Powered properties, modal continuity, and the patchwork principle

Abstract

The principle of modal continuity has become an increasingly popular bit of modal epistemology, featuring prominently in debates about mereology, value, causation, and theism. It claims, roughly, that degreed properties are modally unified. So, if the property of being three inches tall is exemplifiable, so is the property of being four inches tall, and five inches tall, etc. Despite its plausibility, in this paper I show that there is a class of counterexamples to modal continuity: what I call ‘powered properties.’ More surprisingly, I show that an instance of these powered properties is entailed by another widely popular family of modal principles: the Lewisian patchwork principles, also known as cut-and-paste, or recombination, principles. Thus, despite appearing to be similar, and motivated by plenitudinous intuitions about the nature of modality, it turns out that the continuity and recombination approaches to modality rely on crucially different pictures of plenitude.

1 Introduction

Much has been said about how to construct principles that extend our modal knowledge. Authors have proposed that we can develop guides for what is possible by appealing to conceivability (Yablo, 1993), counterfactuals (Williamson, 2007), essences (Lowe, 2012), and patchwork, or cut-and-paste, principles (Lewis, 1986). More recently, the principle of modal continuity––developed most notably by Rasmussen (2014)––has gained traction. Modal continuity is built on the intuition that differences in degree do not make for modal differences. For instance, if it is possible for a cup to be 4 inches tall, continuity tells us it is also possible for a cup to be 5 inches tall, and 6 inches tall, and 7 inches tall, etc. It would be surprising if it were possible for a cup to be 4 inches tall, but metaphysically impossible for it to be, say, 8 inches tall.

The intuitiveness of modal continuity has led to its application in a number of areas, including debates about the possibility of an infinite past (Schmid & Malpass, 2023), value (Rasmussen, 2018), causal essentialism (Gibbs, 2018), mereological universalism (Rasmussen, 2014), arguments for theism (McIntosh, 2022), and much more.

In this paper, I show that there is a rather large class of properties to which modal continuity does not apply, which I have termed powered properties. Even more interestingly (and troublingly! ), one of the instances of these powered properties is entailed by the Lewisian patchwork principles. Thus, in addition to constructing counterexamples to modal continuity, this paper evinces a surprising tension between two prima facie similar approaches to modal epistemology: the continuity and cut-and-paste approaches. While both are motivated by what might be called “plenitudinous” intuitions about the nature of modality, it turns out that these principles rely on subtly different conceptions of plenitude––i.e., of what counts as a ‘gap’ in modal space.

I will proceed as follows. In § 2, I explicate more formally the principle of modal continuity. In § 3, I develop the notion of a powered property and in § 4 I show how it operates as a counterexample to modal continuity. In § 5, I show how the patchwork principle leads to a powered property and thus conflicts with continuity. A few objections are considered in § 6, before a brief concluding remark in § 7.

2 Modal continuity

In a single sentence, modal continuity claims that there is no unified class of degreed properties with a modal gap. I take this definition from Rasmussen (2014: 528-9), who defines ‘unified class of degreed properties’ and ‘modal gap’ as such:

C is a unified class of degreed properties iff there is a transitive and asymmetric relation R such that (i) for all properties x, y ∈ C, either Rxy or Ryx, and (ii) property x is a finite distance from property y.

G is a modal gap in C iff C is a unified class of degreed properties such that (i) at least one member of C is exemplifiable, and (ii) G is a finite, proper subset of C such that no member of G is exemplifiable.

To put it another way, modal continuity is a claim about how degreed properties (properties that differ in mere degree, such as the properties being 3 inches tall and being 4 inches tall) relate to one another. In particular, the claim is that any class of degreed properties cannot have a modal gap: if one of the properties is exemplifiable, all of them must be. So, suppose I weigh three pounds. Then, modal continuity tells us that it is also possible I weigh four pounds, or five, etc., because the class of properties of the form being n-pounds is such that all of its members are exemplifiable.

It is important to note that Rasmussen develops continuity as a defeasible principle of modal epistemology, as it does not apply to all degreed properties. For instance, the principle, as stated herein, has to be refined to exclude blatantly inconsistent properties (e.g., being 3 pounds and 4 pounds) or properties that violate essential limitations (e.g., being human and eating n pounds of food). Thus, continuity comes with certain provisos, limiting what domains the principle applies to. However, once so-refined, continuity seems to be on firm ground qua modal principle, as evidenced by (i) its application in far-reaching metaphysical debates, and (ii) Rasmussen’s (2014: 531) remark that the refined continuity principle “is a continuity principle that has no clear-cut exceptions.”

However, the fact that continuity might still be regarded by some as defeasible means that we will have to contend with the possibility of the following challenge¹: even if I am successful in arguing that certain classes of properties––what I shall call ‘powered properties’––have modal gaps (and thus do not exemplify continuity), all this means is that modal continuity requires a further proviso, i.e., a proviso restricting the continuity principle’s application to non-powered properties. Naturally, one might then wonder: what, exactly, is the upshot of the ensuing investigation between modal continuity and powered properties?

This is a valuable question, for it helps clarify the purpose of our investigation. Let us suppose that such a proviso would be well-motivated, not ad hoc, and not detract from the all-things-considered plausibility of the modal continuity principle. First, it is important to recognize that, even still, the value of the cases to be explored herein would be that they reveal us to a novel domain in which continuity does not apply. As we will see, whether a property counts as ‘powered’ depends on the structure of the entities in question. Thus, powered properties reveal that, when doing modal epistemology, we cannot freely apply modal continuity to any property simply because it is coherent (and does not involve the essence of particular individuals). We must pay attention to an additional factor as well, namely, whether the entities in the property’s extension have a certain structure––a way of interacting with the world that is more common than one might initially realize. Thus, our investigation here is meant to shed light on the nature and applicability of modal continuity, irrespective of whether one takes the examples developed herein to be counterexamples to the truth of continuity, or instead as indicating a further domain of inapplicability. Indeed, I must confess that I myself am quite sympathetic to the application of modal continuity to certain classes of properties, such as weighing n pounds––my purpose is not to undermine those particular inferences. My goal is to show that there is an additional dimension, or factor, we must be attentive to in these contexts; one common enough that it is present in a range of extant metaphysical views, such as mereological universalism, and a particular finitism about space. This factor is even present in the application of other principles of modality, e.g., Lewisian patchwork principles.

Let’s get on, then, with our investigation.

3 Powered properties

I begin first with a few preliminaries about the notion of a powerset. Let S be a set with n-many members. The powerset of S, written as P(S), is the set of all subsets of S. For instance, let S be the following set: {a, b, c}. P(S), then, is: {{}, {a}, {b}, {c}, {a, b}, {b, c}, {a, c}, {a, b, c}}. Since, in most standard set theories, the empty set is a subset of every set (though not necessarily an element), P(S) includes the empty set. However, since this will be orthogonal to our purposes, I propose we work with a slightly non-standard notion of powerset: let P*(S) designate the set of all non-empty subsets of S.

An interesting feature of powersets is that the number of members they have is a function of the number of members in the original set. Where n is the number of members in S, and N the number of members in P*(S), we have: 2ⁿ– 1 = N. This entails that the number of members in a powerset has many “jumps,” or “gaps”. There are certain natural numbers x such that, necessarily, if Q is a powerset, then Q can never have x members. For instance, the number of elements in a powerset can never be even: for all naturals n, there is no even natural N which is equivalent to 2ⁿ − 1. Similarly, powersets can never have 5, 11, 13, or 29 members (since, for any natural n, 2ⁿ − 1 cannot ever be equal to any of these values). Thus, if Q is a powerset, then, necessarily, there are only a certain number of members Q can have.

It is because powersets cannot have certain numbers of members that they are ripe for a source of counterexamples to modal continuity.² Roughly, the idea will be that if the extension of some possible property P can be put into one-to-one correspondence with a powerset, then the class of degreed properties being such that there are n-many x’s such that P(x) is riddled with modal gaps––for instance, any even instance of n. Thus, continuity fails (to apply).

To see how this style of counterexample works, it helps to consider a direct application. A wonderful example comes from Comesaña (2008), who uses this feature of powersets to mount an argument against mereological universalism. For our purposes, we can gloss mereological universalism as the view that, for any two (distinct) material objects, there is a third material object (namely, the fusion of the first two objects). If we assume that there are atoms, then mereological universalism entails that the number of objects in a world w is just the cardinality of the powerset of the set of the atoms in w.³ To see this, suppose that O is the set of objects in w and A is the set of atoms. Let A = {atom₁, atom₂, atom₃}. Because atoms are material objects, atom₁$\in$O. The same applies for atom₂ and atom₃. Additionally, by our statement of universalism, we have that the fusion of atom₁ and atom₂––denoted ⊕(atom₁, atom₂)––is itself a material object, and thus a member of O. But if ⊕(atom₁, atom₂) is a material object, and so is atom₃, then, by universalism, ⊕(atom₁, atom₂, atom₃) is also a material object, and thus also in O.⁴ By repeated application of universalism, we will find that every subset of A can be put in one-to-one correspondence with a member of O. Thus, assuming universalism and atomism, the number of objects at any world cannot take on certain values (e.g., it is metaphysically impossible for there to be a world with exactly 5 material objects). So, if one accepts these positions, then they ought to reject the application of modal continuity, as the class of properties of the form being such that there are n-many objects has a modal gap (assuming there could be some n of objects).

Another way to see how this works is to see that universalism is claiming something about the property being an object: namely, universalism is claiming that its extension can be put into one-to-one correspondence with a powerset––i.e., that being an object is a powered property. And (exemplifiable) powered properties violate modal continuity.

