Part I. Introduction

Section 1 Generic definitions and basic modal realism

Modal assertions involving possibility and necessity are a part of our ordinary languages as well as of our philosophical patrimony. Confining ourselves to the non-modal, there are many things we could not say. We could not mark the difference between a unicorn[1], which could exist, and the square circle, which could not. Modality is a natural way of marking the difference between, on the one hand, the relation of Smith’s being a bachelor to Smith’s being unmarried, and, on the other hand, the relation of Smith’s being fifty-feet tall to Smith’s not being a mammal. Someone could not fail to be unmarried if he is a bachelor, but he could be a mammal even if he is fifty feet tall—though in fact no mammal is that tall.

It is important for ethical purposes to say what could have been done but was undone, and what would have happened were it done. It is generally agreed that in some sense of “possible” a human being can only be held responsible for an act if it was at least logically possible that he avoid it. When we say that moral worth supervenes on actions and non-moral circumstances, we are saying that it could not be the case that someone’s moral worth was different though his actions and the non-moral circumstances were the same.

When we discuss the problem of evil, we sometimes wonder whether it is possible for God and evil to co-exist, a different problem from the de facto question of whether the evils of this world make the existence of God probable or not.

When we talk of natural objects, we often cannot specify the kind that the object falls into without talking of dispositional properties, that is properties that would be actualized were circumstances different. Something might in fact live all its life just like a horse, but if it is true that were it poked in the underbelly, where in fact it never was poked, it would suddenly and naturally sprout wings and fly away, it is not a horse.

Our expressive capabilities would be greatly impoverished without can be, might be, must be, is possible, is necessary, would be and their ilk. We need these terms to talk of the reality around us. Yet, paradoxically, talk involving possibility often does not appear to be about anything real. The unicorn that is possible does not exist, I have not done otherwise than I have, and the actions and non-moral circumstances are as they are.

One way to conceptualize modal notions is to think of a “possible world”, a way (with “way” understood so broadly as not to prejudice the ontological question of what possible worlds are) that a cosmos could have been. Different possible worlds are different ways that our world could have been.

The main alternative to thinking of modality in this global sense is thinking of it in a local sense, of thinking of alternative ways that portions of this world could have been. Such piecemeal modality is what ordinary language normally engages in. When we say that Hitler might never have been born, we do not generally just mean that there is some possible world in which he doesn’t exist—e.g., a world at which the universe is and has always had an unchanging constant energy density. We mean that that portion of this world which corresponds to the birth of Hitler might not have been even though much of the rest of the world, especially at least the distant past prior to Hitler’s birth, was pretty much the same, and the laws of nature were those that we have. What exactly is to be kept fixed in this “might never have been born” claim depends on the context. Thus, while apparently speaking only of portions of worlds, the context determines what whole worlds we are speaking of, namely what portions of the actual world are supposed to be imagined as remaining in that possible scenario in which Hitler had never been born. To disambiguate our ordinary piecemeal talk of possibility we bring in whole possible worlds.

The need to talk of whole worlds is shown particularly clearly when we make counterfactual utterances. For we are wont to ask questions like: “How might or would have the course of history gone had Hitler never been born?” And on a plausible account of how to answer such questions, we should think of whole worlds in which Hitler was not born, and to say what holds in such worlds.[2] Given what our context fixes, namely most events prior to Hitler’s birth and the laws of nature, we can say certain things about what happens in those worlds. For instance, the course of events in other galaxies is the same as in the actual world—whether the awful events of the 20th century occurred or not is not going to affect what happens in other galaxies, if only because the information about them, traveling at the speed of light, has not yet arrived there. But the course of local history would have been at least somewhat different, and we can speculate about how it might have been different. Our ease in saying in the same breath that events in other galaxies would have been the same but the events here would have been different does indicate that it is appropriate to analyze counterfactual situations holistically.

Moreover, what is possible in a portion of the world may well depend on the rest of the world. For instance, what happens categorially here presumably depends on what the laws of nature are that hold. It is impossible for there to be a world with exceptionless laws of nature like ours but where things don’t fall when dropped under appropriate conditions; however, apart from such laws, it is certainly possible. It is impossible that there be unjustified evil in a local portion of the universe if there is an all-powerful, all-knowing and all-good deity in the universe. Moreover, when there is such a deity, then what evils can exist in a portion of the world may well depend on what happens elsewhere in the world, since the justification of some evil in one portion of the world may depend on events elsewhere. Our ordinary modal claims need to be contextually disambiguated, and when thus disambiguated are seen as involve whole possible worlds. Because of all this, possibility and necessity prima facie require reference to be made to whole possible worlds, and so one should try to make sense of possible worlds.

In conversation, Rescher has objected that the holistic intuitions apply to physical (or at least, I suppose, causal) and not necessarily logical possibility. However, at least we should leave open the option that logical necessity might involve some holistic aspects. For instance, suppose that the essentialist intuitions are correct that something could not be water were not certain laws of nature in place. Then the claim that it is possible that, say, it is possible for the water in a glass to fail to be gravitationally attracted to the center of the earth might be a global claim for it is a claim that there could exist certain counterfactual laws of nature, which could be global ones, and that water could logically co-exist with these laws. (Note, by the way, that the truth value of this claim is unclear.) Of course maybe laws of nature will turn out not to be global, but then specifying that they lack global modal oomph will itself be a global claim.

Given a basic notion of possible worlds, whatever their ontology, we need some correlative notions. By “the (or our) cosmos” I shall mean the aggregate (i.e., mereological sum) of all actually existing things. By “the (or our) universe” I shall mean the aggregate of all actually existing spatio-temporal things. Each world represents or corresponds to a way the cosmos could have been. In what way this representation works is open at this point of the investigation. One of the worlds shall be distinguished as “the actual world”, i.e., the world that represents the way our cosmos in fact, or actually, is. An individual “exists in” a world w if, were that world actual, that individual would exist, or, equivalently, if w represents the cosmos as containing that individual. A proposition is “true at” a world w if, were that world actual, that proposition would be true, or, equivalently, if w represents the cosmos as satisfying that proposition. Occasionally, the term “domain” will be used for the collection of all possible individuals that exist in a given world.

What the notions of “represents”, “actual”, “exists in” and “true at” really signify will depend on what our ontology of possible worlds is. There are many possible such ontologies. There is the crazy one, which nonetheless will be conceptually useful at times to keep in mind, that there necessarily is a Platonic library somewhere which contains physical books, of infinite size, each of which gives a maximal consistent description of a cosmos in some fixed language.[3] On this view, a world is one of these books. A world represents some possible way of being a cosmos if the book that the world is describes the way that cosmos would be correctly. A world is actual if everything written in it is true. A proposition is true at a world if it is expressed by some sentence in the book. An individual exists in a world if the world describes the individual as existing.

Other theories will have other renderings of the basic notions. For instance, David Lewis thinks that all possible ways that the universe could be is a way that some concretely existing universe really is. Moreover, cosmoi and universes are the same for him. Thus, worlds are concrete universes. A world represents some cosmos if it is that cosmos. The actual world is the world we inhabit. A proposition is true at a world if it truly describes a state of affairs obtaining in that world. An individual exists in a world if it inhabits that world.

A propositional Ersatzist may take a world to be a maximal collection of compossible propositions. The actual world is the collection all of whose propositions are true. A world corresponds to a cosmos by having as its members propositions true of that cosmos. A proposition is true at a world if it is a member of it. An individual exists in a world if some proposition in that world says that the individual exists.

Leibniz, on the other hand, thinks that worlds are maximally consistent ideas in the mind of God. The actual world is the idea that God has chosen to actualize. An idea corresponds to a universe by being a mental representation of it. A proposition is true at a world if it is a part of, or maybe represented by, that world. An individual exists in a world if the idea represents him as existing.

We can now give a possible worlds semantics for possibility and necessity claims. It is possible that p providing there is a world w at which p is true. It is necessary that p providing p is true at every world. Having possible worlds lets us consider “local” and “global” modalities in a uniform way. When I say “Hitler might not have existed” in an ordinary way, I am saying that the proposition that Hitler does not exist is true at some world which matches ours in various relevant respects. When I say “It is logically possible that unicorns exist”, I may just be making the claim that the proposition that unicorns exist is true at some world, without putting any restriction on which worlds are relevant here.

Some further terms will be useful. A proposition is contingent providing it is true at some but not all worlds, i.e., providing neither the proposition nor its negation is a necessary truth. An individual x is a necessary being if it exists in all worlds. An individual is a contingent being if it exists at some but not all worlds. There is no relevant sense that I am aware of in which one can say that there “are” impossible beings, so I shall not define them.[4] Occasionally, I shall use ðp and àp to mean “necessarily p” and “possibly p”, respectively.

Section 2 Metaphysical versus logical possibility?

The modality in connection with which the possible worlds are possible is what is often called “metaphysical” possibility, with the paradigmatic example being that if Kripke is right, then it is metaphysically impossible that water fail to be H₂O. Some have argued that there are in fact two different kinds of modality. Some propositions, such as the proposition that H₂O contains hydrogen atoms are logically necessary since it is logically necessary that anything that has two atoms of hydrogen and one atom of oxygen in each molecule (and that, after all, is the definition of “H₂O”) contains hydrogen atoms. But that water is H₂O is a different kind of necessity, since it is not one that follows from the logic of the terms involved.

But consider the claims:

(1) H₂O contains hydrogen atoms.

(2) Water contains hydrogen atoms.

The defender of a distinction between logical and metaphysical possibilities claims that (1) and (2) have different modal status.

As an opening gambit, the Kripkean can reply that they cannot have different modal status, because modal status belongs to propositions, not to sentences, and (1) and (2) express the same proposition, and hence have the same modal status by Leibniz’s law. The defender of the distinction between necessities can either deny that (1) and (2) express the same proposition, or claim that they differ in modal status as sentences. The latter claim I have no need to dispute, since I can simply confine my account to that of the modal status of propositions.

But in fact the claim of a variance in the modal status of the two sentences is dubious. What does it mean? That sentence (2) could have expressed a false proposition? Yes, doubtless, but so could (1): after all, it might have been uttered in a language where H₂O means “two electrons and one photon”. Nor will it do to say that (2) might have been true in our language. For in a language in which (2), or rather the proposition expressed by (2), is true, “water” was defined by pointing to some other liquid, and hence the meaning of the word is different, and hence the language is different.

One might also say that the difference between the sentences is that we can know a priori that (1) expresses a true proposition. If this is what is meant by claiming that the modal status is different, I concur, but note that the difference is epistemological not ontological. Perhaps there are truths of arithmetic we cannot know a priori. It is not implausible to suppose we have an intuitive grasp of only finitely many axioms of arithmetic, and perhaps we only have first-order resources available to us in connection with arithmetical truths. But then by Gödel’s incompleteness theorem, some truths of arithmetic cannot be known a priori by us. But it is by no means obvious that it follows that these truths of arithmetic have an ontological and modal status different from the others. Only their status relative to us is different.

Consider now the alternative of claiming that (1) and (2) express different propositions. I shall argue that nonetheless there is reason to think, if Kripke’s ostensive account of the naming of natural kinds is correct, that the two claims have the same modal status. For, suppose I point to Cicero and say:

(3) Cicero is Cicero.

(4) Tully is Cicero.[5]

(5) This is Cicero.

Is there a difference in the modal status of what these sentences express? Surely not in the case of (3) and (4). “Cicero” and “Tully” are just as synonymous as “Sh’lomo”-in-Hebrew and “Solomon”-in-English or as “automobile” and “car” are, and hence (3) and (4) evidently express the same proposition. Admittedly, it takes greater ignorance to deny sentence (3). But if one understands the meaning of “Cicero” and “Tully” in context, it involves no greater self-contradiction. Sincere denial of (4) involves a failure to grasp that in this context “Tully” and “Cicero” are synonyms. But sincere denial of (3) involves a failure to grasp that the two inscriptions “Cicero” are synonyms--which in a different context they might not be (“Aristotle [Onassis] is not Aristotle [the Stagirite]”). Thus, the propositions expressed by (3) and (4) have the same modal status.

What, then, of (4) and (5)? Surely they, too, have the same status. For the use of “This” with a pointing finger renders it into a temporary name for Cicero, no different from “Tully” except in respect of the fact that we use the same inscription “This” in connection with many more meanings than we do “Cicero”. But with no change in propositions expressed, we might subscript all our demonstratives with unique symbols, and evidently things would stand no differently with the new version of (5) (e.g., “This₁₇₃₇₃ is Cicero”) than with (4) in terms of the modal status or proposition expressed. Hence, neither does (5) differ from (4), and hence from (3), in modal status. Note that I could imagine someone denying that (3) and (4), and also (4) and (5), express the same proposition, but the claims about identity of modal status do no seem open to question.

But, then, if “water” functions as a demonstrative pointing to the natural kind of that paradigm body of water that was involved in a Kripkean baptism thereof, and if that natural kind just is H₂O, then the difference between (1) and (2) is precisely that between (3) and (5), and hence involves no change in modal status.

Therefore, if we accept a Kripkean account of natural kind names, there is no distinction between logical and metaphysical necessity that could be used to distinguish (1) and (2). But we can do justice to our intuition that there is some sort of a difference between the two sentences by adverting either to the epistemological difference or to the following distinction. Some terms in English are defined by ostension and some verbally. “Bachelor” is defined verbally as an “unmarried man”. “Water” is defined ostensively as that natural kind. For any sentence S, let V(S) be the sentence obtained from S by first replacing each unquoted word that is verbally defined by its definition, iterating as many times as possible, and then replacing every remaining item of non-logical vocabulary by an undefined logical constant, a different constant for each word defined by a different ostensive act. Then, we can say that S is verbally necessary if and only if V(S) is a tautology. Thus, (1) is verbally necessary. To see this, suppose for simplicity that “H₂O” is defined as “a chemical constituted by molecules containing two atoms of hydrogen and one of oxygen” where each non-logical term here is not itself verbally defined.

Then, V(1) is something like “A C constituted by Ms containing two As of H and one of O is a C constituted by Ms containing two As of H and one of O”, where capital variable letters are logical constants, which is a tautology. But “water” is not a verbally defined term, so V(2) is “W is a C constituted by Ms containing two As of H and one of O”, which is plainly non-tautologous. So we can do justice to both the intuition that there is a difference in the logical status of (1) and (2) and the argument that the propositions they express have the same modal status. Note that the same approach will show a distinction between the logical status of (3) and (5). V(3) is “C is C” while V(5) is “X is C”, so that only the former is verbally necessary. Interestingly, if “Tully” and “Cicero” are independently bestowed names, neither being verbally defined, V(4) is “T is C”, and hence (4) is not verbally necessary.

I shall not use the concept of verbal necessity further. It depends too much on historical accidents, such as whether a second name was defined expressly a synonym for the first or was independently ostensively bestowed. These are important issues for the philosophy of language, but have little ontological significance in them for the structure of possible worlds or the modal status of propositions. I will talk of logical necessity, necessity simpliciter and metaphysical necessity as synonymous, for I do not think useful ontological distinctions can be made between them. None of these necessities are verbal. They are all “real necessity”, to use Kant’s term.

Section 3 S5

The modal logic assumed through most of this dissertation is S5, i.e., a logic satisfying the axioms:

(6) ð(p É q) É (ðp É ðq)

(7) ðp É p

(8) àp É ðàp,

for all propositions p and q, and where à is short for “it is possible that” while ðp is short for ~à~p, together with the “rule of necessitation” that if a formula is an axiom or theorem, then that formula prefixed by ð is also an axiom or theorem.[6] This system is known as S5 and is characterized by an accessibility relation that is reflexive, symmetrical and transitive.

The most controversial axiom here is (8) which says that if something is possibly true, then it is impossible for it to fail to be possibly true. In modernity, the axiom goes back at least to Leibniz’s discussion of Descartes’ ontological argument. Descartes had defined God as a being that has all perfections, and one perfection is the property of necessary existence. Leibniz noted that Descartes’ argument was missing a crucial premiss, namely that it was possible for God to exist, and argued that once that premiss was added, the argument became valid. To show the essential use of S5, simplify the argument by supposing simply that God by definition a necessarily existent being. Then, Leibniz’s point is that once one adds the premiss that it is possible for God to exist, then it follows that God exists. For, then àð(God exists), which implies ð(God exists). But this implication is of course just an application of the contrapositive of (8).

Now, there certainly are kinds of modality for which (8) fails. For instance, suppose we consider a forward branching temporal structure, and say that p is possible at some point z in the structure providing p is true at z or at some future point that can be reached from z. Then, p is necessary at z providing that p does not fail at z or at any future point that can be reached from z. Then, it is possible (here and now) that I will at some point in my life run a marathon. But it is certainly not necessary that this is possible, because there is a future I can reach where my legs are cut off before I run a marathon and at a point in the future of that accident there will no longer be any reachable points at which I run a marathon.

One way to argue for (8) in our setting, however, would be to start with two intuitions. The first is that things could not be have been such that it would have been impossible for things to have been as they in fact are. However things might have gone, it still would have been true that they might have gone the way they in fact have gone. If things could have gone a certain way, then had they gone that way it would have been true that they could have gone the way they in fact went. This is the Brouwer axiom: p É ðàp. It tells us that the accessibility relation is symmetric.

The second intuition is that we when we talk about metaphysical possibility, we are talking about “ultimate” possibilities. Now, if we have a possibility operator à such that àp can hold without ààp holding, then this operator does not tell us about ultimate possibilities. If it could have been that it could have been that p was true, then there is a real sense in which p could have been true. If we then deny that àp, we are saying that à does not tell us of the ultimate possibilities there are, but of possibilities relativized to some way that things have been. Indeed, in such a case there is a reasonable more ultimate possibility operator, namely àà. Thus, if we are talking of ultimate possibilities, it is reasonable to require that ààp should imply àp. This is the S4 axiom; it tells us that the accessibility relation is transitive.

But of course the Brouwer and S4 axioms, together with (6), imply (8). (Just apply the Brouwer axiom to àp to conclude that àp É ðààp; then use (6), S4 and the rule of necessitation to conclude that ðààp É ðàp.)

Alternately, one can argue that broadly logical possibility cannot have been different, since it is a matter of what propositions follow from what propositions (a proposition is possible if and only if its negation does not follow from it), and what follows from what could not have been different. Therefore, if àp, then it could not have been the case that ~àp, i.e., àp É ðàp.

The S5 system of modal logic will be in the background for most of this dissertation. It is worth noting that the most prominent views of possibility to be considered, namely the Lewisian and ersatzist ones, are such as to leave little room for the denial of S5. The theistic account sketched at the end of the dissertation will also be such. One could thus give the following argument for S5: the best metaphysical accounts of possible worlds that we have require S5. However, I shall not give this argument here. Instead, I shall feel free to use S5 in my arguments for and against various metaphysical views of possibility, on account of the plausibility of S5 holding in the case of a notion of ultimate possibility.

Section 4 Six kinds of views on possibility

4.1 Parmenides, Spinoza, Leslie and Rescher

In his poem On Nature, Parmenides learns from the goddess that there are only two

ways of enquiry that are to be thought of. The one, that [it] is and that there is no non-being [ouk esti mê einai], is the path of Persuasion (for she attends upon Truth); the other, that [it] is not and that it needful that there be non-being [esti mê einai], that I declare to you is an altogether indiscernible track: for you could not know [gnoiês] what is not [to ge mê eon]—that cannot be done—nor indicate it.[7]

What is there to be said and thought must needs be: for there is being, but nothing is not [esti gar einai, mêden d’ ouk estin].[8]

The argument, insofar as it is more than just an assertion, is that non-beings plainly do not exist, and while if we speak and think, we are speaking of something.

We can put this argument in a more modern form by thinking about the controversial theory of the truthmaker of a proposition, a theory that I will in fact assume in a number of the arguments in this thesis. Realism requires that propositions be made true by something real. The proposition that there are horses is made true by the horses of this world. The proposition that Socrates is sitting is made true by Socrates’ sitting, or the sitting Socrates qua sitting. The item in the world that a proposition is made true by is called its truthmaker. What exactly the truthmakers of propositions are depends on one’s ontological system. For instance, if one is committed to an Aristotelian worldview on which all there is are substances and their attributes, broadly construed, then the truthmaker of every true proposition will ultimately be substances and their attributes. An event ontology, on the other hand, may have the truthmakers be mereological sums of primitive events. But whatever the truthmakers are in one’s ontology, in the case of propositions giving concrete facts about concrete entities, the truthmakers are made up of concrete things: tables, chairs, dogs, cats, sittings, shoutings, or the like.[9]

Moreover our language provides a way of giving the truthmaker of a proposition in a way that is neutral between ontological systems. To every declarative sentence there corresponds a participial nominalization. To “Socrates is a philosopher, was a war hero and taught Plato” there corresponds “Socrates’ being a philosopher, having been a war hero and having taught Plato”. To “Brutus betrayed Caesar” there corresponds “Brutus’ having betrayed Caesar”. To “There are horses” there corresponds “There being horses.” If a sentence expresses a proposition, then the denotation of its participial nominalization is the truthmaker of that proposition. But what kind of an item “Brutus’ having betrayed Caesar” denotes, whether it is ultimately a complex ultimately of substances and their attributes, or of events, or a fact in a world that is all that is the case, this is left open.

In any case, then, a proposition is true if and only if it has a truthmaker that really exists. This gives us a sense we can attach to Parmenides’ cryptic remarks. If we know or speak truly, there must be an object of our knowledge or speech, namely the truthmaker of the proposition we know or express. It is this object that we know or speak of. The assertion that we cannot know or speak of what is not, then, becomes the claim that if we are to be right, there must be something we are right about: something that makes our affirmations true. Where the truthmaker is not, neither is there anything true.

Of course the notion of a truthmaker is going to be pointless unless we have some substantial theory about what kinds of entities can play that role. I can always say that the truthmaker of p is just its being the case that p, and if I do this for every true proposition, every true proposition will have a truthmaker in an apparently trivial way.[10] However, saying this would not be so trivial. It is after all a substantial ontological claim that there are such things as its being the case that p. But this trivialization of the truthmaker theory does show that a criticism that some theory cannot provide a truthmaker for some proposition is short hand for an argument that we are not satisfied with just this trivial truthmaker for the proposition. One way to be dissatisfied is if one has certain ontological intuitions that do not fit with the idea of its being the case that p being a basic entity not to be further reduced. For instance, if one is an Aristotelian who thinks that all there are substances, their modifications and their relations, one will insist that truthmakers be nothing but substances, their modifications and their relations. This kind of an Aristotelian picture will in fact underlie a number, though not all, of the truthmaker-based arguments I will give. Moreover, it is in general preferable in a philosophical theory of some proposition p that one be able to say more about the truthmaker of p than that it is its being the case that p. Being able to say more about this truthmaker is itself a reason in favor that theory. Thus, even if we do not want to insist that always more can be said, we will ceteris paribus prefer a theory that says more.

