1. Shiftiness

Newton famously believed in “absolute space [which] remains homogeneous and immovable” (Reference Newton and Janiak1687/2004, 64). Leibniz gave a famous argument against Newton’s view (Reference Leibniz, Clarke and Ariew1717/2000, third letter, sec. 5): if there is absolute space, then all material things could have been in different absolute places, while bearing exactly the same spatial relations to one another. But it would violate the Principle of Sufficient Reason for there to be a different possible world so much like our own. Thus, there is no absolute space.

More recently, a certain style of response has gained popularity to the point that it might be considered the “textbook” view.Footnote ¹ It turns on a doctrine of modal metaphysics called antihaecceitism. The antihaecceitist rejects “transworld identity” and uses this rejection to support the view that there are no possible worlds that differ with respect to how particular things are without also differing qualitatively. So, the reply goes, possible worlds that apparently differ by Leibniz’s shift are really the very same world. But it is a bit difficult to understand what this response really amounts to. Some have worried that the antihaecceitists have just changed the subject and that this doctrine about possible worlds is irrelevant to the reality of space (sec. 2; Skow Reference Skow2007; see also Dasgupta Reference Dasgupta2011, sec. 4). There is a version of antihaecceitism about space that commits no such mistake—but stating it carefully depends on articulating a distinctive theoretical role for possible worlds to play, one that is importantly different from the familiar role it plays in possible worlds semantics and in most of the metaphysics literature (sec. 3). I’ll recommend this doctrine as a good candidate for what the antihaecceitists about space should mean. But it is not obviously what they do mean—for it amounts to conceding much of the Leibniz shift’s argument’s force, and in particular it requires revisionary metaphysics of space (sec. 4). Still, there is no better candidate view in the neighborhood that avoids this consequence (sec. 5).

First, though, a bit more detail about the argument. For the sake of fixing ideas I will focus on simplistic classical physics (although the Leibnizian argument and antihaecceitist replies apply to more modern theories as well: see, e.g., Earman Reference Earman1989, chap. 9). The shift argument I’ll discuss targets one particular theory of the nature of the physical world: a Newtonian theory of interacting particles under a central force law in absolute (“full Newtonian”) space and time (e.g., sec. 2.5). Here is an outline of what the theory says.

Call this theory Newt. The conclusion of the shift argument is that Newt does not give the right metaphysical picture of the world. It turns on the fact that uniform translation is a symmetry of Newt. What this means is that, however things could be according to Newt, there is another way they could be, just like the first except that all material objects are shifted uniformly in space. Let be a (nonzero) displacement vector. According to Newt, if it is physically possible for each particle to be at point at each time , then it is also possible for each particle to be at the point at each time . This is a consequence of Newt, because if the trajectories obey the law of motion, then so do the displaced trajectories—and according to Newt obeying the law of motion is all physical possibility requires. For short: shifts are possible.

(Note when I say “possible” without qualification, I mean physically possible—consistent with the laws of physics. I am not assuming any particular story about what it is to be a law, but I am assuming that laws are something physical theories aim to describe and can do better or worse at describing.)

But, as Leibniz reasoned, two possible worlds that were shifted like that would be “indiscernible,” and so we should not think that there are two really different physical situations shifted this way. So Newt distinguishes between physical possibilities with no real physical difference.

Let us say a theory gets things right if and only if it tells the right story about both the physical laws and the world’s metaphysical structure. (More on this notion later.) Here’s the main argument:

I will briefly rehearse the motivation for premises D and E. (I will return to S in the next section.) I think the best version is the argument from undetectability, which is developed explicitly by John Roberts (Reference Roberts2008) and Shamik Dasgupta (Reference Dasgupta2009, sec. 1.1; Reference Dasgupta2011, sec. 8). (But note that Dasgupta avoids putting things in terms of possible worlds in order to sidestep some of the issues I am heading into.)

By “empirically equivalent worlds” I mean worlds that agree on everything that can in principle be empirically detected. I take it to be obvious that things’ absolute positions are not open to direct observation.Footnote ³ Even so, it is not obvious that absolute positions are empirically undetectable—this requires argument. Lots of things are empirically detectable without being directly observable. For example, Newton’s bucket experiment showed (surprisingly!) that absolute rotation is detectable, at least if Newton’s physics is correct (Reference Newton and Janiak1687/2004, 68ff.). We can directly observe the curvature of water’s surface, or the tension in a cord, and the laws guarantee that in suitable circumstances these observable matters covary with absolute rotation.

