Standard approaches to ontological simplicity focus either on the number of things or types a theory posits or on the number of fundamental things or types a theory posits. In this paper, I suggest a ground-theoretic approach that focuses on the number of something else. After getting clear on what this approach amounts to, I motivate it, defend it, and complete it.

W

We should contrast ontological simplicity with elegance (

It is standard to distinguish between

Telling us what to measure when it comes to simplicity is one thing. Telling us that we should, all else being equal, prefer simpler theories is another. So, in arguing for what we should be counting, I am not

Do not multiply what counts against simplicity without necessity!

Indeed, invoking this command only makes sense when we are trying to decide between two or more

Since grounding is integral to this approach to simplicity, some words about it are in order. As I am understanding it, grounding is

In keeping with orthodoxy, I treat grounding as irreflexive, transitive, and asymmetric. Taking grounding to be primitive, fundamentality and partial grounding are defined in the standard ways: _{df}. x_{1}, …, _{n}_{df}. x_{1}, …, _{n}_{1}, …, _{n}_{1}, …, _{n}, z_{1}, …, _{n}

The relation needed for this paper’s approach to simplicity is that of something

_{df}. x

Think of independence as a way of capturing what it means for something to be wholly unconnected, free, and distinct from some things. Of course, there are other ways of capturing these notions (modal, mereological, and spatiotemporal). But here, grounding is given pride of place.

Let me illustrate independence. Where the solid arrow represents grounding and the dashed partial grounding, consider the following three grounding structures (the last of which has _{1}, _{1} collectively grounding _{2}):

Now in the first structure, nothing is independent of anything. In the second, each of the _{1} partially grounds _{2} and _{2} is grounded in a partial ground of _{2}, each of the _{2}. But then each of the

In order to get a better handle on independence, let’s look at some of its formal features. Where in what follows, ‘_{1}, …, _{n}_{1}, …, _{n}

Since independence implies non-identity, nothing is independent of itself.

We should also accept

To see why, assume that

We should not accept

Symmetry and Transitivity have it that if

We should accept

_{1}, …, _{n}_{1} & … & _{n}

To see why, assume that _{1}. So _{1}, one of _{1}’s partial grounds, partially grounded in _{1}, or partially grounded in some of _{1}’s partial grounds. So _{1}, …, _{n}_{2}, …, _{n}

Where ‘_{1}’, …, ‘_{n}

_{1} & … & _{n}_{1}, …, _{n}

To see why, assume that _{1}, …, _{n}_{1}, …, _{n}_{1} and … and _{n}_{1} &…& _{n}_{1}, …, _{n}

Where _{1}, …, _{m}_{1}, …, _{n}

_{1}, …, _{n}_{1},…, _{m}

To see why, assume that _{1}, …, _{n}_{1} and … and _{n}_{1} and … and _{m}_{1}, …, _{m}

We should not, however, accept the converse of Contraction

_{1}, …, _{m}_{1}, …, _{n}

To see why, assume that _{1}, …, _{m}_{1}, …, _{m}, x

These features of independence help give us a grasp on the logic of independence and so on independence itself. But they also help drive home some interesting results and point to differences between independence and related notions. For some of these results and differences, see the appendix.

I have defined independence, illustrated it, and listed a number of its formal features. I now want to state this paper’s approach to simplicity by using it.

Consider a plurality which meets the following condition: for any

For each theory under consideration, consider those largest pluralities of independent things that are also maximal: for any such plurality, there can be nothing in the theory that is independent of it._{T}’ and in theory T* with the variable ‘_{T*}’, here is this paper’s approach to ontological simplicity:

_{df}. X_{T} is smaller than _{T*}.

Now in order to be an informative approach, we need to know what makes it that one plurality is smaller than another. For now, assume that the largest number of independent things a theory posits is finite. (In §4, I drop this finitist assumption and show what happens when we permit pluralities of independent things that are infinite in number.) Given this, we can state our approach as follows:

T is simpler than T* ↔_{df}._{T} is less than the number of things in _{T*}.

So T is simpler than T* just in case the largest number of independent things T posits is less than the largest number T* posits.

Applying the approach, let us compare the simplicity of three theories, the first of which has the first grounding structure depicted in §1.1, the second the second structure, and the third the third structure. In the first theory, the largest number of independent things is one: _{1} and _{2} are these largest pluralities (recall that pluralities of one are, vacuously, pluralities of independent things). In the second, the largest number is two: _{1}, _{1} and _{1}, _{2} and _{2}, _{1}, and _{2}, _{2} are these largest pluralities. And in the third, the largest number is three: _{1}, _{1}, _{1} and _{1}, _{2}, _{1} and _{2}, _{1}, _{1} and _{2}, _{2}, _{1} are these largest pluralities. So the first theory is simpler than the second, which is simpler than the third, which seems right given their grounding structures.

The independence approach allows us to give sense to the notion of the

There are four important features of the present approach. First, given that simplicity is a theoretical virtue, we get the following command

(so called because it tells us to shave, if we can, things each of which is independent of the others). Second, this approach does not require a fundamental level in order for claims of relative simplicity to hold. So it is consistent with grounding never bottoming out and our thinking that it never bottoms out (more on this later). Third, the approach is consistent with

There is more that can be said. For example, we can define a notion of partial independence that, interestingly enough, bears on the epistemology of simplicity given the present approach. And it would be an oversight if something were not said about how this approach to simplicity relates to the notorious “nothing-over-and-above” relation. