1. Introduction

Philosophers call on possible worlds to perform different kinds of jobs. One of these jobs is that of explaining what the truth of modal truths and the falsity of modal falsities consists in. “What does the possibility that p consist in?”, you ask. “It consists in the existence of a possible world at which p”, someone might reply.

Possible worlds are also used as semantic machinery. The semanticist needs entities for her quantifiers to range over, and possible worlds -or, more generally, possibilia- can be used to construct them. One might take a proposition to be a set of possible worlds (structured or not); one might take the intension of a predicate to be a function that assigns each world a set of objects in that world (or some complication thereof); one might use possible worlds to characterize the semantic values of modal operators; and so forth.

(It is sometimes assumed that these two jobs are best performed by a single notion of possible world, as in ^{Lewis 1986}. But it is worth keeping in mind that such an assumption is not mandated. It would in principle be possible to develop two distinct notions of possible world, one for the foundationalist project and another for the semantic project.)

The main objective of this paper is to defend the claim that a specialized modal ontology is not needed as semantic machinery. (The further claim that it is not needed to explain what modal truth consists in is defended in chapters 2 and 5 of Rayo forthcoming.)

Some philosophers might be interested in this sort of project because of ontological scruples. Not me: my own view is that ontology is cheap. I am interested in the project because I think it helps clarify the role of possible worlds in our philosophical theorizing.

My proposal is an instance of what David Lewis called “ersatzism”. I will argue that the needs of the semanticist can be fulfilled by using representatives for possibilia in place of possibilia. Although there are other ersatzist proposals in the literature,¹ I hope the machinery developed here will earn its keep by delivering an attractive combination of frugality and strength.

The proposal is frugal in two different respects. First, it is metaphysically frugal: it is designed to be acceptable to modal actualists, and presupposes very little by way of ontology. (I help myself to settheory, but do not assume a specialized modal ontology, or an ontology of properties.) Second, the proposal is ideologically frugal: it does not presuppose potentially controversial expressive resources such as infinitary languages or non-standard modal operators. The point of developing machinery that presupposes so little is that philosophers of different persuations can put it to work without having to take a stance on difficult philosophical issues.

As far as strength is concerned, one gets a qualified version of the following claim: anything that can be said by quantifying over Lewisian possibilia can also be said by using the machinery developed here. The result is that the proposal can be used quite freely in the context of semantic theorizing, without having to worry too much about running into expressive limitations. An especially useful feature of the proposal is that it allows one to enjoy the benefits of quantification over sets of possibilia, which are often appealed to in the course of semantic theorizing.²

Possible-worlds-theorists sometimes claim that the same individual exists according to distinct possible worlds. (There is a world according to which I have a sister who is a philosopher, and a world at which that very individual is a cellist rather than a philosopher.) Ersatzist representatives for such worlds might be said to be linked. Much of the paper will be devoted to the phenomenon of linking. An effective treatment of linking is crucial to the success of the ersazist program.

2. Possibility

Sadly, I don’t have a sister. But I might have had a sister. In fact, I might have had a sister who was a philosopher. And, of course, had I had a sister who was a philosopher, she wouldn’t have been a philosopher essentially: she might have been a cellist instead. The following is therefore true:³

SISTER

◊(∃xSisterx,ARΛ Phil x Λ◊Cellist xΛ¬Philx) (Read: I might have had a sister who was a philosopher and might have been a cellist rather than a philosopher.)

On the most straightforward possible-worlds semantics forfirst-order modal languages, SISTER will only be counted as true if there are worlds w ₁ and w ₂ with the following properties: according to w ₁, there is an individual who is my sister and a philosopher; according tow ₂, that very individual -as one is inclined to put it- is a cellist rather than a philosopher. It is therefore tempting to say the following:

(Read:There is anxsuch that: (i) according to w ₁, x is my sister and a philosopher, and (ii) according to w ₂, x is a cellist rather than a philosopher.)

But is there anything to make this existential quantification true? A merely possible sister could do the job. But modal actualists believe there is no such thing. The job could also be done by an actually existing object who is not my sister but might have been my sister. (See ^{Williamson 2010}.) But the claim that such objects exist would be a substantial metaphysical assumption -an assumption that it would be best to avoid, if at all possible.

