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Possible Worlds

Possible Worlds

To merely set the stage for possible worlds theory, we may say that possible worlds are whatever entities play a certain role in the quantificational analysis of the modal operators `Necessarily' and `Possibly'. The quantificational analysis of `Necessarily P' is `For all worlds w, P is true in w'; and `Possibly, P' is analyzed as `For some world w, P is true in w'. Thus the only ways of denying that there are possible worlds are (a) to reject the quantificational analysis, perhaps taking `Necessarily' and `Possibly' as primitive sentential operators, or (b) claiming that all statements beginning with the word `Possibly' are false, and all sentences beginning with `Necessarily' are vacuously true. I doubt anyone has ever taken option (b), although perhaps some have taken what might be called (b*): there is exactly one possible world, viz. the actual world, so that every truth is necessarily true and every falsehood impossible. A philosopher who takes any of these options is off the hook: she need say no more about the nature of possible worlds than an atheist need say about the nature of God. (Or, for one who takes option (b*), she need say no more about the nature of God than one who claims that God is identical with the universe.)

Possible worlds theory is thus done only by those who accept the quantificational analysis of the modal operators and who do not take option (b*).

Possible worlds are put to other theoretical uses as well. One of the most significant is in analysis of counterfactual statements: If it were the case that P, then it would be the case that Q. Such statements notoriously resist truth-functional analysis: `If Bush had been re-elected, a Republican would be president' has a different truth-value than `If Bush had been re-elected, a Russian would be president', even though the corresponding component sentences have the same truth-values. These sentences may be analysed using an ontology of possible worlds, however: `In the closest world in which Bush was re-elected, a Republican is president.'

In addition to possible worlds, some analyses require an ontology of possible individuals. An example of Lewis's: `A red thing could resemble an orange thing more closely than a red thing could resemble a blue thing'. Lewis analyses it as follows, where the quantifiers range over all possible individuals: `For some x and y (x is red and y is orange, and for all u and v, if u is red and v is blue then x resembles y more closely than u resembles v'. The analysis requires comparing individuals across worlds, thus requiring of possible individuals. This is an important point, for many of Lewis's criticisms of rivals to his modal realism rely on their inability to give an adequate account of possible individuals.

Lewis's modal realism may be seen as a conjunction of five theses.

(1) There are possible worlds and possible individuals. That is, Lewis accepts the quantificational analysis of modality. 'Nuff said.

(2) Possible worlds are "of a kind" with what we would normally call "the universe," the mereological sum of me and everything related to me in space and time. They have spatial extent, and have material objects as parts. The parts of possible worlds are possible individuals. (To use terminology that Lewis rejects as uninformative, possible worlds and possible individuals are concrete.)

(3) Possible worlds are spatiotemporally isolated from one another. They do not exist a long time ago in a galaxy far, far away; nor for that matter are they nearby. In fact, two distinct possible worlds x and y bear no spatio-temporal relation to one another; and no part of x bears any spatio-temporal relation to any part of y. [With the possible exception of immanent universals which recur as parts of more than one possible world.] So two things being spatiotemporally related is sufficient for their being worldmates, i.e. being part of the same world. Furthermore, Lewis maintains, it is necessary: thus Lewis holds that a possible world is a maximal sum of spatio-temporally related individuals.

(4) If x<SUB>1. . . x<SUB>n (allowing ourselves even infinite ns) are possible individuals, then there is a possible world in which distinct duplicates of x<SUB>1. . . x<SUB>n co-exist, size and shape permitting. Thus there is a possible world in which a duplicate of the Eiffel tower and a duplicate of the Sydney Opera House stand next to one another, and even a possible world in which there is nothing but duplicates of the Eiffel tower forever in every direction, like an infinite lattice. (Size and shape permitting: presumably there wouldn't be room for uncountably many duplicates of the Eiffel tower in one world.) This is Lewis's "principle of recombination."

(5) The modal term `actual' is an indexical term, meaning `part of the world of which I am a part' or `part of the world of which this utterance/inscription is a part'.

Many objections to Lewis's modal realism have been put forth. I will discuss what I take to be the three most interesting objections.