Of course, mereological universalism is a controversial view, but the fact that it entails that modal continuity cannot be applied to objecthood is both interesting and suggestive. Depending on our views about how a certain property interacts with the world––in particular, whether its extension can be put into one-to-one correspondence with a powerset––that property may turn out to be riddled with modal gaps. And there are, I hope to show, many such cases, supplied by views less controversial than mereological universalism.

To develop a wider blueprint for constructing powered properties, let us get a better grip on the nature of powered properties. Where $\mathbb{e}$(P)_w designates the extension of a property P in a world w, I propose the following account of a powered property^5,6:

A property P is a powered property iff for any metaphysically possible world w such that|$\mathbb{e}$(P)_w|< $\aleph$₀, there is a set S such that there is a bijective function f: $\mathbb{e}$(P)_w → P*(S).

However, to develop a useful blueprint for constructing powered properties, it would be helpful to know what sorts of conditions on P could obtain for the above to be satisfied. In other words, it would be illuminating to find a sufficient condition for powered properties. To find this condition, let us consider what it was about the combination of universalism and the existence of mereological atoms that made being an object a powered property. First, the existence of mereological atoms supplied us with a ‘base’ set, the set of atoms, A, from which the set of objects, O, was generated. Second, universalism supplied us with a certain relation on O, fusion, that worked in a particular way. Namely, any non-empty subset of atoms from A corresponded with a fusion, which was itself a member of O. And it was the existence of this relation that allowed us to see that O could be put in one-to-one correspondence with the powerset of A––for any subset of A would have a corresponding member in O, given by fusion. Thus, the real key to identifying P as a powered property is finding a certain relation over $\mathbb{e}$(P).

To express this sufficient condition more precisely, let us make use of one piece of notation. Following the conventions of infinitary union notation, where $\bigcup _{x\in S}\left\{x\right\}$ designates the union of the singletons of each element x in a set S, let R_S(x) designate a two-place relation, R, applied across each element x of some set S (and if|S| = 1, R_S(x) = Rxx). For instance, where S = {a, b, c}, R_S(x) = R(a, R(b, c)).⁷ With this in hand, I propose the following sufficient condition for powered properties:      

P is powered if, for any metaphysically possible world w such that|$\mathbb{e}$(P)_w|< $\aleph$₀, there exists a two-place relation R on $\mathbb{e}$(P)_w such that, for any non-empty set S ⊆ {m| (m = Rxy) → (x = y)}, there is a z ∈ $\mathbb{e}$(P)_w such that z = R_S(x).

Very roughly, the claim here is that P is powered if its extension is such that any subset of the “atomic” instances of P corresponds to, vis-à-vis R, an instance of P––just as, on universalism, any set of objects itself corresponds to an object, vis-à-vis fusion. It is also important to add that, to mimic the operation of the fusion relation, the relation R must exemplify a few additional formal features:

Commutativity:

If Rxy = z, then z = Ryx.

Associativity:

R(Rxy, z) = R(x, Ryz).

Idempotence:

Rxx = x.

Atomic Equality:

For any sets S₁ and S₂ such that (i) S₁ ≠ S₂, and (ii) for any m ∈ S₁$\cup$S₂, m = Rxy → x = y, it is the case that R_S1(x) ≠ R_S2(x).

Unary Absorption⁸:

If Rxy = z, then Rzx = Rzy = z

The attentive reader may have noticed by this point that the relation R defined here is simply (or, also) the formal analog of the union operation in set theory––indeed, the conditions we have imposed on R just are the paradigmatic features of the union.⁹ And this should come as no surprise; for one way of defining powersets is as the set of all unions of all possible combinations of some set, in a structurally identical way to the present account.¹⁰ Indeed, it is helpful to think of the above blueprint for powered properties in the following way: a property is powered if its extension is necessarily closed under something like a union operation. For the mereological universalist, there exists something like a union operation on the extension of being an object: the fusion relation.

4 Powered counterexamples to modal continuity

With this blueprint in place, we are now ready to start developing direct counterexamples to modal continuity vis-à-vis powered properties. Our first counterexample is the property being an action I have performed¹¹:

(CX1): Let (ϕ_t) designate an action ϕ performed at t. For any two actions I have performed, (ϕ_t) and ($\psi$_t*), there is a third action I have performed: the action of ϕ-ing at t and $\psi$-ing at t*, i.e., (ϕ_t and $\psi$_t*).

Here, the property being an action I have performed is powered––the set of actions I have performed can be put into one-to-one correspondence with the powerset of what we might call the “atomic” actions I have performed.¹² In terms of our sufficient condition for powered properties, we can see that what allows this to be a powered property is that there is a certain relation over actions, the performing relation, which is plausibly such that: (i) performing actions x at t and y at t* is itself an action, (ii) performing actions x at t and y at t* is the same as performing y at t* and x at t [Commutativity], (iii) performing the performance of x at t and y at t*, and z at t** is the same as performing x at t and the performance of y at t* and z at t** [Associativity], (iv) performing x at t and x at t is the same as the action x at t [Idempotence], (v) performing distinct collections of atomic actions amounts to performing distinct actions [Atomic Equivalence], and (vi) performing the performance of x at t and y at t* and x at t is the same as simply performing x at t and y at t* [Unary Absorption]. I won’t spell out the relation in every example, but it is worth explicating R at least once to see that what makes, e.g., actions, powered is this relation.

Thus, the number of actions I have performed cannot take on certain values––e.g., it cannot be that I have performed an even number of actions. So, if we assume that it is possible that I perform a finite number of actions¹³, then the class of properties of the form being such that there are n-many actions I have performed has a modal gap.

It is worth being explicit about this last point. We require the assumption that it is possible for me to only have performed a finite number of actions because, if this is impossible, then the class of properties of the form being such that there are n-many actions I have performed doesn’t actually have a modal gap. For a class of properties only has a modal gap if one of the properties in the class is exemplifiable, and one of the properties in the class isn’t. Thus, if there is no natural number n for which being such that there are n-many actions I have performed is exemplifiable, then we have not shown that there is any gap in the class. For any powered property P to be a counterexample to modal continuity, we will need this kind of assumption––call these possible-finitude assumptions. Of course, I do find the possible-finitude assumption for being an action I have performed rather plausible, as I did for being an object, but the requirement of this assumption is noteworthy.

Additionally, with this example on the table, note that, because actions have this structure (roughly, that any two actions compose an action), a similar example can be constructed with the actions I will perform. Several other examples can also plausibly be constructed using particular kinds of actions I have, or will, perform: e.g., all the actions I have performed related to cleaning my room, all the actions I have taken related to studying, and, perhaps, the good (and bad) actions I have taken. Again, at their core, what makes all such examples work as counterexamples to modal continuity is the fact that actions have a particular kind of structure: any two distinct actions “compose” a further action, and this “composition” relation is commutative, associative, idempotent, etc.

Let’s walk through another example. Let us say that a group of people is any non-empty collection of persons (in the same world as one another). Thus, for any two groups of people, g₁ and g₂, there is another group, the people from g₁ and the people from g₂. So, being a group of people has a similar “composition” relation as actions, where any two distinct groups “composes” a further group, and the “composition” of groups of people is plausibly also commutative, associative, idempotent, etc. Thus, since being a group of people is a powered property, and we can construct all sorts of counterexamples concerning properties that distribute over groups of people. For instance:

(CX2): For any two groups of people that ought to be respected, there is a third group of people that also ought to be respected.

Here, being a group of people that ought to be respected is powered, for its extension can be put into one-to-one correspondence with the powerset of the individual people that ought to be respected. And a myriad of other examples can be produced using other properties that distribute over groups of people: e.g., being a malicious group of people, being tall, being in debt, etc. And, since it is plausible to think that it is possible for finitely many groups of people to exist, all these properties could serve as properties with modal gaps. Furthermore, there is of course nothing particularly special about groups of people; many x’s are such that properties centering around groups of x’s will be powered. The key feature is simply that groups––whether groups of people, animals, cookies––have the compositional structure we have been stressing.

A few more examples are worth exploring. Let a ‘coursed meal’ be any meal with n courses, where n > 0.¹⁴ Now consider:

(CX3): For any two coursed meals m₁ and m₂ I know how to cook, there is a third coursed meal that I also know how to cook––the courses of m₁ and the courses of m₂.

Thus, being a coursed meal I know how to cook is powered. The coursed meals I know how to cook can be put into one-to-one correspondence with the powerset of the individual meals I know how to cook––there is no possible world where I know how to cook, e.g., an even number of coursed meals. As such, assuming there is a possible world where I know how to cook finitely-many coursed meals, we have yet another counterexample to modal continuity.

Here is a similarly spirited example: let a fleet of ships be any non-empty collection of boats (in the same world). Then, we have:

(CX4): For any two fleets of ships f₁ and f₂ I could deploy, there is a third fleet I could deploy––the ships of f₁ and the ships of f₂.

And, of course, a similar point can be made about squadrons of aircraft, or packs of animals.¹⁵ In all such cases, I find the relevant possible-finitude assumptions quite plausible.

One last example is worth discussing, given that it involves more contentious metaphysics (similar to mereological universalism). Consider the familiar notion of a spatial region. Following the spatial logics of, inter alia, Randell et al. (1992) and Aiello et al. (2007), we can claim:

(CX5): For any two spatial regions s₁ and s₂, there exists a unique spatial region, the sum of s₁ and s₂, which is just the region such that it is connected to all and only those regions connected to either of s₁ or s₂.