3. A Kripke-Semantics for Actualists

There is a certain sense in which it is straightforward for an actualist to give an adequate Kripke-semantics for modal sentences. The trick is to have one’s semantics quantify over representations of possibilities, rather than over the possibilities themselves. In this section I will describe one such semantics. It is an elaboration of an idea introduced in ^{Roy 1995} and further developed in ^{Melia 2001}.

Let L be afirst-order language, and let L◊ be the result of enriching L with the sentential operator “◊”. An ɑ-world (short for “actualist-world”) for L◊ is an ordered pair ‹D,I›, consisting of a domain and an interpretation function, such that:

The notions of truth and satisfaction at an ɑ-world are characterized along standard lines, with the proviso that “x=x” is only satisfied at an ɑ-world by objects in the domain of that ɑ-world, with the result that “x=x” can be used as an existence predicate. See Appendix A for details.

The best way to see how ɑ-worlds are supposed to work is by considering a simple example. Take the ɑ-world ‹D1,I1› from figure 1. The domain of consists of the pairs ‹Agustín, “actual”› (which represents me) and ‹Socrates, “nonactual”› (which represents not Socrates, but a merely possible object), and I ₁ assigns “Sister” a set containing the pair ‹‹Socrates, “nonactual”›, ‹Agustín, "actual"››, The result is that ‹D ₁,I _1› makes “∃x(Sisterx,AR)” true, and therefore represents a possibility whereby I have a sister. (Why use the pair ‹Socrates, “nonactual”› to represent my merely possible sister rather than some other pair -‹Plato, “nonactual”›, say? No good reason. Just like one can write a story according to which I have a sister using any font one likes, so one can define an ɑ-world according to which I have a sister using any pair ‹z, “nonactual”› one likes.)

It is useful to compare ɑ-worlds to Lewisian worlds. Like ɑ-worlds, Lewisian worlds can be thought of as representing possibilities. Here is Lewis:

How does a world, [Lewisian] or ersatz, represent, concerning Humphrey, that he exists? [ . . . ] A [Lewisian] world might do it by having Humphrey himself as a part. That is how our own world represents, concerning Humphrey, that he exists. But for other worlds to represent in the same way that Humphrey exists, Humphrey would have to be a common part of many overlapping worlds [ . . . ]. I reject such overlap [ . . . ]. There is a better way for a [Lewisian] world to represent, concerning Humphrey, that he exists [ . . . ] it can have [ . . . ] a Humphrey of its own, aflesh-and-blood counterpart of our Humphrey, a man very much like Humphrey in his origins, in his intrinsic character, or in his historical role. By having such a part, a world representsde re, concerning Humphrey -that is, the Humphrey of our world, whom we as his worldmates may call simply Humphrey- that he exists and does thus-and-so.⁴

(It is easy to lose track of Lewis’s representationalism because he also subscribed to a different claim: what it is for it to be possible that p is for there to be a Lewisian world at which (i.e. representing that) p. For further discussion of these issues, see Rayo forthcoming.)

Whereas Lewisian worlds represent by analogy, ɑ-worlds represent by satisfaction. A Lewisian world represents the possibility that I have a sister by containing a person who is similar to me in certain respects, and has a sister. An ɑ-world, on the other hand, represents the possibility that I have a sister by being such as to satisfy the formula “∃x(Sisterx,AR)”.

From the perspective of the Lewisian, an individual with a counterpart in the actual world represents its actual-word counterpart, and an individual with no counterpart in the actual world represents a merely possible object. From the present perspective, a pair of the form “‹x, ‘actual’›” represents its first component, and a pair of the form “‹x, ‘nonactual’›” represents a merely possible object (even though the pair itself, and both of its components, are actually existing objects).

Two representations are linked if -as one is inclined to put it- they concern the same individual, even if the individual in question doesn’t exist. In order for a Kripke semantics based on ɑ-worlds to verify SISTER, there must be linking amongst ɑ-worlds. In particular, some ɑ-world must represent a possibility whereby I have a sister who is a philosopher and another must represent a possibility whereby -as one is inclined to put it-that very same individualis a cellist rather than a philosopher.