Why aren’t there similar experiments we could perform to detect absolute positions and, thus, distinguish between shifted worlds? The reason is that if Newt’s account of the laws is correct, then shifts (unlike “twists” that change everything’s absolute rotation) preserve physical possibility: shifts are a symmetry of Newt’s laws. So shifted worlds do not agree just on everything that is directly observable, but also on the causal explanation in terms of the laws for anything that is directly observable. Any device we might build to detect absolute position will work in exactly the same way and give exactly the same observable reading in both of two shifted worlds. So shifted worlds are empirically equivalent. (The undetectability defense of D draws together two of the main strands of the shift argument’s development: both directly observable structure, as in the verificationist strand, and also causal explanatory structure, as in the determinist strand.)

Leibniz defended E (or something similar) by appeal to the Principle of Sufficient Reason (PSR): “that nothing happens without a reason why it should be so rather than otherwise” (Reference Leibniz, Clarke and Ariew1717/2000, second letter, sec. 1). God would have had no grounds for a rational choice between creating either of two shifted worlds—so God could not have created either of them. But the theological premise is not especially convincing. “Eliminating God from this scenario leaves no problem at all; since no such choice was ever made, there need never have been rational grounds to make it” (Maudlin Reference Maudlin1993, 191). Leaving God in the scenario, it still is not clear that there could not be arbitrariness in God’s choice to create, as Clarke maintains: “But this sufficient reason is often times no other than the mere will of God” (Reference Leibniz, Clarke and Ariew1717/2000, second reply, sec. 1). (Peter van Inwagen offers a version I like better [Reference van Inwagen and Morris1988]. A good God might make “disjunctive decrees,” willing general features but not every detail of what creation is like. Then some facts would have no deeper explanation than that the world had to be some way or other in order to fit God’s general purposes.)

I take the best defense of E to be an appeal to Occam’s Razor. If possible worlds agree with respect to everything empirically detectable, then any distinction our theory makes between them is empirically idle. So it is better to do without the distinction. Of course, this kind of principle is defeasible. As Tim Maudlin puts it, “Man is not the measure of all things, and there is no reason to believe that all real properties must fall within the power of human observation. Still, one should be made at least uncomfortable by the postulation of empirically inaccessible physical facts. Ceteris paribus, one would prefer a theory without them” (Reference Maudlin1993, 192). So really we should hedge the premise E and the conclusion of the argument:

(But I will not generally pronounce the hedge.) This means that the argument only has any real bite if Newt has some rival without commitments to absolute positions, which is otherwise about as good.Footnote ⁴ Still, the embarrassment of empirically superfluous physical structure is at least reason to look for such an alternative theory.

A theory with absolute position facts is less parsimonious than a similar theory without them would be. But while we want simpler theories, we do not want them too simple. After all, a theory without absolute rotation facts would also be more parsimonious, but Newton’s bucket showed us that these do play an essential empirical role in Newtonian physics. What shifts show is that absolute positions are “unbucketable”: if Newt gets the laws right, then absolute positions are empirically idle, fair game for elimination.

2. Cheapskates

Newt is committed to the physical possibility of shifts: its laws do not rule them out. But maybe the move is too quick from this possibility to the premise

Some hold that there is a gap between possibility and possible worlds, so that S does not follow from the possibility of shifts.

David Lewis is the chief defender of this sort of gap (Reference Lewis1986, sec. 4.4). The key to his defense is a rejection of transworld identity in favor of counterparts. According to Lewis, for a particular thing to be possibly is for it to have a counterpart which is in some world. The crucial way in which counterparthood differs from identity is that a single thing can have more than one counterpart in any particular world. Thus a single world can provide multiple ways a thing could have been, one for each of ’s counterparts in . (And in particular, an object can have a distinct counterpart in its own world.)

Lewis’s counterpart theory allows him to hold two doctrines that may at first seem incompatible. Lewis is an antihaecceitist in that he holds that there are no possible worlds that differ with respect to how particular things are, without also differing qualitatively. But he also accepts certain haecceitistic possibilities: for example, you might have lived in the first of two duplicate epochs of history, or you might have lived in the second epoch. There is no qualitative difference between these two ways things might have been. Lewis accepts that both are possible but holds that they do not correspond to different worlds. There is just one world that underlies both possibilities, and you have two different counterparts in that world. This combination of antihaecceitism about what worlds there are, with haecceitism about what might have been, Lewis gives the label cheap haecceitism.