Thus, being a spatial region is a powered property––the set of all spatial regions in a world w can be put into one-to-one correspondence with the powerset of the set of all “atomic” spatial regions (those spatial regions with no proper parts). Now, assuming there are such atomic spatial regions, and that there is a possible world with finitely many spatial regions, it follows that there are modal gaps in the number of spatial regions that could exist. Of course, the possible-finitude assumption this time around is by no means trivial: the atomic spatial regions at a world w are standardly taken to be (infinitely many) spatial points, and it strikes me as somewhat plausible to think that this constraint on space holds necessarily. Nevertheless, there are detractors to this standard (see, e.g., Pratt-Hartmann, 2007: 13–14), and it is at least defensible to maintain that finitism (about spatial regions) is metaphysically possible. At any rate, the relevant, and intriguing, point is that if one has metaphysical views of this stripe, then we have on our hands yet another powered property that violates modal continuity.¹⁶ Additionally, this sort of example plausibly has a temporal analog, using spans of times.

Hopefully, the structure of the counterexamples here is clear: find an entity E such that, for any two E’s that have P, by some suitable relation R, there is another E which itself exemplifies P. As long as it is possible for finitely many E’s to exemplify P, we have a property that is modally discontinuous.

Before moving on, I will offer here a few more counterexamples to modal continuity using powered properties. It is worth noting that it may well be controversial in some of these cases whether the entities in question really form a third, whether the property always extends to the third, whether the relevant relation R is really commutative, associative, idempotent, atomically equivalent, and absorptive, or whether the possible-finitude assumption is plausible. I leave this open and offer multiple examples in hopes of appealing to a wider crowd.

(CX6)

Let an ‘s-bundle’ be a bundle (non-empty collection) of item(s) from a store s. For any two s-bundles, there is a third s-bundle––the bundle of the items from the first two. Thus, the s-bundles can be put into one-to-one correspondence with the powerset of the items in s. Plausibly, being an s-bundle my son would desire is powered.¹⁷

(CX7)

Suppose sets exist, such that for any material object(s) o₁,…, o_n in a world w, there is a set {o₁,…, o_n} belonging to world w. Call such sets “material sets.” It follows that the set of material sets belonging to a world w is the powerset of the objects o in w.

(CX8)

For any two events, e₁ and e₂, that have occurred at t and t*, there is a third event, the event of e₁ occurring at t and e₂ occurring at t*. [And similarly for ‘will occur’].

(CX9)

Let a ‘real-world artifact-collection’ be a collection of artifacts, all of which are from the actual world (non-rigidly designated). For any two real-world artifact-collections such that I would profit from selling them, there is a third collection I would profit from selling––the artifacts of the first and the artifacts of the second.

(CX10)

Let a ‘TV binge-list at w’ be a collection of TV shows from w. For any two TV-binge lists at w that I couldn’t finish today, there is a third binge-list I couldn’t finish today––the shows of the first list alongside the shows of the second list. More generally, the set of TV binge-lists at a world w can be put into one-to-one correspondence with the shows in w.

I doubt that these are near the total number of properties that, as applied to certain entities, become powered. Indeed, many entities exhibit the sort of identity conditions conducive to producing powered properties, whereby any plurality of e’s is itself an e.¹⁸ This, in many ways, is one of the central insights I take powered properties to provide: the application of modal continuity to a given property P depends not merely on whether P is, e.g., coherent, but also on whether the world interacts with P in a certain way (e.g., whether there exists a certain relation over which P distributes), which makes P discontinuous.

5 Patchwork principles as a powered property

A particularly interesting feature of powered properties is that, in addition to shedding light on another factor which we must pay attention to in our applications of modal continuity, powered properties reveal that another popular family of modal principles––recombination, or patchwork, principles––entails the existence of a powered property, and is thus in tension with modal continuity. I leave it open just how strong my argument for the tension is, as the inference that what follows is a powered property is not, by my lights, as secure as our previous examples, as it requires a few additional assumptions and a contrived proof. My hope is to signal that powered properties might be a good source for developing this unforeseen tension.

Patchwork principles, adapted from Lewis (1986: 88–91) and endorsed by, inter alia, Koons (2014: 258), are Humean modal principles meant to capture the plenitudinous nature of modal space. Patchwork principles are supposed to do justice to the modal intuition that “anything can coexist with anything else, at least provided they occupy distinct spatiotemporal positions…if there could be a dragon, and there could be a unicorn, but there couldn’t be a dragon and a unicorn side by side, that would be an unacceptable gap in logical space, a failure of plenitude” (Lewis, 1986: 88). The motivation here is that there are no gaps in modal space (very much like modal continuity! ). As such, patchwork principles make claims about what sorts of spatiotemporal arrangements of objects are possible––in particular, that we should be able to recombine certain arrangements to produce others. Simplifying a bit, one version of the patchwork principles often claims something like the following¹⁹:

(PP)

For any possible worlds w₁ and w₂ with spatiotemporal regions R₁ and R₂ respectively, containing contents c₁ and c₂ respectively, if there is a possible world w₃ with spatiotemporal region R₃ with enough ‘room’²⁰ to contain R₁ and R₂ and their contents without overlap, then there exists a possible world w₄, with spatiotemporal region R₄, that has parts p₁ and p₂ that exactly resemble c₁ and c₂, in some arrangement a.²¹