Let us first see how linking gets addressed from a Lewisian perspective. l ₁ and l ₂ are Lewisian worlds: l ₁ contains an individual α1 who bears the right sort of similarity to me and an individual s ₁ who is α1’s sister and a philosopher; l ₂ contains an individual s ₂ who is a cellist. Accordingly, l ₁ represents a possibility whereby my sister is a philosopher, and l ₂ represents a possibility whereby someone is a cellist. But nothing so far guarantees linking. Nothing so far guarantees that -as one is inclined to put it- the individual l ₁ represents as my sister is the very individual that l ₂ represents as a cellist. What is needed for linking is that s ₁ and s ₂ be counterparts: that they be similar in the right sorts of respects.

The same maneuver can be used when it comes to ɑ-worlds. Like the Lewisian, we shall use counterparthood amongst representations to capture linking. For Lewis, representations are counterparts just in case they are similar in the right sorts of respects. From the present perspective, we shall say that representations are counter-parts just in case they are identical (though other ways of defining the counterpart relation could be used as well). Here is an example. The ɑ-world ‹D1,I1› represents a possibility whereby I have a sister who is a philosopher. Now consider ɑ-worlds ‹D2,I2› and ‹D3,I3› from Table 1 (p. 9). Each of them represents a possibility whereby someone is a cellist rather than a philosopher. But only ‹D2,I2› is linked to ‹D1,I1›. For ‹D1,I1› and ‹D2,I2› both employ ‹Socrates, “nonactual”› as a representation, and it is this that guarantees that -as one is inclined to put it- the individual who ‹D1,I1› represents as my sister is the very individual that ‹D2,I2› represents as a cellist. On the other hand, since ‹D3,I3› represents a possibility

These examples assume that the only non-logical expressions in L are “Philosopher”, “Cellist”, “Sister” and “AR”, and that the domain of Lis {Agustín}.

Table 1: Examples of A-worlds

whereby someone is a cellist by using ‹Plato, “nonactual”› rather than ‹Socrates, “nonactual”›, what one gets is that -as one is inclined to put it- the individual who ‹D ₁,I ₁› represents as my sister is distinct from the individual that ‹D ₂,I _2› represents as a cellist.

It is important to keep in mind that an ɑ-worlds-semantics is not a way of improving on the informal characterization of linking that I supplied a few paragraphs back. (Representations are linked if -as one is inclined to put it- they concern the same individual, even if the individual in question doesn’t exist.) In particular, it is not a way of dispensing with the qualifying phrase “as one is inclined to put it”.⁵ What an ɑ-worlds-semantics delivers is an (actualistically acceptable) device for representing possibilities, which enjoys the following feature: it is clear when two representations are to be counted as linked. The reason this is helpful is that, as we shall see below, much of the theoretical work that can be carried out by quantifying over possibilities can be carried out by quantifying over representations of possibilities instead. So an ɑ-worlds-semantics puts the actualist in a position to get on with certain kinds of theoretical work without having to worry about giving a proper characterization of linking.

4. Admissibility

There are ɑ-worlds according to which someone is a married bachelor, and ɑ-worlds according to which there might have been a human who wasn’t essentially human. Such representations need to be excluded from our semantics, on pain of getting the result that sentences like “◊(Ǝx(Married(x)Λ Bachelor(x)))” and “◊(Ǝx(Human(x)Λ◊(Ǝy(y=x Λ ¬Human(x)))))” are true. What we need is a notion of admissibility. Armed with such a notion, one can say that ˹◊Ø˺ is true just in case Ø is true at some admissible ɑ-world, and that ˹◊Ø˺ is true just in case Ø is true at every admissible ɑ-world. ⁶

It is important to keep in mind the distinction between the semantic project of developing a Kripke-semantics for modal languages, on the one hand, and the project of accounting for the limits of metaphysical possibility, on the other. A semantics based on ɑ-worlds is meant to address the former, but not the latter of these projects. Accordingly, the notion of admissibility that is presupposed by the semantic project should be thought of as a placeholder for whatever limits on the metaphysically possible turn out to be uncovered by a separate investigation. (On reasonable assumptions, one can show that a suitable notion of admissibility is guaranteed to exist wherever the limits turn out to lie.)⁷ A semantics based on ɑ-worlds is compatible with different views about how tofix the limits of metaphysical possibility. One might appeal to a Principle of Recombination, for example, or to a set of “basic modal truths”.⁸ (My own account is spelled out in chapters 2 and 5 of Rayo forthcoming.)