These Lewisian doctrines look like they might present an escape route from the shift argument. Following Lewis, the “cheap haecceitist” about absolute space says that shifts are possible, but there are no possible worlds that differ merely by a shift. The possibility of particles being uniformly displaced from where they actually are does not correspond to a distinct possible world. Instead, it corresponds to the actual world, with shifted points considered as counterparts: each displaced point is a counterpart of the point . The shift does not give us a different world—just our own world reconsidered. So we can accept the possibility of shifts without accepting S.

(I should note that not all of those I cited in note 1 are cheap haecceitists: some of them deny even the possibility of shifts—and others seem to sometimes affirm the possibility and sometimes reject it. I will address some alternative views like this in sec. 5. Oliver Pooley is the most explicit defender of cheap haecceitism about space.)

Brad Skow objects to this reply: if worlds come apart from what is possible, then worlds are beside the point. All the shift argument really relies on (he claims) is contingency in where things are.

Skow’s claim is that if possibility and possible worlds come apart, possible worlds are irrelevant to the reality of space. The Leibnizian argument can be restated without mention of possible worlds:

The cheapskate’s antihaecceitism does nothing to diminish the force of the restated argument.

On behalf of the cheapskates, Pooley resists Skow’s charge: “But this is baffling. Leibniz’s PSR decrees that God must have acted for a reason when creating the actual world. Surely it is the possible worlds argument that is most relevant to this scenario. It is only if there are two distinct yet indiscernible worlds that God is forced to make an arbitrary choice in actualising one but not the other. On the other hand, if we suppose that there is just a single possible world corresponding to each way things might be qualitatively, there is no dilemma” (Reference Rickles, French and Saatsi2013, 102). He might offer the same speech about the detection version of the argument: it is different worlds that really raise the problem of empirically idle structure, not mere contingency.

But why would this be? The traditional role for possible worlds to play, both in the possible worlds semantics and in most applications to metaphysics, is as ways things might have been. Worlds are supposed to obey the “Leibniz biconditionals”:

But the cheapskate’s “worlds” do not do this. Say is some complete qualitative description of a Newt-world, and is some absolute position claim consistent with it. The cheapskate says that -and- and -and-not- are both possible. The first possibility would imply by P that is true at some -world. The second possibility implies by uncontroversial modal principles that if--then- is not necessary, which would imply by N that is not true at every -world. So there would have to be at least two -worlds—one where is true and one where is not true—which conflicts with antihaecceitism.Footnote ⁵

If the cheapskate’s worlds are not doing the traditional job outlined by P and N, then it is not clear what job they are doing. So it is not clear what distinctness of shifted worlds adds beyond contingency of absolute position. To satisfactorily answer Skow’s objection, the cheap haecceitist needs to meet two challenges. The first challenge is to articulate a role for possible worlds to play that goes beyond the traditional role of tracking what could have been the case. The cheapskate needs to say what distinctive work a “world” does that something that obeyed P and N would not be fit to do. The second challenge is to explain why the shift argument’s premise E only plausibly applies to the things that play this distinctive world role. Whatever the cheap haecceitist says is special about worlds as opposed to mere ways things could have been, it must make it plausible that there is something distinctively bad about being committed to distinct empirically equivalent worlds, without putting the same constraint on our commitments just to what is possible.

Lewis has a clear answer to the first challenge. According to Lewis, possible worlds are maximal spatiotemporally connected parts of reality—genuine concrete universes like our own. Lewis thinks these things do not exactly correspond to all the different ways things could have been because things of that sort could not have the patterns of overlap required for genuine transworld identity (Reference Lewis1986, sec. 4.2). But this way of meeting the first challenge is of little help: first, because not many of those who are attracted to antihaecceitism about space will accept Lewis’s peculiar doctrine about what worlds are; and, second, because this view of what worlds are provides no obvious way to meet the second challenge. Why would Occam’s Razor tell us to dispense with empirically idle differences between distinct spatiotemporally connected regions of concrete reality, as opposed to distinct ways things could have been? (Well, maybe the Razor cuts out all those Lewis-worlds—but why just the empirically equivalent ones in particular?)