The first of these is the epistemic objection. This objection trades on the abstract/concrete distinction. According to this objection, some sort of causal acquaintance is required to know of the existence of concrete objects. To know of the existence of a particular donkey requires me stand in some sort of causal relation to the particular donkey: either I have seen it, or have talked to someone who has, etc. The same thing goes for knowledge of the existence of sorts of things. I might know of the existence of donkeys from having seen a donkey, or having spoken to someone who has seen them, etc. On the other hand, we need have no causal acquaintance with abstract objects such as numbers in order to have knowledge of their existence.

Lewis claims to know that there are other-worldly talking donkeys: he has modal knowledge that it is possible that there are talking donkeys, and he analyses this modal proposition as "There is a world that has a talking donkey as a part". (No need to worry with Kripke about whether these other worldly things really are donkeys: what's important is that they are intrinsically very similar to this-worldly donkeys.) But these other-worldly donkeys bear no causal relation whatsoever to David Lewis: so how can he know of their existence, since donkeys are paradigmatically concrete objects?

Lewis's reply to this objection is to deny that the distinction between abstract and concrete subject matter (insofar as we can make sense of the distinction!) is what makes the difference between knowledge that requires causal acquaintance and knowledge with objects and knowledge that doesn't. Lewis concedes that our knowledge of the existence of this-worldly donkeys requires causal acquaintance and out knowledge of numbers does not, but not because the first sort of knowledge has concrete subject matter and the second doesn't. The reason that our knowledge of this-worldly donkeys requires causal acquaintance and our knowledge of numbers does not is that our knowledge of other worldly donkeys is a knowledge of contingent facts, whereas our knowledge of numbers is knowledge of necessary facts.

Thus, whereas my knowledge that there are this-worldly donkeys requires causal acquaintance with donkeys, being a contingent matter, my knowledge that there are other-worldly talking donkeys is knowledge of a necessary matter (since what is possible is necessarily possible, at least in the broad metaphysical sense of possibility), and thus requires no causal acquaintance.

Whereas the epistemic objection is an objection to Lewis's concreteness thesis, the axiological objection, advanced by Robert M. Adams, is an objection to Lewis's indexical account of actuality. Normally, Adams points out, we do not take it that is bad that in some possible world Hitler (or his counterpart) is successful in killing all the Jews, nor do we think it is a good thing that in some possible world the Holocaust never happened. What matters, what is good and bad, is what happens in the actual world. But Lewis, with his indexical analysis of actuality, must deny this, for there is nothing special about the actual world, according to Lewis, in virtue of which only the actual events have any axiological significance. On this, Lewis bites the bullet. Lewis doesn't deny that other-worldly goods are indeed goods, and that other-worldly evils are indeed evils.

The objection may be extended, however, to one that is not purely axiological, concerning only good and evil independent of our actions, but to one that is ethical in character. What, one might ask, is wrong with committing some heinous crime, given that whether or not I do it in this world, I (or one of my counterparts) will commit it in some world, and will refrain from committing it in another world. Regardless of my actions, the same overall balance of goods and evils throughout the sum of all the worlds will remain the same: so why not commit the crime? Lewis replies that if this is a problem, it is a problem only for consequentialists of a particularly strict sort. "Only if morality consists of maximizing the total of good, absolutely regardless of where and to whom the good may accrue, can it lose its point because the sum total of good throughout the plurality of worlds s non-contingently fixed and depends not at all on what we do." Modal realism subverts only a "truly universal ethics," and this is something Lewis claims we should do without anyway.

The final sort of objection is the most common: what Lewis calls "the incredulous stare." An incredulous stare is, in itself, not an objection, but it may be indicative of a certain objection: modal realism disagrees, to a great extent, about what we would pre-theoretically take there to be. Whereas this is a legitimate objection, and does count something against the theory, it is only decisive if "common sense" is some sort of final authority in philosophy, which it is not. The power of "common sense" is imposed by theoretical conservativeness, which demands that we not alter our theories, philosophical and scientific, as little as possible when we formulate a new theory to do some sort of theoretical work. (In the case of theories of possible worlds, to state what sorts of individuals play the certain roles in quantification analyses of modality; given that this is the best way to analyze modality.) Thus if there is some other theory that works just as well as modal realism, but which has an ontology with a less extravagant ontology, Lewis is ready to embrace it. But he thinks that no such theory has been offered.

