …, σ _n s.t. \(\sigma_{1}(x'_{1})=x_{1}\), \(\sigma_{n}(x'_{n})=x_{n}\) and \({\mathcal{M}}^{0}\models R^{j}(x'_{1},\ldots, x'_{n})\);

   4. d.

      there is no model of \({\mathcal{L}}_{{\bf C}}\) that satisfies a-c and is a proper submodel of \({\mathcal{M}}^{1}\);

3. n+2)

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

   1. a.

      \({\mathcal{M}}^{n+1}\) is a submodel of \({\mathcal{M}}^{n+2}\);

   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 \(\{\tau_{i,j}({\mathcal{M}}^{n+1}[w])\}_{i\leqslant\mu,j\leqslant\kappa_{n+1}}\).

   3. 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 σ₁, …, σ _n s.t. \(\sigma_{1}(x'_{1})=x_{1}\), \(\sigma_{n}(x'_{n})=x_{n}\) and \({\mathcal{M}}^{n+1}\models R_{n+1}^{j}(x'_{1},\ldots, x'_{n})\);

   4. d.

      there is no model of \({\mathcal{L}}_{{\bf C}}\) that satisfies a-c and is a proper submodel of \({\mathcal{M}}^{n+2}\).

4. ω)

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

Observation: the condition that \({\mathcal{M}}\) be an intended model ensures that the relations 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.

It suffices to show that if \({\mathcal{M}}^{0}\models C(x',y')\) then \({\mathcal{M}}^{0}\models C(x'',y'')\). Since \({\mathcal{M}}^{0}={\mathcal{M}}\), 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

It suffices to show that \({\mathcal{M}}^{*}\) satisfies P1-P4. The demonstration proceeds by induction on the construction of \({\mathcal{M}}^{*}\).

1. (0)

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

2. n+1)

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

   1. [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, \({\mathcal{M}}^{n+1}\models W(y)\).

   2. [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.

   3. [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. \({\mathcal{M}}^{n+1}\models I(x,z)\).

   4. [P4]

      Likewise. \(\square\)

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

The cases in which ϕ is not governed by modal operators are obvious. Hence, it suffices to show that the claim holds when \(\phi=\diamondsuit\psi\). The direction from left to right is trivial, as \({\mathcal{M}}\prec{\mathcal{M}}^{*}\). To see that the converse holds, suppose that \(({\mathcal{M}}^{*},\xi^{*})\models\diamondsuit\psi(\bar{x})^{w^{*}}\). This means that for some world \(v^{*}\in{\mathcal{M}}^{*}\):

So, there is a variable assignment \(\lambda^{*}:Free(\psi)\rightarrow {\mathcal{M}}^{*}[v^{*}]\) s.t.

The construction of \({\mathcal{M}}^{*}\) (points c and d) guarantees the existence of a world \(v\in{\mathcal{M}}\) s.t. \({\mathcal{M}}[v]\simeq_{\tau}{\mathcal{M}}^{*}[v^{*}]\). Thus, by inductive hypothesis:

It remains to show that, for each 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:

that is \(({\mathcal{M}},\xi)\models[\diamondsuit\psi(\bar{x})]^{w}\). \(\square\)

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).

Given a \({\bf CT}^{\bf TB4}\)-model \({\mathcal{M}}\), let \(\widehat{{\mathcal{M}}}\) be a choice model derived from \({\mathcal{M}}\). If \(C\) and \(\widehat{C}\) are their respective counterpart relations, \(\widehat{M}\) is said to be 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

My principal claim is the following:

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 \({\mathcal{M}}\).

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

• (\(\varepsilon1\)) 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).

The next result takes care of \(\varepsilon1\).

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

This fact can be proven by induction. If 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 \(w'\in{\mathcal{M}}^{n}\).

1. (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·κ₀ 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.

   [α₀] 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 ):

   1. 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 α₀. Being minimal, it picks only the individuals in \({\mathcal{M}}^{1}\backslash{\mathcal{M}}^{0}\) specified by α₀. 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 α₀.

1. 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 ):

   1. i.

      X _i (x, y)   iff   \(\widehat{C}'_{n+1}(\rho_{i,s}\cdot\tau(x),y)\)

   2. 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 \(\alpha_{n+1}\). \(\square\)

It remains to show that \(\varepsilon2\) is true. In order to do so, we need to refine Lewis’ translation scheme. Namely, the translation of a formula ϕ governed by a modal operator must specify whether ϕ is true in virtue of an instance of splitting or merging counterpart relation. First, let us define:

The 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:

1. (T8′)

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

2. (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

Since the cases where ϕ is not governed by a modal operator are trivial, it suffices to show that the claim holds when \(\phi(\bar{x})=\diamondsuit\psi(\bar{x})\). Assuming \(({\mathcal{M}},\xi)\models[\diamondsuit\psi(\bar{x})]^{\ddot{w}}\), there exist \(v, a_{1},\ldots,a_{n}\in{\mathcal{M}}\) s.t.

So, the individuals \(\bar{a},\xi (\bar{x})\) define a functional and 1–1 set \(X_{i}\in\overline{C}_{\langle{v,w}\rangle}\). Let 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 \(i\,\leqslant\,n\):

Hence,

which is \((\widehat{{\mathcal{M}}},\xi)\models\diamondsuit[\psi(\bar{x})]^{w}\). The converse is trivial. \(\square\)

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.

For clarification’s sake I want to consider a few examples of modal facts expressed by statements involving nested operators. Take first the sentence “Al has a possible twin who could have been fond of him”. If x refers to Al, the formal regimentation ϕ of this sentence is:

Now assume that \([\phi]^{\ddot{u}}\) is true in our intended \({\bf CT}^{\bf TB4}\)-model \({\mathcal{M}}\). Thus, there are two possible twins B and C at some world v such that C is a counterpart of Al and

is true of the pair \(\langle C,B\rangle\). There must then be two individuals D and E at some world w such that E is a counterpart of C, D is a counterpart of B and

is true of the pair \(\langle E,D\rangle\). Since the truth of \([\phi]^{\ddot{u}}\) in \({\mathcal{M}}\) does not involve splitting or merging counterpart relation, the perfection principle (Π) guarantees that [ϕ]^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).

Nested modalities in an intended \({\bf CT}^{\bf TB4N}\)-model

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Here is another example: “There is a possibility that Cicero was friends with Catiline and that he could even attempt to overthrow the Roman Republic.” Provided that x and y refer to A = Cicero and B = Catiline respectively, the paraphrase ψ is:

One way to make \([\psi]^{\ddot{v}}\) true in the intended \({\bf CT}^{\bf TB4}\)-model is as follows. Cicero and Catiline have distinct 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 ^*.

Nested modalities in an intended \({\bf CT}^{\bf TB4N}\)-model

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It is noteworthy that, whereas \([\psi]^{\ddot{v}}\) is true of \(\langle A,B \rangle\), the sentence χ

which reads “There is a possibility that Cicero was friends with Catiline and that he could even be Catiline”, is unsatisfiable in \({\bf CT}^{\bf TB4}\). It follows that [χ]^v is false in any intended model of perfect counterpart theory.

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^{Footnote 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^{Footnote 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.^{Footnote 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.^{Footnote 27}