Modal sentences deal with things that can or must or cannot be. For example, we say that a triangle can be drawn, must be three-sided, and cannot be round. What makes a modal sentence modal? Short answer: its inclusion of a modal term like can (possibly), must (necessarily), and so forth. Such terms register that a claim is being qualified in such a way that the conditions of its truth are not limited to the way things actually are. But what is this modal talk about? Over the past two and a half millennia, answers have varied. Relatively recently, we have come to think of modes in terms of quantification over worlds: what is possible is true in at least one world, and what is necessary is true in all. Call this the worlds-reading (WR) of modal sentences. David Lewis (1941–2001) famously understands WR ontologically: these worlds really exist as spatiotemporal isolates, and are every bit as real as our own.

Contrast WR with a much older—and for a long time prominent—understanding of what modes are: terms whose operation on sentences expands (or ampliates) the extension of their terms, so that the terms range over possible objects, including non-existent ones. The modal properties of these objects are grounded in the causal powers of existing things: a triangle can be drawn because you or I can draw one; it is necessarily three-sided because there is no causal power (not even God) capable of making a triangle to be otherwise—at least, not without depriving it of its triangularity. Call this the objects reading (OR) of modal sentences. This is the view of John Buridan (c.1300–1361).¹ A careful examination of these views reveals that (i) they are incompatible, so that the Lewisian plurality is not a possible object or collection of objects; and accordingly that (ii) the worlds-reading, at least in its Lewisian form, is incompatible with a basic tenet of classical theism.

Why compare Buridan and Lewis? I have three reasons. First, Lewisian modal realism is well-known, and therefore provides a convenient off-the-shelf foil for Buridan’s modal ontology. Second, Lewis has clear ontological commitments, and so he is easy to pin down. Compare the ontologically agnostic Kripkean modal semantics and syntax: you and I may have very different views on what worlds are, but nevertheless agree on a Kripkean reading of the claims of WR. So the Kripkean account does not provide a clear and illuminating contrast for Buridan’s modal ontology, the way Lewis’s approach does. Third, contrasting Lewis and Buridan illuminates latent aspects of both. It gives us an insight into Lewis, hitherto unrecognised in the literature; and it reveals Buridan’s own views on the limitations on divine power—limitations he does not explicitly discuss at length. After all, placing restrictions on God’s power would have been a hazardous thing to do at the fourteenth-century University of Paris.² All the more so for an Arts Master who, as he explicitly acknowledges, is not qualified to teach theology.³ All the same, we can tease out the consequences of the views Buridan does express. And there is more here than meets the eye.

Let’s begin with WR, which is relatively familiar, and has two important shortcomings that point to two strengths of OR.

1 Possible Worlds

Nowadays, we tend to think of modality in quantificational terms: a modal is a sentence with a modal operator like “\(\square\)” or “\(\lozenge\),” for necessity and possibility, respectively. Such operators quantify across possible worlds. On these lights, \(\square\varphi\) just says that \(\varphi\) holds in all possible worlds, and \(\lozenge\varphi\) says that \(\varphi\) holds in at least one. The parallel, then, is with the ordinary first-order quantifiers: (\(\square\)-like) “\(\forall\),” and (\(\lozenge\)-like) “\(\exists\).”⁴

There is much to be said for WR, but here I will limit myself to two points. First, it’s versatile: we can use the apparatus of worlds to construct a wide variety of systems of alethic modal logic—that is, modal systems dealing with necessary truths, possible truths, and so on. We can characterise an astonishing number of systems in this way, and haggle about which one is best (or best for what). We can also characterise non-alethic systems to model knowledge and belief (epistemic logic), past, present, and future time (tense logic), and morality (deontic logic). WR, then, is extremely fruitful.⁵

Second, the WR is precise: can we give clear quantificational definitions of terms like necessarily and possibly, which might otherwise seem qualitative and murky. And, using Kripke’s apparatus of frames, we can characterise our systems with mathematical precision. But beyond all this, we might wonder: what are these worlds, anyway?

1.1 Lewisian Worlds

David Lewis’s answer to this question is famous and bold: all possible worlds exist, and they are just as real as ours. As he tells us (1986, 2):

The other worlds are of a kind with this world of ours. To be sure, there are differences of kind between things that are parts of different worlds [\(\ldots\)] but [\(\ldots\)] the difference between this and the other worlds is not a categorical difference. Nor does this world differ from the others in its manner of existing.

According to Lewis, there are many worlds—as many, in fact, as there are ways things can be. This ontological account of WR prompts two questions: how are these worlds externally distinct from each other, and how are they internally unified? Answers to both questions turn on spatiotemporal relations. To the former, Lewis tells us (1986, 3):

There are no spatiotemporal relations at all between things that belong to different worlds. Nor does anything that happens at one world cause anything to happen at another. Nor do they overlap; they have no parts in common.