There are two sorts of alternatives to Lewis's modal realism which I will explore. Both of these theories are what Lewis calls ersatzist theories: rather than full-blown, "concrete" possible worlds, isolated in space and time, ersatzist theories attempt to give us something else suitable for playing the role of possible worlds and possible individuals in quantificational analyses of modality. Both sorts substitute full-blown possible worlds for entities which represent the world, rightly or wrongly: the actual world is the ersatz world that represents the universe as it is, and the other possible worlds are ones which misrepresent the universe in some way, or according to which the universe is such-and-such a way. Thus an ersatzer will analyze "It is possible that there be talking donkeys" as "There is a possible world according to which there are talking donkeys". The crucial difference between the two sorts of ersatzism I will explore is in how the possible worlds are purported to represent.

The possible worlds of linguistic ersatzism represent in the way that sentences, or sets of sentences, represent. Indeed, we might as well go ahead and say that the possible worlds of linguistic ersatzism are sets of (interpreted) sentences: maximal, consistent sets of sentences. Maximal, in that the description of the world given by the set is in some sense complete; every possible world must either represent that there is a talking donkey, or represent that there is no talking donkey. Consistent, in the sense that the no contradiction or other impossibility can be true according to the world; we can't have a possible world according to which 2+2=5, or according to which there are married bachelors. We needn't limit ourselves to natural languages such as English, of course; indeed we had better not, for surely there are possibilities that English cannot express, if only because there are things that have no name in English, and our possible worlds need to represent alternative possibilities for these things as well. The linguistic ersatzer would do well to use what Lewis calls a "Lagadonian language," in which sentences are set theoretic constructions having individuals as "words," each individual naming itself in the language.

One of Lewis's objections to linguistic ersatzism is that it cannot do the work that it is supposed to do, viz. provide a quantificational analysis of modality, for modality will creep into the analysis at one of two places, depending on whether the language of the possible worlds is a rich one or a poor one.

Suppose that the language of the theory is a poor one, e.g. having only names for all the space-time points and predicates that say what intrinsic state each point is in. Thus the possible world may be said to explicitly represent that such and such a spacetime point is in such and such a intrinsic state, but must implicitly represent that there is a talking donkey, by implication. But implication, at least when taken as primitive, is a modal notion, for one sentence implies another iff it is impossible for the first to in such and such a intrinsic state, but must implicitly represent that there is a talking donkey, by implication. But implication, at least when taken as primitive, is a modal notion, for one sentence implies another iff it is impossible for the first tospace-time continuum and to say that there is a talking donkey, no human philosopher could state such a definition, and thus the linguistic ersatzer must resort to primitive implication, a modal notion, in order to say how his possible worlds (that explicitly only represent the distribution of intrinsic states across space-time) represent that there is a talking donkey.

On the other hand, if the language of the possible worlds is a rich one, capable both of describing the distribution of intrinsic states across space-time, and saying things like "There is a talking donkey," the linguistic ersatzer is forced to appeal to primitive consistency, a modal notion, in order to guarantee that the possible worlds are consistent. For there are some arrangements of intrinsic states over the space-time points that are incompatible with the existence of a talking donkey; but this sort of inconsistency cannot presumably be defined syntactically or model theoretically. So the ersatzer who uses a rich language must resort to primitive consistency, a modal notion, to guarantee that there is no possible world according to which it is both the case that such and such an arrangement of intrinsic states are realized by the space-time points, and also according to which there is a talking donkey (where the arrangement is in fact incompatible with there being a talking donkey).

A second objection of Lewis's to linguistic ersatzism is that it cannot give an acceptable of the possible individuals needed for certain analyses. Just as linguistic ersatzism substitutes maximal sets of sentences for Lewis's possible worlds, linguistic ersatzism substitutes maximal sets of open sentence (having one free variable) for Lewis's possible individuals. A problem arises though with the possibility of indiscernible individuals, alike in both their intrinsic nature and their extrinsic relations. One would think it a general principle that if according to a world there are n individuals of such and such a nature, then there are n possible individuals of that nature associated with that world. But consider a world in which (or according to which) there are but two material beings, intrinsically identical: everything else is empty space. According to that world, there are two individuals of such and such a nature. But there are not two possible individuals, if possible individuals are what the linguistic ersatzer says they are: since every description true of one individual is true of the other, there is but one set of open sentences, only one possible individual.

A second sort of ersatzism that Lewis explores is what he calls "magical ersatzism": according to linguistic ersatzism, the possible worlds represent in the way that sentences do, in virtue of their syntactic structure. The possible worlds of magical ersatzism just represent, and there's an end on't.