While Parmenides did not deal with modality per se, we may be able to find in his writings an argument against change, which argument easily generalizes to an argument against modality.

And how could something that is [to eon] be in the future? How could it come to be? For if it came into being, it is not: nor is it if it is ever going to be in the future.[11]

A claim about the future must be made true by a truthmaker that is in the future. But then there is nothing now by which the claim is to be made true. Hence, when we say something will be, we are perforce speaking of something that is not, and thus not speaking truly.

The argument construed in this way may be criticized by a B-theorist for conflating existence simpliciter with merely present existence, but it is much more interesting in the modal case. We can say that the proposition that there will be a sea-fight tomorrow is made true by tomorrow’s seafight, which exists simpliciter. But there is a much deeper problem in the case of modal propositions. What makes true assertions of mere possibility? Suppose no sea-fight in fact occurs tomorrow. What, then, makes true the proposition that there can be a sea-fight tomorrow? If there will be a sea-fight tomorrow, then maybe the sea-fight that is tomorrow can make the proposition reporting its future occurrence true now. So, in parallel with this, the sea-fight that is merely possible maybe makes true the proposition that there can be a sea-fight. But this will not do, because “is merely possible” is truth-canceling in a way that “is tomorrow” is not. A merely possible sea-fight is not anything that exists. If it is not anything that exists, it cannot make anything true. But what else could the assertion that there can be a sea-fight be about, one asks, other than the future sea-fight?

Parmenides, not having a clear notion of modality, merely claims that his one reality is atemporally unchanging. But he could have used the same arguments to arrive at the further claim that this one reality must be as it is and can be no other, and doubtless if he were asked the modal question clearly, he would say this. And this is the Parmenidean puzzle of modality. It comes as a paradox and a problem. It seems that the proposition that there can be unicorns is, if anything, about unicorns—its truthmaker would have to be comprised, at least in part, of unicorns or their existing. Thus, its truthmaker does not exist, there being no unicorns and no existing of unicorns, and so the proposition is false. But it is paradoxical to admit that only the things that are could be.

If we are to avoid this paradox, we need to explain what the truthmakers of modal propositions are, and what it is about these truthmakers that makes them suitable to be such. Otherwise, we cannot have any realistic theory of the truth of modal propositions. It will be the purpose of this thesis to attempt an answer to the problem. Moreover, the attempt will be made within the confines of a broadly Aristotelian ontology, where the basic entities are substances and their modifications (properties and relations), an ontology which thus will tend to be unfriendly to the idea that such entities as the state of affairs of it being possible that unicorns exist could be primitive. However, although the intuitions behind this kind of an ontology drive much of the project, they are not presupposed by the individual arguments that will be given for the preferred answer to the Parmenidean problem and against the non-preferred answers.

Parmenides actually has two arguments against change. The second argument is an invocation of the Principle of Sufficient Reason (PSR) that says that every true proposition has an explanation. If what is should have come to be, then Parmenides asks:

[W]hat need would have driven it later rather than earlier, beginning from the nothing, to grow?[12]

This argument presupposes that the existent must have come from the non-existent, and Aristotle will later dispute this presupposition. But on this presupposition, the argument is a very interesting one. Later, St. Augustine in his Confessions would ask in the same way why God created the world when he did and not any earlier, and supply the answer that there was no time prior to the creation of the world so the question is malformed. Augustine’s answer is also a good reply to Parmenides’ argument for the eternity of the world.

But now that the PSR is on the table, one can use it to formulate a similar argument against alternate possibilities. Everything that is has a reason why it is the way it is. For a reductio, suppose that there are contingent propositions, and consider the conjunction C of all of them. This conjunction then has an explanation. This explanans is a proposition that is either contingent or necessary. But a necessary proposition cannot give a complete explanation of a contingent proposition, since the necessary proposition is just as true at possible worlds at which the contingent proposition fails and hence cannot supply the complete explanation of the contingent proposition. Hence, the explanans must be a contingent proposition. But then the explanans is one of the conjuncts in the explanandum C. Hence the explanans must be capable of explaining itself, which is absurd.

This is the PSR-based argument against modality. The argument can be attacked at three points. First, one may reject the PSR, in which case the argument has nothing to stand on. I shall not take this route for the very good reason that the theory of possibility that I shall end up endorsing as the only reasonable candidate theory of possibility entails the PSR. Secondly, one may allow that there be necessary propositions that can explain contingent propositions. After all, the explanans need not explain the explanandum, so that the mere fact that the explanans holds in worlds where the explanandum does not is no problem. That Smith developed paresis is explained by his having syphilis and by certain statistical laws of nature, but these laws and the proposition that he has syphilis do not entail that he will develop paresis. Or, more to the point, if the libertarians are right, then that Smith made a free choice between alternatives A and B might well be a complete explanation of Smith having done A. If this is so, then it is epistemically possible that should there be a necessarily existent God who necessarily chooses between possible worlds to create, and the proposition that God chose between worlds to create might explain the existence of this world. Or, thirdly, one might allow that there are contingent self-explaining propositions. The only candidate for a contingent self-explainer that I know of is the claim that some person freely chose something.[13]

The PSR-based argument against modality goes hand-in-hand with a deterministic view of physical explanation, since such a view would make plausible the claim the explanans entails the explanandum, which I have rejected above. Something like this PSR-based argument is found in Spinoza’s argument for the necessity of everything. It is essential to Spinoza’s argument that for any thing, there must be a cause, namely ultimately the action of God, that determines not just that the thing is but the precise manner in which it is. For, otherwise, given the PSR and the insistence that the explanans entail the explanandum, the thing itself would have to explain why it is in the manner it is rather than in another possible manner, which explanation it cannot supply, Spinoza thinks. [14]

Recently, John Leslie and Nicholas Rescher have defended views that entail that there is only one possible world. Leslie proceeds through an “axiarchic principle”, or principle of ethical requiredness (Leslie, 1997; Leslie, forthcoming). This principle corresponds to Plato’s Form of the Good, and imposes on the world the necessity of satisfying certain conditions that make it be for the best. One argument for this principle would be through the considerations that fall under the head of “the anthropic principle” (cf. Leslie, 1990). The constants in the laws of nature (masses and charges of elementary particles and strengths of basic forces) appear to be calibrated in such a way as to make life possible. If they were somewhat different, and physics gives us no reason to think they could not be, life like we know it would not be possible. This provides evidence for the axiarchic theory, in that if the axiarchic theory is true, such finetuning is unsurprising, while if it is false, it is more surprising. However, obviously, this also provides evidence for other alternate theories, such as traditional theism, or Carter-type theories that claim that there is an infinite number of universes so it is probable that in some life would exist and those ones are the only ones that we can observe.

Rescher, on the other hand, has argued for a metaphysically necessary principle of optimality as a theory that explains why we find orderly laws of nature that can be mathematically formulated and understood by us. This principle ensures that, necessarily, things are for the best, understood in a Leibnizian sense as a balance between variety and lawlike unity. Of course, there are other theories that, if true, explain the same explanandum. Theism provides one such theory.[15] Another would be a more limited version of Rescher’s theory that merely claims that the laws of nature are necessarily for the best, while at least some of the contents of the world are contingent. This more limited version does not overturn modality, and rather than counting as a general view of possibility simpliciter, I shall simply take it to be a view about what possibilities are in fact there.

Rescher’s view in its unlimited form appears to be subject to the following objection. First of all, if there is only one possible world, then saying that our world is the best of all possible worlds is not saying anything interesting. One could say with equal propriety that it is the worst of worlds. Consequently, the optimality principle cannot explain why the laws of nature are orderly, because if, per impossibile, the only possible world were one where they were disorderly, that world would also be the best.

Rescher’s own reply (personal communication) is to distinguish a notion of logical possibility from a notion of metaphysical possibility. There is more than one logically possible world, and of these the best one is the one that is metaphysically necessary. One might make non-trivial sense of the claim that the one and only possible world is optimal, e.g., by considering worlds that are metaphysically impossible recombinations of things in this world (cf. Armstrong, 1989) but nonetheless are modeled by mathematically coherent structures and hence capable of comparison to our world. But then the problem of evil shows its ugly head. The argument from evil against the existence of an omnipotent, omniscient and omnibenevolent deity, difficult enough as it is, takes a particularly difficult form if it is claimed that this world is in fact not just worthy of being made by such a God, but is the best conceivable world. Even if we could answer the original argument from evil,[16] claiming that this world is the best one is a yet further task. Moreover, the evidence from the apparent non-optimality of this world weighs against the evidence from the lawlike orderliness of the world. And there are theories that are better supported by the conjunction of these two pieces of evidence than Rescher’s full theory: e.g., the more limited theory that says that the laws of nature are necessary and optimal, but the events in the universe, including freely done human actions, are contingent.[17]

Rescher also has another argument, not against possibility simpliciter, but simply against the existence of possible worlds. That argument is based on the impossibility of us individuating possible worlds, and I shall discuss it in due course in Part V.

4.2 Leucippus, Democritus, Meinong, Lewis and Aristotle

One extreme holds that merely possible worlds do not exist in any way, because our world is the only possible one. The other extreme view is that all possible worlds must exist. Leucippus’ and Democritus’ atomism could be an early representative of this view.

Leucippus holds that the whole is without bound…part of it is full and part void… Hence arise unboundedly many [apeirous] worlds, and are resolved again into these elements.[18]

If one takes the “unboundedly many” in the most extreme sense as involving all possibilities, then indeed we do get a view that all possible worlds exist.

This view would have been of interest merely to historians were it not for Alexius Meinong and David Lewis. Meinong sought to explain the intentionality of thought by invoking objects that correspond to all of our ideas, even those ideas that in our world are not exemplified. Thus, there are some things that don’t exist.

David Lewis does this, too, at least for possible objects, but further organizes the things that don’t actually exist into worlds. More precisely, Lewis posits that every possible world exists, and that these worlds are ontologically on par with one another. What makes two entities be a part of the same world is that they are spatio-temporally related. Thus, the different worlds are not spatio-temporally related, presumably unlike the worlds of Leucippus and Democritus. But, because of the ontological parity thesis, as in Leucippus and Democritus, the worlds are concrete entities just like our world, albeit ones that we could not possibly come in contact with. The horses in the other worlds are horses in exactly the sense in which the horses we know are, except that they are not spatio-temporally related to us.

Material reality is for Lewis much richer than we normally think. There exist dog-headed “men”, and chimeras and unicorns—but not in our world. Fortunately, most of our language is relativized to our world, the actual world, which for Lewis is set apart from other worlds only indexically: the actual world is nothing but the mereological sum of all things that are spatio-temporally related to us. When I say, speaking ordinarily, that there are no unicorns, I mean that no unicorns are actual, that the actual world does not contain unicorns, i.e., according to Lewis that no unicorn is spatiotemporally related to us. In ordinary non-modal language, quantifiers are restricted to the speaker’s world.

But of course, and this is the point of the theory, we can also speak with unrestricted quantifiers. Thus, we can translate the assertion “Unicorns are possible” to say “There is a world w such that unicorns exist in w”, involving ourselves in a quantification over all worlds. These sorts of quantifications give sense to modal language. Moreover, Lewis believes his theory of possible worlds makes it possible to give an account of various other philosophical notions. Thus, a proposition is a set of possible worlds—those worlds that it is true at—and a property is a set of individuals, with the set being allowed to extend beyond one world if desired.

This theory is elegant, solves many problems and appears coherent. But why should we think it true? Why should we think that reality is so much richer in material objects than we had thought? Lewis’s answer follows in the footsteps of Leibniz’s answer to Lady Masham’s worry about Leibniz’s system. Lady Masham wrote:

But it appears not yet to me that [your system] is more than a Hypothesis; for as Gods ways are not limited by our conceptions; the unintelligibleness or inconceivablness by us of any way but one, dos not methinks, much induce a Beleefe of that, being the way which God has chosen to make use of.[19]

Leibniz replied inter alia with the following methodological observation:

Pour y joindre mes remarques, suivant vos ordres, je diray (1) qu’il semble que c’est quelque chose de considerable qu’une hypothese paroisse possible, quand toutes les autres ne le paroissent point, et (2) qu’il est extrement probable qu’une telle hypothese est la veritable. Aussi at-on tousjours reconnu dans l’Astronomie et dans la Physique, que les hypotheses les plus intelligibles se sont trouvées veritables enfin: comme par exemple celle du mouvement de la terre, pour sauver les apparences des Astres … .[20]

The very fact that a theory gives a coherent account of difficult problems, where other theories have failed, is evidence for its truth, in philosophy as in science. Thus, Lewis thinks, we should believe his theory because it is elegant, solves many problems and, Lewis thinks, appears coherent.

It should not come as a surprise that such a drastic revision of the account of what we think there is as Lewis provides carries with it counterintuitive consequences. For instance, as we shall learn in Part III, were we to believe Lewis, we would have to become inductive sceptics and revise basic moral notions. This price would be too high. For just as the fact that a theory gives a coherent account of one thing provides evidence for the theory’s truth, likewise the fact that the theory gives rise to seemingly absurd consequences elsewhere, e.g., in issues of induction or morality, gives evidence against it. The theoretical benefits of Lewisian possible worlds theories will be critically considered in Section 5. If another theory can be found that has all or most of the benefits that survive this critical examination, but lacks the demerits of contradicting induction or morality, then that other theory is to be preferred. In Part VI it shall be argued that there is such a theory.

One final theorist who uses the same strategy as Lewis for grounding modality in a larger totality of existent things should be mentioned. This is Aristotle. The inclusion of Aristotle here may seem surprising, but in fact in Aristotle we find two different threads of thinking about modality. One of these threads involves a modal logic based on time. A proposition is necessary if and only if it holds at all times and possible if it holds at some time. Observe that in order to avoid the absurd consequence that all true propositions are necessary, we need to take an analysis of indexical sentences which makes a sentence like “It is now noon” express one and the same proposition at different times, which one proposition is true at noon but false at other times. (This is not the only way one could analyze indexical sentences. We might take “It is now noon” to express a different proposition at each different time during which it is uttered, and there is support in ordinary language for this.[21] But if we did this, then any proposition, however seemingly contingent, will always have the same truth-value at all times, and hence on Aristotle’s theory will either be necessary or impossible, which is absurd.)

This Aristotelian theory is actually very similar to Lewis’s. Just as in Lewis’s theory, modalities are analyzed in terms of quantifications over things that have the same kind of reality as the things we meet. The dog that will be born in ten years has the same kind of reality as the dogs that we meet today, just as the dog in another Lewisian world does. It is also true on Aristotle’s theory that the difference between mere possibility and actuality is indexical. The actual is what is now, just as for Lewis the actual was what is here, i.e., in our world. There is, however, a difference here. Aristotle does not see, as far as I can tell, the full ontological parity between other times and the present that Lewis sees between other worlds and ours.

And there is a single objection that can be made both against Aristotle’s theory and against Lewis’s. Both theories share a crucial feature with the accounts of Parmenides, Spinoza, Leslie and Rescher. The whole of reality could not be different than it is. In the case of Lewis’s theory, the whole of reality, i.e., the mereological sum of all universes, is fixed. There are no semantic resources in his theory for making it possible for this sum to be different. For were it possible for this sum to be different, it would be different at some world, whereas all worlds are parts of the same total reality. In the case of Aristotle’s theory, contingency is only possible in propositions that change in truth-value. Hence, a proposition that reports in a timeless way the sum total of what happens over time would be, for Aristotle, necessary—though Aristotle is apparently unaware of such propositions.

But surely the sum total of reality could have been different. I am not making the controversial claim that it could have been radically different, i.e., that it could have lacked all the things it has[22], but only the more modest claim that it could have been different in some respect. If Lewis is right, there are infinitely many universes.[23] But it seems quite coherent to suppose that there was in fact only one. This coherence Lewis must reject as merely apparent. Similarly, it is quite coherent to imagine the possibility that in fact there never was any change, a possibility Aristotle must, and does (in Metaphysics L.6), reject as merely apparent. Both Lewis and Aristotle thus go against common sense modal notions. Lewis, however, can argue that the theoretical benefits of his theory are worth it. But if an alternate theory were found which had the same benefits and fewer paradoxical conclusions, then Lewis will be the first to prefer it (cf. Lewis, 1986a, p. 5).

4.3 The linguistic view

Possible worlds have much theoretical value. Thus, it would be nice to have them without paying the price of Lewis’s extravagant ontology. One suggestion that has been made in many forms (see, e.g., Roper, 1982 and Jeffrey, 1983) is that possible worlds can be taken to be maximal sets of compossible sentences. A proposition holds at a world if it is entailed by the propositions expressed by the sentences that the world consists of. The actual world is the world all the sentences of which are true. (This does not mean that all the sentences uttered “in that world” are true. We need to distinguish between two senses of a sentence s being in a world: first, in virtue of s’s being a member of the set of sentences that are in that world, and, second, in virtue of there being a sentence that is “in” that world in the first sense which claims that s is uttered.) In doing this, we ontologically commit ourselves to sentences, which we already probably believe in, and sets, which have various theoretical benefits and which Lewis, too, needs in order to reap all the benefits of his possible worlds theory. The price is low.

Of course one needs to be more precise about what one means by sentences. It will not do to limit ourselves to actually uttered sentences. That would lead to the absurd conclusion that were there in fact no speakers, nothing would be possible. But if we speak of possible sentences, then our account of possibilia becomes circular, since we were supposed to be clarifying the ontological status of possible individuals. Fortunately, there is a simple solution to this dilemma. By “sentences” we mean types of sentences. Now, there is no great ontological extravagance in positing such types. Human-type languages can, to a good approximation, be reduced to sequences of discrete symbols, and types of sequences of symbols can easily be modeled set-theoretically. So this account in fact needs nothing more in the ontology beyond set theory. Since we may well want set theory for independent reasons, this is a very cheap price.

But, Lewis has argued, you get what you pay for, and we shall see in Section 2 of Part IV that we don’t get enough. If we use an actual language, we have the problem of alien properties: basic properties for which our language has no words, but which are instantiated at other possible worlds. But if we use a non-actual language, then we need to have some way of specifying what that language is, and that is impossible for us unless by specifying the language we create it, thereby contradicting the fact that it was supposed to be non-actual. Moreover, unlike Lewis’s account, this account does nothing to illuminate the meaning of modal propositions, because it presupposes modality in the requirement that we talk of maximal consistent sets of sentences, whereas of course a set of sentences is consistent if and only if the conjunction of the sentences is possible. I shall also press against the linguistic view an arbitrariness objection. There are many actual and infinitely more non-actual languages. Which language is to be the privileged one? In Section 2.3 of Part IV, I shall argue that no solution to this problem is satisfactory.

4.4 The propositional primitive modality view

An approach that escapes the concern about the arbitrariness and limited expressiveness of linguistic representations is through abstract propositions. For various theoretical purposes, it is useful to introduce entities knows as abstract propositions which are what our sentences express. Two sentences are synonymous if and only if they express the same proposition. Moreover, it is the proposition which is the carrier of truth, because truth does not belong to a sentence, which is language relative, but to the language-invariant proposition expressed by the sentence.

Now that we have introduced propositions by ostension as entities which sentences express, we can speak of the whole collection of propositions. Not all members of this collection are expressed by some sentence actually uttered. Nor even are all the members actually expressible by some sentence of a human language. This is so because propositions are something that is invariant not just between actual languages, but between actual and possible languages. So whatever can be expressed by any possible language is a proposition. And, conversely, any possible proposition p can arguably be expressed by some language, e.g., by the language that uses the sequence “All mimsy were the borogoves” to mean p. Note, however, that we escape the circularity objection because we are not actually defining propositions in terms of possible languages. We are defining them by ostension in terms of our language. But then we realize, on theoretical grounds, that propositions have a life of their own going beyond our actual language, rather as the electrons we posit on theoretical grounds to explain some actual phenomena have a life of their own and possess dispositional properties not exhausted by the actual circumstances of this world. Admittedly, the kind of explanatory role the two serve is different: electrons play a causal role while Platonic entities such as propositions do not. But nonetheless the propositions do explain various facts about sentences and propositional attitudes.

The above should have given us a grasp of the notion of a proposition. Propositions, moreover, enter into logical relations. This is so because logical relations between sentences are supposed to be invariant under paraphrase, and hence must be reducible to logical relations between propositions. We can therefore talk of propositions being consistent or not. Now we can define a possible world: it is simply a maximal consistent collection of propositions, assuming there are such maximal collections (I shall argue in Section 3 of Part IV that this assumption is in fact justified). Or, alternately, we can define a possible world as a class of logically equivalent maximally strong propositions, where a proposition is “maximally strong” if it entails every proposition compatible with it. These approaches have been championed most notably by Robert M. Adams and Alvin Plantinga.

Of course both of these definitions presuppose modality whether in the notion of consistency or in that of entailment, and so we will not get a reductive Lewis-type analysis of modality. But possible worlds may still be a useful construct to have, even if they do not give such an analysis. In Section 3 of Part IV, I shall argue that approaches to modality along these lines fail to answer the Parmenidean objection to modality. Presumably, on this account, the truthmaker of the proposition that it is possible that there are unicorns is the having of some property by the proposition that there are unicorns. But how does the having of some abstract property by some abstract proposition relate to the possibility of there being unicorns? I shall argue that this is an insoluble problem if one limits oneself to the resources of this theory.

Moreover, there is a mystery as to the ontological status of propositions. What are ideas that no one is thinking of? Are they substances? What sorts of substances are they? Someone who is not enamored of Platonism will shrink from propositions.

But on the positive side, propositional and linguistic approaches both avoid the paradoxes that plague Lewis’s theory.

4.5 Aristotle again and branching

While one of Aristotle’s notions of modality was seen to be unsatisfactory, there is also another implicit in his work to choose from. Parmenides was worried that change involved something’s coming to be out of nothing. For when A comes to exist, then earlier A did not exist. To answer this concern, Aristotle developed his tripartite account of change. There is a substance, a form and a privation. In the case of generation, the substance goes from having a privation of a form to having that form. Thus, a man may go from having a privation of beardedness to having a beard. But the beard does not come from nowhere. Rather, the man at the beginning of the process of change was potentially bearded, though actually clean-shaven. The privation that he had was a potentiality for beardedness.

On this account, there is something in the substance which can be identified as a potentiality for the alternate states of the substance. If we further accept Aristotle’s general thesis that potentiality is grounded in actuality, we have to say that there is something actual in the substance in virtue of which that substance can change. But this account not only helps to solve the Parmenidean puzzle about change—it may also help with the Parmenidean puzzle about modality. Even if I never grow a beard, it is true to say it is possible for me to grow a beard because there is in me and in the environment around me something in virtue of which the growing of a beard is possible, i.e., a power (of course further scientifically analyzable) in the hair-follicles on my chin to produce hairs together with the capability, not restricted by the environment, for refraining from shaving. The truthmaker, on an Aristotelian account, of the proposition that it is possible for me to have a beard is to be found in these worldly actual powers and capabilities.