Lewis frequently treats causation as the paradigmatic spatiotemporal relation. Since the worlds have no spatiotemporal relations to one another, there can be no causal interactions between them. They are therefore not like planets that are too far removed to interact with each other. They are, rather, spatiotemporal isolates. Call this Lewis’s isolation doctrine.

Importantly, Lewis does not say that different worlds cannot interact, as if blocked from doing so. Rather, they just do not: the notion of interaction between different worlds makes no sense within his theory. This requirement has a stipulative flavour—and, indeed, it is precisely that: a stipulation. This point is important, and we will return to it in section 3.

In like manner, Lewis accounts for the unity of worlds in terms of spatiotemporal relations (1986, 71):

If two things are spatiotemporally related, they are worldmates [\(\ldots\)] things are worldmates iff they are spatiotemporally related. A world is unified, then, by the spatiotemporal interrelation of its parts.

Again, this is presented in a stipulative way, though it is a corollary of the doctrine of isolation: worlds are spatiotemporally isolated, and therefore any spatiotemporally related things belong, eo ipso, to the same world. Here, whether or not causal interaction actually occurs is less important than immediately above: there does not need to be any obvious causal relation between two things for them to belong to the same world. A long-dead star too distant from Earth to interact with it nevertheless has spatiotemporal relations to us: it is some distance away in time and space, and it came into being at some time relative to us. It is, therefore, our worldmate.

The foregoing considerations can be distilled into a precise account of Lewisian worlds or possibilia, to wit:

The spatiotemporal relation is, in its most general sense, Euclidean. Let \(\textrm{R}\) be the spatiotemporal relation, so that \(\textrm{R}xy\) says that \(x\) is spatiotemporally (though not necessarily causally) related to \(y\). Then, by possibilia_L,

\[\forall xyz (\textrm{R}xy\land\textrm{R}xz\rightarrow\textrm{R}yz)\]

For clarity, we can also represent this diagrammatically, as follows:

Here, \(\textrm{R}\) is represented by arrows; if the relation represented by the solid arrows between \(x\) and \(y\), \(x\) and \(z\) hold, then the relation represented by the dotted arrow between \(y\) and \(z\) also holds.

This fact makes the case that the Lewisian plurality is impossible (set out in section 3) much easier to make, so let’s linger on it for a moment. Let \(\textrm{R}xy\) and \(\textrm{R}xz\). It follows that \(\textrm{R}yz\). If it didn’t, then \(x\) would be worldmates with two objects that are not themselves worldmates with each other. So there would be partial but incomplete overlap among at least two worlds. And this goes against both possibilia_L, and against commonsense thinking about spatiotemporal relations: if, for example, \(x\) is some spatial or temporal distance from both \(y\) and \(z\), then there must be some distance, however great, between \(y\) and \(z\) themselves. Therefore, the spatiotemporal relation R is Euclidean.

At the beginning of this section, I noted two significant advantages to the WR of ordinary modal language: WR is precise, and fruitful. Before we turn to the possible objects of Buridan, it’s worth asking whether WR has any drawbacks. For present purposes, I want to highlight two: WR does not represent what is going on in ordinary modal language, and taken on its own, it is uninformative about what grounds the modal properties of things.

To begin with the latter: the extensional account furnished by WR does not capture the ordinary notion of necessity for or as. For example, triangles are necessarily three-sided; three-sidedness is necessary for triangle-hood. Whereas you can paint a triangular object blue without removing its triangularity, you cannot, say, rearrange its parts in such a way that it gains (or loses) a side, and yet remains a triangle. This fact is not directly expressible on WR; all it can tell us about this (or any other) necessary claim is that it is true in every world. Fair enough, but such claims do not account for the inseparability of three-sidedness and triangularity.

Probably for this reason, most ordinary modal talk is not about worlds at all, but rather about things, and the ways they can be in this world. Scott Soames gives some remarks that support this point in his discussion of reference to non-existent objects (2010, 128):

Although this is controversial, the idea that we can refer to, and quantify over, only things that exist is, I believe, an unfounded philosophical prejudice at variance with our ordinary thought and talk. For instance, imagine that I have all the materials to build a doghouse, plus a plan specifying every detail of the design and construction, including how each of the materials will be used. From studying the plan and materials, I know exactly which structure I intend to create. Having identified it uniquely, I can refer to it, predicate properties of it, and even name it.

Soames’s dog house is a possible, non-existent object. What makes it possible is what he can do with materials and plans in this world. A lot of our day-to-day modal talk is like this: when, for example, someone says they can paint their house green, they are talking about themselves, and what they can do with their house—not about their counterpart, in a relevantly similar world in which their counterpart’s house is green.