Even if one appeals to an ontology of possible worlds infixing the limits of metaphysical possibility, one might have good reasons for using ɑ-worlds rather than possible worlds for the purposes of semantic theorizing. The easiest way to see this is by distinguishing between sparse and abundant conceptions of possible worlds.⁹ A sparse conception countenances worlds according to which there are objects that don’t actually exist, but not worlds according to which it is true of specific non-existent objects that they exist. There is, for example, a possible world w ₁ according to which I have a sister who is a philosopher and might have been a cellist rather than a philosopher, but no possible world according to which it is true of the specific individual who would have been my sister had w ₁ obtained that she exists. (Not even w ₁ is such a world, for even though w ₁ is a world according to which I have a sister, it is not a world according to which it is true of some specific individual that she is my sister.) On an abundant conception of possible worlds, on the other hand, there are possible worlds according to which it is true of specific non-existent objects that they exist. There is, for instance, a possible world w ₂ according to which it is true of the very individual who would have been my sister had w ₁ been actualized that she is a cellist rather than a philosopher.

On the sparse conception of possible worlds, the existence of a world is conditional on the existence of the objects the world represents as existing, in the same sort of way that the existence of a set is conditional on the existence of its members. Had w ₁ been actualized, I would have had a sister, and all manner of sets containing that very individual would have existed. But as things stand, my sister doesn’t exist, and neither do sets having her as a member. Similarly -the story would go- had w ₁ been actualized, I would have had a sister, and a world according to which that very individual is a cellist would have existed. But as things stand, my sister doesn’t exist, and neither do possible worlds according to which she herself exists.

The absence of w ₂ does not prevent a defender of the sparse conception from using possible worlds to determine a truth-value for SISTER. For, on the assumption that possible worlds track metaphysical possibility, the existence of w ₁ is enough to guarantee that SISTER is true. But the absence of w ₂ does mean that the sparse worlds do not by themselves deliver the ontology that would be needed to give a Kripke-semantics for a sentence like SISTER. For, as emphasized above, a Kripke-semantics will only count SISTER as true if the range of one’s metalinguistic quantifiers contains both w ₁ and w ₂.

Fortunately, a sparse ontology of possible worlds is enough to guarantee the existence of a notion of admissibility relative to which an ɑ-worlds semantics assigns the right truth-value to every sentence in the language (see footnote 7). So it is open to the sparse theorist to use admissible ɑ-worlds, rather than possible worlds, for the purposes of semantic theorizing. The upshot is not, of course, that one has done away with one’s specialized modal ontology, since possible worlds may be needed for the project of pinning down the crucial notion of admissibility (or for the project of explaining what modal truth consists in). But by using ɑ-worlds as the basis of one’s semantics, the requirements on one’s modal ontology are confined to needs of these non-semantic projects.

A related point can be made with respect to mere possibilia. The availability of ɑ-worlds means that there is no need to postulate mere possibilia (or specialized surrogates, such as Plantinga’s individual essences) as far as the project of developing a Kripke-semantics for modal languages is concerned.¹⁰

5. Interlude: The Principle of Representation

In previous sections I have made informal remarks about the ways in which ɑ-worlds represent possibilities. The purpose of this interlude is to be more precise. (Uninterested readers may skip ahead to section 6.)

When ‹D,I› is considered in isolation from other ɑ-worlds, anything that can be said about the possibility represented by ‹D,I› is a consequence of the following principle:

REPRESENTATION PRINCIPLE (Isolated-World Version)

According to the possibility represented by ‹D,I›, ,p

if and only if

there is a sentences of L such that s says that p and s is true at ‹D,I› .

Accordingly, when considered in isolation from other ɑ-worlds, ‹D ₂,I _2› and ‹D ₃,I _3› represent the same possibility. It is the possibility that there be exactly one thing and that it be a cellist but not a philosopher.

We shall normally assume that L (and therefore L ^◊ ) contains a name for every object in the domain of L. With this assumption in place, the following is a consequence of the Representation Principle:

Suppose z is in the domain of L. Then z exists according to the possibility represented by ‹D,I› just in case ‹z, “actual”› is in D.

In particular, one gets the result that none of the objects in the domain of L exists according to the possibility represented by ‹D ₂,I _2› (since ˹Ǝx(x= C)˺ is false at ‹D ₂,I _2› for any constant C in L), and that I exist according to the possibility represented by ‹D ₁,I _1› (since “Ǝx(x= AR)” is true at ‹D ₁,I _1›).

So much for considering ɑ-worlds in isolation. When they are considered in the context of as pace of ɑ-worlds, linking plays a role. So there is slightly more to be said about the possibilities that they represent. Let A be a space of ɑ-worlds and let ‹D,I› be in A. Then anything that can be said about the possibility represented by ‹D,I› in the context of A is a consequence of the following principle:

REPRESENTATION PRINCIPLE (Official Version)

1. According to the possibility represented by ‹D,I› in the context of A, p

if and only if

there is a sentence s of L ^◊ , such that s says that p and s is true at ‹D,I› in the Kripke-model based on A.

2. Let ‹D*,I*› be an arbitrary ɑ-world in A. Let ˹Ǝx ₁ …x _k Ø (x ₁ …x _k )˺ and ˹Ǝx _{1 …} x _k ϒ (x ₁ …x _k )˺ be sentences of L ^◊ which say, respectively, that x ₁ …x _k are F and that x ₁ …x _k are G. Assume that ˹Ǝx ₁ …x _k Ø (x ₁ …x _k )˺ is true at ‹D,I› in A and that ˹Ǝx _{1 …} x _k ϒ (x ₁ …x _k )˺ is true at ‹D*,I*› in A. Then:

as one is inclined to put it, some of the individuals that are F according to the possibility represented by ‹D,I› in the context of A are the very same individuals as some of the individuals that are G according to the possibility represented by ‹D*,I*› in the context of A

if and only if

one of the sequences of pairs that witnesses ˹Ǝx ₁ …x _k Ø (x ₁ …x _k )˺ at ‹D,I› in A is identical to one of the sequences of pairs that witnesses ˹Ǝx _{1 …} x _k ϒ (x ₁ …x _k )˺ at ‹D*,I*› in the Kripke-model based on A.

When A includes both ‹D ₁ ,I ₁ › and ‹D ₂ ,I ₂ ›, the first clause yields the result that, according to the possibility represented by ‹D ₂ ,I ₂ › in the context of A, there is a cellist who might have been my sister. And the two clauses together yield the slightly stronger result that -as one is inclined to put it- the individual who is a cellist according to the possibility represented by ‹D ₂ ,I ₂ › in the context of A is the very same object as the individual who is my sister according to the possibility represented by ‹D ₁ ,I ₁ › in the context of A.

A consequence of the Principle of Representation is that the possibilities represented by ɑ-worlds are not maximally specific. Suppose, for example, that the property of tallness is not expressible in L. Then the possibility represented by ‹D ₁ ,I ₁ › is compatible with a more specific possibility whereby my sister is tall and is also compatible with a more specific possibility whereby my sister is not tall. On the other hand, the possibilities represented by ɑ-worlds are maximally specific as far as the language is concerned: one can only add specificity to the possibility represented by an ɑ-world by employing distinctions that cannot be expressed in L ^◊ .

The Principle of Representation can be used to determine which properties of an ɑ-world are essential to its representing the possibility that it represents, and which ones are merely artifactual. It entails, for example, that ‹D ₂ ,I ₂ › and ‹D ₃ ,I ₃ › represent the same possibility when considered in isolation, so any differences between them are merely artifactual. In particular, the use of ‹Socrates, “nonactual”› in ‹D ₂ ,I ₂ › is merely artifactual. On the other hand, ‹D ₂ ,I ₂ › and ‹D ₃ ,I ₃ › represent different possibilities when considered in the context of {‹D ₁ ,I ₁ ›, ‹D ₂ ,I ₂ ›, ‹D ₃ ,I ₃ ›}. For whereas according to ‹D ₂ ,I ₂ › there is a cellist who might have been my sister, according to ‹D ₃ ,I ₃ › there is a cellist who couldn’t have been my sister. So the use of ‹Socrates, “nonactual”› in ‹D ₂ ,I ₂ › is essential in the context of {‹D ₁ ,I ₁ ›, ‹D ₂ ,I ₂ ›, ‹D ₃ ,I ₃ ›}. This is not to say, however, that a possibility whereby there is a cellist who might have been my sister can only be represented by an ɑ-world if the ɑ-world contains ‹Socrates, “nonactual”›. For the possibilities represented by ‹D ₁ ,I ₁ ›, ‹D ₂ ,I ₂ ›, and ‹D ₃ ,I ₃ › in the context of {‹D ₁ ,I ₁ ›, ‹D ₂ ,I ₂ ›, ‹D ₃ ,I ₃ ›} are precisely the possibilities represented by ‹D ₄ ,I ₄ ›, ‹D ₃ ,I ₃ ›, and ‹D ₂ ,I ₂ ›, respectively, in the context of {‹D ₄ ,I ₄ ›,‹D ₃ ,I ₃ ›,‹D ₂ ,I ₂ ›}.