As an account of modality in general, this is insufficient. For one, at first sight it only applies to local de re modalities. This approach will not give us possible worlds in any obvious way. Moreover, the account is not reductive, since it accounts for modality in terms of ability, and ability is a modal term. However, while ultimately not reductive, the account is illuminating. For in ordinary language, the notion of ability is arguably more basic (cf. Place, 1997), and from it general notions of possibility are obtained by extension. We have personal knowledge of ability, e.g., in the way Kant outlines in the second Critique through recognizing ourselves as morally responsible for an evil act and thus as having been capable of doing otherwise. There is also less mystery about capability than there is about modality in general since capabilities are actual properties of actually existing things, and so the account is indeed helpful. And at the very least, if this approach worked, it would reduce modal talk in general to a particular subset of it.

Aristotle’s account when generalized in a moral global way may lead to branching theories of modality (see, e.g., Mackie, 1998). When a substance has more than one alternative before it, these alternatives can be thought of as presenting a world-branch—though unless we want to make Lewis’s move of making all worlds concretely existent, we should not think of there concretely existing worlds corresponding to all branches. If we look at all modality as induced by such branchings, so that we see a proposition as possible if it is true somewhere on the full tree and necessary if true everywhere, then we will in fact be generalizing Aristotle’s account of the change in a single substance.

There is, however, still a problem: in fact, the same problem as was the decisive consideration against Aristotle’s temporalized approach. It is, it seems, possible for all events in time to have been different—while there might be necessary entities such as mathematicals, there surely are no necessary temporal events. This intuition will be a metatheoretical constraint on theories of possibility, and one that it seems difficult to accommodate in a branching theory—for where should the branching point corresponding to this possibility be put?

4.6 Leibniz

Leibniz gave life to the notion of possible worlds. On his view, God necessarily exists, and possible worlds are maximal self-consistent ideas or concepts in God’s mind. One could also talk of these worlds as maximal self-consistent thoughts entertained by the divine mind, and this would for all practical purposes be equivalent since any such thought corresponds to the concept of a truthmaker of that thought, whereas any maximal self-consistent concept corresponds to the thought of this concept’s being instantiated.

Leibniz had in fact given an argument for the existence of God from the existence of necessary truths, and hence from the existence of modal truths (since assertions of necessity and possibility are necessary truths by S5, which Leibniz accepts). Necessary truths, Leibniz argues, must be grounded in some reality, and the only reality Leibniz can see as capable of this is a necessarily existent mind. Of course, the argument leaves much for discussion. Why can’t the necessary truths be grounded in the thoughts of different minds in different worlds? Why can’t they be self-subsistent in a Platonic way?

Positing divine ideas as possible worlds gives one the benefits of the linguistic and propositional theories of possible worlds. Like the linguistic theory, this approach allows something that we all have some ordinary pre-theoretical understanding of, the ideas of a mind, to constitute the collection of possible worlds. Admittedly, mystery is introduced by the fact that these are the ideas of a divine mind and by the maximality involved in them. But at least we do not have the dark mystery of the Platonic propositions, whose ontological status is almost completely opaque—for what, indeed, does it mean to be “abstract”? And like the propositional theory, the possible worlds have a representational power that a merely human-like language will not.

Of course, the theory does carry the ontological commitments of theism. But it is not revisionary of our ethical and epistemological notions in the way that Lewis’s theory will be seen to be. Moreover, there is independent evidence for these commitments, namely all the evidence for theism.[24]

Unfortunately, Leibniz’s account fails to answer the Parmenidean worry. Granted, the proposition that such and such a world is possible is an idea in the mind of a necessarily existent God. But what makes this idea true if the world in question is non-actual? Where is the idea’s truthmaker?[25] And, in virtue of what is the idea that is identical with a possible world self-consistent?

In Part VI, I shall argue that the limitations of Leibniz’s approach can be precisely supplied by the merits of the Aristotelian approach in the previous section. The resulting theory will be the most satisfactory account of the nature of possible worlds and the meaning of modal propositions.

But first we need to consider impossible worlds. Then, why possible worlds are useful at all. Then, we will consider Lewis’s approach. Next, in turn, we will consider linguistic and propositional approaches. Next, I will reply to arguments of Rescher against the very existence of possible worlds. And finally I will argue for and sketch the new Aristotelian-Leibnizian theistic view. The exposition of this view will, necessarily, be an outline needing further elaboration.

Section 5 Impossibilia

But before turning to the detailed consideration of possible worlds, something needs to be said about impossible ones. It has been argued (see, e.g., Lycan, 1991), that philosophical and explanatory utility arguments made for possible worlds also apply to impossible worlds. However, as follows from the analysis in Part II, none of the applications of possible worlds that survive critical scrutiny are ones that would give one reason to posit impossible worlds. The application most cited in this connection is that of using possible worlds to analyze beliefs and propositions, since after all people often believe impossible things. Indeed, I know that the conjunction of all my beliefs is logically inconsistent because I believe that the conjunction of all my beliefs is false, which belief is logically inconsistent with that conjunction![26] However, the application of possible worlds talk to beliefs and propositions will be seen to fail, and hence will give no reason to posit impossibilia.

The two surviving applications are the analysis of modality, for which possible worlds suffice, and the analysis of counterfactuals. One might try to make an argument for impossible worlds on the grounds that they allow for an analysis of per impossibile counterfactuals. However, per impossibile counterfactuals are so contextually bound that it is by no means clear that a unified account of them is possible or even desirable. One might think that a sufficient condition for a genuine subjunctive conditional to be true is that the antecedent entails the consequent, and per impossibile counterfactuals fail this criterion, since an impossible proposition entails all propositions.

But other than saying that impossible worlds are not needed, is there any argument for why there in fact are not any such worlds? David Lewis has argued that in his setting there is a simple such argument. For Lewis, the “at” in “p is true at w” only restricts the scope of quantifiers in p and ensures that expressions referring to “the actual world” get evaluated indexically with w in place of “the actual world”, and similarly for cognates. Consequently,

(9) (not-p) is true at w

entails:

(10) not-(p is true at w).

But it is plain, then, that if there is a world w at which p and not-p are both true, then it is also true that (p is true at w) and not-(p is true at w) which is a contradiction not relativized to any world and hence unacceptable by anybody’s account. One might think that this works only for logical impossibility, but since I have argued that there is no substantial distinction between logical and metaphysical necessity, by the same token there are no impossible worlds.

Of course, the Lewisian account only works because his worlds are concreta like the actual world. If our worlds are books in a heavenly library, then there is no contradiction at all in the fact that one book might contain a sentence expressing p and another sentence expressing not-p. So there is no logical contradiction in supposing impossible worlds.

There is, however, a serious disanalogy between possible worlds and impossible worlds. A good account of possible worlds will closely relate possible worlds to the ontological ground of alethic possibility, i.e., it will be closely related to an answer to the Parmenidean problem of what we are talking about when we make objectively true assertions about possibilities. But there is no parallel claim to be made for impossible worlds because arguably we are not talking or thinking of anything when we talk or think about impossibilities. If one thinks that a thought counts as an of-X conception if and only if were X to exist, the conception would be a conception of the X,[27] then the notion of a conception of something contradictory is not viable.[28]

This does not deny that one can describe impossible worlds, and one can produce a logic for handling the contradictions within in them in a way that prevents every description from being true of those impossible worlds. But not every description corresponds to a concept that has intentionality. I would submit that when we appear to be talking of impossible worlds, we are either talking about possible worlds in disguise (e.g., “A world with non-mammalian horses is an impossible world” can be paraphrased as “No possible world has non-mammalian horses”) or else we are talking about words, with it being completely up to us how we assign truth values to our assertions. There are presumably many possible ways of individuating impossible worlds, and many different logical systems for handling contradictions in a controlled way, each yielding different truth value assignments.

A parallel statement can be made for possible worlds as well, of course, so this argument is not yet complete. There evidently are many systems of modal logic and many ways of individuating possible worlds. However, statements about possibility and necessity matter to our ordinary life and our ordinary concepts in a way in which I submit statements about impossible worlds, except when paraphraseable into statements about possible worlds, do not matter. For instance, it matters ethically speaking whether there is a possible world in which a child who was actually conceived through rape was not conceived through rape, since knowing the fact of the matter about this should affect, perhaps though only to a very small degree, the child’s attitude towards her mother’s rapist (see Section 4.2.1.b of Part III for more discussion of how modality matters). We would like to know what is the objectively true fact about many issues dealing with possible worlds.

I would conjecture, and arguing for this conjecture in detail would go beyond the scope of this thesis, that any true assertion about impossible worlds can either find a truthmaker within the totality of possible worlds or is an assertion whose terms do not have the requisite connection with aspects of our ordinary language where we have reason or need to suppose there is objectivity, and hence is an assertion whose truth value is to be assigned by convention. If this conjecture is correct, then in an ontological study of what underpins the truth of statements about possible worlds, we have no need to worry about impossible worlds. Though it is worth noting in passing that even if the conjecture is false, there is still some hope of making sense of impossible worlds in the Aristotelian-Leibnizian setting I will sketch in Part VI, though there is none at all in Lewis’s.

Part II. Applications and pseudo-applications

Section 1 Modality

1.1 Box and diamond

The most obvious application of the theory of possible worlds is, of course, to furthering the understanding of modal claims. This can be either as a useful logical device to make it easier to grasp complex modal assertions and to express assertions that cannot be otherwise expressed, or one may more ambitiously see possible worlds theories as giving an analysis of all modal claims. Whether one can take the more ambitious approach or not depends on whether one’s construction of possible worlds presupposes modality or not. If it does, then obviously possible worlds cannot provide an analysis of all of modality.

The only serious account of possible worlds that does not appear to presuppose modality and hence that supports the more ambitious use of possible worlds is that of David Lewis. Unfortunately, I shall argue in Part III that (a) it, too, must presuppose some modality (see Section 4.2.1.b of Part III), and (b) it leads to too many paradoxes for it to be at all acceptable. I am not claiming that having counterintuitive consequences is enough to refute a view; but the sheer number and weight of these in the case of Lewis’s system is enough. Just as Lewis’s case for his account is a cumulative one based on the multiplicity applications, my case against his account is a cumulative one based on the multiplicity of serious paradoxes.

The general way in which modal claims are expressed in terms of possible worlds is by quantifying over all worlds: for instance, ðp holds if and only if "w (p is true at w), while àp holds if and only if $w (p is true at w). But the expressive power of possible worlds goes beyond box and diamond operators as Lewis (1986a, Section 1.2) claims and Melia (1992) proves.

1.2 The global nature of modal claims

Moreover, as mentioned in Section 1 of Part I, the notion of a possible world is correlated with our intuition that even the box and diamond modalities have a global component. To tell whether some proposition is possible, one has to have some idea about whether it could be made to fit into a story of a whole world. Of course, if one has Humean intuitions that a world is made up out of parts such that any part is compatible with any other part, considerations of shape, space and time permitting, then this is not so important, since to tell whether a proposition is possible it is then only enough to examine the putative local state of affairs that would make the proposition true, and to decide whether this is possible.

However, there is good reason to reject a view of possibility that does not have the resources for discussing global possibilities of some sort. Many ordinary language modal claims are of an apparently local nature and for disambiguation require globalization. If I say “I might have been a physicist”, I am apparently making a local claim: this proposition, that I might have been a physicist, is logically possible. But this claim is not the one I mean to express. I do not mean, for instance, that there might have been a world radically different from ours in which every person has innate knowledge of the laws of physics from conception. That world is not a part of the truthmaker of the claim I mean to express with “I might have been a physicist.” Rather, I mean to say that much of this world might have been as it is, with me having roughly the mental capacities I do, and yet with me being a physicist. It is this claim that is non-trivial and interesting. But it is a claim that requires one to talk of worlds as a whole rather than piecemeal of the possibility or necessity of an isolated proposition. For one is asserting that there is a possible world that matches the actual world in such-and-such respects, but in which I am a physicist. And to disambiguate the claim, I will have to point out, perhaps contextually, what those respects of world-match are, and this is just to specify the set of possible worlds that I am quantifying over in the claim.

Nor will it do to avoid possible worlds by just taking a proposition that describes the way the possible worlds I am talking about are supposed to be (contain me, have such-and-such laws of nature, have such-and-such a history prior to my conception, etc.), and say that all I am claiming is the compossibility of this proposition with the proposition that I am a physicist. For the possible worlds that I am quantifying over when claiming “I might have been a philosopher” cannot be described in a finitary way, for to do so would be to describe most of the history of the actual world, since all the history prior to my conception is arguably supposed to be fixed by the “might have been”. Of course, one might believe in a rich store of propositions, among which there is an infinitary proposition that describes all those features of the actual world that I wish to keep fixed. But if one has such a belief, then presumably likewise there will be an infinitary proposition that describes all the features of the actual world. And since the actual world should not be taken to be exceptionally fortunate vis-à-vis the expressive power of propositions and since propositions have necessary existence, likewise for any possible world there will be an infinitary proposition that describes all of it. But possible worlds under the name “consistent infinitary propositions describing all of a world” smell as sweet, or as ontologically heady, as those under the name “possible worlds”: so one hasn’t avoided possible worlds.

Moreover, there is good reason to believe essentialism is true (some of the reasons will be discussed in 4.2.1.b of Part III), or at least good reason to have a theory of possibility that at least has the expressive capability for making sense of essentialist claims—one does not want to rule them out from the outset. If Kripke is right, then, for instance, the claim that horses are possible involves a claim about a possibility of certain laws of nature, which is a global claim—a whole universe satisfying such-and-such laws is claimed to be possible. Or, even without essentialism, we can find a theological case. It is surely possible that there be a God, where “God” is defined as an abbreviation for the definite descriptor “the unique perfectly benevolent, all powerful and all good creator of all other concrete beings.” But then the proposition that it is possible that God and some evil exists could well be a global claim about a possible world: it is a claim that there is some world which on the whole has properties that justify God in allowing this evil.

Both in the essentialist case and in the theological case, it is very natural, then, to consider modal claims as bound up with quantifications over whole worlds (or large parts of them, but that would be no gain, since a world itself is, trivially, a part of itself).

1.3 Supervenience

As another example, we can define the notion of supervenience using possible worlds: A-type states of affairs supervene on B-type states of affairs (the locus classicus being the claim that goodness supervenes on natural facts—see Hare [1964, p. 80ff]) if and only if any two worlds which are indistinguishable in respect of B are indistinguishable in respect of A. David Lewis (1986a, Section 1.2) has argued that in fact such claims cannot be expressed with ordinary box and diamond operators. If so, then possible worlds are indeed a useful tool.

Of course one could also do the same thing with quantifications over “aspects” and occurrent states of affairs:

(11) ð("a"b((a is an A-type state of affairs and a obtains and b is the B-aspect of the actual world) É ð(a does not obtain É b does not obtain))).

However, if we take (11) to be an analysis of the claim that A-type states of affairs supervene on B-type states of affairs, we have not gained anything over a possible worlds analysis. For we have admitted to our ontology complete aspects of worlds, and after all there is a one-to-one correspondence between a world and the collection of all of its aspects.

1.4 Transworld comparison

One might wish to define the notion of, say, x’s a being an entity than which no greater is possible or of a picture being such that no picture can be uglier or the like. I shall confine myself to the more hallowed Anselmian case. As Lewis (1970) has demonstrated, the notion of maximal greatness is prima facie ambiguous. At the least, one could reasonably understand it as claiming one of the following:

(12) "w"y((y exists in w and x exists in w) É (y is not greater in w than x is in w))

(13) "w"y((y exists in w) É (x exists in w and y is not greater in w than x is in w))

(14) "w"y((y exists in w) É (y is not greater in w than x is in the actual world)).

And it is in fact (14) that is the best interpretation in an Anselmian context. Of course if one allows oneself quantification over greatnesses, then one can do without possible worlds even in (14), just as if one allows quantification over aspects, one can do without possible worlds in analyzing supervenience. Thus, (12)–(14) are respectively logically equivalent to:

(15) ð(x exists É "y(y is not greater than x is))

(16) ð("y(x exists and y is not greater than x is))

(17) "g(g is the greatness of x É ð("y(y does not have greatness exceeding g))).

Note, however, that introducing quantifications over greatnesses or aspects is moving to a second order logic. Given possible worlds on the ground level, one can do all this in first order logic. Moreover, one may plausibly argue that (14) is not only easier to understand than (17) but is closer to what is meant by the assertion that nothing greater than x is conceivable. For, it seems more natural to say that we are comparing not greatnesses but individuals-in-respect-of-greatness.

Section 2 Counterfactuals and causality

Perhaps the bigger feather in the possible worlds theorist’s cap is the Lewisian analysis of counterfactuals:

A counterfactual “If it were that A, then it would be that C” is (non-vacuously) true if and only if some (accessible) world where both A and C are true is more similar to our actual world, overall, than is any world where A is true but C false.[29]

How one measures similarity of worlds may depend on the context, though Lewis does have a preferred measurement method. Thus, I, though not Lewis, will allow that in some context one might weight similarity in the past more strongly than similarity in the future: When I say “Were I to eat this piece of rotten bread, I would be sick”, the worlds where the past differs from the past of my world are too relevantly dissimilar from the actual world to count, just as the worlds in which the laws of nature are different are too dissimilar. (Lewis thinks he can do without something like temporal weighting, but it shall be seen in Section 3 that this is not so.)

One might think that the amount of arbitrariness possible in defining similarity of worlds makes this definition unacceptable. On different accounts of similarity, different counterfactuals will come out true. But, surely, there is an objective context-free matter of fact whether some counterfactual is true or not.

However, there is no such context-free matter of fact in general. Take the joke: “Were Queen Victoria alive today, what would she be doing? Clawing at the inside of her coffin.” Consider two worlds, both of which have laws of nature more or less like ours, except that in each a miracle occurs: in one, Queen Victoria today comes back to life, and in another she never died. If we weight similarity in the past heavily, then the world in which Queen Victoria tomorrow comes back to life is closer to ours. And in some contexts we do need to weight the past more heavily: specifically, when we are making a rational decision between actions and considering counterfactuals of the form “Were I to f, A would result”. On the other hand, if we weight similarity more in terms of closeness of laws of nature, then it is arguably a lesser departure from the actually holding laws of nature to suppose Queen Victoria had never died than to suppose her coming back from the dead after her body has been rotting for many years.

The only reason I am aware of for thinking that counterfactuals should be context-free would be a conditional principle of bivalence that claims that for any p that is possible and any q, either were p true, then q would be true or were p true, then q would be false. But this conditional principle of bivalence is false. It is neither true that were the moon made out of cheese then it would be made out of blue cheese nor that were it made of cheese then it would not be made of blue cheese. For further discussion of conditional bivalence, see Section 3.1 of Part IV.

The counterfactual account provided by Lewis is thus not invalidated by the contextuality of the measure of similarity. However, Lewis’s account of causality fares worse as we shall see in Section 3. For if we think that causality is an objectively existing relation in the world with explanatory oomph, then we will be much less inclined to accept a largely context-dependent analysis of causality. (Though of course it is open to say that there are distinct but related senses of the word “cause” in the way that, say, Aristotle talked of “four causes”.)

Section 3 The direction of time

David Lewis in his 1979 paper “Counterfactual Dependence and Time’s Arrow” (Lewis, 1979a) has argued that according to his possible worlds analysis of counterfactuals, “backtracking” counterfactuals of the form “If event A were to happen at t_A, then event B would happen at t_Bwhere t_B precedes t_A”, are usually false if B does not actually happen at t_B. On the other hand, there are plenty of such counterfactuals true with t_B following t_A (such as: “Were I to drop the glass now, it would hit the ground at some point in the future”). This time-asymmetry, Lewis claims, follows from his possible worlds analysis of counterfactuals despite the fact that this analysis of counterfactuals is entirely time symmetric. The asymmetry is, however, a contingent fact about the arrangement of this universe. Lewis argues, further, that this asymmetry gives meaning to the common notion of the future as “open” and the past as “closed”—even if determinism both of the future by the past and of the past by the future were true, which for the purposes of the analysis he assumes and which assumption I shall accept (only) for the purposes of the argument.

Much of the argument of Lewis’s paper is a reply to an objection against his analysis of counterfactuals. I shall argue that this reply succeeds in some interesting special cases but fails in others. There are many common events that do not exhibit the kind of asymmetry Lewis argues for—indeed, enough of them to ensure that Lewis’s analysis of what it is that constitutes the difference between the openness of the future and the closedness of the past fails.

But most seriously, I shall argue that any asymmetry Lewis finds, if there is one, is actually grounded in the preselection in the kinds of events that tend to figure as antecedents of ordinary language counterfactuals. This preselection I shall suggest is based on the common-sensical notion that it is past events that are the causes of future ones, with it almost[30] never being the other way around. Hence, the asymmetry that Lewis finds through his analysis is parasitic on people’s time-asymmetric intuitions. Therefore, Lewis’s analysis fails to give independent objective grounding for the counterfactual arrow of time.

3.1 Lewis’s argument

Recall Lewis’s account of counterfactuals:

It does not matter for the purposes of Lewis’s account of time asymmetry whether worlds are concretely existing physical objects (as Lewis of course thinks they are), or whether they are to be understood in some “ersatz” way as maximal compossible sets of propositions or in the Leibnizian way as ideas in the mind of a God.

Lewis’s notion of counterfactuals is, of course, practically useless without a measure of similarity of worlds. Moreover, it gives rise to the following objection stated by Kit Fine and also supported by a number of other people[32]:

The counterfactual “If Nixon had pressed the button there would have been a nuclear holocaust” is true or can be imagined to be so. Now suppose that there never will be a nuclear holocaust. Then that counterfactual is, on Lewis’s analysis, very likely false. For given any world in which the antecedent and consequent are both true it will be easy to imagine a closer world in which the antecedent is true and the consequent false. For we need only imagine a change that prevents the holocaust but that does not require such a great divergence from reality.[33]

To get out of Fine’s objection, Lewis proposes a measure of similarity of worlds that has four factors ranked as follows:

(1) It is of the first importance to avoid big, widespread, diverse violations of [physical] law.

(2) It is of the second importance to maximize the spatio-temporal region throughout which perfect match of particular fact prevails.

(3) It is of the third importance to avoid even small, localized, simple violations of law.

(4) It is of little or no importance to secure approximate similarity of particular fact, even in matters that concern us greatly.[34]

These factors are rigged to make sure that Lewis gets the right answer to Fine’s objection. One might of course have serious objections to these four factors and/or to their mutual ordering.[35] But even if they are implausible, it would be very impressive if Lewis could derive a time-asymmetry from them and from his definition of counterfactuals, since (1)–(4) are clearly time-reversal symmetric as is the definition of the counterfactuals.

Lewis’s argument against Fine is then as follows. We need to evaluate Fine’s counterfactual that if Nixon had pressed the button, the world would have been blown-up. Lewis exhibits four different kinds of possible worlds where Nixon pressed the button at t:

(i) In worlds of the kind of w₁ (and I shall sometimes for short talk of just the world w₁ rather than of kinds) everything happens as in the actual world w₀ until shortly before time t, but then the worlds begin to diverge.

The deterministic laws of w₀ are violated at w₁ in some simple, localized, inconspicuous way. A tiny miracle takes place. Perhaps a few extra neurons fire in some corner of Nixon’s brain.[36]

And so Nixon presses the button and the nuclear holocaust follows. No further divergences from law happen, but the miracle is necessary given the assumption of determinism if the pasts of w₀ and w₁ are to coincide.

(ii) In worlds of the kind of w₂, on the other hand, physical laws are never violated, and Nixon presses the button. However, both the past and the future are different, because of the assumption of bi-directional determinism. At no time is w₂ the same as w₁.

(iii) Then, in worlds of the kind of w₃, two small miracles, i.e. violations of the natural laws of the actual world, happen. First the same kind of miracle as in w₁ happens. But then a second miracle prevents the nuclear holocaust from stopping. However, the world is already very different. Nixon will write different memoirs, the wire has heated up, the movement of the finger has changed the gravitational gradient in China, etc. Indeed, it is plausible that nowhere in the whole of the future light cone with apex at the first miracle will the universe be exactly the same.

(iv) And, finally, in w₄-type worlds two miracles also happen. The first of these is the same as in w₃, but the second is a lot more impressive than it was in w₃. Not only does the nuclear holocaust not happen, but all the traces of the button pressing are removed, and so after the second miracle, w₄ looks just like w₀.

Which of these four kinds of worlds is closest to ours? To have an answer to Fine, Lewis must argue that it is w₁, since it is only there that the nuclear holocaust happens. Now, w₁ is definitely closer to our world than w₃ by Lewis’s criteria, because the only advantage of w₃ is that it lacks a nuclear holocaust in its future and hence there is more approximate future agreement between w₃ and w₀ than there is between w₁ and w₀. And indeed this agreement is merely approximate in the future light cone with apex at the event that caused the pressing of the button. However, w₃ has an extra small miracle, i.e. deviation from physical law. Avoiding small miracles is Lewis’s third most important similarity factor. Therefore, w₃ is better in terms of the fourth most important factor, and w₁ in terms of the third important factor, and so w₁ is to be preferred to w₃ as a candidate for a closest world.

What of w₄? It is true that w₄ matches w₀ in a very large spatio-temporal region: all of the future of the second miracle and all of the past of the first. This is Lewis’s second factor. However, w₄ must have a rather large miracle. The gravitational gradient has to be corrected throughout a large region of space. The particles shifted around (admittedly by a tiny distance) in China by the change in gravitational gradient caused by Nixon’s hand-movement towards the button have to be shifted back. Nixon’s apparent memories have to be altered. The wire has to be cooled. The vibrations from the click of the button have to be stopped from propagating. This would violate the first criterion for closeness of worlds in a way that w₁ does not. Hence, w₁ is closer than w₄.

On the other hand w₃ is just about nowhere in space-time identical with our world. The lack of exact match anywhere in w₃ means that w₃ violates the second criterion of closeness, whereas w₁ only violated the third by having a small miracle. Hence, indeed, w₁ is the closest of the worlds (or, more precisely, types of worlds) in which the button is pressed—at least if no other worlds are candidates which for the nonce I shall grant Lewis, though in Section 3.4, below, I shall argue that there is an important candidate that Lewis has passed over. And since the nuclear annihilation of humankind does happen in w₁, it follows that Fine’s counterfactual “If Nixon had pressed the button, there would have been a nuclear holocaust” is indeed true on Lewis’s account.

Moreover, the analysis does display a past-future asymmetry. For, given that the closest world is w₁, it follows that counterfactuals of the form “If Nixon had pressed the button then C would happen at t_C”, where in the actual world C does not happen at t_C, can only be true if t_C is after to the time of the “small miracle” in w₁, which time is slightly before pressing the button. So, as Lewis admits[37], there may be a modest amount of backtracking in the counterfactual—but only back to the time of the miracle.

What grounds the above analysis is the fact that an event like the pressing of a button has a lot of disparate effects—but a fairly localized cause. It is this temporal disanalogy that, on Lewis’s account, grounds the counterfactual arrow of time that gives meaning to our intuitions about the openness of the future and the closedness of the past. In Sections 3.3 and 3.4, however, we shall see that this analysis is hopelessly flawed because of the failure to consider a fifth class of worlds. But first consider a different counterexample.

3.2 The pulled plug

(C) Suppose that in our world, in fact Nixon has pressed the button. However, Captain Smith working in a computer room saw that the doomsday device was being activated, and he pulled out the plug of the computer that controls the doomsday device, thereby saving the world from certain destruction. Moreover, whereas Nixon’s decision to press the button was a highly conflicted one such that there was a single neuron which happened to fire, and had it not fired the decision would not have been made, Captain Smith was a man of high moral caliber who had many times thought to himself what he would do if the doomsday device was activated, and had gone over mental scenarios of pulling the plug, so that his decision to pull the plug was highly overdetermined. Captain Smith had many reasons to pull the plug, and many neurons fired in his brain, any one of which would have been sufficient to make him pull the plug. Moreover, many different muscles fired up simultaneously, any one of which would have been enough to pull the plug (he pulled with both hands while kicking at it with a leg), and since he had a time window of several minutes, if that move had not succeeded, he could be expected to have tried again and again.

Now, consider the obviously true counterfactual: “Were Captain Smith not to have pulled the plug, the world would have been blown up.” But on Lewis’s account, to affirm the counterfactual on the grounds on which Lewis affirms Fine’s counterfactual would require that the closest world to the actual one is a world where a miracle happens in Captain Smith’s brain, which miracle prevents him from pulling the plug. But given the structure of Captain Smith’s brain, the miracle would actually be a pretty big one. Not only would it have to prevent him from pulling the plug at the time he did, but also at all other times in the time window. Such a thing would require a fairly large scale reorganization of Captain Smith’s brain. Call a world where this happens w₁’ and denote the world that (C) describes as “our world” by w₀’.

But now consider the alternate world w₅. This is the world in which a small miracle happens in Nixon’s brain. This small miracle prevents from firing that one neuron whose firing was necessary for the pressing of the button. In w₅’, the urge to press the button might come over Nixon, but quickly passes. It was never a very decisive urge anyway. And not surprisingly, in w₅’, Captain Smith doesn’t pull the plug—because he has no reason to do so!

But now it can be argued that w₅’ is closer to w₀’ than w₁’ is. For, w₁’ involves a much greater miracle than w₅’ does, namely a complete reorganization of Captain Smith’s brain. It is true that w₁’ matches w₀’ exactly spatio-temporally for a while longer than w₅’ does. However, this is only for a while—more precisely, for the amount of time between Nixon’s pressing of the button and the time when Captain Smith’s brain gets reorganized in w₁’. And Lewis, we know, is willing to allow worlds to mismatch for a short while in order to ensure that the miracle that happens is a much smaller one. In the original case of Fine, after all, the miracle might have happened a fair amount of time prior to the pressing of the button. For, maybe, Nixon had in the actual world never even gone near the button, so in order to avoid the large miracle of transporting him bodily to the button, a modification of his brain when he stood a distance away from the button, in order to get him to walk towards it would have to happen in w₁, so that w₁ and w₀ would have been mismatched for a significant amount of time prior to the pressing of the button. But if Captain Smith’s reflexes are quick enough, then the plug might have been pulled quite soon after Nixon’s deciding to press the button and further one can imagine that Nixon was hovering indecisively over the button, so the miracle needed in his brain could have been very soon before the button’s pressing. So the amount of time of mismatch between worlds brought about by having the miracle happen in Nixon’s rather than Captain Smith’s brain could be small. And given how much smaller the miracle would be in Nixon’s brain than the miracle that would be needed in Captain Smith’s brain, w₅’ will be closer to w₀’ than w₁’ will be.

But if so, then Lewis’s counterfactual analysis, assuming (following Lewis’s discussion of Fine’s case) that nothing closer than w₁’ and w₅’ can be found, implies that the counterfactual “Were Captain Smith not to have pulled the plug, the world would have been blown up” is false, since its consequent is false in w₅’. Thus Lewis’s analysis fails here.

However, note that there is still a temporal asymmetry. For, although the counterfactual backtracks to Nixon’s decision, which is surely in this case too far, it does not backtrack very far back, whereas it forwardtracks arbitrarily far: in the future there would always be different consequences of Nixon’s not having pressed the button (e.g., history would look at Nixon very differently) than of Nixon’s having pressed it but Captain Smith’s having saved the world.

It might be objected that “Captain Smith not pulling the plug” is an illicit event description since it is the complement of an event. However, Lewis intends his account of counterfactuals to give a counterfactual relation between propositions, not events, and hence the objection is one that he cannot make. Moreover, Lewis must allow propositions reporting the complement of an event as antecedents of counterfactuals, because after all he proposes to analyze the sentence “event A caused event B” as “A and B occurred, and were A not to have occurred, then B would not have occurred”, where the italicized phrase is precisely a counterfactual with a negative antecedent. Thus, the objection under consideration is one that Lewis cannot possibly make.

3.3 The button on the laser

Consider another simple example. For a long time, an activated laser has been sitting pointed into the sky. In the actual world, at time t₀ the moon happens to be where the laser is pointed, and a short while later, at t₁, a place on the moon is illuminated with the light emitted by the laser at t₀. The laser is equipped with a switch which has the property that when it is quickly depressed and released, the laser beam gets turned off.

Our counterfactual is: “Were the button to quickly move to the depressed state and then to the released state at t₀, the spot on the moon would not be illuminated at t₁.” I shall abbreviate “to quickly move to the depressed state and then to the released state” as “to be pressed”, but with it being understood that no inference from “the button is pressed” to “someone pressed the button” is thereby warranted. It may well be that the button gets pressed without a person doing it (e.g., by a sudden rise in air pressure over it). Indeed, I shall suppose that persons are far away from the laser at t₀ so any pressing in a counterfactual world that does not involve a big miracle of bringing a person will be a non-personal pressing.

Let w₀ be the actual world, and let w₀* be its time-reverse. Let t₀* and t₁* be the points in the reversed time of w₀* corresponding to t₀ and t₁ respectively, so that t₁* is strictly earlier than t₀*. In w₀* we have various light rays converging on our spot on the moon at t₁* (corresponding to light rays that were scattered off that spot at t₁ in w₀), and then being focussed precisely into a coherent beam arriving at the laser at t₀*. Likewise, heat radiation precisely converges on the laser at that time. And then after t₀* the button on the laserbeam is pressed in and released. A possible mechanism by which this button moves could be that various influences such as the sound waves, mechanical vibrations and heat in w₀* corresponding to what was given off in the pressing of the button in w₀ combine to reverse the motion. Such a story can be given because we’re working ex hypothesi with time-reversible laws of physics.

But now consider a world w₁* which matches w₀* precisely up to shortly before t₀*. Then a miracle happens: of itself, the button depresses and releases at t₀* (note that the time reversal of the button being pressed is the button being pressed, since “being pressed” was defined as consisting of a depression followed by a release). And after that the laws of nature continue their usual course. Thus, in w₁* as in w₀ various light-rays converge on a spot on the moon at t₁*, and then are focussed into a coherent beam pointed at the laser. This coherent beam arrives at the laser at t₀*, which happens to be shortly after the button was miraculously pressed.[38] Likewise, heat radiation converges on the laser at that time, as do the various sound waves and mechanical vibrations mentioned in the description of w₀*. What happens when all these things meet the laser at t₀* is difficult to predict. Perhaps the light converging on the laser beam scatters off somewhere from the turned-off laser. But it does not really matter for our purposes what exactly happens.

Now, let w₁** be the time reverse of world w₁*. What exactly happens in w₁** prior to t₀ we cannot really say. It will, however, be different from what happens in w₀. But what matters is that in w₁** the spot on the moon is still going to be illuminated at t₁ even though the button was depressed at t₀. How will this spot be illuminated despite the laser being turned off at the relevant time? Presumably the mechanism will be roughly as follows. In w₁*, we had a beam of light converging from the moon to the laser, and then scattering off the deactivated laser all around. Thus, w₁**, we will first have light from all around the laser converging on the laser at just the needed angles so as to be reflected off the deactivated laser into a coherent moon-pointed beam. And shortly after t₀, w₁** matches the actual world.

Thus, w₁** perfectly matches the actual world throughout all of the future of t₁ but has a small miracle in it which presses the switch at t₀. Prior to this miracle, there is a mismatch between w₀ and w₁**.

Lewis’s analysis of Fine’s counterfactual, as applied here, would produce a world w₁ which is like w₀ until shortly prior to t₀, but then the button is miraculously depressed, and the laws of physics then again come into effect, ensuring that the spot on the moon is not illuminated at t₁. In order to uphold the truth of the counterfactual that “were the switch pressed at t₀, the spot on the moon would not be illuminated”, Lewis has to argue that w₁ is closer to w₀ than w₁** is, since in w₁** the counterfactual’s antecedent is verified while its consequent is not. But there is no reason to think w₁ is closer than w₁** to w₀. Both worlds contain a small miracle. World w₁ matches ours exactly in the past, but not in the future, whereas w₁** matches ours exactly in the future, but not in the past. The only way Lewis could claim that w₁ is closer than w₁** would be if he could argue that the past is longer than the future. But such an assumption is highly dubious as we shall see in the next section where we return to the original case of Fine.

3.4 The general case

We have seen in the previous two sections that there are rather natural cases where the Lewisian analysis (a) does not yield a correct evaluation of the truth values of these counterfactuals, and (b) does not disallow backtracking in the way Lewis would like to do so.

A major worry is that perhaps Lewis’s analysis will in fact fail even for the original case of Fine. After all, there, too, one can form a world w₁** in which Nixon presses the button, and which exactly matches the actual world from a time shortly after the button press forever after. Shortly after the time t₀ of the button press, there is a small miracle in w₁** which ensures that if we deterministically trace back the conditions from the time of the miracle to t₀, the button is indeed pressed by Nixon at t₀. The way one is to find such a little miracle is to imagine time running backwards, and then try to figure out how to change things slightly prior to the time corresponding in our reversed sequence to t₀ so that at t₀ Nixon’s finger is releasing the button (a release of the button in the time-reversed world corresponds to a press of it without time-reversal). And then we can deterministically backtrack to define all the history of the world prior to t₀. The major and inexplicable short-coming of Lewis’s analysis is the failure to consider w₁**.

In order for Lewis’s analysis of “Were Nixon to press the button, the world would blow up” to work, the world w₁, which, recall, matches our world until shortly before the button press, then has a little miracle causing the button press and then continues deterministically forever, must be closer to the actual world w₀ than w₁** is. Recall, Lewis’s four criteria, which together produce a lexicographic ordering, for closeness of a world to a ours world: (1) no large-scale deviations from laws of nature, (2) extent of spatio-temporal region of exact match, (3) size of small-scale deviations from laws of nature, and (4) extent of spatio-temporal region of approximate match.

If one were to dogmatically assume that the future is shorter than the past, then since w₁ matches w₀ exactly in the past and w₁** matches w₀ exactly in the future, by Lewis’s criterion (2), w₁ would trump w₁**. However, the assumption is unjustified. It is false on those theories according to which time has always existed and will always exist. Likewise, it is false on those theories according to which time had started at some point, say the Big Bang, and will continue forever, and it is false on those theories according to which time started at the Big Bang and will continue to the Big Crunch, because on such theories we are presumably less than half-way to the Big Crunch from the Big Bang in light of the fact that the universe is still expanding. There are some eschatological religious views on which the assumption of a future shorter than the past is true, but not even all eschatological views claim this since many religious people believe that the believers will live forever in time.

Moreover, if the future is longer than the past, then Lewis has a major problem on his hands. For then w₁** will automatically count as closer to w₀ than w₁ does, unless it could be argued that the miracle in w₁** is a large-scale departure from the laws of nature, so that by criterion (1), w₁ will win. However, it is not clear that w₁** need be a large-scale departure. If we imagine time running backwards, it would seem reasonable to imagine that with clever engineering a fairly small miracle shortly prior to the time-reversed version of t₀ will ensure that at t₀ Nixon is releasing the button (and hence shortly before this, he was made by this miracle to depress it). We may have a hard time imagining how to engineer such a miracle, but this may be because we’re not used to engineering things in time-reversed universes—human engineering is always concerned about bringing about effects in our future. The miracle in question may be bigger than that in w₁, but there is little reason to suppose it will have to be one of the large-scale miracles that criterion (1) of world-closeness is talking about it.

Lewis has three options at this point for trying to save his analysis. He can argue that in w₁** there must indeed be a large-scale miracle. This would require a careful physical argument, and since we don’t know about the engineering of backwards-running brains, it is a difficult task. Alternately, he can try to demote criterion (2), extent of perfect spatio-temporal match, so that it is less important than criterion (3), size of small miracles. But he cannot afford to do this, because then world w₂ where Nixon does press the button and where neither the past nor the future matches our world but which world is close to ours in terms of (3) as there are no violations of laws of nature, will turn out to be closer to our world than w₁. This will not do, since the temporal asymmetry Lewis is looking for will then disappear, since w₂ is different in the past, even the distant past, from the actual world, and so there will be plenty of backtracking counterfactuals that will hold if w₂ is the closest world in which Nixon presses the button.

The remaining option for Lewis is to insist that the future is not longer than the past. Without invoking apocalyptic scenarios, the only feasible way for Lewis to do this would be to insist that time is infinite in the past and infinite in the future. I shall assume that this is done. Lewis, however, is still not home-free. He cannot argue on the grounds of criterion (2) that w₁ is to be preferred to w₁** as a match for the actual world. It is dubious whether he can argue it on the grounds of (4), since w₁ will differ quite radically from the actual world in the future—there is a nuclear holocaust in w₁. What remain are (1) and (3), and these come to the same thing here: Lewis must argue that the miracle in w₁ is smaller than that in w₁**.

There is some plausibility on his side. After all, we have a fairly good idea of what kind of an event could produce the miracle in w₁, namely something happening in Nixon’s mind—thus, a fairly local event—which determines him to press the button. However, we cannot say what kind of a miracle is needed in w₁**. The reason for this is that the miracle in w₁** must be one put in after the button press which has the property that if we backtrack deterministically from the time of the miracle to t₀, Nixon ends up pressing the button at t₀.

Our rational activity constantly requires of us that we calculate what kind of a present state will forwardtrack according to the laws of nature to determine a desired event in the future, and this is the kind of calculation that is involved in figuring out what the miracle in w₁ is to be. However, we rarely strive to figure out what kind of a present state backtracks according to the laws of nature to determine a desired event in the past, and so we are hard pressed to find a small future miracle which will determine the button press in its past. We must thus beware of concluding from the fact that we cannot figure out what miracle to put in w₁** that that miracle must in fact be greater than that in w₁.

But there actually is a reason to think that it is likely that the miracle in w₁** is greater. To see this, note that we can divide up non-actual events into four non-disjoint classes. Class A consists of all non-actual events E which could have been produced by a modification of the actual world that keeps the past fixed up to shortly before the time of E and then inserts a relatively small miracle shortly prior to the time of E which miracle deterministically forwardtracks according to the laws of nature to yield E. Class B consists of all non-actual events E which could be produced by a modification of the actual world that keeps the future fixed from shortly after the time of E, and inserts a relatively small miracle shortly after the time of E which miracle deterministically backtracks according to the laws of nature to yield E. Class C is the intersection of classes A and B, and class D consists of all non-actual events outside of A and B. It is important here that the notion of “a relatively small miracle” be kept constant between the definitions of these classes.

Now, intuitively if E is a non-actual event chosen at random, it is highly probable that E lies in D. To see this, suppose for example that I am now at position x in the universe, and consider all counterfactual events of the form “Pruss being at y now”, where y ≠ x. For most choices of y, a relatively large miracle would be required, whether in the future (with the past shortly before and back fixed) or in the past (with the future shortly after onwards fixed), to produce this. Most choices of y would place me very far away from where I am right now, indeed even outside this galaxy, and certainly large miracles would be required to quickly transport me there. Intuitively, there a lot fewer non-actual events that are close to a state of the actual world than there are ones that depart wildly from the actual world, and it is the ones that are close to a state of the actual world that are much more likely to lie in classes A or B, so the vast majority of non-actual events will be in D. Of course, we cannot be considering these likelihoods merely in terms of the cardinalities of the classes, since after all A, B and D are all uncountably infinite. However, we know that cardinalities are not the right way to consider probabilities: a random point in the United States is unlikely to be in Pittsburgh, even though Pittsburgh contains the same number of points—continuum many—as the rest of the United States.

That an event falls in the union of classes A and B already says, thus, that the event is atypical. This is enough to damage Lewis’s position in general. For, if E is a non-actual event neither in A nor in B, then any counterfactual world containing E and having either its past or its future match the actual world will have to contain a massive departure from the laws of nature, and so such a world will be by Lewis’s criterion (1) further away from the actual world than a world containing E, satisfying the laws of nature always, but differing from ours throughout the past and the future. But if such a world is the closest to ours that contains E, then backtracking counterfactuals are licensed, and this Lewis wanted to avoid. Therefore, if Lewis’s account is to have any hope, it must work only for a modest subclass of the class of all non-actual events. There is, however, still hope that it will work for events such as Nixon pressing the button.

To see where this hope lies, consider a random event from A that in fact significantly differs (by a measure of distance similar to that involved in saying that the miracles involved in producing A-events are “relatively small”) from the state of the actual world (Nixon’s pressing of the button is an example, because it would presumably involve his arm, a significantly large macroscopic object, being in a different place from where it in fact was). It is unlikely that this event is in class B as well. To illustrate this, consider an almost frictionless physically-isolated billiard ball system in our world, with billiard balls moving on it according to the laws of physics. Intuitively, most variant configurations of billiard balls would require a relatively large miracle to produce, either by a miracle in the past that forwardtracks to the desired configuration or by a miracle in the future that backtracks to that configuration. But now consider a random non-actual configuration E of the billiard balls and their velocities at time t₀ that happens to be in A, but that in fact is significantly different from the actual configuration at time t₀. Configuration E thus can be produced by making a small miraculous modification to the actual world in the recent past of E, say at t_–1, and then forwardtracking deterministically to produce E at t₀. Let B₀ be the actual world, and let B₁ be the world thus produced. Now, because the miracle at t_–1 was small, it follows that right after the miracle the state of the counterfactual world is still fairly close to the state of the actual world. But because E significantly differs from the actual configuration at time t₀, it follows that as we move deterministically from after the miracle to t₀, we diverge more and more from the actual state of the world.

Now, our physical intuitions are that if two worlds are diverging under the influence of deterministic laws between t_–1 and t₀ then after t₀ they are likely to continue diverging at least for a while, unless things have been rigged in some special way. Thus, it is likely that after t₀, worlds B₀ and B₁ will continue to diverge, or at least are not likely to reconverge, unless the choice of the worlds has been rigged to ensure such convergence. Therefore, it is likely that at a time t₁ shortly after t₀, the configuration E₁ of the billiard balls formed by forwardtracking from E will be significantly different from the actual configuration F₁ at that time. Therefore, E is not likely to be a member of B, since if it were, then it would occur in a world B₁** whose future matches the future of our world after some t₁ which is shortly after t₀, but which has a small miracle at t₁ that backtracks deterministically to yield E at t₀. But then it would follow that the configuration of that counterfactual world at t₁ would be close to the configuration of the actual world at that time, since only a small miracle at t₁ would change one configuration to the other. But the configuration at t₁, prior to the miracle, of the billiard balls in B₁** will in fact have to be close to E₁, because E₁ is produced from E by forwardtracking deterministically.[39] However, we have already argued that E₁ is probably significantly different from the actual configuration at t₁, and so it cannot be that the miracle in B₁** is relatively small. Therefore, configurations that are significantly different from the actual ones and that are in A, are unlikely to be in B. The same statement will, note, hold with A and B reversed, however, since the laws governing billiard balls on an approximately frictionless isolated table are approximately reversible.

In general the intuition is that it takes rigging to make sure that an event significantly different from what is in the actual world should be in A, since presumably most events significantly different from actual ones cannot be produced by a miracle that is relatively small. But there is no reason to think that such rigging would also rig the event so that it should also be in B. Therefore, intuitively, C, the intersection of A and B, is significantly smaller than A. And likewise, intuitively, C is significantly smaller than B.

Suppose that the above plausibility argument that C is significantly smaller than A is right. Then, Lewis’s analysis of counterfactuals may work for Nixon’s case. Nixon’s pressing of the button does lie in A, even if we set the fixed measure of the size of miracles used in defining A to be quite small. By the plausibility argument, however, there is little reason to think that the pressing of the button is in C. Most members of A aren’t. If this is right, then the miracle in w₁** will have to be greater than that in w₁, since the miracle in w₁ will not exceed the measure used in defining the classes A, B, C and D, but that in w₁** will, if Nixon’s pressing of the button is in A but not in C and hence also not in B.

If, however, the plausibility argument fails, then it is very unlikely that Lewis will be able to make any case for the miracle in w₁** being greater than that in w₁. He might be able to establish this on an ad hoc basis for some events, but the prospects for a general argument look grim. Henceforth, I shall assume that the plausibility argument does indeed work, and C is significantly smaller than A. By parity, we would expect C to be significantly smaller than B, since the same intuitions are in play there.

Consider, now, yet another modification of the original situation:

(D) The button is located in a sealed container which it is beyond the technical power of us human beings to open or to manipulate the insides of. As a matter of fact, the button is never pressed. The counterfactual is now: “If the button had been depressed, then the world would have been blown up.”

In (D), we have a harder time picking out what “small” miracle prior to the depressing of the button would let one forwardtrack to a depression of the button. Perhaps the air density over the button would miraculously increase and thereby depress the button. This is not so small a miracle, of course: a lot of air molecules would have to be moved. But most importantly, it is by no means obvious that a miracle located after the pressing of the button could not be equally small while backtracking to the depression of the button. In fact, it could even be a very similar sort of miracle: just imagine time running backwards starting with the actual future of the world, and add a miraculous increase in the density of the air over the button. This case is one where the pressing of the button actually falls in class C, and Lewis’s analysis fails, because the miracles involved in w₁** and w₁ could be of the same, or very similar, magnitude—and so w₁** would be the better match for w₀ given that it matches throughout a future that is significantly longer than the past that w₁ matches w₀ in.

We are now in a position to see what is special about the Nixon case. In the Nixon case, there is an intuitively canonical miracle prior to event A. When asked to imagine the scenario of Nixon (contrary to fact) pressing the button, we imagine him thinking differently from the way he actually was, and the miraculous inducing of this act of him thinking is the small miracle we are after. We know roughly where and how in the past of A to locate the miracle. But when asked to imagine the scenario of the button in (D) becoming depressed, our intuition does not provide us with any such easy answer. Maybe the right miracle is an increase in air density. Maybe, if the button and the console are made of metal, a spontaneous magnetization of the button. There is no canonical location for the miracle in our world prior to the event.

Now, if there is such a canonical location for the miracle prior to the event, then the event in the counterfactual’s antecedent will be in class A. By the plausibility argument given above, it is likely then that A will not be a member of B, and hence Lewis’s analysis will work.

Moreover, counterfactuals of this kind are arguably much more common in our reasoning than counterfactuals of form (D). We often use such counterfactuals, and more generally such subjunctive conditionals, in our practical reasoning:

Were I to choose A, C would result; but were I to choose B, D would result; I prefer D to C, so I will choose A.

And in all counterfactuals that deal with practical reasoning, there is a clear canonical place to put the “small miracle”—in the mind of the agent.[40] The counterfactuals and subjunctive conditionals in fact already come with a story about where the divergence in worlds can easily start.

It is true that we do sometimes ask hypothetical questions of merely theoretical interest like: “What would have happened had that button become depressed in case (D)?” Of course, we ask such counterfactual questions more rarely than we do in the cases of practical reasoning. But I take it that even when we do ask such theoretically-oriented counterfactual questions, we have in mind some story as to how the antecedent of the counterfactual “could have come about”. In cases where the antecedent is a human action, the story typically involves the person’s having made a different act of will. But even in other cases, we tend to have a story in our minds as to how the antecedent “could have come about”. Otherwise, we would (a) be liable to think the counterfactual question to be ill-defined, and (b) have little reason to ask the question in the first place. For, to ask the question, we have to think of the antecedent in the conditional as having been a real possibility. And thinking of something as a real possibility involves having some story about how it “could come about”. Moreover, without such a story the relevant truth value of the counterfactual might be impossible to determine. In the case of (D), if our story about how the button is most easily pressed involves a massive earthquake that shakes up the box, it might not follow from the pressing that the world blows up—for the earthquake could also destroy the wire connecting the box to the doomsday machine.

If we ask counterfactual questions where the listener has no story in mind as to how the antecedent could have come about, we are liable to get puzzlement. “Were Queen Victoria to be alive today, what would she be doing?” The joke answer is: “Scratching at the inside of her coffin.” But actually, the meaning of the question is not clear unless it is specified what miracle story about Queen Victoria’s survival is the relevant one for consideration. E.g., is it that she became rejuvenated in the coffin, or is it that she just never died but lived an extra long time? The answer will be very different in both stories. And an intelligent questioner knows this. That is why he is unlikely to ask a counterfactual question without specifying some process by which the antecedent could come about, unless of course the context clearly singles out a process in virtue of that process being particularly simple—as in the case of the process of the button being pressed by Nixon through the firing of the neurons.

It follow that there are two kinds of intelligent counterfactual questions that might come up in everyday discourse. The first will specify in the antecedent what process for making the antecedent true is meant. The second will be one where there is an “obvious” simplest process, which simplest process is presumably the one that requires the least difference from this world. Actually, the two kinds of counterfactuals are not disjoint. The process in the first kind will almost always be specified incompletely, with the gaps being left to be filled in some “obvious” simplest possible way. And so most if not all cases of the first kind will in fact be cases of the second kind, with the partial process description being counted as a conjunct in the antecedent (e.g., “If Queen Victoria never died and Queen Victoria was alive today, what would she be doing?”). Moreover, both kinds of counterfactuals will have a common feature. The process, whether implicit or explicitly stated, will be one that is in the past of the event that is the primary concern of the antecedent. If from looking at the antecedent one cannot get an obvious process—necessarily in the past, since it is only at the past that our intuitions look for an efficient cause—that could have effected the antecedent, then the intelligent questioner, in every-day cases, is unlikely to ask the counterfactual.

Hence, those counterfactuals about which we are most likely to ask are such that there is an “obvious” process in the past of the antecedent (or perhaps partially overlapping with temporal parts of the antecedent—though even then, the process is likely to be set back in time) which process would bring about the antecedent A. The “obviousness” of the process means that the process cannot deviate too wildly from physical law. Hence, a Lewisian miracle bringing about this process would not be too big, and so w₁ will be close to w₀. Thus, our event will lie in class A, and since the intersection of A and B is much smaller than A if the plausibility argument given above works, likely it will not be in B, and hence the Lewisian analysis will work for this event.

This shows that Lewis’s analysis probably works for most “everyday” counterfactuals in virtue of these counterfactuals’ antecedents, for practical reasons, having been preselected so as to lie in class A, the class of all non-actual events that can be produced by a relatively small miracle prior to the event.

Without such preselection, Lewis’s account fails because it is by no means guaranteed that w₁ is a better match for w₀ than w₁*. Given an event A without the described preselection, it seems prima facie just as likely that a small miracle in A’s future (or, more precisely, in the future of the point in time where one would like A to occur) would backtrack to A as it is that a small miracle in A’s past would forwardtrack to A. So there is no temporal asymmetry, or if there is one, then for events happening in our time it points the wrong way because the fact that the future of the universe is longer than the current past gives one a strong consideration in favor of w₁* as a closer match to w₁, as already discussed. We see this in case (D), and we can also see it in the case of event N discussed in Section 3.3.

We also see a partial lack of such preselection, though in a somewhat different way, in the example discussed in case (C). There, given how great a miracle would be required to override Captain Smith’s resolute plans, the Lewisian will be at a bit of a loss as to where the miracle should be placed—in Captain Smith’s brain or in Nixon’s—and will end up having to countenance some counterintuitive backtracking to Nixon’s own decision. Had the counterfactual in (C) been posed, however, in a more natural way as

(E) Had Captain Smith not pulled out the plug after Nixon’s pressing of the button, then the world would have been blown up

there would have been no problem with a Lewisian analysis. There would then have been an obvious and unambiguous place where the miracle should be located—in Captain Smith’s brain, prior to the time where in the actual world he pulled out the plug. And then the counterfactual would not exhibit the extended backtracking that case (C) gave. However, counterfactual (E) is preselected in the way described above—it is formulated in such a way as to force the process that would make the antecedent true to involve Captain Smith having a different mental state.

3.5 Another counterexample

It might be argued that examples (A) and (B) are rather contrived and do not represent most counterfactuals. For a more natural case, consider the following more natural example. It is highly likely that relevantly similar cases have in fact occurred.

(F) In the actual world, for several months prior to t₀ a one kilogram steel ball has been hanging from a strong plastic cable at a height of one meter above a highly elastic rubber pad. At t₀, the cable is severed in a highly overdetermined way (e.g., several oxyacetylene torches are turned on it), and the ball falls; moreover, things are set up so that it would have been very difficult for the cable to be cut miraculously prior to t₀ (e.g., the only cutting/burning implements were not available prior to t₀), and the whole experiment was quite overdetermined. The pad is so elastic that the ball will rebound to a height of 0.9 meters. For simplicity of calculation, suppose all this is done in a vacuum.

Assuming an acceleration of gravity of 9.8 meters / second², the ball will take about 0.45 seconds to fall. Now, let t₁ be a certain specific time which I will specify more precisely later, but about which I will now only say that it is a time between t₀ and the ball’s impact on the pad. Let A be the (temporally extended) event of the ball being suspended at a height of 0.8 meters at t₁ and of not having been above height 0.8 meters at any time in the two seconds preceding t₁.

Observe that whatever the time t₁ between t₀ and impact is, event A is a rather difficult one to produce by a miracle in its past. One way to produce the event would have been to cut the cable prior to t₀ and then to arrest the ball at height 0.8 meters and suspend it miraculously there for two seconds. However, it was assumed that it was hard to produce a miraculous severing of the cable prior to t₀. Alternately, one might backtrack several months, tie the ball at a lower height of 0.8 meters, and modify the whole set-up so the cable is cut at t₁ instead of at t₀. However, that would involve a miracle quite long before t₀ and might require a large one if the whole set-up was overdetermined.

On the other hand, if t₁ is chosen appropriately, it is easy to find a small miracle in the future of A that backtracks to A. Let t₂ be the time of maximum impact of the ball on the pad in the actual world, that is the time at which the pad is maximally compressed. Since the ball will rebound to 90% of its original height, the compressed pad at t₂ stores about 90% of the gravitational potential energy the steel ball (assuming the steel ball does not itself become compressed) had at its release. The rest of the energy went into such things as the vibrations of the laboratory floor and heat. Consider the time-reverse, v₀, of the actual world w₀. In v₀, we have heat energy and vibrations of the laboratory floor which together with energy stored in a compressed pad conspire to propel a steel ball that at t₂* is lying on the compressed pad up to a height of one meter at a later time t₀*. Suppose that in v₀ we added a small localized miracle prior to t₂* that decreased the energy stored in the compression of the rubber pad by 22%, or redirected this amount of energy away from the steel ball (e.g., by directing it into the ground). Call the modified world v₁; this world matches v₀ in its whole past up to shortly before t₂*. Then, the heat energy and vibrations of the laboratory floor which contain 10% of the energy needed to lift the steel ball to height one meter are still going to transmit energy to the ball, but the pad will now only transmit (0.78)(90%) » 70% of the energy the steel ball would need to be raised to that height. Hence, the ball in v₁ will be raised merely to a height of 70%+10% of one meter, namely to a height of 0.8 meters. It will achieve this height at some time t₁* shortly prior to t₀*, and it will then presumably start falling after t₁* so that it will never be at a height other than 0.8 meters.

Now, let w₁* be the time reverse of v₁. In this world, at a time t₁ shortly after t₀ the ball will have zero velocity and be at height 0.8 meters (note that this construction defines what t₁ is). It will then fall on the rubber pad. Moreover, in the two (or even more!) seconds prior to t₁, the ball will not have been above height 0.8 meters. Hence, A occurs in w₁*. There is a small miracle after t₂ in w₁* which backtracks to the occurrence of A, and after that miracle w₁* matches the actual world exactly. The miracle in question is a small one: it is just the decrease of energy in a compressed rubber pad (note how the compression of the rubber pad makes the miracle even more localized). So, A can be produced by a small backtracking miracle but it does not appear likely it can be produced by a small forward-tracking miracle. Therefore, Lewis’s framework for counterfactuals will produce many non-trivial true backtracking counterfactuals with antecedent A, since the past of w₁* presumably is never exactly like the past of w₁. This is not only absurd, but exhibits the opposite of the arrow of time Lewis tried to exhibit.

It might be objected that there were some idealizations in this account. For instance, I assume the experiment was done in a vacuum and that even if we decreased the energy of the rubber pad in the time-reverse world v₀, still the energy in the vibrations of the floor and in the heat would go to propel the steel ball. These assumptions, however, are not essential to the example. Suppose that these idealizations are false. Nonetheless, miraculously decreasing the energy of the rubber pad in v₀ by 22% around t₂* and then forward-tracking deterministically will have to (by energy conservation) produce a world where the steel ball will achieve a maximum height lower than one meter at a time t₁* in the future of t₂*. Let the height that will be achieved be h. Perhaps h is not exactly 0.8 meters. But whatever it is, we can then define the event A as being the event of the ball being suspended at a height h at t₁ and of not having been above height h meters at any time in the two seconds preceding t₁, where t₁ is the time corresponding to t₁* in the non-time-reversed world, and the analysis continues to go through. All that matters is that in the time-reversed world if we decrease the energy in the compressed pad, which is a small localized miracle, we will ensure the ball will achieve a lower height.

3.6 Conclusions

3.6.1 The problem with Lewis’s approach

The most serious defect that Lewis’s analysis suffers from is its complete neglect of consideration of worlds of the form w₁** which match ours in respect of the future shortly after the event of Nixon’s pressing of the button, but which include a miracle shortly after that pressing from which one can backtrack to the pressing itself. While this neglect might not affect his analysis of the case Fine challenged him with, it does affect the analysis of other cases.

Lewis (1979a, pp. 473–475) thought that the temporal asymmetry that he had found was based on the contingent fact that overdetermination of the past by the future was more common than overdetermination of the future by the past. This he found by comparing world w₁ with world w₄ which included two miracles, one of which was designed to erase the future effects of the first, and noting that this second miracle would have to be comparatively very large. Price (1997, p. 151) has suggested that at the microscopic level the asymmetry may disappear. In fact, we can see that Lewis’s intuitions are vitiated by the choice of worlds to look at. For the most relevant worlds to compare to the actual world in a Lewisian setting are not w₁ and w₄, but w₁ and w₁*, and so what one requires to get the temporal asymmetry is a condition to the effect that a lesser miracle is needed if we put the miracle in the past of the event described in the antecedent of the counterfactual, from which miracle we can deterministically forwardtrack to the antecedent’s event, than if we put the miracle in the future of the antecedent’s event and backtrack.

Lewis brings in here an overdetermination asymmetry according to which a single actual event is overdetermined by a number of future events but not by a number of past events.

Whatever goes on leaves widespread and varied traces at future times. Most of these traces are so minute or so dispersed or so complicated that no human detective could ever read them; but no matter, so long as they exist. It is plausible that very many simultaneous disjoint combinations of traces of any present fact are determinants thereof; there is no lawful way for the combination to have come about in the absence of the fact. (Lewis, 1986a, p. 50.)

As an example, Lewis alludes to the phenomenon of the spreading of spherical waves spreading from a point: “Countless tiny samples of the wave each determine what happens at the space-time point where the wave is emitted or absorbed” (ibid.) If Lewis was right, then a miracle that keeps the future constant would have to cut each of the nomic connections between a past event and its future overdetermining events, and that would involve a large-scale miracle. However, Lewis must be wrong, since we have already seen in case (B) that a single small miracle in the future of the button-press can backtrack to a button-press.

But one can do better than just saying “Lewis must be wrong”. In fact, one can show that the overdetermination does not occur even in the paradigmatic case of the spreading of a spherical wave. Suppose that some event happens at t₀ in the actual world such that an amount E of energy is released from which a spherical wave starts spreading, and for simplicity suppose that in the region in question this is the only relevant source of energy. Lewis’s claim is that the release of the energy is overdetermined by various disjoint samples of the spherical wave at some time t₁ after t₀. But this is false due to energy considerations. For consider the allegedly overdetermining disjoint parts S₁,S₂,…,S_n of the spherical wave at t₁. The S_i are events occurring in disjoint areas of space-time, and their energy comes from the originating event at t₀. According to Lewis, none of these samples can be present without the originating event. Now, consider the system as a whole at t₁. By conservation of energy, the total energy of the system is E. Each of the S_i carries some non-zero portion E_i of the energy released by the originating event.

If Lewis is right about his overdetermination claim, then it is nomically impossible that S₁ occur at t₁ without the originating event occurring at t₀, and in particular without amount E of energy being released at t₀. However, consider a world whose state at t₁ is just like the state of the actual world, except that S_n is replaced by an event T_n of strictly smaller energy. Then by energy conservation, what happens at t₁ cannot backtrack nomically to the release of amount E of energy at t₀, since we have ensured that the amount of energy at t₁ is less than E.

We can be more explicit about the construction of the world in question. Take the time-reverse of the actual world, letting t₀* and t₁* be the analogues in the time-reversed world of times t₀ and t₁, respectively, so that t₁* < t₀*. At t₁* we have a number of events S₁*,S₂*,…,S_n* which are the time-reverses of S₁,S₂,…,S_n. At t₀* an amount E of energy is absorbed. Now suppose that by a miracle occurring shortly before t₁* we replace S_n* by an event T_n* of lower energy, and then we evolve the system until t₀*. Since all the energy in the events S_i comes from the originating event at t₀, likewise all the energy in the events S_i* is absorbed in the event at t₀*. If we replace S_n* by a lower energy event, then the amount of energy available to be absorbed at t₀* will be lower. Hence, if we replace S_n* by T_n*, then at t₀* there will be an amount E₁ < E of energy absorbed. Now take the world, w*, where this replacement happens, and reverse time once again to get a new world w. Then, in w, at t₀ there will be a release only of an amount E₁ of energy, and events S₁,S₂,…,S_n_–1 will occur but S_n will be replaced by a lower energy event T_n. Moreover, all the right nomic connections hold in the interval between times t₀ and t₁ since in w* all the right nomic connections held between times t₁* and t₀*. Therefore, it is not the case that the occurrence of S₁ nomically necessitates the occurrence of a release of amount E of energy at t₀. Indeed, not even the occurrence of all of the events S₁, S₂ through S_n_–1 necessitates that, simply because these events do not carry enough energy to necessitate this.

We have argued that in fact the asymmetry in Lewis’s analysis of everyday counterfactuals, if and insofar as there is any asymmetry, comes from the fact that everyday counterfactuals are usually preselected for having antecedents which can be effected by easily imaginable processes acting in the antecedent’s past. But this does not reveal an objective asymmetry that would be independent of people’s intuitions that it is past processes that cause future events rather than future processes that cause past events. For those very intuitions are the psychological ground of the preselection of everyday counterfactuals, which preselection yields the asymmetry in the counterfactuals if we analyze the counterfactuals in a Lewisian way. If Lewis’s argument could be taken to be an analysis of our everyday intuitions, this would not be so bad. However, it cannot be thought of as such an analysis, because it crucially depends on the fact that any localized event normally has many, often tiny, effects throughout a large spatio-temporal region (e.g., Nixon’s button press, even if no nuclear holocaust happens, affects the gravitational field in China), whereas our everyday intuitions about counterfactuals do not depend on this.[41]

And once we depart from the realm of everyday counterfactuals, we can find a number of cases where Lewis’s analysis breaks down. If the antecedent of the counterfactual is something like a neuron firing (the case in Section 3.3), or a button depressing in a locked box, the Lewisian analysis fails—or at least may well fail—to yield a past-future asymmetry grounding our view of the future as open and of the past as closed.

None of this contradicts the facts (a) that there is an asymmetry in our counterfactuals, even though Lewis’s analysis has failed to give a proper objective grounding to it, (b) that this asymmetry may well be responsible for our view of the future as open and the past as closed, and (c) that there may be an asymmetry in causal overdetermination, even though Lewis’s analysis in the end fails to connect this up with the asymmetry in (a). The asymmetry in counterfactuals remains unexplained—unless one could reduce it to the asymmetry in causation where past events are causes of future events but not the other way around. If one is satisfied with taking the asymmetry in causation as basic—as opposed to wanting to make sense of it in terms of the counterfactual asymmetry of time—then perhaps the fact of there being such preselection rules for antecedents of counterfactuals as ordinary usage puts in place is adequately grounded in the objective reality of world, since as we have seen it is grounded in our intuition of the causal asymmetry. But a person satisfied with this would be running a project that is the opposite of Lewis’s here.

3.6.2 A fix

Fortunately, we can make Lewis’s account yield the right values for counterfactuals by simply building the arrow of time explicitly into the definition of the counterfactual by demanding that any worlds invoked in the analysis should closely match the actual world in the past (cf. Davis, 1979), or the past light cone, perhaps, of the event described in the antecedent except perhaps for the very recent past.[42] But if we do this, then we no longer have a derivation of an arrow of time.