Thus for all its versatility and precision, WR does not provide a full and accurate report of what is going on in ordinary modal language. Such language, judging by Soames’s example, is about possible things, at least some of which do not exist, whose modal properties are grounded in existing causal powers. I have called this the objects reading (OR) of modal language; it is the approach taken by John Buridan. It turns out that objects like Soames’s doghouse are precisely what Buridan has in mind in his analysis of possibilia.

2 Possible Objects

In the WR of modal language, modes operate on whole sentences, quantifying over possible worlds. In contrast, Buridan’s modal logic is not propositional but terminist; he thinks of modes as acting on sentences’ terms.⁶ Hence in his treatment of modal semantics in Tractatus de Consequentiis (2.4), he tells us that:

A sentence (propositio) [\(\ldots\)] about possibility has a subject term that is ampliated (ampliatum) by the modal term that follows it, so that it stands (ad supponendum) not only for those things which exist, but also for those things which can exist even though they do not. Hence in this way it is true that air can come from water, although this is not true of any air that presently exists.⁷

Air from water is, as Paul Thom (Thom 2003, 170) has observed, a simple account of boiling. The water in this pot could boil; but since it is not boiling, it is not true of any actual air that it came from this water. Hence this water is possible—but not actual—air. Elsewhere, Buridan gives the example of vinegar that could be produced from this wine, but will not, simply because I am going to drink the wine first (de Caelo, 1.25).⁸ These are the non-existent possible objects—or possibilia—to which the modal terms expand—or ampliate—the terms of a sentence.⁹

What are these non-existent possibilia?¹⁰ Buridan deals with possibilia obliquely in his logic and metaphysics, and so we will have to reconstruct his view from these discussions. Here, I present three key passages: one dealing with necessity, one with impossibility, and the last with possibility. Approaching Buridan’s account of the possibilia from these three angles will allow us to build up a consistent and robust picture of his views on what they are.

2.1 Necessity in the Prior Analytics

If \(\mathrm{S}\) is necessarily \(\mathrm{P}\), then (by modal duality) it is not possible for \(\mathrm{S}\) not to be \(\mathrm{P}\). Yet this analysis faces a problem. As Buridan asks in his Quaestiones super libros “Analyticorum Priorum” (QAPr 1.25), what is the modal status of the following sentence?

Is (1) necessarily true? In Prior Analytics 1.9 (310^a31), Aristotle clearly thinks so. And indeed, (1) serves as a stock example of a necessary truth in medieval logic.¹¹ Yet (1) is falsifiable, since God could annihilate all human beings. As Buridan tells us (QAPr 1.25, arg. 3):

If it were supposed that (1) were not necessary, it would be because God is capable of annihilating every human being. And in such a case, no human would exist, and so no human would be an animal.¹²

For Buridan, all affirmative sentences, including universals, have existential import, in contrast with negative sentences (both universal and particular), which do not. Thus Buridan would reject the reading of (1) given by classical FOL \((\forall x[\textrm{Human}(x)\rightarrow\textrm{Animal}(x)])\), which is capable of vacuous truth.¹³ Since there is no vacuous truth for affirmatives, (1) can be rendered false by the annihilation of its subject matter. Therefore, since (1) is falsifiable, it expresses a contingent truth.

Nor is this sort of contingency limited to sentences which, like (1), are taken from the natural sciences. It is also a problem for geometry:

If this were so, then no claim of geometry would be necessary either, since God can just as well annihilate all magnitudes as all human beings. And then it would follow that geometry would not be a science, which everyone would regard as false and unsuitable. (QAPr 1.25, arg. 3).¹⁴

God can annihilate everything with magnitude, and therefore magnitude itself. If God were to do that, then all the affirmative claims of geometry would be false, since the things they deal with would not exist. This is a consequence of Buridan’s anti-realism, which extends even to the objects of mathematics and geometry: if it so happened that there were no triangular arrangements of matter, then there would be no triangles (though it would still be possible to think and talk about them, like the roses of yesteryear). The same holds for all other geometric and mathematical objects.

Worse, even if God never gets that destructive, a crisis remains: the mere fact that geometric claims could be falsified by an act of divine will entails that these claims are contingent. If the truth of any claim is contingent, so is its subject matter. Since the subject matter of any science (scientia) must be necessary, it follows that even geometry is not a science. We can expect the other sciences—with the obvious exception of theology—to fare no better, given that God could annihilate their subject matter, too. So can there be any science (apart from theology) at all?

Buridan’s answer is yes: the claims of geometry (and of the other sciences) are necessary, but their necessity is attenuated: they are not necessarily true simpliciter. Rather, they are true “so long as” or “just when” (de quando) the things their subject and predicate terms stand for exist. Assuming no annihilation of their subject matter occurs, they will remain true—indeed, necessarily true:

Necessity “just when” (de quando) comes about from the fact that, whenever the subject and predicate terms do stand for anything, they stand for the same thing (I am here speaking of affirmative sentences). And in this way I say that the following are necessary: “Humans are animals,” or also “Horses are animals.” Indeed, even “A rose is a flower” is necessary in this way, even if there are no roses now. And although there is not a lunar eclipse happening right now, still the following is necessary: “An eclipse is an obstruction of the moon by the sun.” (QAPr 1.25, co).¹⁵

So a sentence like (1) is necessarily true, assuming the existence of the things it deals with, namely humans. Likewise, the claims of astronomy are true even when the events they describe are not presently occurring, since any time they do occur, the sentences are true. Thus, according to the account set out by Buridan in QAPr 1.25, a sentence like (1) can only be falsified by the annihilation of the things it deals with. There is no way to falsify (1) that leaves humans intact. So whenever humans exist, (1) is true.

Thus the contrast between necessity and contingency in terms of modality simply construed (simpliciter) is the contrast between unfalsifiability and falsifiability. The contrast between necessity and contingency in terms of de quando modality is the contrast between falsifiability only by annihilation (de quando necessity) and falsifiability by alteration (de quando contingency). That humans are animals is de quando necessary, because it can only be rendered false by the removal of its subject matter. On the other hand, the fact that some humans are bearded is de quando contingent, since shaving them alters the fact, but leaves the subjects essentially intact.

From these observations, we can give the following Buridanian definition of necessity:

This provides a good starting point for Buridanian modality; however there are crucial ambiguities that must be sorted out, if the above definition is to be consistent with the others we will look at below. Its adoption here is, therefore, tentative.

2.2 Impossibility in the Peri Hermeneias

In Peri Hermeneias 2 (16^a19), Aristotle tells us that nouns (ὀνόματα; Aristoteles Latinus: nomina) have signification. But Buridan asks, what about nouns like chimera, which do not signify anything at all?

We ask: does every noun (nomen) signify something?

Objection: it does not, because the term chimera signifies nothing apart from a chimera. And yet a chimera is nothing. Therefore, it signifies nothing whatsoever.¹⁶

A chimera not only does not exist, like the roses of yesteryear; it is, in fact, impossible. Buridan makes this point several times: the chimera is made of incompossible parts.¹⁷ In this respect, we may take it to be just like Schopenhauer’s wooden iron or Frege’s square circle (Schopenhauer (1819)), vol.1, §53; Frege (1884), §74). Because the chimera cannot exist, it cannot be signified. And this seems to present a semantic counterexample to the Peri Hermeneias definition of nouns, even though syntactically, chimera functions like any other noun.

Buridan’s solution here is to treat chimera as equivalent with the phrase “animal made up of parts that cannot be combined,” and to note that, although this whole phrase does not signify anything, it has significative parts (namely animal and part). The details of this solution need not detain us here. What is significant for our purposes is the role of the chimera as an impossible object, whose impossibility is a function of its putative combination of incompossible parts. We can use such impossibilia for our next definition:

This relatively straightforward definition will figure prominently in an important exegetical problem in section 2.4.

2.3 Potency in the Metaphysics

Buridan’s most detailed discussion of modal properties of possibilia is in his Questions on the “Metaphysics” of Aristotle (QM) 9.5. There, Buridan asks whether everything that something will do can be said to be what it is able to do. If so, we get some strange results, as Buridan points out:

A horse can come from wool. For earth comes from wool [by decomposition], and grass come from the earth, and from those grass which perhaps a horse will eat there can come horse semen, and, at length, another horse. And so even a horse can come from wool. And the same holds for all other modes of transmutation.¹⁸

Here the problem is whether or not the relation between \(\mathrm{S}\) and \(\mathrm{P}\) expressed by “\(\mathrm{S}\) is possibly \(\mathrm{P}\)” is transitive: if \(\mathrm{S}\) can be \(\mathrm{P}\), and \(\mathrm{P}\) can be \(\mathrm{Q}\), does it follow that \(\mathrm{S}\) can be \(\mathrm{Q}\)?

No, says Buridan: when we say that \(\mathrm{S}\) can be \(\mathrm{P}\), we are generally speaking in terms of a proximate potency, rather than a remote one: \(\mathrm{S}\) is proximately possibly \(\mathrm{P}\) if \(\mathrm{S}\) can become \(\mathrm{P}\) in no more than one transmutation. In this way, wool is possibly earth, because it can become earth in one transmutation (i.e., decay); similarly, earth can become grass, and so on. Any other potencies that require multiple transmutations are remote—as is, for instance, the potency of wool to become a horse. Hence Buridan tells us that:

Aristotle concludes the opposite. For he asks, when should something be said to be in potency, and when should it not? And he says that something should not be said to be in potency with respect to some form, except when only one transmutation is required, by which that form may be imparted on it.¹⁹