When I speak of the possibility that an ɑ-world represents what I will usually have in mind is the possibility that is represented in the context of the space of all admissible ɑ-worlds.

6. The Dot-Notation

I would like to introduce a further piece of notation: the dot. The dot is a bit like a function that takes representations to the objects represented. Suppose I am seeing a play according to which I have a sister; applying the dot-function is like shifting my attention from the actor who is representing my sister to the character represented (i.e. my sister).

Consider the following two formulas:

For w a fixed representation, the undotted formula is satisfied by all and only objects z such that w represents a possibility whereby z is an F; the dotted formula, on the other hand, is satisfied by all and only objects z such that z is used by w to represent something as being an F. Thus, if p is a performance of a play according to which I have a sister, the actor playing my sister satisfies “[Sister(AR,ẋ)]_p" but not “[Sister(AR,x)]_p” (since the performance uses the actor to represent someone as being my sister, but the performance does not represent a scenario whereby I have that actor as my sister). And I satisfy “[ƎySister(x,y)] _p ” but not "[ƎySister(ẋ,y)]_p" (since the performance represents a scenario whereby I have a sister, but -unlike the actors and props- I am not used by the performance to represent anything).

Now consider how the dot-notation might be cashed out from the perspective of a Lewisian. Let l ₁ be a Lewisian world representing a possibility whereby I have a sister. Accordingly, l ₁ contains an individual a ₁, who is my counterpart, and an individual s ₁, who is a ₁’s sister. Now consider the following two formulas:

[Sister(AR,x)]_l1[Sister(AR,ẋ)]_l1

From the perspective of the Lewisian, no inhabitant of the actual world satisfies the undotted formula. For no inhabitant of the actual world could have been my sister; so -on the assumption that Lewisian worlds track metaphysical possibility- no inhabitant of the actual world is such that l ₁ represents a possibility whereby she is my sister. The dotted formula, on the other hand, is satisfied by s ₁, since she is used by l ₁ to represent something as being my sister.

Here is a second pair of examples:

[ƎySister(x,y)]_l1[ƎySister(ẋ,y)]_l1

The undotted formula is satisfied by me, since l ₁ represents a possibility whereby I have a sister. But it is not satisfied by a ₁ For although it is true that a ₁ has a sister in l ₁,l ₁ represents a possibility whereby I have a sister, not a possibility whereby my counterpart has a sister. The dotted formula, on the other hand, is satisfied by a ₁, since a ₁ is used by l ₁ to represent something as having a sister (i.e. me). But the dotted formula is not satisfied by me, since it is only the inhabitants of l ₁ that do any representing for l ₁, and I am an inhabitant of the actual world.

Let me now illustrate how the dot-notation works from the perspective of the modal actualist, with ɑ-worlds in place of Lewisian worlds. (A detailed semantics is given in Appendix A.) Here is the first pair of examples (where w ₁ is the ɑ-world ‹D ₁ ,I ₁ › from section 3):

[Sister(AR,x )]w ₁ [Sister(AR, x˙) ]w ₁

Since I ₁ (‘‘Sister”) = {‹‹Agustín,‘‘actual” › , ‹Socrates, “nonactual” ››}, w ₁ represents a possibility whereby I have a sister who doesn’t actually exist. Accordingly, from the perspective of a modal actualist, there is no z such that w ₁ represents the possibility that z is my sister. So, from the perspective of the modal actualist, nothing satisfies the undotted formula. The dotted formula, on the other hand, is satisfied by ‹Socrates, “nonactual” ›, since ‹Socrates, “nonactual” › is used by w ₁ to represent something as being my sister.

Now consider the second pair of examples:

[ƎySister (x,y )]w ₁ [ƎySister (x˙,y )]w ₁

Since the pair ‹‹Agustín, “actual” › , ‹Socrates, “nonactual”›› is n I ₁ (“Sister”), w ₁ represents a possibility whereby I have a sister. The undotted formula is therefore satisfied by me. But it is not satisfied by ‹Agustín, “actual” › because w ₁ does not represent a possibility whereby any ordered-pairs have sisters. The dotted formula, on the other hand, is satisfied by ‹Agustín, “actual”›, since ‹Agustín, “actual”› is used by w ₁ to represent something as having a sister (i.e. me). But it is not satisfied by me, since it is only ordered- pairs that do any representing in w ₁, and I am not an ordered-pair.