The Lewisian reply is no help. So the cheapskate needs some other way of answering these two challenges. This is what I will now provide.

3. Facts and Worlds

Here is my proposed answer to the first challenge (see also Russell Reference Russell2013).

Some ways that we represent the world correspond to reality and others do not, although they may still succeed at some other job—they may be useful placeholders in our reasoning, or signal conventions, or do something else. It is not generally obvious at the outset which representations have which status. Distinguishing them is a project of metaphysics.

It would be nice to say more precisely what the difference between “metaphysically first-rate” and “second-rate” representations consists in, but this is not easy (cf. Field Reference Field1994; Fine Reference Fine2001; Dreier Reference Dreier2004; Sider Reference Sider2011, chap. 11). Consider an example. According to the Minkowski interpretation of special relativity, claims about absolute simultaneity are “meaningless”—not in the sense that they are unintelligible gibberish, but in the sense that they do not correspond to any genuine, determinate, objective way for the world to be. The (nowadays out of favor) Lorentz interpretation holds the opposite, that absolute simultaneity is objectively meaningful. Suppose the Lorentzian says:

(He might add: and I do not just mean simultaneous-in-my-frame.) What should Minnie the Minkowskian say about claim 4? The claim is intelligible, and Minnie may even want to use similar claims herself in suitably hygienic contexts. (For example, it is common practice to define “meaningful” notions like the space-time interval in terms of “meaningless” notions of spatial and temporal distance. This practice seems to rely on the intelligibility of the latter.) But in that case some of the things that might at first seem tempting to say about 4 will be difficult to sustain.

Minnie should not just say

This would be taking sides on a question which she thinks has no objective answer. She might want to say 4 is not true. But “true” has a “thin” deflationary use, on which it is axiomatic that

Minnie cannot say that neither 4 nor 5 is true, consistently with 6 and 7—unless classical logic fails (see Williamson Reference Williamson1996, 187ff.; Field Reference Field2003). (She might go on to distinguish “disquotational truth” from “correspondence truth,” like McGee and McLaughlin [Reference McGee and McLaughlin2000]—but this relabels the difficult distinction rather than explaining it.) Minnie might want to say that 4 does not express a proposition. But as with truth, we have a “thin” conception of propositions as mere “shadows of sentences,” according to which it is clear that 4 expresses the proposition that A and B are simultaneous.

Rather than search further for some account of what Minnie should say 4’s “factual defectiveness” consists in, I will just assume that we can make sense of the distinction somehow or other and put a label on it. Using “proposition” in an inclusive, lightweight sense, we can say that some propositions are factual.Footnote ⁶ It is hard to see how we could get by without some distinction of this sort—since debates with the same general structure as the one between the Lorentzian and Minkowskian are everywhere in philosophy of physics. I will mention just a few of the many examples. Theories of space-time structure are distinguished by whether they are committed to objective facts about which events happen at the same location, or about how fast things move, or about which things accelerate—and so the question of what kind of structure space-time has is partly a question of which of these are genuinely factual matters (e.g., Earman Reference Earman1989, 30ff.). A related debate is about whether there is any objective difference between a universe like ours and its time reversal (e.g., Maudlin Reference Maudlin2002). Another is about whether there are objective facts about the values of quantities that vary under gauge symmetries (e.g., Healey Reference Healey2001). Yet another is about whether quantum mechanics teaches us that there is really no objective fact about whether Schrödinger’s cat lives or dies (as opposed to merely living-at-a-branch-of-the-wave-function; e.g., Saunders Reference Saunders1995). These debates are difficult to understand without some notion like factuality.

If we accept a distinction between two sorts of propositions, factual and nonfactual, then we can also make sense of a parallel distinction between two sorts of world. There is a “thin” conception of possible worlds as mere ways things could have been—possible specifications of all ordinary “thin” propositions—and then there is a “thick” conception of possible worlds as ways the real world could have been—possible specifications of the factual propositions.