So although remote potencies can be discussed transitively, proximate potencies cannot. If the two are conflated, as in the wool-into-horse example, then, according to Buridan, the result is an equivocation.²⁰ Thus, although wool can decompose into earth, grass can grow from earth, and so forth, it does not follow that wool can become grass—much less a horse. Hence in speaking of possible horses, we are not speaking of all the things that, through multiple transmutations, could become a horse. If we were, then everything would be a possible horse, since, as Buridan observes, “anything can come from anything—albeit through several transmutations.”²¹

So much for possibilia arising from natural causes, like possible dirt that can be generated from wool. But a problem remains: why couldn’t God just rearrange the matter in a horse, say, to make it into a pile of dirt? So then a horse is possibly dirt (and vice-versa).²² And if so, then our main problem comes roaring back: everything is possibly everything.

Buridan himself does not consider this problem, but there is indirect textual evidence that he would reject such a claim: after all, he frequently tells us that the following is impossible:

Granted, it is not beyond divine power to transform the matter of a human being into a donkey by imparting on it the appropriate form. But again, (2) is impossible. How?

The solution is to appeal to the notion of change entailing annihilation (or destruction—more on this in a moment), which we saw above in connection with de quando necessity. For example, consider the following sentence:

Any formulation of (3) is true whenever Socrates exists. And while (3) can be rendered false, this can only happen by the destruction of Socrates. Similarly if, instead of being served a hemlock cocktail, Socrates met his demise by having his matter suddenly morphed into the form of a donkey, (3) would become false. But so would the claim that Socrates himself is a donkey, since Socrates himself would no longer exist. So Socrates is not possibly a donkey.

We have limited ourselves to transmutation in talking about things-possibly-being-other-things, and to one transmutation at that. Granted, then, God can morph Socrates’ matter into a donkey. But this morphing does not count as a transmutation in the natural sense, nor is it a potency belonging to Socrates. And so this fact no more entails that Socrates is a possible donkey than does the fact that Socrates can die and decay into soil, which then nourishes a plant, which a donkey eats, etc.

Here, then, we return to the original claim that impossibilia are incompossible combinations: donkey-Socrates, chimaeras—anything, in short, made up of parts that cannot be combined. Soon, we will see that Lewisian possible worlds, too, are Buridanian impossibilia. But first, we have to find a way of making the foregoing definitions consistent.

2.4 What are Buridanian Possibilia?

In a seminal (1989) paper, G.E. Hughes raises several questions about Buridan’s modal logic and its underlying ontology. Concerning the latter, he tells us (1989, 97):

For a long time I was puzzled about what Buridan could mean by talking about possible but non-actual things of a certain kind. Did he mean by a “possible A,” I wondered, an actual object which is not in fact A but might have been, or might become, A? My house, e.g., is in this sense a possible green thing because, although it is not in fact green, it could become green by being painted. But this interpretation won’t do; for Buridan wants to talk, e.g., about possible horses; and it seems quite clear that he does not believe that there are, or even could be, things which are not in fact horses but which might become horses.

Here Hughes makes no mention of the Metaphysics discussion—about horses, too!—which we just considered. This comes as no great surprise: that text is, to this day, neither edited nor translated.²³

Here, Hughes’s initial proposal is quite close to Buridan’s own account: a house is a possible green thing, because there are powers in the world capable of making it so. The issue of substantial change—things becoming horses—is somewhat more thorny, since it seems odd to speak of things which are not horses, but which could become horses, as Hughes observes. And yet this is precisely what we are warranted to do, as Buridan explicitly tells us, provided we limit ourselves to at most one transmutation: horse semen is not a horse, but it is a possible horse.

Frustrated by his version of the horse puzzle, and unaware of Buridan’s QM discussion, Hughes falls back on the familiar framework of possible worlds:

What I want to suggest here, very briefly, is that we might understand what he says in terms of modern “possible world semantics.” Possible world theorists are quite accustomed to talking about possible worlds in which there are more horses than there are in the actual world. And then, if Buridan assures us that by “Every horse can sleep” he means “Everything that is or can be a horse can sleep,” we could understand this to mean that for everything that is a horse in any possible world, there is a (perhaps other) possible world in which it is asleep. It seems to me, in fact, that in his modal logic he is implicitly working with a kind of possible worlds semantics throughout.

Here, Hughes first claims that Buridan’s modal logic can be understood using the modern apparatus of possible worlds semantics. But then he strengthens that claim: Buridan is in fact working with possible-worlds semantics, however implicitly.