7. Inference in a Language with the Dot-Notation

The semantics fora-worlds that is supplied in Appendix A guarantees the truth of every instance of the following schemas:

Schemas 2-5 are enough to guarantee that any sentence in the actualist’s language is equivalent to a sentence in which only atomic formulas occur within the scope of “[…] _w ”. For instance, the actualist rendering of “◊(Ǝx(Phil(x)Λ ◊ (¬Phil(x))))”:

Ǝw[ Ǝx( Phil(x) ΛƎw' ([¬Phil (x)]_w' ))]_w'

is equivalent to

ƎwƎx([ Phil(ẋ)]_w ΛƎw' (¬[Phil (ẋ)]_w' ))._'

As a result, the dot-notation allows a language containing the modal operator “[…:℄ _w ” to have the inferential behavior of a (non-modal) first-order language.

8. The Expressive Power of the Dot-Notation

In this section we shall see that a suitably qualified version of the following claim is true: anything the Lewisian can say, the modal actualist can say too -by using the dot-notation.

Here is an example. The Lewisian can use her mighty expressive resources to capture a version of the following thought:

LINKING

There are possible worlds w ₁ and w ₂ with the following properties: according to w ₁, there is an individual who is a philosopher; according to w ₂, that very individual is a cellist.

It is done as follows:

Ǝw₁Ǝw₂Ǝx₁Ǝx₂(I(x₁,w₁) Λ I(x₂,w₂) Λ Phil (x₁) Λ Cellist (x₂) Λ C(x₁,x₂))

(Read: There are Lewisian worlds w ₁ and w ₂ and individuals x ₁ and x ₂ such that: (a) x ₁ is an inhabitant of w ₁ and x ₂ is an inhabitant of w ₂, (b) x ₁ is a philosopher and x ₂ is a cellist, and (c) x ₁ and x ₂ are counterparts.)

How might this be emulated by a modal actualist equipped with the dot-notation? Consider what happens when one treats the variables in the Lewisian rendering of LINKING as ranging over (admissible) ɑ-worlds rather than Lewisian worlds, and carries out the following replacements:

I(x _n ,w _n ) → [x˙ _n = x˙ _n ]w _n

Phil(x _n ) → [Phil(x˙ _n )]w _n

Cellist(x _n ) → [Cellist(x˙ _n )] w _n

C(x _n ,x _m ) → x _n =x _m

The result is this:

Ǝw₁Ǝw₂Ǝx₁Ǝx₂([ ẋ₁ = ẋ₁]_w1 Λ [ẋ₂ = ẋ₂] _w2 [Phil(ẋ₁)]_w1 Λ [Cellist(x˙₂ )]w ₂ Λ x₁ = x₂ )

(Read: There are admissible ɑ-worlds w ₁ and w ₂ and objects x ₁ and x ₂ such that: (a) x ₁ is used by w ₁ to represent something and x ₂ is used by w ₂ to represent something, (b) x ₁ is used by w ₁ to represent a philosopher and x ₂ is used by w ₂ to represent a cellist, and (c) x ₁ = x ₂.)

or equivalently:

Ǝw₁Ǝw₂Ǝx([Phil(ẋ)]w₁ Λ [Cellist(x˙)]w ₂ )

(Read: There are admissible ɑ-worlds w ₁ and w ₂ and an object x such that: x is used by w ₁ to represent a philosopher and x is used by w ₂ to represent a cellist.)

What gives the actualist’s method its punch is the fact that it generalizes: one can show that there is a systematic transformation of arbitrary Lewisian sentences into dotted actualist sentences which preserves truth-values and inferential conections.¹³(See Appendix B for details.)

The actualist’s transformation-method does not preserve meaning -where Lewisian sentences quantify over Lewisian possibilia, their actualist transformations quantify over ɑ-worlds and ordered-pairs.