In general, for any particular kind of proposition—or subject matter—there are various specific ways for things to be with respect to propositions of that kind. Take the matter of what color snow is: the propositions that snow is white, that snow is blue, that snow is light blue, and so on. A way for this matter to be ascribes some specific color to snow. If is a way for a certain matter to be, then we say that a -proposition is true at if (intuitively speaking) is one of the specific ways for to be true. Two -ways agree on a matter if and only if each -proposition is true at both ways or neither. A subject matter is a way of dividing up logical space—a way of making a certain kind of distinction between alternative ways for things to be and ignoring other kinds.Footnote ⁷

There are two important subject matters for our purposes. First, there is the subject matter that includes all propositions, including the “thin” ones. Among the ways for this matter to be, we should restrict our attention to whose which are possible, that is, those at which every physically necessary proposition is true. These are ways things could be, the “thin” sort of possible world. They obey the Leibniz biconditionals P and N. I will call them possibilities.

Second, there is the subject matter consisting of just the factual propositions. Again, among the ways for this matter to be, we want to restrict our attention to those at which every physical necessity is true. These are the “thick” sort of possible world—ways the world in itself could be—which generally I will just call possible worlds. (I am following Lewis’s usage in keeping “possibilities” rather than “possible worlds” hooked up to P and N. But this terminological choice is a bit arbitrary.)

Possibilities are more specific than possible worlds. A world settles every factual question, but mere possibilities can be even more specific than that: they can make merely “conventional” distinctions as well. Two thin possibilities may be merely “verbally different” ways of representing the exact same objective reality—that is, the same way for the facts to be. We say a possibility represents the (unique) possible world that says the same thing as about each factual proposition. So this is the characteristic principle for individuating worlds:

In this account, possibilities and possible worlds are devices for thinking about two different questions. We tell some story (about absolute positions, or which events are simultaneous, or which duplicate epoch you live in, or whatever). The first question is: could things have been the way the story says? If so, we have a possibility, and if two alternative stories are each possible, we have two distinct “thin” possibilities. The second question is: if things were the way one story says, would the world be genuinely different from the way the other story says, as opposed to merely a different way of representing what is really the same situation? If so, we do not just have distinct possibilities, but distinct worlds.

(I am not committed to anything about what internal structure worlds and possibilities might have, if any—the important thing is just their different roles. But one might take the view that the things that play the possibility-role are mathematical structures—the models of a physical theory. At any rate it is a common view that, like these possibilities, models stand in a many-one representation relation to physically possible worlds.Footnote ⁸ But the question of how best to interpret models is marginal to the issues at hand.)

Let us suppose that we accept this distinction between metaphysically “thin” and “thick” propositions and, thus, between metaphysically “thin” possibilities and “thick” possible worlds. This meets the first challenge, to sketch a role for possible worlds to play that makes sense of how they come apart from what is possible. With this understanding of the role of worlds, we can now meet the second challenge as well, to defend the cheapskate against Skow: it is really distinct worlds that matter for the shift argument, rather than mere contingency.

Skow’s unworldly version of the shift argument relies on the premise (2) that shifts are not possible. How might this be defended? First there is Leibniz’s own defense. Skow writes:

But even those who are convinced of Leibniz’s PSR should not be convinced by this, if they recognize contingency in the nonfactual. Nonfactual matters should not be thought of as part of the world that God creates. If there is no genuine difference between a -world and a not--world, then there is no arbitrariness in God’s creation—even if neither nor not- is necessarily true. (This might have been the thought motivating Pooley’s reply to Skow.)

What about the Occamist defense? There is a good reason to reject distinctions between empirically equivalent worlds. The reason comes from principle F: distinct worlds disagree on some fact. Empirically equivalent worlds agree on every empirically detectable fact, so if they are distinct, then they disagree on some undetectable fact. Thus a proliferation of distinct empirically equivalent worlds, of the sort that obey F, means a proliferation of empirically undetectable physical facts. And this is the sort (or anyway, one sort) of proliferation of metaphysical commitments that Occam’s Razor cuts down. On the other hand, mere contingency of nonfactual matters—proliferation of alternative metaphysically equivalent representations that are consistent with the laws—does not raise any such problem. Saying “there is no fact of the matter” is a way of rejecting metaphysical commitments, not taking them on. Occam’s Razor does not cut the nonfactual, so there is no empiricist objection to mere contingency of this sort.