More precisely, one possible modification of the counterfactual is to build into the similarity relation between possible worlds a requirement that, in the context of evaluating a counterfactual whose antecedent is the report of an event at t, the worlds that one compares the actual world to should match shortly up to time t. Alternately, and perhaps preferably, one can say that two worlds that match from their beginnings until t₁ are always counted as closer together than two worlds that match from their beginnings until a time t₀<t₁. This has the advantage that we do not need to specify the time of the antecedent event.[43]

There is, however, one possible pitfall with this approach. If the order of time is dependent on the order of causality, as in fact I will argue in Section 2.4 of Part VI, then we can no longer analyze causality in terms of counterfactuals and hence in terms of possible worlds. I take it, however, that counterfactuals and possible worlds are useful whether we can analyze causality in terms of them or not. It is worth noting at this point that possible worlds themselves will in the final analysis be elucidated in terms of causality (see Part VI, below). I do not do not at all mind explanation or causation being ontologically basic.

3.6.3 McCall’s approach

Storrs McCall (1984) has suggested a different approach to counterfactuals, also in response to shortcomings in Lewis’s approach and also building an asymmetry into the definition of a counterfactual. Moreover, McCall hopes to make the notion of the similarity of worlds simpler and more precise than Lewis does.

Say that a physically possible world w branches relative to the actual world at t₀ if it matches the actual world throughout space and time prior to t₀ and if there are times arbitrarily close to t₀ (obviously, they will have to be after t₀) at which w does not match the actual world. To evaluate a counterfactual, McCall suggests that we consider those physically possible worlds at which the antecedent holds which branch relative to the actual world and which have the property that no physically possible world in which the antecedent holds branches relative to the actual world any later than they do. If the consequent holds in these worlds, then the counterfactual is true.

Of course, there is a technical difficulty in the account caused by the fact that there may be a sequence of worlds in which the antecedent holds and which branch relative to the actual world at later and later times that asymptotically approach some time t₀ with the property that no world that branches relative to the actual world at t₀ or later satisfies the antecedent of the counterfactual. But this is easily handled: if there is such a sequence with the property that the consequent fails to hold at infinitely many of the worlds in an asymptotic sequence with this property, then the counterfactual is false; else, it is true.

More seriously, the sheer amount of indeterminism in the world destroys McCall’s account entirely. McCall’s example of how his account works is the counterfactual: “If Napoleon had won the battle of Waterloo, he would not have died on St. Helena” (p. 466). Admittedly, McCall says, there is a physically possible world where Napoleon wins at Waterloo and dies on St. Helena, because St. Helena had earlier become a popular vacation spot for French officers. But the branching of that world relative to our world is earlier than that of a world that branches during the battle. However, for all physical worlds w which branch during the battle and which have Napoleon dying off St. Helena, with the exception of those worlds in which Napoleon receives a fatal wound during the battle (and there is no reason to think that those worlds would be the ones where victory would be snatched by a later branching), we can find a branching relative to w that has Napoleon dying on St. Helena. After all, it is physically possible that Napoleon might on a whim go to St. Helena and have a fatal accident there, or that someone might on a whim persuasively suggest it to him as a vacation destination and then murder him there, or even that due to quantum randomness he would be randomly teleported to St. Helena and promptly killed there. Worlds like this need not branch any earlier from the actual world than the ones where Napoleon is victorious and never goes to St. Helena.

Lewis himself will not be bothered by such worlds because these worlds exhibit a qualitative dissimilarity from the actual world—e.g., Napoleon acquiring a different taste in vacation destinations than in fact he does, or a weird quantum phenomenon occurring whereas no such phenomena occur in the actual world. This shows a superiority in Lewis’s similarity-based account, though as I’ve argued the similarity needs to be weighted temporally or in terms of causal antecedents.

For another counterexample to McCall, consider the following counterfactual: “Were Napoleon to have won at Waterloo, his victory would have come about out of something very weird.” This is false, albeit vague. Were Napoleon to have won at Waterloo, presumably it would have been because his soldiers fought a little more bravely than they did, or the English faltered a little more. This kind of a variation is not a particularly weird event. But any world where victory is snatched by such non-weird means will have to branch from the actual world at some time significantly prior to the end of the actual battle. However, there are physically possible worlds which branch very close to the end of the actual battle and where Napoleon wins a victory. These are worlds where the victory is won because of an extremely weird event. For instance, the quantum phenomena in the hearts of all the English ensure that all but one of them faint at almost the end of the battle.[44] Since these worlds branch later, they are the ones that on McCall’s account need to be considered when evaluating the counterfactual. Presumably there are no physically possible not-very-weird victories that could be snatched seconds before the actual defeat, and so indeed it will be true on McCall’s account that were there a victory of Napoleon, it would have come about in a very weird fashion.

Section 4 Propositions

4.1 Unstructured propositions

On Lewis’s first take, a proposition is a set of possible worlds (Lewis, 1986a, Section 1.4). We say a proposition p is true at w if w is a member of p. This won’t quite work, because as we shall see later (Part III.7.2), on no reasonable account of possible worlds is there a set of all possible worlds. However, it is open to us to say that a proposition is a class or collection of possible worlds, and for all intents and purposes this will be just as good.

Providing we know what collections are and have an account of possible worlds, we thus have an account of propositions. Unfortunately, as Lewis certainly realizes, this account does not distinguish between propositions that are logically equivalent. But the standard criticism of Lewis here, made forcefully by people like Plantinga, is that for many purposes such a distinction is necessary. For instance, if one thinks that propositions are both the bearers of truth and what sentences express, then one may be uncomfortable with saying that all the necessary truths are one and the same proposition which is identical with the collection W of all worlds. In particular, all mathematical truths are the same. Matters are even worse if we accept essentialist claims that genus-species relations are necessary: the necessary a posteriori proposition that horses are mammals will turn out to be the same as the proposition that spiders are invertebrates, and both will be identical with the a priori necessary truth that Fermat’s Last Theorem is.

If one thinks that knowledge is of propositions, one then wonders what we have learned when we learned that Fermat’s Last Theorem was true that went beyond the proposition we already knew that 1=1. One might want to say that we just learned something about our language, namely that when English speakers use the words

(18) “There are no positive integers a, b, c and n such that n>2 and aⁿ+bⁿ=cⁿ”

they express a necessarily true proposition, namely the proposition W. But this answer won’t do on either of the two reasonable interpretations of the word “English.” Either “English” is a rigid designator here of the language which I am now using or it is a definite sociological description of a language spoken by a group of people who are qualitatively described. In the first case, the proposition that when English speakers use the words in (18) they express a necessarily true proposition is itself a necessary truth, and hence on the above account of propositions saying that we have learned this proposition isn’t saying anything more than that we have learned that 1=1. For if “English” is a rigid designator of our language, then what (18) means in English is an essential property of (18). On the other hand, consider the case where “English” is a definite sociological description. Then there is a possible world w₁ where “English” does uniquely pick out a language but where (18) means that water is H₂O or, if we don’t like Kripkeanism, that 1234 ´ 4321 = 5332114. Evidently then what we have learned in discovering Fermat’s Last Theorem is different from what the English speakers in w₁ have learned upon discovering that the words in (18) express a necessary truth. But if what we learned was that (18) expresses a necessary truth in English, then we have indeed learned nothing other than those people have. And this is absurd.

Nor are propositions that are logically equivalent the same proposition, as this Lewisian account would make it out. We can see this when we observe that explanation is a relation between propositions. Now let w be a forwards- and backwards-deterministic world. Let L be a proposition reporting the laws of nature of w. Let S_t be a proposition reporting the complete physical state of w at time t. Then, two-way determinism ensures that S_t and L jointly entail S_u for all times t and u, so that the conjunctions (S_t and L) and (S_u and L) are always logically equivalent. Now, the conjunction of S₀ and L evidently explains S₁ in a deductive nomological way. Hence, if propositions that are logically equivalent are to be identified, likewise (S₁ and L) explains S₁, since (S₀ and L) and (S₁ and L) are equivalent. But this is absurd, because the claim that “(S₁ and L) explains S₁” could only possibly make sense if S₁ were a self-explanatory proposition[45], which in this case it is not. We can also apply similar reasoning to conclude the absurdity that (S₁ and L) explains S₀ since the logically equivalent proposition (S_–1 and L) does.

4.2 Structured propositions

Realizing the need for an account of propositions that allows for differences between logically equivalent propositions, Lewis calls those propositions that he identified with collections of possible worlds “unstructured propositions”, and suggests that we also define “structured propositions” as set theoretic constructions out of the unstructured ones.

For instance we could associate the modifier ‘not’ with the unstructured relation [collection of all pairs of individuals in all possible worlds that fall under the relation] N that holds between any unstructured proposition and its negation, that being the set of all worlds where the original proposition does not hold. Then a negative structured proposition could take the form áN, Pñ, where P is a (structured or unstructured) proposition. (Lewis, 1986a, p. 57)

This process can be continued with other connectives. If A is the relation that holds of a triple áp, q, rñof unstructured propositions if and only if r is the conjunction of p and q, then in the context of the previous section we can say that áA, áS₀, Lññ explains S₁ but áA, áS₁, Lññ does not, even though the unstructured propositions corresponding to áA, áS₀, Lññ and áA, áS₁, Lññ are identical. Explanation is a relation where structure matters. Similarly, this approach does let one say what it is that one learns when one learns that Fermat’s Last Theorem is true: one learns that a certain complicated structured proposition is true.

However, there are many set-theoretic constructions that will work equally well or equally badly for these purposes. Now, why should we call áN, Pñ “the proposition which is the negation of P” instead of bestowing that title on the pair áP, Nñ? Moreover, even if we choose an order for the ordered pair, there are multiple set-theoretic constructions for ordered pairs. For instance, we can use { N, {N, P} } to set-theoretically represent the ordered pair áN, Pñ or we can use { { Æ, N }, { {Æ}, P } or even { N, {N}, P } . It is up to us—all do the job. So which constructions should we choose?

One might wonder: What is the fuss about since if all the constructions do the job equally well, then can we not just choose whichever one we want? Concerning a somewhat related issue with properties, Lewis writes:

It’s not as if we have fixed once and for all, in some perfectly definitive and unequivocal way, on the things we call ‘the properties’ … (1986, p. 55).

But to dismiss the fuss over which construction is the right one is to forget what job propositions were supposed to do. Propositions are theoretical entities introduced as that which we mean by our language. They are supposed to provide criteria for synonymy within and between languages: two sentences (in the same or different languages) are synonymous if and only if they express the same propositions. It is essential that propositions be language independent—otherwise, for most intents and purposes we could just define propositions as sentences of some fixed language, say Latin. The multiplicity of set theoretic constructions that “do the job” mirrors the multiplicity of languages, and hence Lewisian set-theoretically structured propositions are no great improvement over the situation we have when we just stick to languages, except for the advantages of formalization and the availability—for many semantic purposes unnecessary—of alien properties that our languages have no terms for.

There are two possible responses here. The first is that we should choose a particular set theoretic approach, and call the resulting constructions “the propositions”. These propositions will do the job that propositions are supposed to do. However, if this is done, then it becomes mysterious how it is that propositions are supposed to be what we mean by language. Suppose I affirm the negation of an unstructured proposition. Let us grant that I affirmed áN, Pñ. But how could we possibly find out that when I affirmed this, my meaning in fact used the one privileged set theoretic construction for ordered pairs rather than another? Indeed, it is not even plausible to suppose that it did use one construction rather than another. What kind of a queer fact would it be about our language that when we affirm negations our meanings are one kind of set theoretic construction of ordered pairs rather than another?

It is tempting to say that this is a misunderstanding of the role that the set theoretic constructions are supposed to play. Take a particular axiomatic rendition of the general theory of relativity. This theory models our space-time as some kind of a Riemannian manifold. But there are, of course, many set theoretic ways of expressing Riemannian manifolds, just as there are many set theoretic ways of expressing real numbers (one can express them as pairs of lower and upper Dedekind cuts, or just as lower Dedekind cuts, or just as upper Dedekind cuts, or as equivalence classes of Cauchy sequences, and so on). Which one of these constructions is “the right one”? Well, surely, the question does not matter—the same physical reality is modeled. And likewise, it might be suggested, it does not matter which set theoretic construction is used to model propositions. But this suggestion forgets that the construction of propositions was not supposed to give a model of propositions—it was supposed to give us the propositions themselves, the things we mean by our assertoric sentences. If Lewis were only giving a model of propositions, then that would mean that there would be propositions out there, of which the set theoretic constructions are mere models, and which propositions presumably would not be the constructions, just as, on some interpretations of relativity, there really are entities in the world that are points in space-time. But if this were so, then Lewis’s account of propositions would no longer provide an ontological reduction of propositions to set-theoretic constructions out of possible worlds. One could just as well consider the objectively existent propositions themselves, instead of his model. The model might be theoretically helpful, but it would not remove the ontological puzzlement that Lewis himself admits to feeling (Lewis, 1986a, Section 3.4) about what propositions are.

The other possible approach is to go in the opposite direction. A proposition now is not just some set-theoretic construction, but all of them in some sense. Suppose that C₁ and C₂ are systems of set-theoretic methods for modeling propositions, and p₁ and p₂ are variables that range over the set-theoretically constructed entities within these respective systems that model propositions. Then, there is an equivalence relation that holds between the pair C₁ and p₁ and the pair C₂ and p₂ if and only if p₁ and p₂ play the same role in their respective systems (e.g., if C₁ is the system where the negation of P is represented by p₁=áN, Pñ and C₂ the system where it is represented by p₂=áP, Nñ). Perhaps, the suggestion goes, the propositions are equivalence classes of pairs under this relation. And perhaps there is some underlying intuitive and natural notion of a “relation” and “equivalence class of pairs” that does not require a specific set-theoretic construction (given that these are the only two notions we are dealing with, this is more plausible than the suggestion that there are always such notions for all the set-theoretic constructions that would be involved within a single system C for constructing structured propositions)—since otherwise we haven’t gotten rid of the arbitrariness in the definition.

But the difficulty here is in the equivalence relation. If we can help ourselves to “the relation” that holds between two system-entity pairs when the entities “play the same role”, then by the same token we could help ourselves to “the relation” that holds between two language-sentence pairs when the sentences are synonymous. Moving from languages to quasi-linguistic set-theoretic constructions does not gain one much here. There is little hope that we could specify the relation in either case adverting to the notion of a proposition and saying, in the linguistic case, that we are talking of the relation that holds when the sentences express the same proposition, and, in the set-theoretic case, of the relation that holds when the two constructions model the same proposition.

Even more serious than the mere fact of arbitrariness in the construction is that different constructions give different truth values to semantic claims. On the account of conjunctive propositions that was introduced above, the sentences “The sky is blue and grass is green” and “The grass is green and the sky is blue” are not synonymous, because the former expresses the proposition <A, that the sky is blue, that grass is green> and the second expresses the proposition <A, that grass is green, that the sky is blue>. But one might also give an alternate construction for the structured conjunction of p and q: one might define this conjunction as áA,{ p,q }ñ. On this construction, the two sentences will end up expressing numerically the same proposition. The choice between áA,p,qñ and <A,{ p,q }ñ cannot thus be arbitrary, because it has substantive philosophical consequences.

My own intuition is that the conjunction of two sentences expresses the same proposition regardless of the order of conjoining[46]; I also think the parallel point is clear in the case of properties (surely there is no difference between an electron’s spinning and being charged and an electron’s being charged and spinning). But whether this is so should not be legislated by constructional fiat. If as realists we think there actually is a fact of the matter as to whether the conjunction of two propositions is the same regardless of the order of the conjuncts, then this fact of the matter will be grounded in some reality independent of our construction of “structured propositions”. But if so, then Lewis’s theory of structured propositions, if it be correct, actually presupposes as primitive a fact about whether conjunction depends on the order of conjuncts, and hence is no longer a reduction of propositions to collections of worlds and to things built up out of worlds.

One might not want to be a realist about synonimity or identity of propositions, of course and hold that there is no fact of the matter that determines whether two sentences are synonymous or two propositions identical. For various theoretical purposes one might choose various accounts of “propositions”. This may even be the parallel of the point that Lewis is making above for properties. But it is significant to note that this is a point that Lewis in the large structure of On the Plurality of Worlds cannot make. Lewis is offering an argument for why we should believe that there are possible worlds by arguing that they have great philosophical utility, and offers his accounts of propositions and properties as examples. But in order for this argument to provide justification for realism about possible worlds it is surely necessary that the possible worlds should figure in an explanation of the philosophical phenomena in question. That possible worlds are useful for producing an account of propositions is only going to give one reason to be a realist about worlds if one is a realist about propositions or at least about the sort of talk that propositions are used to clarify. If the propositions are arbitrary constructions useful contextually, the worlds can be such, too. Admittedly, no one ersatz construction of worlds works equally well for all cases, but we can use different ones for different contexts, just as we have different constructions of propositions for different purposes.

There is another serious difficulty with the arbitrariness involved in the construction of structured propositions. The purpose of propositions is to be language-independent analogues of assertoric sentences, one per equivalence class of synonymous sentences. But Lewisian structured propositions are basically linguistic entities, albeit ones constructed in a language whose symbols are various sets, classes or collections, instead of marks on pages or vibrations in the air. Indeed, the fact that they are linguistic entities is seen from the fact that there are different languages that all provide constructions of propositions, e.g., a language that specifies that the grammatically correct form is áp,q,Añ instead of áA,p,qñ.

But if moving from sentences to propositions is just moving from sentences in a natural languages to ones in a more rarified language whose symbols are abstracta, then the move fails to provide the language-independence that the notion of propositions was supposed to yield. And, indeed, it threatens a classical third-man regress. If we introduce propositions to be that whose expression synonymous sentences have in common, and if propositions are themselves sentences, albeit ones in a rarified language whose terms are Platonic abstracta, then either:

(a) there should be a yet third class of sentences to explain the synonymity between the sentences of natural language and the propositions which on this account are sentences, or

(b) we have no principled reason to introduce propositions, at least for this purpose, since if (a) is false then at least sometimes synonymity between sentences can be explained without reference to any entities other than the sentences themselves (at least in the case where one of the sentences is a proposition) and hence the possibility is opened up of such an account being given of synonymity in general.

The first option leads to a vicious regress while the second removes a good part of the epistemic ground for supposing there to be propositions—and hence removes much of the warrant that the existence of Lewis’s account of propositions gives to the real existence possible worlds. Of course, one might have other reasons for positing propositions. But if propositions are really sentences, albeit in a rarified language, then one doubts whether there will ever be a principled argument for why one needs something other than the “more ordinary” sentences.

It might be argued that there is a gain. Lewisian structured propositions have as their ultimate constituents, along with set-theoretic constructions, the unstructured propositions and properties, which are arguably relevantly different in kind from ordinary language sentences and predicate expressions. Hence, there is a significant conceptual difference between a Lewisian structured propositions and properties and parts of natural language—the structured propositions, for instance, include as parts collections of those possible worlds that satisfy those parts, while no such thing can be said about ordinary language expressions. That is true, but it does not help Lewis much. For if one of the main theoretical goals of an account of propositions is to provide an account of synonymy, then the part played by the unstructured propositions is to account for logical equivalence. But that two sentences are logically equivalent (i.e., they cannot differ in truth value) is only a very small and least troublesome part of their being synonymous. The structuring of propositions that Lewis engages in is supposed to allow them to play the more difficult role. But this structuring is essentially linguistic and hence is no gain—else the third man regress arises.

Finally, observe that Lewis’s account of structured propositions presupposes that there are simple or atomic propositions that cannot be analyzed as connected combinations of other propositions and that out of these simple propositions all other propositions are composed—call this thesis general atomism. For the Lewisian structured propositions are built up out of unstructured propositions, and the unstructured propositions cannot be analyzed into other propositions, since they are just sets of worlds.[47] If general atomism is false, some proposition p is analyzable into a combination of simpler propositions, some of which in turn are analyzable into a combination of yet simpler propositions, and so on without termination. To model p in his system, Lewis would have to cut off the infinite complexity of p at some point, making some level of analysis consist entirely of unstructured propositions, thereby destroying the finer structure of p and hence failing to model it. If we think that the existence of propositions of such infinite propositional complexity is a genuine possibility, then we will have reason to be dubious of Lewis’s theory of propositions prejudging against this (observe how Lewis, 1986b, himself worries about the epistemic possibility of infinite complexity and uses it as an argument against Armstrong’s view of structural universals).

If Lewis’s account were to give one an intelligible ontology for propositions in terms of possible worlds, this would be a big asset for those possible-worlds that do not construct possible worlds out of propositions. But, alas, Lewis’s account has proved unsatisfactory, whether in its unstructured or structured incarnation. However, the unstructured incarnation may have some limited theoretical uses. For some intents and purposes, the differences between logically equivalent propositions can probably ignored. For these purposes, Lewis’s account may be a useful tool. But since the “unstructured propositions” are not really propositions, the anti-realist about possible worlds can accept unstructured propositions just as much, considering them to be mere useful fictions.

We can observe that this attack on Lewis’s account of propositions takes much of the wind out of Lewis’s sails when he assails those accounts that construct possible worlds out of propositions for their failure to give an account of what propositions are. For, Lewis, too, surely needs propositions, albeit for different purposes, and he cannot give a satisfactory account of them either.

Section 5 Properties

Lewis offers an account of properties that is similar to his account of propositions. Thus, a property may be seen in an unstructured way as the collection of individuals in all possible worlds that has that property. And, similarly to the way structured propositions, if we wish to distinguish properties that are necessarily co-existensive, we can use set-theoretic constructions to produce “structured properties”. Lewisian propositions can then be seen to be properties of worlds.

The difficulty with the unstructured properties is the same as the one in the case of unstructured properties. However, there is a serious problem with structured properties as well. Granted, one can in a quasi-linguistic way distinguish a number of co-extensive properties. For instance, if Ù is the ternary relation between unstructured properties which holds if and only if the third relatum is the intersection of the first two, then we can distinguish the conjunctive property of being a horse and a mammal from the necessarily co-extensive property of being a horse by using á Ù, á H, M ñ ñ for the first property and H for the second, where H and M are the properties of being a horse and a mammal respectively. The main objection against structured propositions was the arbitrariness of the set-theoretic construction. One might level a similar objection against structured properties now. If one thinks of the properties of an object as those entities in virtue of possessing which various predicates are predicated of the object, then one might wonder why a horse should possess á Ù, á H, M ñ ñ as opposed to, say, á á H, M ñ, Ù ñ. Properties are those aspects of reality that are pointed out by an attribution of a predicate to a subject. These should thus be uniquely defined, just as propositions should.