But meaning-preservation is not what the actualist wants, since she doesn’t want to countenance Lewisian possibilia. What she wants is a way of enjoying the theoretical benefits of quantification over Lewisian possibilia within the sober confines of an actualist frame- work. Here are two examples of ways in which she is able to do so:

1. Firstorderizing Modal Sentences

By quantifying over Lewisian possibilia, the Lewisian is able to render any sentence in the language offirst-order modal logic in (non-modal)first-order terms. The sentence

◊(Ǝx(Phil(x) Λ ◊ (Cellist(x))))

(read: there might have been a philosopher who might have been a cellist),

for example, gets rendered as the (non-modal)first-order sentence:

Ǝw ₁ Ǝw ₂ Ǝx ₁ Ǝx ₂ (I(x ₁,w ₁ ) Λ I(x ₂,w ₂ ) Λ Phil(x ₁ ) Λ Cellist(x ₂ ) Λ C(x ₁,x ₂ ))

And ^{Lewis 1968} shows that it can be done in general.¹⁴ The (non-modal) firstorderizability of modal sentences brings two immediate advantages. The first is that it allows one to think of the inferential connections amongst modal sentences in terms of the inferential connections amongst the corresponding non-modal sentences; the second is that it allows one to read off a semantics for modal sentences from the semantics of the corresponding non-modal sentences. The actualist transformation-method allows actualists equipped with the dot-notation to enjoy both of these advantages.

2. Characterizing Intensions

On a standard way of doing intensional semantics for natural languages, characterizing the semantic value of an expression calls for quantification over possibilia.¹⁵ Oversimplifying a bit, the semantic value of, e.g. “philosopher” might be taken to be the set of pairs ‹w,z› where w is a possible world and z is an (actual or merely possible) individual who is a philosopher at w.

As emphasized in ^{Lewis 1970}, the Lewisian is able to do the job by quantifying over Lewisian possibilia:

[[“philosopher”]] = {‹w,z› : I(z,w ) Λ Phil(z)}

But the actualist transformation-method allows actualists armed with the dot-notation to follow suit, by quantifying over ɑ- worlds and ordered pairs:

[[“philosopher”]] = {‹w,z› : [Phil(ż= ż)_w Λ [Phil(ż)]_w}

or, equivalently,

[["philosopher"]] = {‹w,z› : [Phil(ż)]_w}

Since the actualist transformation-method preserves inferential connections, the actualist semantics is guaranteed to deliver the same theorems as its Lewisian counterpart. And since the transformation-method preserves truth-value the actualist’s axioms will be true just in case their Lewisian counterparts would count as true from the perspective of the Lewisian.

In particular, one can expect the actualist semantics to deliver every instance of the (world-relative) T-schema. For instance:

True(“ƎxPhil (x)”,w ) ↔ ([ƎxPhil (x)] _w )

or, equivalently,

True(“ƎxPhil (x)”,w ) ↔ Ǝx([Phil (ẋ)] _w )

(Read:The object-language sentence‘‘ƎxPhil (x)” is true at admissible ɑ-world w just in case there is an individual which is used by w to represent a philosopher.)

9. Limitations of the Proposal

A-worlds are subject to an important limitation.¹⁶ Whereas differences between ɑ-worlds are no morefine-grained than is required to make distinctions expressible in one’s language, there might be differences amongst Lewisian worlds toofine-grained to be expressed in one’s language.¹⁷

Because of this limitation, some of the metaphysical work that the Lewisian gets out of Lewisian possibilia cannot be replicated by an actualist equipped with the dot-notion. Here is an example. Lewis 1986 treats properties as sets of worldbound individuals. Up to a certain point, the actualist is able to follow suit. When the Lewisian claims that the property of being a philosopher is to be identified with the set of philosophers inhabiting actual or non-actual Lewisian worlds, for instance, the actualist could claim that the set {‹z,w› : [Phil(z˙)] _w } is to be used as a surrogate for the property of being a philosopher. But the strategy breaks down when it comes to properties making finer distinctions than can be expressed in one’s language.¹⁸

In general, whether or not the actualist’s limitation turns out to get in the way will depend on whether the job at hand calls for using possibilia to makefiner distinctions than can be expressed in one’s language. When the job at hand is a piece of semantic theorizing the extra resources are unnecessary: since a semantic theory is ultimately an effort to explain how language is used, it need not be concerned with distinctions toofine-grained to figure in our explanations. But when the job at hand is metaphysical reduction, matters are other- wise.¹⁹

Appendices