Here's an example of how a magical ersatzist theory might go: There are these things called states of affairs, that obtain or fail to obtain, depending on how the world is. Some states of affairs imply one another: A implies B iff necessarily, A obtains only if B obtains. Some states of affairs are maximal, in that they are not implied by any other state of affairs. Necessarily, exactly one maximal state of affairs obtains. A state of affairs represents in the following way: S represents that P iff necessarily, S obtains iff P. These maximal states of affairs are the possible worlds. (The one that obtains is the actual world.)

Furthermore, to turn this into more than a mere theory-schema, the ersatazer must say something about the nature of the states of affairs. (For Lewis holds a theory conforming to this schema as well, if "state of affairs" means "set of possible worlds", "obtains" means "has the actual world as a member", "implies" means "is a subset of".) So the ersatzer will probably say that states of affairs are "abstract", and have no internal structure, set-theoretic or mereological.

The first thing to note is that the magical ersatzer, like the linguistic ersatzer, cannot analyze modality, for it has been appealed to in his definitions of "maximal" and "represents". But, the magical ersatzer might say, what's a little primitive modality if it lets you dodge those incredulous stares?

Lewis has a stronger objection to magical ersatzism. The magical ersatzer takes as primitive the relationship of obtaining that holds between the world and exactly one of the maximal states of affairs. While it's not fair to ask a philosopher to analyse what she takes as primitive, it is fair to ask her to classify that primitive: if it is a one-place predicate, is it intrinsic or extrinsic? If it is a two-place relation, is it an internal property, or an external one?

It is this last question that Lewis asks about the relationship of obtaining that holds between states of affairs and the world. Is it an internal relation, like `is larger than', i.e. does it depend solely on the intrinsic features of the relata? If so, Lewis objects that it is a mystery how we ever come to know that one state of affairs obtains and another doesn't, for to know that an internal relation holds, we must know something about the intrinsic natures of the relata. But we know next to nothing about the intrinsic natures of states of affairs as the magical ersatzer has introduced them, for all she has said is that they lack internal structure, set-theoretic or mereological, and that they are "abstract", whatever that means.

Perhaps the ersatzer will reply that the relationship of obtaining is external one instead, like a relationship of distance. But the relationship of obtaining must be a modal relation, that is, it must necessarily hold between certain states of affairs and the world, given that the world is of such and such a character. Our paradigm examples of external relations are relationships of distance, that can hold or fail to hold independently of the nature of the relata. It would be ridiculous to assert that for every x, there is a y such that necessarily y is 3 feet from x iff x is green, and 4 feet from x iff it is not green. (Greenness being an intrinsic property.) But this is exactly the sort of thing that the ersatzer asks us to swallow if she claims that the relationship of obtaining is an external one. Somehow, it is necessarily the case that if the world is has such and such an intrinsic character, it stands in the external relation of obtaining to certain states of affairs and fails to stand in this relation to others.

Van Inwagen, an admitted magical ersatzist, has responded to this dilemma with a tu quoque. The relationship of membership between a thing and a set, a relationship that Lewis accepts and that plays a crucial role in Lewis's system, is equally mysterious, van Inwagen claims. Although it is clear that membership is not an internal relation, since it is at least possible that there be intrinsic duplicates, and there would be some set containing one of these and not the other, we may ask if the relation of membership is range-internal: a relation R is range-internal iff necessarily, if something x bears R to something y, x bears R to all duplicates of y. (Being ten feet from something the same color as is range-internal, although not internal simpliciter.) Now, van Inwagen asks, is set membership range-internal or range- external?

If it is range-internal, then by reasoning parallel to that of Lewis, it seems that we could not understand the relation, since we know little about the intrinsic nature of sets. On the other hand, if membership is range-external, we may wonder how it is that sets have their members essentially, as they do according to van Inwagen. Why couldn't I have been a member of a set of which I am not actually a member?


Adams, Robert Merrihew. "Theories of Actuality," Noûs 8 (1974), 211-31. Reprinted in Michael J. Loux, The Possible and the Actual. Cornell, 1979.

Lewis, David. On the Plurality of Worlds. Blackwell, 1986.

Van Inwagen, Peter. "Two Concepts of Possible Worlds," Midwest Studies in Philosophy XI (1986), 185-213.

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