From what we’ve seen of Buridan so far, we can see that at least the latter claim is mistaken. Buridan’s view of modality is grounded in causation: if there exists no power to make \(\mathrm{S}\) to be not \(\mathrm{P}\) (at least without annihilating \(\mathrm{S}\)), then \(\mathrm{S}\) is necessarily \(\mathrm{P}\). Likewise, if \(\mathrm{S}\) can be made to be \(\mathrm{P}\) (through at most one transmutation), then \(\mathrm{S}\) is possibly \(\mathrm{P}\). Thus something’s modal properties are grounded in the powers that exist in this world, which are capable of making it to be this or that way. In other words, Buridanian possibilia are, in general terms, objects, some of them nonexistent, whose modality depends on the causal powers of actually existing things. Since one of these existing things is the Almighty, and since the Almighty exists by simple (which is to say strictly unalterable) necessity, the modal properties of the possibilia are stable. There are no other worlds in the picture.

So much for what Buridan’s view is not. But the definitions we’ve distilled from the texts face an important exegetical problem: both necessity, on one hand, and possibility, on the other, are each in their own way inconsistent with the account of impossibility as sketched above. Impossibility, unlike necessity, does not turn on annihilation: a chimaera is made up of incompossible parts, not parts that would be literally reduced to nothing if they were combined. Moreover, there are diachronic possibilities, such as a human turning into a corpse, which are not synchronically possible: a human cannot be inanimate and rational at the same time. Just like chimera, inanimate rational animal therefore picks out an impossible object. The language of transmutations is therefore not applicable to synchronic incompossibilities. These facts call for a re-examination of necessity and of possibility as set out above. We will soon see that (i) these accounts can, happily, be made consistent, and (ii) that the consistent account that emerges gives us a straightforward definition of Buridanian possibilia.

First, the account of necessity, which turns on annihilation (rather than destruction) of the subject is too strong: for there is more than one way to make Socrates not a human: through (divine) annihilation—literal reduction to nothing—or through (divine or natural) destruction—undergoing a change that entails removal of his (human) essence. After all, following his death, Socrates is no longer a human, but this fact does not turn on any annihilation of Socrates.

Why then does Buridan discuss necessity in terms of annihilation at all? Recall that, in the QAPr, Buridan is (inter alia) worried about the falsification of geometry: if all magnitudes were annihilated, then the propositions of geometry would be rendered false. But this would not follow if everything with mass were simply destroyed—that is, if everything now existing were reduced to an undifferentiated soup. Even in that soup, there would be at least some dimension, surface, and so on. Conversely, the claim that humans are animals would be falsified if all humans were destroyed—that is, if everyone died all at once. Hence it seems that the reliance on annihilation is stronger than it needs to be for the definition of humans as animals, though perhaps not for the propositions of geometry taken collectively. I therefore propose a weakening of this requirement, at least for our definition of possibilia: \(\mathrm{S}\) is necessarily \(\mathrm{P}\), just in case \(\mathrm{S}\) cannot be made other than \(\mathrm{P}\) without destroying \(\mathrm{S}\).

The second exegetical problem is that the definition of possibility is quite weak: supposing that \(\mathrm{S}\) is possibly \(\mathrm{P}\) just in case \(\mathrm{S}\) can become \(\mathrm{P}\) through at most one transmutation, it follows that Socrates, while still alive, is possibly a corpse. Fair enough; but, as we observed, the combination of Socrates, qua rational animal, and corpse, qua inanimate object, is impossible.²⁴ Therefore, the most straightforward reading of impossibility, set out in section 2.2, clashes with the weak sort of possibility set out in section 2.3. What do we do?

It is true that Socrates is possibly a corpse. And it is also true that Socrates, while alive and barbate, is possibly clean-shaven. In the former case, Socrates loses his essence; in the latter he does not. We should therefore distinguish two kinds of change: one which involves loss of essence, but only through one transmutation; and another which leaves the subject intact.

Which kind of possibility is relevant to our purposes? Impossibilia are incompossible combinations; possibilia then should be possible ones. Since at least some transmutations involve change into something incompossible with the essence of the subject, as our example of rational animal and inanimate object shows, possibilia cannot comprise contrary diachronic states considered synchronically. We should, therefore, take the stronger reading of possibility, suggested by the account of impossibility: \(\mathrm{S}\) is possibly \(\mathrm{P}\) iff \(\mathrm{S}\) can be \(\mathrm{P}\) in a way that does not entail the destruction of \(\mathrm{S}\).

From these considerations, we can give the following definition of possibilia, which balances out the accounts in Buridan’s texts:

This definition casts a pretty wide net: possibilia will include not just the various natural kinds and subkinds we see in the world, but also anything else which could be produced by any power—including God—without destruction of the subject. So horses larger than planets are, presumably, (divinely) possible; as are humans capable of walking on water, virgin mothers, and so on. But conspicuously absent from this jungle of possibilia is the Lewisian plurality of worlds with which we began.

3 Are Lewisian Possible Worlds Possible?

—Or, to put the question in Buridanian terms: can God create a Lewisian plurality of worlds? First, the argument pro: it seems that God can indeed create as many worlds as God pleases. Recall our account of the unity of Lewisian worlds, set out above (section 1.1). So long as we conceive of a world as just a cluster of spatiotemporally interrelated possibilia, there seems to be no barrier in principle to clustering them. Here is why: some—and probably most—possible objects are made up of interrelated possible parts. Consider, for example, a possible watch that does not now exist. Such a possible watch will not be undifferentiated all the way through, like pâté, but will have interrelated possible parts—possible gears, possible springs, etc.

Now it would be arbitrary and just plain wrong to place a limit on how large such a possible object could be, at least in terms of what God can create: if a watch can be made the size of a tower clock, why not a watch the size of Manhattan? Likewise, it would be arbitrary to place a limit on their complexity: if a watch the size of Manhattan is permissible, why not a huge and complex astronomical horologium—one as large and complex as our universe, even?

From these considerations, we can distill two principles, namely:

From (i) and (ii)—so the argument runs—it follows that God could make worlds, roughly construed as manifolds of interrelated objects.

In fact, we can strengthen this claim: the possibilia just have to be in some possible world. Consider a possible object, say a fork: can such an object exist outside a world or manifold? Or must any such possible object exist within some kind of manifold? The existence of a fork outside some spatiotemporal manifold seems, if not impossible, then at least a little weird. A fork in the absence of other objects is one thing, but a fork in the absence of space-time is quite another. And so, it seems, possible objects only ever inhabit worlds. Thus a metaphysics of possible objects must, if it is to be coherent, collapse into a metaphysics of possible worlds.²⁶

So much for the argument pro; now for the argument contra. These worlds are either actual, in the sense that God has made them, or they are possible but non-existent, in the sense that God has not made them, but could. In either case, the question is: could God make an actual plurality of worlds? If so, then the Lewisian plurality is possible; if not, then it is impossible.

Following Lewisian doctrine, these worlds will have to be isolated: if they are not, they no more count as distinct possible worlds than do planets in different galaxies or cities in different epochs. They must not be at any spatiotemporal distance from each other. So can God create worlds that are not worldmates in this way?

Suppose God made these worlds. What does it mean to say such worlds are causal isolates—i.e., that they cannot interact? Distance will not do the trick: worlds are not causally isolated by any spatiotemporal distance, the way you and I are isolated from a long-dead star in Andromeda. Space is not what separates the worlds. Nor is time. Lewis has been clear.

Perhaps we can say that God stipulates that the worlds cannot interact: there is just an impermeable barrier between the worlds, analogous to the glass plates separating different tanks in a divided aquarium, or the walls splitting off different theaters in a cineplex. Perhaps it is physical, perhaps it is by divine fiat. Either way, we face three problems.

First, what happens when two things in different worlds interact with the dividing barrier or fiat that separates them? Suppose, for instance, that there is a barrier between worlds \(\mathrm{A}\) and \(\mathrm{B}\); and \(a\) and \(b\), which are possible objects in \(\mathrm{A}\) and \(\mathrm{B}\) respectively, are blocked from interacting by the barrier/fiat (imagine fish bumping into the opposite sides of a glass aquarium divider). Then a barrier that prohibits causal interaction between the two worlds, \(\mathrm{A}\) and \(\mathrm{B}\), nevertheless causally interacts with both of them. Therefore, that barrier will be a member of both worlds, according to Lewis’s definition: it has worldmates on both sides. But preventing such world-straddling was precisely what the barrier was supposed to do. We can try adding barriers so that the two barriers on the \(\mathrm{A}\) and \(\mathrm{B}\) sides are separated, a bit like parallel sheets of glass in a double-paned window. But then we get a regress: what keeps the barriers themselves apart? What would happen if one barrier collided with whatever separates it from the other? In any case, the barriers must both interact with whatever separates them.

Second, even if God could somehow separate \(\mathrm{A}\) and \(\mathrm{B}\) causally from each other, it would still make sense to think of them as related temporally: just as we can speak of one movie in a cineplex starting at the midpoint of another, so we can speak of a universe being half as old as another—that is, as being created midway along the life cycle of another universe. For instance, we could reasonably ask whether, from God’s perspective, the timeline of \(\mathrm{B}\) is half as long as that of \(\mathrm{A}\), whether \(\mathrm{B}\) already existed when \(\mathrm{A}\) was created, and so on.

Third, and most importantly, even if such worlds could be isolated from each other in a way that circumvents the foregoing two problems, they will still still be causally related via their causal dependence on God. Recall, from section 1, that the general spatiotemporal relation (though not necessarily causation) is Euclidean: if \(x\mathrm{R}y\) and \(x\mathrm{R}z\), then \(z\mathrm{R}y\). Thus although two worlds may not causally interact, they are not spatiotemporally independent, since they have the same cause. They are, then, causal siblings, even if they never interact. And if they are produced by the same cause, then they are causally related, if only in virtue of being created by the same God.