There may be thought to be another objection against Lewisian structured properties. Let us suppose the structured account is correct. Now, a structured property that a thing has is built up set-theoretically out of other properties. At the pain of a regress, we must eventually come to basic properties which are not themselves set-theoretic constructions out of other properties.[48] These basic properties will be unstructured. But now here is the problem: Is it not possible that there be two basic properties that are necessarily co-extensive? If there are such properties, then Lewis’s account fails. For these properties in light of their basicness cannot be distinguished as different set-theoretic constructions, since they are not set-theoretic constructions at all.

But can we distinguish necessarily co-extensive basic properties by some set-theoretic construction that involves, say, attaching numbers to them? Thus, supposing F and G are necessarily co-extensive basic properties that are distinct, we could let S be the (unique) unstructured property corresponding to them and then label F as á 1, S ñ and G as á 2, S ñ. But there is no reason why we should not label them the other way. Ex hypothesi, they are different properties, and it will not do to say that we might as well treat “á 1, S ñ“ as F and “á 2, S ñ“ as G—one might as well say that it doesn’t matter whether we treat F as F and G as G rather than treating F as G and G as F. We should not be just modeling—for then we could just use that reality we are modeling instead of the Lewisian properties—but are supposed to be identifying the properties, surely, perhaps pace Lewis’s intent.

Lewis thus needs to deny the existence of extensionally equivalent distinct basic properties. But that should not be a great burden, since it is difficult to point to any.[49] The arbitrariness of construction objection remains, but it is only serious if one thinks, pace Lewis, that there is an objective fact of the matter as to what the properties are.

The problem of infinite complexity reappears in a particularly clear way. There is a plausibility to the logical possibility of a property having infinite complexity, i.e., of it not being capable of analysis into basic properties that cannot themselves be further analyzed. On one account of properties, when we learned about molecular motion, we learned that the property of being hot is nothing but the property of having molecules that move about rapidly. When we learned that molecules are built up out of atoms, we learned that the property of having molecules is the property of having atoms that are bonded together. When we learned that atoms are built up out of electrons, protons and neutrons, we learned that the property of being an atom is the property of having electrons, protons and neutrons appropriately arranged. When we learned about quantum mechanics, we learned that “being appropriately arranged” is a certain mathematical property of wavefunctions (as opposed to, as we had wrongly thought, being a mathematical property of sharp spatial positions). Is it absurd to suppose that this process could continue without an end, without there being an ultimate analysis of the property of being hot as a combination of properties that themselves on conceptual grounds cannot be analyzed further? It is traditional to call this scenario the “possibility of infinite complexity”, though the term is misleading: the property of “having F₁ or having F₂ or having F₃ or …” is not “infinitely complex” in this sense if the F_i are basic non-analyzable properties because there is an ultimate, albeit infinitary, analysis into properties that are not themselves further analyzable.

If infinite complexity is possible, then Lewis’s account of properties is wrong. For Lewisian unstructured propositions cannot be further analyzed, whereas his structured propositions can all be analyzed into unstructured ones. Hence, the Lewisian account is at base an atomistic one, and therefore incompatible with the possibility of infinite complexity. Now perhaps Lewis could say that the fact that the best account of properties, namely his account, leaves no room for infinite complexity gives one good reason to think that infinite complexity is impossible. But this would only work if Lewis’s account was the correct account of propositions and properties understood in a realistic fashion. If it is simply a somewhat arbitrary account of one particular way of modeling properties, then its inability to model infinite complexity should not reflect on the logical possibility of infinite complexity—it might simply reflect on its inadequacy. If the account were a realistic one that gave an account of what the properties really are, then by inference to best explanation we would have reason to believe the account to be realistically true and hence to believe that what is incompatible with it is false.

One might also object to the Kripkeanism in the above account. I am not merely claiming that those things that are molecules are made of atoms. I am claiming that to be a molecule is to be made of atoms arranged in a certain way (and I allow here the possibility that we may not be able to specify reductively what this way is other than as “that way of arrangement of atoms that things of this kind have”, with the “this” pointing to atoms), which entails the Kripkean claim that, necessarily, whatever is a molecule is made of atoms. It would, however, go beyond the scope of this paper to argue for this claim, beyond pointing out that the naturalness of telling the story I did. It is quite natural to say that we have learned something about the property of heat when we learned that heat is the rapid movement of molecules, and not just something about the hot things that exist.

Of course, one might come up with an argument against the possibility of infinite complexity.[50] But without such an argument, the Lewisian construction of properties should not prejudge the issue.

Section 6 Overall assessment

Possible worlds let one formulate in a uniform way various modal notions that seem to intrinsically involve consideration of and comparison between more than one world: e.g., supervenience, transworld comparison of individuals, and counterfactuals. Moreover, ordinary modal claims like “I might have been a physicist” or “Hitler might never have been born” are naturally disambiguated against a background of possible worlds, with the context determining which worlds we are quantifying over. Since there is good reason to think that all these modal claims make sense, and since a very natural way to make sense of them is possible worlds, this gives us good reason to believe there are possible worlds.

But of course there is more than one theory of possible worlds. If we should find that only one extant theory is a serious option that withstands all criticism, then the fact that there is good reason to believe there are possible worlds will provide us with good reason to believe that this theory is true. I shall argue that of the theories under consideration, only one has the hope of being satisfactory: an Aristotelian modification of Leibniz’s theistic theory. Until a better theory should be found, the fact that this theory fits the theoretical data while the others do not gives us some reason to think it true (and, by implication, to think that God exists). But first we must consider the alternatives.

Part III. The Lewisian ontology of extreme modal realism

Section 1 The Lewisian account of possible worlds

For purposes of discussion of Lewis, I will call a maximal mereological sum of objects that are spatio-temporally related to one another a “universe”. I shall assume that being spatio-temporally-related-to is a transitive and symmetric relation.^{^[51]} By definition, if there are two distinct universes, there are no spatio-temporal relations between them. David Lewis’s extreme modal realism (EMR) then claims that each possible world is an existing universe ontologically on par with the universe we in fact inhabit and every possible way for a universe to be is a way that some concretely existing world is. In a Lewisian context, thus, “possible world” and “universe” are interchangeable. Not so, of course, in other contexts: if possible worlds are maximal sets of compossible propositions, say, then a possible world is not a universe, since there are no spatio-temporal relations between propositions or sets of propositions.

A proposition, then, is true at a world providing it truly describes a state of affairs obtaining in that world. Quantifiers in many ordinary language propositions are restricted to the world at which we are concerned with their truth value. The tokening of a proposition is truthful if and only if the proposition is true at the world at which it is tokened. It is true at w that there exists a unicorn if and only if in the concretely existing universe that w is, there is an individual that is a unicorn. My ordinary utterance of “There exists a unicorn” is truthful if and only if in our universe, i.e., in the maximal aggregate of things that are spatio-temporally related and that includes me, there is an individual that is a unicorn. It is irrelevant whether there are such individuals elsewhere. Thus, it is false that there exists a unicorn. However, context may indicate a wider scope for quantifiers, allowing them to range over individuals in multiple worlds. Thus, speaking as a Lewisian philosopher, I could say “There exists a unicorn”, provided the context makes clear that I allow the quantifiers to extend over the domains of all worlds.

For Lewis, ontologically all worlds are on par. Our world and universe is just one among infinitely many. We say it is actual, but this only says it is home—being actual is not an absolute non-relational property of our world, any more than something’s being home (as opposed to being a home) is of a house. Like something’s being home, a world’s being actual is nothing but an indexical claim. The actuality of the actual world consists in nothing but its being the maximal spatio-temporally connected aggregate that contains us. Moreover,

at any world w, the name “the actual world” denotes or names w; the predicate “is actual” designates or is true of w and whatever exists in w; the operator “actually” is true of propositions true at w, and so on for cognate terms of other categories.[52]

In the standard Lewisian model each individual exists in only one possible world. For if some individual x is a member of worlds w₁ and w₂, then all items in w₁ are spatio-temporally related to x and likewise all items in w₂. But since being spatio-temporally-related-to is symmetric and transitive, it follows that all items in w₁ and w₂ are spatio-temporally related to one another, and since the worlds are supposed to be maximal, w₁ and w₂ are identical.

But if every possible way for a world to be is a way that some world is, then surely there is a world at which ARP has dyed his hair green. Since it is false at the actual world that he has done so, one might think that it follows that ARP exists in some other world, a world where he has green hair. However, on Lewis’s account, it is not ARP himself who exists at that other world, but someone very much like him, a counterpart to him, albeit with green hair. Lewis can then enrich the semantics of “… is true at …” by saying that a proposition about some particular individual x is true at some other world (where x does not exist by the above argument) if and only if a counterpart to x exists at that world and what the proposition had asserted of x is true of the counterpart. On this semantic move, one can say that indeed there is a world at which ARP has green hair. That is a world at which ARP’s counterpart has green hair.

To say that one individual is a counterpart of another is to affirm that there is a certain contextually-determined similarity between them. If we consider being-a-counterpart-of as a relation that, for any given individual x in a world, picks out the unique individual, if there is one, in another world who most closely resembles x, assuming this resemblance is “close enough”, then being-a-counterpart-of will be a non-symmetric non-transitive relation for Lewis. The non-transitivity will be discussed below, in Section 4.2.1.b. The non-symmetry can be seen as follows. Suppose that I have an almost identical twin in the actual world. The only significant difference between him and me is that he has a big birthmark on his arm which I do not have. Now, there is a world, w, where there is a perfect copy of this twin of mine, but where there is nobody else even remotely like me or my twin. The copy of my twin is then a counterpart of mine and makes it true to say that I have a big birthmark on my arm at w. But I am not the counterpart of that copy of my twin—my twin is. For my twin resembles him more than I do, since it is he who has the birthmark.

Actually, for the interpretation of many propositions Lewis (1979b) thinks we will need to speak not just of truth at a world but truth at a world and at an individual in that world. This is how Lewis makes sense of the fact that there can be a world w containing me with a big birthmark I don’t in fact have and containing another person just like me but who does not have that birthmark. If we imagine such a universe, my counterpart will be the fellow without the birthmark since he is more similar to me.[53] Hence, that universe is not one at which I have a birthmark. Here, we need to interpret the statement that at w it is true that I have a birthmark and there is someone just like me but without the birthmark by assigning it a truth value at the ordered pair (w, the fellow in w just like me with the birthmark) and not just at w. This consideration shows that a Lewisian needs to allow that a given person has more than one counterpart at a world. The fellow without the birthmark can be a counterpart of mine as the fellow with the birthmark. The context constrains who could count as a counterpart of whom. It is certainly not true, except in a really strange context, that I have a counterpart with a birthmark at a world at which no human beings have any birthmarks but all donkeys do.

Since Lewis wants to reap the benefits for modality of the available of possible worlds, he will define possibility as truth at some world and necessity as truth at all worlds. To take care of the sorts of issues mentioned in the previous paragraph, one might have to sometimes modify this by saying that possibility is truth at some world-individual pair (or even world-individual₁-…-individual_n (n+1)-tuple), where the individual (or n-tuple of individuals) is a counterpart to the individual (or individuals) that the proposition is ostensibly about. Thus, we can say it is possible that I have a birthmark which actually I don’t have but there be someone just like me, but distinct from me, and without the birthmark. This is possible, because it is true at (w, the fellow in w with the birthmark) where w is as in the previous paragraph.

Section 2 Identity vs. counterpart theory

If we are going to believe that all the ways that some world actually is a way that some world really is, which I shall call “basic EMR”, we still have a choice whether to accept identity theory or counterpart theory. If we accept identity theory, then the same individual ends up existing in multiple worlds. Lewis refers to this as an “overlap of worlds”, in that different worlds overlap in having the same individuals in them. If we accept counterpart theory, then each individual exists in exactly one world but may have counterparts in some others.

2.1 Arguments for counterpart theory

The only positive argument for counterpart theory that I am aware of is an argument, perhaps implicit in Leibniz’s reasoning behind his view that each individual existed in only one world, to the effect that (a) there evidently are such things as essential properties, and (b) there is no intelligible way of drawing a distinction between essential and other properties (cf. Leibniz’s “Discourse on Metaphysics”). As a result of (a) and (b), it is concluded that all properties are in fact essential. But “existing in world w” is a property, and hence an essential property, and thus no individual that has it can lack it, and so if an individual exists in world w it exists in no other world.

This argument is not the usual reaction to someone’s claiming (b). The natural reaction is not to claim that all properties are essential, but that all properties are inessential. However, I shall not take the route of this objection to the argument, because there are very good reasons to believe in essential properties as we shall see in Section 4.2.1.b. Instead, I shall object to the claim that there is no intelligible distinction between essential and non-essential properties. For if basic EMR is true and if identity theory is true, there is a very simple and highly intelligible way of drawing the distinction: a property of an individual is essential if and only if the individual has this property in all worlds. To deny the intelligibility of this is to beg the question against the identity variant of EMR and not to give an argument against it. Moreover, even if we have a theory of possible worlds other than basic EMR, then the observation that there are clear cases of essential properties, e.g., being a point in space-time, and there are clear cases of inessential properties, e.g., having hair that is dyed green, outweighs the worry that there is no distinction. The fact that we do not always know how to draw the distinction does not militate against the existence of a distinction.

Lewis has given a more formidable argument. We normally distinguish between relational properties, such as being a father, and non-relational or innate properties, such as being square. But if the identity variant of EMR is true, then something (e.g., a sponge or an amoeba) may be square in one world and round in another (e.g., a sponge), and so being square is a property we must relativize to a world. Otherwise, we violate the law of non-contradiction. But, generalizing, it follows that every non-essential property is relative to a world.

For similar reasons, Lewis rejects the theory of numerical identity over time, preferring a theory of temporal stages that make up an individual considered as a space-time worm. However, it is certainly open for an identity theorist to insist on the older Aristotelian understanding of the law of non-contradiction according to which things are barred from having contradictory properties at the same time, but are allowed to have them at different times. By the same token, perhaps, the same being can have contradictory properties in different worlds.

But what does it mean to say that something is square relative to one world and circular relative to another? What is it really like? (One cannot give the answer that what things really are is their individual essences, whatever those are. For the essence is not the answer to the question of what things are like, but what they are simpliciter.)

The mystery about what properties like shape are, given that they turn out to be relational, does not even disappear if we say that space itself is relational. For in a given extended object, there will presumably be internal spatial relations as well as external ones. The internal ones will differ depending on the shape of the object. Thus, a circular disc has the internal property of containing a point such that all peripheral points are equidistant from that point. But since the disc could have been square-shaped, it follows, assuming basic EMR and identity theory, that these internal distances in the disc are not merely binary relational properties between points in the disc, but are ternary relations between the two points and the world (or maybe other items in the world). This seems to be an unpleasant complication.

But a complication is not a contradiction. It makes a theory more expensive, but does not knock it down. Maybe we are wrong about distances being binary relations. Maybe we are wrong about electric charges being innate properties. Perhaps everything is more holistically relational.

But now here is a more serious problem. What are the innate properties relational to? First to show the problem more clearly, suppose our identity theorist accepts a non-relational view of space-time. She will then presumably accept identity theory for space-time as well, since the same arguments as she advances for identity theory in other cases apply[54], so that a given point will exist in the space-time of several different worlds, and different worlds might have the same space-time. But if she does this, then she cannot say what might be the most natural solution to the problem, namely that shapes are relational properties with the other relatum being space-time. For if something exists in more than one world, it might be that both worlds have numerically the same space-time. So when we say that something is square relative to one space-time and circular relative to the other, we are affirming something contradictory—since there is only one space-time in the two universes, and nothing can be square and circular relative to the same space-time, at least not in the same way.

If we think space-time is relational, the same problem reappears. For unless we think that “the world” is something over and beyond the mereological sum of its parts, something that could itself stand in relations, we will want to say, e.g., that this amoeba is square in relation to the other individuals in the world, while in another world it is circular in relation to the individuals of that world. But it is conceivable that the two worlds have the same individuals, the only difference between them being the shape of this amoeba. Then, we have said that the amoeba is square in relation to the same individuals in relation to which it is circular. And this is absurd.

The remaining solution for the advocate of the identity variant of EMR is to insist that “the world”, which basic EMR insists is nothing else than “our unvierse”, is something that things can stand in relation to. Thus, the amoeba is something that is square in relation to one world and circular in relation to another. But now another problem appears. If the world is itself a concrete individual, as for Lewis it is, then the same kind of counterfactual reasoning that would make one think that this very amoeba which is actually circular could be square apply to the world at large. This very world, which in fact is populated by people, might have been populated by mere hydrogen gas. This very world or universe, which in fact is populated by our circular amoeba, might have contained this same amoeba which, though, was square. But then our amoeba is circular and square in relation to the very same world. The identity theorist who accepts basic EMR thus needs to exempt worlds from his identity theory. But this is not satisfactory, given that basic EMR equates worlds with universes, and hence with concrete individuals, and identity theory should surely hold either for all or for no individuals.

Another EMR-based argument against identity theory is that based on Lewis’s analysis of actuality. If for a world to be actual is for it to be the world in which I exist, and if I exist in more than one world, then more than one world is actual, which is absurd.

However, without EMR, we have no good argument against identity theory, this time construed as the assertion that when we are saying that something could have had that property we are not saying this in virtue of any individual other than the one we’re talking about, except the dubious one that proceeds by denial of the difference between the essential and the accidental.

2.2 Arguments for identity theory

2.2.1 General arguments

As has often been argued, someone very much like me becoming a spelunker in another world cannot be a truthmaker of the proposition that it is possible that I become a spelunker. How indeed, it is asked, is it at all even relevant to the proposition? This way of putting the issue is not sharp enough since that someone in this world who is very much like me becomes a spelunker is good evidence for the proposition that I could become one, too. But it is not because someone like me becomes a spelunker in this world that it is possible for me to become a spelunker. Indeed, the explanation goes the other way. Because it is possible for me to become a spelunker, it is possible for someone like me to become one, and those capabilities that ground that possibility in that other person also enter into the explanation of his actually becoming a spelunker. Why, then, should things be different when that fellow who is like me in fact dwells in another world? Why should he then become involved in the truthmaker? His example may inspire my imagination, thinking about him might lead me to regret me not having taken his exciting road in life, but his example plays an essentially different role than the role of making it true that I could have become a spelunker.

An analogy might help. Suppose we have a system which emits a light when a button is pressed. We certainly would not want to say that the fact that light was emitted at t₀, t₁, t₃, and t₄ when the button was pressed, with the situation these times being closely analogous to t₂ in terms of setup, is what makes it true to say that at t₂ the system also had the dispositional property that it would respond with light were the button pressed. Certainly, the facts about the activation of a like dispositional property t₀, t₁, t₃ and t₄ are evidence, perhaps even conclusive evidence, that the property was had at t₂. But one must not confuse the evidence with the truthmaker. In this case, such a confusion would lead one to buy into a Humean reduction of dispositional properties (and laws, too) to occurrent states. Lewis appears to be guilty of a similar reduction, albeit the occurrent states in his account are in other worlds. At the same time, the Humean comparison shows that a respected philosopher can hold a view of this sort, and so more of an argument is needed than just pointing out the kind of reduction that is going on.[55]

2.2.2 Attributions of ability

A person is only free to do something if she can do it. A Lewisian analysis of the modal claim that someone can do something will involve statements about what some of the person’s counterparts in other worlds in fact do. For instance, if one’s notion of ability is of the sort incompatibilists bring in, then a necessary condition for my now being able to do something is, in Lewisian terms, that some counterpart of me who shares a copy of my past up to now in fact does it.

However, here the objection discussed in Section 2.2.1 comes in: That I can do something is surely a statement reporting a fact specifically about me, not about another person such as a counterpart of me. After all, my being able or unable to do something has normative import for me: if I am unable to do something, then I was not free to do it, and hence I cannot be held responsible for not having done it. The objection, however, as given appears to merely beg the question. After all, the statement that my counterpart does something is, on Lewis’s theory, a statement about my modal properties, and hence in a straightforward sense of “about” a statement about me.

If the objection is not to beg the question, a more precise sense of “about” must be brought in. The notion of a truthmaker helps. We can say that a true proposition is about some existent thing, if that thing is one of the items involved in the proposition’s truthmaker[56], where the notion of being “involved” in the kind of thing which is the truthmaker of a proposition is taken as primitive and illustrated by propositions such as:

1. The butler, the master, the stabbing and the knife all are involved in the butler’s having stabbed his master to death with a knife.

2. Socrates is involved in the truthmaker of the proposition that Socrates’ existed.

3. Clinton is involved in the truthmaker of the proposition that there was a 42nd President of the United States.

4. If[57] pain is (reductive identity) nothing but a firing of one’s C421, then my C421 is involved in my being in pain.[58]

The objection now is that surely I must be involved in the truthmaker of the proposition that I am now able to, say, jump. But the truthmaker of that proposition on Lewis’s account, it seems, is simply the jumping of relevant counterparts of mine, and this involves merely the actions of certain counterparts of mine and not me.

This objection, however, fails because it commits a de dicto / de re modal fallacy. Suppose x, y and z are rigid designators of those counterparts of mine whose actions are involved in making it true that I can jump. Then, it is not only x’s, y’s and z’s jumping that makes it true that I can jump, but also their being relevant counterparts of mine. It is true that for Lewis that I can jump is made true by the jumping of certain counterparts of mine, but here “counterparts of mine” must be read de dicto: one cannot substitute the rigid designators “x, y and z” for “certain counterparts of mine”. But I am involved in the truthmaker of the proposition that x, y and z are relevant counterparts (or even just counterparts) of me, since the truthmaker of that proposition is the relevant similarity between me and x, y and z, so that likewise I am involved in the truthmaker of the proposition that I can jump.

But despite this analysis, one may have a feeling that x, y and z are interlopers in the truthmaker of the proposition that I can jump. Why should their actions be at all relevant to the truth of this proposition? The worry now is not that I am uninvolved in the truthmaker, but that some people who have no business being involved are. One may think that in some sense the proposition that I can jump should only be about me, my intrinsic properties and my immediate surroundings (including my relations to my immediate surroundings). For why should the actions of other people causally isolated from me have any bearing on the normative claims that follow from my being able or unable to jump?

There is, however, a way in which Lewis can accommodate some of the background of the intuition that I, my intrinsic properties and my surroundings are the only things that are involved into the truthmaker of the proposition that I can jump. Lewis can say that I, my properties and my surroundings are the only actual things that enter into this truthmaker. But this would only help if one thought, as Lewis vehemently denies, that to be actual and to exist are the same thing. Given Lewis’s distinction between the actual and the existent, the mystery as to why some non-actual but nonetheless existent persons should be involved in the truthmaker of the proposition that I can jump remains.