4. Sophisticated Substantivalism?

This way of understanding cheap haecceitism about absolute position meets both challenges and escapes Skow’s objection. It says that shifts are possible, but shifted possibilities do not represent genuinely different ways for the world to be. I also think the world role I have sketched is a pretty good candidate for what the cheapskate might be implicitly relying on in order to make sense of the difference between worlds and possibilities.Footnote ⁹

But even though this kind of cheap haecceitism avoids the inference from “shifts are possible” to “there are distinct shifted worlds,” it does not escape altogether. For it does not amount to a way of rejecting the original premise S, that if Newt gets things right, there are distinct shifted worlds. The reason is that the particular theory Newt is committed not only to the possibility of shifts but also to there being a genuine difference between shifted possibilities.

I will spell out the argument for this carefully. Recall that for a theory to get things right is for it to tell the right story both about what is physically possible and also about what the world is ultimately like: both to get the laws right and to get the facts right. I noted in section 1 that Newt’s law of motion is compatible with shifts, and thus if Newt gets the laws right then shifts are physically possible. Applying principle P, then,

That is the first step.

To get the facts right, a theory needs to describe the world in factual terms. Newt describes the world in terms of the absolute positions that particles have at particular moments. If Newt gets the facts right, then these claims should correspond to reality: they should be factual. Consider the propositions: is at at (for each particle and position and time ). Call this matter where things are.Footnote ¹⁰ In short, then:

Now consider two shifted possibilities. These tell different stories about where things are: they give different answers to every question about the absolute position of a particular particle at a particular time. If one of them says that is at at , the other says is instead at at . If these propositions are factual, then the possibilities disagree on a factual matter. This implies, by F, that they represent distinct possible worlds. That is the second step.

Summing up:

The Newtonian substantivalist says that there are absolute position facts: where things are is a factual matter. But this implies that a difference in where things are is a genuine metaphysical difference, and thus a genuine difference between worlds, in the sense that sets them apart from mere possibilities.

Cheap haecceitism, spelled out the way I have suggested, says that shifts are possible, but they do not represent genuinely different ways for the world to be. This amounts to metaphysical antihaecceitism about space:Footnote ¹¹ it implies that there are no genuine facts about where particular things are at particular times. And so it implies that Newt really does not get the facts right, because Newtonian substantivalism is wrong. This looks like an unwanted conclusion: after all, the cheapskates are called “sophisticated substantivalists” and have seen their antihaecceitism as a way of avoiding the shift argument, not conceding its conclusion.Footnote ¹²

Despite this appearance, some “sophisticated substantivalists” may really be happy to accept metaphysical antihaecceitism and thus reject Newtonian substantivalism as I have framed it. This fits with some of their writings—Pooley is the most explicit: “Facts about the world are exhausted by purely general facts. The way the world is, fundamentally speaking, is purely qualitative” (Reference Rickles, French and Saatsi2013, sec. 3.3.3). The absolute position facts I have been discussing— is at at —are singular facts about the positions of particular things at particular times, so evidently Pooley rejects them. These facts are central to Newt’s characterization of the physical world. So evidently Pooley rejects the metaphysical adequacy of Newt, as well.

I tend to think of metaphysical views that reject commitments to absolute spatial positions of particular things as kinds of relationism, broadly speaking, rather than substantivalism.Footnote ¹³ But the label is not worth arguing over. (Dasgupta finesses the issue by distinguishing “thick” and “thin” substantivalism.) Whatever label we give them, the rejecters face the same challenge as relationists: to find some alternative empirically adequate theory of space without those commitments. Neither Newt nor anything like it avoids them. And Dasgupta persuasively makes the case that nothing the sophisticates have said amounts to a serious alternative theory (Reference Dasgupta2011, sec. 5). He also offers some promising gestures in the direction of an account of space-time that dispenses with particular absolute positions, but so far (as he acknowledges) nothing that amounts to a worked-out theory (secs. 10–11).

The puzzling thing is, if the sophisticates are really in this camp, why they have not seemed to appreciate the challenge to produce an alternative theory. Indeed, rather than taking up the relationist’s burden, the sophisticates seem to reaffirm what looks like exactly the sort of commitment that gets Newt into trouble (“a commitment to the real existence of space and its parts as concrete, basic entities in the world”; Pooley Reference Rickles, French and Saatsi2013, sec. 3.3.1, original emphasis). So I am not sure what they have in mind. I have offered a view that makes sense of many of the things sophisticates want to say about shifts and possible worlds. But this view does not escape the shift argument’s conclusion, and sustaining the view requires revisionary metaphysics that does without facts about where things are.

Perhaps the sophisticates simply have not appreciated the consequences of their view. But another option is that they do not understand worlds in the way I have suggested—as tracking differences in genuine facts. Of course, if they are cheapskates, then they also cannot think of worlds as mere possibilities, the things that obey P and N. On that conception, it is incoherent to say that things could have been shifted but there are no shifted possibilities. So, unless they are to fall to Skow’s objection, they must be relying on some other conception of the role of worlds, as distinct from mere possibilities. The last thing I will do is briefly examine a few other candidate views and argue that they do not meet the sophisticates’ needs.

5. Other Antihaecceitisms

Perhaps the sophisticates’ “possible worlds” involve a distinction not between two kinds of proposition, but rather between two kinds of modality. The idea is that while shifts are possible in some “thin” way, even so they fall short of being possible in the full-fledged “thick” way.Footnote ¹⁴

The thought would have to be that the kind of possibility that shifts enjoy is different from the kind which is used in motivating premise D, that shifted worlds are empirically equivalent. Recall that it was crucial to D that shifts are symmetries of the laws: they preserve physical possibility and thus do not interfere with physical explanations of the phenomena. So the kind of possibility which is relevant to D is the kind that our physical theories aim to describe by stating laws.

This means that if shifts are not really possible in this way, then Newt does not succeed at describing what is genuinely physically possible: Newt does not get the laws right. So it looks to me like this too amounts to accepting the conclusion of the shift argument—but in an importantly different way from either the relationist or the metaphysical antihaecceitist. This view does not require revision of Newt’s metaphysical account of spatial structure. Instead it requires some alternative account of what is physically possible, in the “thick” sense involved in physical laws and explanations. I will consider three ways this alternative account might go.

One thought is that physical laws ought to be purely general, rather than involving singular necessities. Newt’s law of motion is de re. The theory begins with an ontological preamble—there are positions, times, and particles, with certain structure—and then it goes on to the law of motion: necessarily, those things satisfy a certain dynamical condition, . In this account, the ontology takes wide scope over the laws. But one might think this is the wrong way around: maybe the laws should take wide scope over the ontology instead. A revised system would say: necessarily, there are positions, times, and particles, which have certain structure and satisfy the dynamical condition. This version makes no requirement that it is the same positions, times, and particles in each situation.

But if this is all the laws say, then they are extremely liberal about singular possibilities. It is consistent with a law of this form for any object to have any possible qualitative profile whatsoever. This is not antihaecceitism, but extreme haecceitism.Footnote ¹⁵ In particular: this view still says that shifts are physically possible—and in fact, so are arbitrary geometric jumbles of points, or for positions to swap roles with particles. We might still say we conventionally ignore some of these possibilities in most circumstances.Footnote ¹⁶ But this view is not a way of escaping the shift argument. Purely general laws put even weaker constraints on absolute positions than Newt’s, so they make it even harder to see how things’ absolute positions could be detectable in principle. In the extreme haecceitist revision, absolute positions dangle even looser than in Newt.

A second way to revise Newt is in the opposite direction: not by stripping down its de re commitments but by strengthening them. One way of doing this is to say that spatial positions have their occupation properties essentially: for each and and , either it is necessary that is at at , or it is necessary that is not at at .Footnote ¹⁷ This will have to be combined with the view that it is contingent which absolute positions there are (as in the extreme haecceitist version), or else this theory would rule out all physical contingency whatsoever. This theory rules out the physical possibility of shifts: it says that if a particle is at , then it could not have been at the position —although it could have had a position that is not spatially related to at all. Insofar as we want to say that, in some sense, could have been displaced from where it is, we will have to gloss this in terms of counterparts.

I have no knockdown argument against the essentialist revision, but there are a few reasons it does not seem very attractive. Shifts show us that there is a mismatch between Newt’s account of the laws and its account of the facts—its facts outrun its laws, in that some of its facts vary under empirical symmetries. This mismatch offends against parsimony. Eliminating these facts would restore balance and remove the offense. But it only adds to the offense to instead tack on new laws that do not have shifts as symmetries. To be sure, this would also restore a kind of balance, ensuring that the facts in question do not vary under symmetries of the laws—but only by matching the excessive complexity of the facts with further complexity in the laws (i.e., in the physical necessities). I have no objection to essentialist principles as such, but in this case they do not seem well motivated. Unlike the law of motion, these extra necessary principles do not play any natural role in physical explanations. Thus they do not make it plausible that absolute positions really are detectable in principle.

