Finally, I discuss and combine two historical approaches. The first is an Aristotelian approach that says a non-actual event is possible is to say that some actual substances could have initiated a causal chain that could lead up to the event in question. However, it can be shown that some plausible global possibility claims can be made true on this account only if there is a necessarily existent first cause (or aggregate of first causes) capable of initiating very different universes. On the other hand, Leibniz made possible worlds be ideas in the mind of an omniscient necessarily existent deity. Leibniz fails to explain what it is that makes these possible worlds possible , but if we were willing to combine his story with the conclusion drawn from the Aristotelian one, we could get the following story: Possible worlds are ideas in the mind of an omniscient deity and what makes them possible is that this deity has the Aristotelian capability of initiating causal chains that can lead to them being actualized.

David Lewis makes possible worlds be concretely existing universes. Unfortunately, I show Lewis’s account involves set-theoretic, ethical, inductive and probabilistic paradoxes, and commits Lewis to an objectionable form of primitive modality that governs the choice of the counterpart relation. The most promising contemporary alternatives to Lewis’s theory have been the worlds of Adams and Plantinga constructed out of Platonic entities such as maximal collections of consistent propositions. However, these approaches fail to provide a satisfactory answer to the question of what makes true modal claims true. I also criticize some alternative accounts.

This thesis examines the alethic modal concepts of possibility and necessity. It is argued that one cannot do justice to all our modal talk without possible worlds, i.e., complete ways that a cosmos might have been. I argue that not all of the proposed applications of possible worlds succeed but enough remain to give one good theoretical reason to posit them. The two central problems now are: (1) What feature of reality makes correct alethic modal claims true and (2) What are possible worlds?

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, 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. 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. 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. Occasionally, I shall use ðp and àp to mean “necessarily p” and “possibly p”, respectively.

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 H 2 O. Some have argued that there are in fact two different kinds of modality. Some propositions, such as the proposition that H 2 O 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 “H 2 O”) contains hydrogen atoms. But that water is H 2 O 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) H 2 O 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 H 2 O 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) 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., “This 17373 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 H 2 O, 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 “H 2 O” 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.

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. 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.

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.

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].

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.

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. 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.

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?

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.

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.

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. 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, 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.

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.

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.

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.

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 … .

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. 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, but only the more modest claim that it could have been different in some respect. If Lewis is right, there are infinitely many universes. 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).

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.

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.

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?

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.

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? 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.

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! 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, then the notion of a conception of something contradictory is not viable.

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.

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.

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).

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.

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.

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.

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”.)

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 B where 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 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.

Recall Lewis’s account 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.

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:

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.

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.

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. 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 w 1 (and I shall sometimes for short talk of just the world w 1 rather than of kinds) everything happens as in the actual world w 0 until shortly before time t, but then the worlds begin to diverge.

The deterministic laws of w 0 are violated at w 1 in some simple, localized, inconspicuous way. A tiny miracle takes place. Perhaps a few extra neurons fire in some corner of Nixon’s brain.

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 w 0 and w 1 are to coincide.

(ii) In worlds of the kind of w 2 , 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 w 2 the same as w 1 .

(iii) Then, in worlds of the kind of w 3 , two small miracles, i.e. violations of the natural laws of the actual world, happen. First the same kind of miracle as in w 1 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 w 4 -type worlds two miracles also happen. The first of these is the same as in w 3 , but the second is a lot more impressive than it was in w 3 . Not only does the nuclear holocaust not happen, but all the traces of the button pressing are removed, and so after the second miracle, w 4 looks just like w 0 .

Which of these four kinds of worlds is closest to ours? To have an answer to Fine, Lewis must argue that it is w 1 , since it is only there that the nuclear holocaust happens. Now, w 1 is definitely closer to our world than w 3 by Lewis’s criteria, because the only advantage of w 3 is that it lacks a nuclear holocaust in its future and hence there is more approximate future agreement between w 3 and w 0 than there is between w 1 and w 0 . 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, w 3 has an extra small miracle, i.e. deviation from physical law. Avoiding small miracles is Lewis’s third most important similarity factor. Therefore, w 3 is better in terms of the fourth most important factor, and w 1 in terms of the third important factor, and so w 1 is to be preferred to w 3 as a candidate for a closest world.

What of w 4 ? It is true that w 4 matches w 0 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, w 4 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 w 1 does not. Hence, w 1 is closer than w 4 .

On the other hand w 3 is just about nowhere in space-time identical with our world. The lack of exact match anywhere in w 3 means that w 3 violates the second criterion of closeness, whereas w 1 only violated the third by having a small miracle. Hence, indeed, w 1 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 w 1 , 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 w 1 , 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 w 1 , which time is slightly before pressing the button. So, as Lewis admits, 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.

(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 counterfa