Moreover, the answer that the other persons involved in the truthmaker are non-actual will not do on Lewisian principles. For we can imagine a case where many of those persons whose actions make it true that I can jump are in fact actual. Imagine I am in an almost deterministic world in which at the beginning of every century one new human being has come into existence, lived for a hundred years, and then disappeared. Then a new human being, qualitatively indiscernible from the previous, appeared, and the cycle went on. This would of physical necessity go on forever in the future, except that there is one and only one bit of freedom and contingency that each human being is allowed. At the beginning of the 27th year of his life he can jump or refrain from jumping. If he jumps, everything will go on as before, a new human being appearing at the end of this one’s life, and so on. But if he refrains from jumping, the laws of nature and initial conditions are thus constituted that after he dies, no more humans will live, and there will no longer be any contingency allowed by the laws of nature. Suppose now I am one of these human beings, and I refrain from jumping. Up to my time, every century was repeated. After me, everything is different and the laws of nature allow no more contingency.

Consider the true proposition that I could have jumped at the beginning of my 27th year. This is made true by the jumping of my counterparts in worlds with the same causal structure. First assume that there are no qualitatively identical but distinct worlds. Then in fact, there are only two relevantly similar different worlds with the same causal structure as the actual world. One of these worlds, w₁, is a world of endless recurrence where in fact every human being from t = –¥ to t = +¥ jumps at the beginning of his 27th year. The other world, w₂, is one where an infinite number of human beings jump, but then one doesn’t, and then everything is different. The proposition that I could jump at the beginning of my 27th year is made true by my counterparts in these two worlds thus jumping (i.e., by the jumping of all the people in w₁, and of all the people in w₂ prior to the first who did not jump). But now observe that w₂ is in fact qualitatively identical with the actual world, and hence is in fact the actual world since we have assumed that qualitative identity implies identity for worlds. Hence, among those counterparts whose jumping makes it true that I could have jumped, are counterparts living in my world, i.e., actual persons. The same conclusion holds even if qualitative identity does not imply identity of worlds, because nonetheless the actual world will be among those worlds that are relevantly similar to the actual world, and my worldmates in the past will still be good counterparts for me. Hence in any case, some of the people involved in the truthmaker of the claim that I could have jumped are actual people who existed prior to me in the actual world—and these are interlopers by any account. These people’s existence and their jumping may illustrate my being able to have jumped, but surely my ability does not even in part consist in their having jumped.

Observe that there is a simple argument that shows that Lewis cannot possibly give a possible worlds analysis of “I can now jump” if, as is plausible, the truthmaker of this proposition only involves me, my properties and my causally relevant surroundings. For if the truthmaker only involves these entities, then the proposition would be intelligible in the absence of any worlds other than this one—since its truthmaker is wholly this-worldly. But if the proposition would be intelligible in the absence of worlds other than this one, then other worlds cannot enter into its analysis.

The best strategy for Lewis here would be the tu quoque. On competing views of possible worlds, entities other than I, my properties and my surroundings are also involved in the truthmaker of the proposition that I can jump. For instance, views which construct possible worlds out of abstract propositions will have abstract propositions involved in the truthmaker. For instance, the truthmaker of the proposition that I can jump might be the-proposition-that-I-jump’s being compossible with circumstances similar to those in which I in fact find myself, which clearly involves the-proposition-that-I-jump, an entity not part of me, my properties or my surroundings, and in no way causally relevant to anything concrete. Lewis can ask: Why should the intrinsic properties of this abstract entity be involved in my concretely being able to jump? Thus, perhaps, whether we be Lewisians or not, we simply must accept that my being able to jump involves things other than me, my properties and my surroundings.

Lewis’s opponents here can argue that the entities they involve in my being able to jump are abstract, and hence one should not complain of their presence here. But why not? Similar intuitions to those that say that the truthmaker of the proposition that I can jump should only involve me, my properties and my surroundings will also say that the truthmaker should be something entirely concrete—which on Lewis’s view it is.

However, tu quoque answers always have the weakness that they become useless when a theory comes into view that does not share the bad features of the old theories. In Part VI, I shall argue for a theory on which the truthmaker of the proposition that I can jump involves only me, my qualities and my surroundings, and is in a relevant sense a concrete part of the world. But this is a distance in the future. At present, however, we can observe that there is a theory in view that almost does this: Lewisian multiple worlds with an identity theory of transworld identity. For, then, although other-worldly properties of me are involved in my being able to jump on this theory, the only substances involved are ones that exist in this world: I and those substances that exist in my surroundings. For, that I can jump, on this theory, just says that I jump—in some world with relevantly similar surroundings. Hence it is more understandable why I could be held responsible for not jumping if I do not.

Further ethical considerations will be brought to bear on this issue in Section 8.

2.3 Conclusions about identity and counterpart versions of basic EMR

Neither the identity nor the counterpart versions of basic EMR are completely satisfactory. However, the identity theory only requires that we revise our beliefs about what properties are relational and what are intrinsic, while allowing an exception for worlds, all of whose properties need to be taken to be essential, since otherwise EMR evidently collapses—multiple worlds become one, as it were. Such an exception is an uncomfortable thing given that basic EMR takes worlds to be themselves individuals, but it is not absurd. The counterpart version of basic EMR does, however, involve one in serious problems as to what the truthmaker of possibility claims is. On the whole, the identity version may be preferable.

The fact that neither version is quite satisfactory provides a dilemma argument against basic EMR. Either the identity or counterpart version is true. In each case problems ensue.

Section 3 Indiscernible worlds?

Some of my arguments, notably those in Section 8.3, Section 9 and Section 10, against EMR will presuppose the thesis that there are no indiscernible worlds, i.e., that no two worlds are copies of each other. The arguments in Sections 8.3 and Section 10 can be made to work in the case where there are only finitely many copies of each world.[59] The real problem for those two arguments, then, is with the cases where there are infinitely many worlds. The argument in Section 9 will work if each world has an equal number of copies, finite or infinite.

Lewis (1986a, p. 84) himself is not committed to either the view that there are indiscernible worlds or to the view that there are not.

First of all, however, there is a simple argument based on the apparent absurdity of the following claim:

(19) It is possible for everything to be as it is while the actual world is not actual.

This claim is true if there are multiple copies of this world. For it is possible that the actual world not be actual, which claim simply says that there is contingency, and this possibility would be verified by another of the copies of this world in which copy everything would be as it is. If one thinks (19) to be absurd, then one will not accept multiple copies of worlds. But Lewis can always answer that we just lack proper intuitions about things like (19) since such claims are not made in ordinary life. Thus, other arguments are needed.

A quick argument against any indiscernible worlds could be based on the Principle of Identity of Indiscernibles which states that no two distinct objects are indiscernible, i.e., share all the properties describable in purely general terms. But one can also argue against indiscernible worlds on weaker assumptions about individuation, using an argument of Donald Turner’s (forthcoming). The argument claims that indiscernible objects can be individuated only if they are in a common space and/or time and thus capable of spatial and/or temporal separation. If one takes this view, then distinct Lewisian worlds could not be indiscernible, because there are no spatio-temporal relations between Lewisian worlds.

One difficulty with this argument is that current physics appears to provide a counterexample in the form of elementary particles that are bosons (French, 1988), e.g., photons. Such particles can have the same wave function. Consequently, they are not spatio-temporally distinguished, and moreover they can be indiscernible in all other respects. The defender of Turner’s argument against indiscernible worlds can proceed in two ways. The first is to adopt a controversial interpretation of quantum mechanics according to which while normally one describes the universe as containing n photons with k degrees of freedom each, it would be more correct to speak of a single “n-fold Photon”, existing in up to n places at once, and having nk degrees of freedom.[60]

The second retort is to weaken the principle that indiscernibles can be distinguished only by actual spatio-temporal relations to a much more reasonable principle that says that individuals can be distinguished only by actual or potential spatio-temporal relations or actual or potential differences in properties. Multiple photons that in fact share a wave-function can be distinguished by virtue of being potentially spatio-temporally separated—namely, by virtue of it being possible that they are thus separated. But even this very weak principle of indiscernibles does not allow one to distinguish indiscernible worlds. For on Lewis’s account there are no alternate possibilities for pairs of worlds, but only for individual worlds. So it is impossible to speak of a joint possibility for the two worlds where they are distinguished, e.g., by having different qualitative properties.

Another argument against indiscernible worlds might be from Ockham’s razor. Why posit multiple worlds where a single will do? But, a plausible reply goes, a single world will not do, since the absurdities resulting from EMR that we shall observe in Section 8.3 and Section 10 to ensue on the assumption that there are no indiscernible worlds show this.

However, there is a plausibility argument based on arbitrariness considerations. If there are multiple copies of some world, one asks: How many? If 227, why not 327? One answer might be that the number is proportional to the objective probability of that world being actual. This might neatly solve the problem of induction that will be raised in Section 9: the worlds where induction hold maybe are more numerous because there are a lot more copies of them. The problem with this probabilistic answer is that surely probabilities of various worlds being actual surely can be irrational numbers (certainly objective probabilities of events can be irrational, if one thinks that there are non-deterministic events of the sort that non-deterministic interpretations of quantum mechanics talk about, while this approach would make them only rational numbers, i.e., ratios of numbers of copies worlds. So this will probably not do.

Multiple finite numbers of worlds thus introduce a very unfortunate element of arbitrariness into EMR. And we could surely just as well conceive of different numbers of copies of worlds. However, this conceptual possibility is one that EMR has no conceptual resources to analyze. For suppose there are 167 copies of this world (counting this world as a trivial copy of itself), and we wish to say that there could have been 168 copies of that very world. How could we express this possibility? First of all, EMR as it stands has no conceptual resources for possibilities for multiple worlds at a time. But maybe we can extend EMR. We can say some proposition about multiple worlds is possible if there is some set of multiple worlds which is aptly described by that proposition. Thus, the claim that there could have been 168 copies of this very world is true if and only if there is a set of 168 worlds which makes it true. But these 168 worlds would all have to be copies of this world, thereby contradicting the assumption that there are precisely 167 such worlds. So, no, even an extension of EMR cannot handle this conceptual possibility. But unless there is a non-arbitrary account of the numbers assigned to each world, this possibility is surely there—and this possibility Lewis cannot admit.

Nor will it do to posit infinite sets of copies of each world. For the cardinalities of these sets will have the same arbitrariness that the finite numbers would. Why should there be À₁₆₇ instead of À₁₆₈ copies of this world? Is there not then a conceptual possibility of this cardinal number being different, a possibility EMR cannot countenance? This arbitrariness makes the theory of multiple copies of Lewisian worlds untenable, even if the Principle of Identity of Indiscernibles types of arguments did not.

There is a solution to the arbitrariness problem, and this is to posit that there is a Cantorian Absolute Infinity of copies of each world. Cantor believed that there was an Absolute Infinity, in which all the paradise of sets was contained. John Leslie (personal correspondence, 1999) used a similar solution to solve a problem for his theory according to which, in order that things be as best possible, there are infinitely many indiscernible omniscient deities. How many deities are there? Since there could always be more, the answer is Absolute Infinity. However, the notion of Absolute Infinity is so very murky that to have to escape to it is unacceptable. And of course if one escapes to it then one presumably can no longer even say that the collection of all worlds (or deities in Leslie’s case) is a proper class. And this is surely absurd if the worlds (or deities) are concrete things.

Another solution to the arbitrariness problem would be to say that the number of indiscernible copies of each world depends on the number of individuals in it. Suppose that a world w₁ contained two discernible individuals, A and B, and that it would be logically possible for A to occupy the role occupied by B while B occupied the role occupied by A. Then, there would be some plausibility to the idea that there might be two copies of w₁ in order to model the distinct possibilities of A and B occupying the roles they in fact occupy in w₁ and of them occupying switched around roles. If instead of accepting counterpart theory one accepted identity theory, then in fact one could say that in w₁ the very individuals A and B occupy certain roles while in the copy-world the two individuals occupy the same roles switched around. This is indeed quite a plausible story given identity theory, and hence the identity theorist might quite reasonably want to have indiscernible worlds.

And perhaps the counterpart theorist might want indiscernible worlds for a similar reason. Were w₁ actual, she might say that a different world could have been actual, one in which A occupied the role of B and conversely. The most natural Lewisian way of making sense of this claim might well be to posit indiscernible worlds. And the number of copies of a given world w₁ is then equal to the number of ways that individuals can be assigned to the roles of the individuals in w₁. Note, however, that the number of ways here is in fact plausibly greater than two. For, plausibly (at least for an appropriate choice of w₁), there will be a world w₂, discernibly different from w₁, which contains an individual C distinct from A and from B, which individual could fill either of the two roles. Thus, we want more copies of w₁ to account for this possibility. In fact, if we want there to be as many copies of w₁ as the number of pairs of possible individuals that could fill the roles of A and of B.

But now we have a problem. For on Lewis’s account, we cannot distinguish between a possible individual that could fill the role of A from a possible individual that is identical with A, since all transworld identity is just a counterpart relation, and those individuals that could fill the role of A are precisely those individuals whose counterpart is A. Thus, in fact, the number of copies of w₁ must be equal to the number of pairs whose counterpart is the pair <A, B> in w₁. This is indeed a non-arbitrary number, but the intuitive reasoning for getting this non-arbitrary number has just been undercut. For that <A, B> is a counterpart of <C, D>, is supposed to be a model of a transworld identity between <A, B> and <C, D>. The properties <A, B> have in w₁ represent from the point of view w₂ the properties that <C, D> itself would have were w₁ actual. Hence if one says that <C, D>’s filling the role that <A, B> has in w₁ is an alternate possibility from <A, B>’s filling that role, one has contradicted the central insight of counterpart theory. Hence, the counterpart theorist is not entitled to the above intuitive account of the number of copies of a world. And even if she were, there would be the problem of individuating such worlds.

And I leave it as an exercise to the reader to check that each of the arguments against Lewis that uses the thesis that there are no indiscernible copies of worlds either is an argument specifically against the counterpart version of Lewis’s theory or else is an argument that can be made to work given the above account of indiscernible worlds on assuming identity theory.[61]

Section 4 Lewis’s arguments for his ontology

4.1 The analysis of actuality argument

Taking actuality to be nothing but indexical is something that Lewis cannot escape. For suppose on the contrary that actuality is an absolute non-relational property that happens to accrue to one and only one possible world. But “[s]urely it is a contingent matter which world is actual. A contingent matter is one that varies from world to world. At one world, the contingent matter goes one way; at another, another. So at one world, one world is actual; and at another, another. How can this be absolute actuality? — The relativity is manifest!” (Lewis, 1986, p. 94). Hence, Lewis cannot allow a non-world relative notion of actuality.

But Lewis does not mind being saddled with the idea of actuality being world-relative and indexical. On the contrary, it can be taken as an argument for his theory that all possible worlds concretely exist as universes. For observe that if, on the contrary, only this world was or corresponded to[62] a concretely existing universe[63], then one could define a non-world-relative notion of actuality: a world would be actual if and only if it is or corresponds to a concretely existing universe. Hence, by modus tollens, if actuality is merely world-relative and a non-world-relative notion is unavailable, then all possible worlds do concretely exist as universes. In other words, Lewisian worlds must be ontologically on par. This shows that pace Lycan (1994), not only is Extreme Modal Realism not independent of the thesis that actuality is indexical, but most likely EMR is true if and only if actuality is indexical.[64]

An analysis of actuality that would make it indexical thus would provide an argument for Lewis’s theory. One might, for instance, be impressed by the fact that each world is actual at itself. This might seem to suggest that being actual is world-relative. But such an inference is fallacious. One might as well conclude that truth is story-relative because each story is true in itself, where we say that a proposition is true in a story if it is implied by the propositions of the story. The qualifier “... in a story” is truth canceling, and the opponent of EMR will insist that so is the qualifier “... at a world”.

One might think that Lewis can give an argument for the indexical account of actuality simply on the grounds of its ontological economy. If we adopt this account, then we do not have to worry about ontological questions of what the mysterious nature of actuality is, of what it is that singles out one world amongst others to be actual (cf. Lewis, 1986a, Section 1.9). But the ontological economy here is perhaps illusory. The opponent of EMR can simply say that a world is actual providing the universe that exists at that world, the universe that corresponds to that world, exists (“simpliciter” one is tempted to add, but the temptation should be resisted since the opponent of EMR does not recognize another kind of existence), or, equivalently, if the universe that exists corresponds to that world. At first sight, nothing has been gained. The notion of existence is as philosophically mysterious as the notion of actuality. Indeed, to say that a world is actual if the universe that exists corresponds to it is not to go very far in analyzing actuality. But what it does show is that the opponent of EMR can analyze actuality into a notion that Lewis’s ontology also has: existence (or, existence simpliciter, if one insists). After all, Lewis himself asserts that all of his worlds are existing universes. If this assertion is comprehensible, then so is his opponent’s insistence that a world is actual if the universe that corresponds to it exists.

This reply is not completely fair to Lewis, since it hides the fact that the anti-Lewisian notion of existence is of an absolute but contingent quasi-property. Many of the things that are might not have been. This is not so for Lewis’s absolute notion of existence, construed as existence in the totality of all worlds. Each world there exists and it makes no sense to talk of a possibility of its not having existed. Thus, it might seem, if we attack the Lewisian account of actuality as above we are committed to a notion of existence that is contrastive in a way that Lewis’s is not: existence is to be contrasted with mere possibility. However, this contrast is no more mysterious than one that Lewis himself must admit: existence (in the totality of all worlds) as contrasted with non-existence, i.e., impossibility. Just as Lewis will explicate impossibility in terms of absolute non-existence (square circles are impossible if and only if there is no world at which a square circle exists), the opponent to EMR will insist that non-actuality is to be explicated in terms of absolute non-existence. One is tempted to reply on Lewis’s behalf here: “How can you talk of some thing’s not existing? Is that not mysterious?” But this should be no more mysterious than talking of impossible “things” not existing.

The only difference is that the opponent to EMR owes one an account of how it is we can say a non-existent thing is possible, what it is that we are talking about when we talk of such things. But this is a different issue: we are no longer debating the question of relative versus absolute actuality, but of whether there is a non-Lewisian account of modal language.

4.2 The cost-benefit argument

Lewis presents a cost-benefit argument for the truth of his theory. On the benefit side, it solves the problem of what modality is and, he thinks, has other useful applications.

4.2.1 What is modality?

4.2.1.a A solution to the Parmenidean challenge and the mystery of modality

Recall the Parmenidean challenge. It seems that when we talk about things that are non-actual, we are not talking of anything at all, and hence our sentences are either false or meaningless. Moreover, even if it is granted that there is some way of rendering talk of unactualized possibilities meaningful by providing it with objects, nonetheless it is not clear what sorts of things modal language affirms of these objects. EMR handles both concerns. What is a possible but non-actual horse? It is nothing but a horse, albeit in a different world. We know what horses are, and the non-actual horse satisfies the description “horse” in exactly the same way the familiar horses in this world do. The non-actual horse exists, though not here, and there is no problem in talking about it. Neither is there any problem in talking of a Socrates who was never a philosopher or of a unicorn. All these things exist elsewhere. There is no difficulty about their ontological status, because they have the same ontological status as the things we are familiar with. They only differ relationally from these things in that they fail to be spatio-temporally related to us.

Moreover, EMR tells us what we are actually asserting of a situation when, e.g., we say it is possible but not actual. Rather than predicating some mysterious shadowy existence of a shadowy object, we are simply affirming that the object exists and lacks the relational property of being spatio-temporally related to us. EMR allows us to reap in the fullest sense the benefits of a possible worlds rendition of modal language (cf. Section 1). On the official story, there is no primitive modality left on the ontological level: everything modal is analyzed in terms of the relations between concrete and unproblematic entities. The mystery of modality is removed.[65]

It would be a mistake to object that what worlds exist depends on what worlds count as possible, since the only way we can know about what worlds exist is in terms of what worlds are possible. Epistemological priority does not imply ontological priority, as Aristotle had already noted. According to EMR, while it may well be that modal claims are “more knowable to us”, the plurality of worlds is “more knowable by nature”.

4.2.1.b Essential properties, counterpart relations and primitive modality revisited

However, there is an aspect of modality that Lewis’s official account does not provide a satisfactory analysis of. I clearly could have been a bank robber rather than a philosopher. But it is evident that I could not have been a set, the number seven, a point in space-time, the space-time manifold itself, a trope (a particular innate quality of a particular, e.g., this-negative-chargedness-of-this-electron), the Bank of England, the Swedish monarchy, the law of gravitation, or a particular act of jumping. It is less clear, but plausible, that I could not have been an electron. But it is not at all obvious whether I could have been a woman, or conceived by other parents, or a chimpanzee. Thus, there are some kinds of things that I clearly could have been, some that I clearly could not have been, and some where it is not obvious either way.

Even if it is not obvious either way, the presumption is that there is a fact of the matter as to whether I could have been something. After all, the question prima facie appears meaningful. It appears to have normative implications. For instance, in connection with one variant of the problem of evil, a theist might readily agree that it was possible for God to create a world populated with beings that are overall happier than we are—e.g., a world populated by an infinitude of angels that lead deeply meaningful lives—but that nonetheless God cannot on this account be said to be less than perfectly kind to us because in that world we would not exist whereas our lives are on the whole better than nonexistence. The debate over this kind of problem of evil presupposes the meaningfulness of the question whether we could have been the kinds of creatures that exist in the hypothetical world populated with happier beings.[66] Or consider the following case. A girl was conceived as a result of this act of rape. Because she was conceived through rape, her life was not perfectly happy. Assume, however, that overall her life was worth living.[67] Suppose then she sues the rapist.[68] Can the rapist’s lawyers claim that she could not have existed except as a result of rape, and hence overall has benefited from the rape since overall her life was worth living? I am not claiming that if the answer to the modal claim that she could not have existed except as a result of rape is positive, the rapist owes her nothing.[69] But I am claiming that the question is at least relevant and meaningful.

Another example where it is relevant what I could have been is the Rawlsian concept of justice. According to this concept, the just society is that which rational self-interested negotiators would come up with while under a veil of ignorance of their actual position and role in the world. Crucial to this account is the question of what exactly should be veiled. If the fact that the negotiators are not mosquitoes is veiled, then the just society will be one where the killing of mosquitoes is forbidden. If, on the other hand, I fail to veil the fact that my hair is dark, then in self-interest I will come up with a society where dark-haired people are especially favored. A natural suggestion as to what is to be veiled is that one should veil everything but the fact that one is a rational being. But that will not do, for then in a “just” society people who are severely mentally retarded for all of their lives will have no rights, which is absurd. It is reasonable to veil all properties that are not purely qualitative but have an indexical or haecceitistic component (if there are such properties). And since the point is to produce societal rules, we need only consider publically observable properties. But there is still the question of which purely qualitative observable properties should be veiled. Perhaps the only reasonable non-arbitrary determination of what is to be veiled is that if Q is a purely qualitative observable property, then it should be veiled whether one has Q if and only if it is both possible that one have Q and possible that one not have Q. In other words, we veil all non-essential purely qualitative observable properties. How well this approach works depends on what properties are in fact essential. If our only essential properties are being human and that which that property entails, then the approach will work as well as a Rawlsian approach can be expected to work.[70]