Roughly, the intuition behind (PP) is that, for any two possible spatiotemporal regions (and their contents), there should be a third possible spatiotemporal region––the region containing exact duplicates of the contents of the first two, as long as there is some possible size and shape of spacetime permitting for the relevant arrangement of the exact duplicates. Let us say, for concision, that a spatiotemporal patch is a spatiotemporal region and its contents. Then, patchwork principles claim that, for any two possible spatiotemporal patches, there is a third possible spatiotemporal patch, the “combination” of the first two, as long as there is a possible spacetime that can accommodate it. At first blush, one might be tempted to immediately claim that the powered property is obvious here: PP entails that being a possible spatiotemporal patch is a powered property. The structure is strikingly similar to the examples explored in § 4. But (at least one) problem with this approach is that it is very plausible that there are infinitely many possible spatiotemporal patches, such that the class of properties being such that there are n possible spatial patches plausibly won’t be a counterexample to modal continuity.

Thus, to generate a counterexample, we will need to hone in on a specific kind of possible spatiotemporal patch––such that it is plausible that there are only finitely many of this kind of patch.

First, it will be helpful to focus on the following (restricted) instance of PP, which concerns finite spatiotemporal regions, allowing us to remove Lewis’ spacetime proviso²²:

(PP*): For any two finite possible spatiotemporal patches ₁ and ₂, there is a third finite possible spatiotemporal patch, ₃, containing duplicates of ₁ and ₂ in some arbitrary arrangement a.

To build this example, we start first with the notion of a batch. A batch of x’s is the minimal²³ finite spatiotemporal patch of x’s such that (i) each of the x’s is spatially related to one another, (ii) there are no x’s outside of the patch, and (iii) none of the x’s are exact duplicates of one another. For instance, since all of the cookies in the actual world are spatially related (and, I assume, none are exact duplicates), they form a batch: the batch of actual cookies. Note that there can be no more than one batch of cookies in any world, as batches are defined to be maximal (no cookie can be outside it). Next, we define the notion of a w-cookie: a cookie is a w-cookie just in case it is an exact duplicate of one (and only one) cookie in w. Thus, note that there may be w-cookies in worlds other than w. Because batches are spatiotemporal patches, the following (independently plausible) proposition is entailed by PP (and PP*)²⁴:

(Cookies): For any two distinct²⁵ possible batches of w-cookies ₁ and ₂, there is a third possible batch of w-cookies, ₃, containing exact duplicates of the w-cookies of ₁ and ₂.

For instance, if there is a possible world, w*, with the batch of w-cookies {cookie_A, cookie_B}²⁶, and another world, w**, with the batch {cookie_C}, then there is a possible world, w***, with the batch {cookie_A, cookie_B, cookie_C}. To produce a counterexample to modal continuity, we will need two more assumptions. First, that there is a possible world with finitely many cookies, not all of which are intrinsic duplicates of one another (call this A1). This is plausible––and probably actually the case. Second, we will need a modal assumption: for any cookie in a world w, there is a possible world where (a duplicate of) that cookie is the only cookie (call this A2). This too, I think, is plausible. Note that this is also supported by modal continuity (if 2 cookies are possible, so is 1).

With these modest assumptions, we now have a derivation from PP to a modal gap. Informally, it is as follows. Let w be a world with the following batch of cookies: {cookie_A, cookie_B, cookie_C}. There is a possible world, w*, with the batch of w-cookies: {cookie_A*} (by A2).²⁷ There is another possible world, w**, with the batch of w-cookies: {cookie_B*} (by A2). By PP, for any two distinct possible batches of w-cookies, there is a third possible batch of w-cookies, containing exact duplicates of the w-cookies of the first two. Thus, by PP, there is a possible world, w***, with the batch of w-cookies: {cookie_A**, cookie_B**}.²⁸ By repeated application of PP, we will have that for any subset of w-cookies, there is a possible world where (duplicates of) the w-cookies in that subset exist, corresponding to a possible batch of w-cookies. In other words, the set of possible batches of w-cookies can be put into one-to-one correspondence with the powerset of the cookies in w. But, because it is possible for there to be a world with finitely many cookies, the class of properties of the form being such that there are n-many possible distinct batches of w-cookies has a modal gap.²⁹ So, Lewisian patchwork principles supply us with a powered property, alongside a class of properties with a modal gap.

To my mind, this modal gap is an interesting discovery, not only because of how it interacts with modal continuity, but because Lewis’ patchwork principle was meant to prevent the existence of “unacceptable gaps in logical space” (Lewis, 1986: 88). Taking this remark seriously, perhaps we ought to conclude that discontinuities do not make for unacceptable gaps in modal space: only failures of combination count as unacceptable gaps. While both Rasmussen’s and Lewis’ principles are based on vindicating the intuition of the plenitudinous nature of the modal landscape, it seems they are undergirded by different conceptions of plenitude––i.e., what counts as an unacceptable gap on these pictures is (crucially! ) different.

Indeed, it is important to note here the nature of the conflict. There is nothing in the nature of batches, qua batches, preventing the following from being true: for any n, and some world w, there could be n possible batches of w-cookies. It is the truth (if it is a truth) of the patchwork principle that prevents this from being true. If one was an ardent defender of continuity, one could easily and freely believe that possible batches come in any number; it would simply require embracing the failure of modal recombination at at least one world. Thus, in at least some cases, those attracted to modal continuity and Lewisian patchwork principles on the grounds of the plenitudinous nature of modality must make a choice: is their commitment to plenitude based on a combinatorial intuition, or a continuousness intuition?

6 Objections

I’d like to close by considering a few potential objections––in particular, concerning whether modal continuity is really threatened by the existence of powered properties, or if, instead, powered properties fall under one of its provisos.

The first objection I’d like to consider is that modal continuity is not threatened by the existence of powered properties because powered properties fall under the coherency proviso. Recall that refined versions of modal continuity are standardly formulated to exclude narrowly logically inconsistent properties like being 3-sided and 4-sided––i.e., they come attached with a coherency proviso. Might it be objected that all of the cases of powered properties I’ve developed herein are not genuine counterexamples, as they are excluded by this proviso? After all, it is logically impossible for there to be, say, an even-number of actions I have performed, for purely set-theoretic reasons.

In response, I find it hard to see how the counterexamples I have given here are narrowly logically impossible. They all require notable metaphysical assumptions about how objects combine, their identity conditions, what the extensions of properties are like, and possible-finitude assumptions. For instance, take the case of the mereological universalist: it seems, indeed, that if one is convinced of universalism, and if it is metaphysically possible that (only) finitely many atoms exist, one has a direct counterexample to modal continuity. Crucially, then, whether being an object is a property which admits of a class with modal gaps is not a matter of the definition of being an object: it depends on, among other things, several contentious claims about the nature of the world. It is no matter of logic whether finitely many atoms could exist. This point is also evident in the case of the patchwork-inspired counterexample––PP is far from a narrowly logical truth. It is a deeply contentious view about the nature of possibility. Thus, while it is true that if a property is powered, it will follow as a matter of logic that the property is a counterexample to modal continuity, whether a property is powered is seldom, by my lights, a narrowly logical truth.

Another way to see this point is to compare powered properties with the kinds of examples Rasmussen (2014) seeks to exclude with his constraint against logically inconsistent degreed properties. For instance, take the class of properties of the form being an n-sided triangle. This class is riddled with modal gaps: indeed, only one property in this class is exemplifiable, because it follows from the definition of being a triangle that it can only have 3 sides. This class is no counterexample to modal continuity, however, precisely because of this feature. Compare, now, the class of properties of the form being such that there are n material objects. If the universalists are right, this class is also riddled with modal gaps. But surely it does not follow from the definition of what it is to be a material object that the property being a material object encodes various powerset-like features. The fact that there cannot be, e.g., 622 material objects––if it is a fact at all––is true in virtue of a particular metaphysical view, that is, in virtue of the particular way the world is structured and the identity conditions of objects. Indeed, if there are counterexamples to modal continuity, shouldn’t they take precisely this form? Some metaphysical feature F of the world is such that F entails that, while a certain property is possible to a certain degree, it is not possible to other degrees. Surely, we cannot exclude such counterexamples merely because F logically entails this fact. Indeed, if the logical constraint on modal continuity is construed this broadly, I find it difficult for the principle to enjoy the wide metaphysical application it has experienced.

A second objection one might press is to claim that, while powered properties are not excluded on the basis of the coherency proviso, they are excluded on the basis of Rasmussen’s essential limits proviso.³⁰ Recall that, in addition to the mandate that the properties within a unified class of degreed properties be coherent, Rasmussen (2014: 531) also requires that, in order for the class to have no modal gaps, the class must not have any properties involving the violation of the essential limits of some particular thing, p. For instance, consider the class of degreed properties of the form being LeBron James and capable of eating n pounds of lentils (a modified example from Rasmussen). Plausibly, there is some mass of lentils such that it is too large for LeBron James to possibly eat. This, however, is no counterexample to modal continuity because there is a property in this class that involves a violation of LeBron James’ essential limitations, to wit, his capacity to eat lentils. More generally, Rasmussen proposes that modal continuity only apply to properties that do not involve the mentioning of any particular individuals, e.g., LeBron James and Keith Lehrer. With this proviso in hand, the following might be pressed: powered properties do not threaten the principle of modal continuity, for it plausibly lies in the very essence of the relevant entity that there cannot be, e.g., an even number of that entity. For instance, if mereological universalism is true, it plausibly lies in the nature of what it is to be an object that there cannot be 22 of them.³¹ Powered properties, then, involve violations of essential limits.

This is a wonderful objection, for it allows us to explore and unveil a number of important points concerning our investigation. As such, I have a few responses to offer.