Lewis considers pseudo-plurialities like these (1986, 72), which, according to him, are not made up of truly isolated worlds. Their constituents are, rather, worldmates, even if locally they look like isolated worlds. Here is the one our cineplex and aquarium examples most closely resemble:

The spacetime of the big world might have an extra dimension. The world-like parts might then be spread out along this extra dimension, like a stack of flatlands in three-space.

But, as Lewis is quick to point out, this is not a true plurality. Thus there is no way, on Lewis’s account, to speak of temporal relations across truly isolated worlds: if there is anything like a God’s eye view of the sort we have been discussing, then the worlds belong to the same manifold. And if they belong to the same manifold, they are not truly isolated.²⁷

Here is the most common objection I have faced to this line of reasoning: it is not that Lewisian worlds cannot interact, in the sense that there is some mechanism keeping them apart. Instead, they just do not. We already noticed (in section 1.1, above) that the isolation doctrine is not a conclusion Lewis reaches by argument. It is, rather, a stipulation. And in fact, this is how Lewis presents it: right up front, on the second page of his (1986) exposition. It is thus more a starting point than a destination.

Accordingly, no criticism of this doctrine can address Lewis’s arguments for it, since he does not give us any. All that can be asked is whether it makes any sense. The answer, on Buridan’s metaphysics (or any metaphysics that posits one First Cause), is no. To anyone who espouses such a metaphysics, then, a Lewisian plurality of worlds must be something like Naive Set Theory: plausible on the face of it, but deep down self-contradictory. Lewis’s worlds simply do not work on Buridan’s framework. And, we might think, so much the better for Buridan.

I am not, by the way, the first to make any claims about the (in)compatibility of Lewisian worlds with classical theism, though the causal one I have been elaborating here is novel. Paul Sheehy (Sheehy 2006) sets out a number of problems for the classical theistic conception of God on Lewisian modal metaphysics. The most significant of these is his argument, suggested by Richard Davis (Davis 2008), that Lewisian possible worlds effectively chop God up, making each counterpart God a world-bound entity—an understanding that runs contrary to classical theism’s commitment to divine unity. Ross Cameron (2009) disagrees: Lewisian metaphysics can countenance abstracta existing outside of any world, as numbers do, so long as these abstracta are pure sets—that is, sets which contain only sets in their transitive closure (sets, sets of sets, sets of sets of sets, and so on, but no elements anywhere but sets, including the empty set). God, it seems, could be such a set—even if it’s doubtful whether such a set is what God’s believers believe in (or, anyway, believe they believe in). Subsequent debate (2021) has dealt with this problem of divine (unitary) existence and world-boundedness, and whether, in these ways, God can be countenanced on Lewisian worlds. Brian Leftow (2012, 541–545) has, moreover, criticised Lewis on the grounds that positing one God is more economical than positing several (more on this in a moment).

For my part, I agree with Cameron and Collier that a Lewisian ontology can indeed countenance an abstract, un-world-bound Necessary Being of sorts. And I agree with Sheehy and Vance that Lewisian worlds are incompatible with classical theism, albeit for reasons different from the ones they examine. After all, it is integral to classical theism that God has a creative—which is to say causal—role to play as well: God “created the heavens and the earth” (Genesis 1:1), is the One without Whom “nothing was made that was made” (John 1:2), the Originator, “Who commands only”Be!” and it is” (Al Baqarah “The Heifer,” 117), and so on. (Countless other sources could be cited to this effect, but you get the idea). This central aspect of God’s activity is incompatible with Lewis’s doctrines about the plurality of worlds. Accordingly, possible worlds of the sort we have considered here will be deeply incompatible with (monotheistic) medieval philosophy in general—even if certain aspects of a given thinker’s modal logic or ontology might remind us of this (by now quite familiar) framework.²⁸

What about Lewisian metaphysics considered in its own right? Even though a unified First Cause is not available on this framework, it does not follow that Lewis and his followers have to be atheists; if there is plurality in the worlds, there can also be a plurality of first causes. There is textual evidence that Lewis recognises this implication of his theory: in the introduction to the first volume of his (Lewis 1983) Philosophical Papers, he remarks in passing that his view is consistent with the claim that “there are countless gods but none of them are our worldmates” (xi). Since the worlds are, ontologically speaking, just like ours, it follows that our worldmates could include a local deity, and Lewis could merely be mistaken about the constituents of our actual world. So the Lewisian can still opt for a kind of polytheism, or mono-poly-theism, to adapt a term coined by Hart (2013, 127). But even basic classical monotheism is, on these lights, impossible. For Lewisian ontology is a jealous god.