At a more abstract level, shifts show us that Newt has an equivalence class of possibilities where just one possibility would empirically suffice. The essentialist fix is to pick a single member to do duty for the whole equivalence class. But this just feels arbitrary. It seems more natural to say that there is no genuine distinction being made among the equivalent alternatives rather than to admit the distinction and privilege one of the possibilities.

One final version is worth considering. Rather than saying the laws are more permissive than Newt about singular possibility, or that they are more demanding, one might want to say that there are just no facts about singular physical possibility. According to this view, the laws are purely general in that physical possibility and impossibility are only really “meaningful” for general propositions, in the first instance.

This way of understanding the sophisticates is strongly suggested by their common rejection of transworld identity. What does this rejection amount to? One could understand it as simply denying that any individual exists at more than one possibility. But David Kaplan seems right to say, “that view is properly reserved for the Haecceitist who holds to an unusually rigid brand of metaphysical determinism” (Reference Kaplan1975, 723)—like the essentialist we just considered. A different way of taking it, the view Kaplan originally called “anti-Haecceitism,” denies that transworld identity even makes sense: “There is no notion of trans-world being” (Kaplan Reference Kaplan1975, 723). In Kaplan’s terms, the view is that the overlap between the domains of individuals at different possibilities is “an artifact of the model” we use to represent what is possible rather than “a feature of the metaphysical reality being modeled,” that is, what is genuinely possible (722). Or as Kit Fine puts it, “anti-Haecceitism states that the identity or non-identity of individuals in distinct possible worlds is a matter of convention” (Reference Fine1978, 125; see also Reference Fine2005). If possible worlds correspond to what is possible in the traditional way, then Fine’s technical results show that the thought that transworld identity is not meaningful lines up with the thought that de re possibility is not meaningful. The things you might say that turn on which transworld identities hold are the very same things that turn on what is possible for particular individuals (at least in the language of first-order quantified modal logic).

So perhaps it is best to understand the cheapskates as advocating this kind of antihaecceitism: conventionalism about de re modality. Now, when applied to physical possibility, I do not see the appeal of the view. Physics teaches us about the nature of physical things. Some philosophers understand physical possibility as fundamentally resting on objects’ “essences” or “powers,” where these are thought of as something that cuts finer than just how they must or can be (e.g., Demarest, Reference Demarestforthcoming). Anyone who accepts that sort of view should accept the basic point that de re physical necessity is part of the realm of scientific inquiry. I do not accept that sort of view—I am thinking mainly of the laws as our primary tool of physical explanation rather than either of these alternatives. But even so, I do think the laws give us insight into the nature or essence or powers of the physical world and its parts at least in the coarse-grained sense of what is physically possible for them. Certainly (if we take physical possibility seriously at all) laws at least constrain the possible distribution of masses and distances and that sort of thing. But if physics gives us this kind of insight into the natures of physical properties and relations, what is so special about physical individuals, that the laws must be silent about them? One possible answer is that there are no facts about such things for the laws to govern. But if we have ruled that answer out, I do not see why we should think that singular physical possibility makes any less sense than qualitative possibility.

Still, regardless of its other merits, it is hard to see how conventionalism about singular physical possibility helps as a reply to the shift argument. It is true that if there are no objective facts about singular possibility, then we cannot straightforwardly say that shifts are a symmetry of the laws—that they take physical possibilities to physical possibilities. But while this escapes the letter of the undetectability argument for D, it does not escape the spirit. The reason we cannot say shifts are a symmetry of the laws is not because there is some causal-explanatory difference between shifted worlds—the sort of difference that might make it possible in principle to detect where things are in absolute space. On the contrary, since the laws are silent about the positions of particular things, these positions do not figure into straightforward physical explanations at all. De re conventionalism makes it no easier for absolute positions to have a role in explaining the phenomena: nomic silence about where things are is just as unhelpful as nomic contingency. So according to this view, absolute positions remain empirically idle superfluous structure—and it is still better to eliminate them, if we can.

None of these three ways of modally revising Newt seems to help. The shiftiness of where things are does not call for a different account of physical modality, but rather a better metaphysical account of spatial structure.