First, it is not clear to me that powered properties actually violate Rasmussen’s essential limits proviso. That proviso, as mentioned above, is formulated as a constraint against properties mentioning particular individuals, so as to rule out the possibility of modal continuity’s applying to a property that involves some particular’s essence. It is, to use Rasmussen’s terminology, a requirement that the properties we apply continuity to be non-haecceitous. But it seems very clear that many powered properties do not involve any such haecceities. Or, at the very least, that they needn’t. For instance, Rasmussen himself (2014: 531)³² claims that the property being such that there are n co-located objects does not violate his requirement of not mentioning any particular individuals––indeed, it is a paradigmatic example of not violating the proviso. But, of course, if being such that there are nco-locatedobjects does not count as invoking a particular essence, surely neither does being such that there are n material objects. And I am tempted to say something similar for actions, groups, spatial regions, meals, sets, fleets, and the like.

Second, putting this last point aside, if we do take a reading of the essential limits proviso on which powered properties involve violations of some thing’s essence, I am worried that this reading might seriously constrain the applicability of modal continuity, in a way that arguably severs continuity more than if we simply conceded that it does not apply to the domain of powered properties. Consider, first, that if it really lies in the nature of a material object that its extension is powered, and if this means that modal continuity does not apply to the number of material objects that could exist, plenty of judgements about other properties––which we originally wanted to apply modal continuity to––can no longer be supported by appeal to modal continuity. For instance, Rasmussen’s opening example of modal-continuity-style reasoning––that, if two objects could be co-located, so could n––may turn out to not be an arena in which we can directly employ modal continuity. For, if mereological universalism is true, perhaps it turns out to lie in the very nature of (co-located) material objects that they are powered, and as suchcontinuitydoesnotapply (depending on whether co-located objects retain the mereological assumptions required, e.g., that they do not involve objects having multiple fusions).. And, as it turns out, there is a more general worry here. Enforcing an essential-limits proviso as wide as the one being pressed herein to exclude powered properties entails that, before applying modal continuity to some domain, we must settle the various modal questions about the natures of the concepts involved in the relevant domain. If, e.g., we are interested in questions about the possibility of certain kinds of causal and nomic relations and hope to apply modal continuity to make headway (Gibbs, 2018), or interested in questions about the possibility of various degrees of value (Rasmussen, 2018), it turns out we will have to first settle questions about the nature of these entities before we apply modal continuity. But this, we might worry, seems to affect the purpose of modal continuity, which is to be a guide to possibility. Of course, we could instead maintain that modal continuity always provides us with a defeasible reason to believe certain modal claims––rather than that it does not apply until we have settled certain questions––and that we must simply keep in mind that this reason is always defeated by concerns about the essences of the entities involved. But, aside from worries about strength, this version of the proviso seems to grant weight back to some of the results concerning powered properties. For instance, it seems to entail that modal continuity gives us a defeasible reason to disbelieve the patchwork principle (in at least some cases). In a nutshell, my worry for this objection is that limiting modal continuity in the way being proposed might inevitably “spill over” to limiting the principle from being applied in debates where it has enjoyed application, thereby restricting the principle anyways.

Lastly, even if (i) the essential limits proviso really does exclude powered properties, and (ii) there are no worries for such a reading of the proviso, I’d like to suggest that our investigation of powered properties remains worthwhile and provides insights concerning modal continuity. Firstly, investigating the existence of powered properties opened our eyes to the fact that modal continuity can fail to apply to properties with a certain structure––a structure notable enough that it is imposed by patchwork principles onto certain properties, revealing a difference between combination-based and continuity-based pictures of plenitude. Secondly, the existence of powered properties might still merit caution about how wide continuity applies. For not only might there be many more powered properties than we initially suspect, but the idea of a powered property which I have evinced herein is only a single instance of a more general fact about modal continuity: any property P with an extension E whose number of members is determined by some function f serves as a property to which we cannot apply modal continuity, as long as, for all naturals n, there is some natural n* such that f(n) $\ne$n*. In this paper, we focused on one such function, f(n) = 2ⁿ– 1, but I find it hard to believe there is no other such function f which appropriately models the extension of some property P. Paying attention, then, to any functioned property is a relevant factor when applying modal continuity. And it is, at the very least, an epistemically live possibility that there are properties that are a function of something else––or, that certain principles impose this sort of structure on some properties.

7 Conclusion

In total, we have made two noteworthy advances with respect to modal continuity. First, we showed that there is a family of classes of properties with modal gaps––powered properties. More surprisingly, we found that another popular principle of modal epistemology, the patchwork principle, is likely in tension with modal continuity, as it supplies us with a direct powered property. I am happy to leave open the extent to which modal continuity applies as a guide to possibility in light of these findings. It may well be that, e.g., continuity and patchwork principles can be reconciled, and made plausible in tandem, with the right kinds of constraints in place. But that investigation is for another day.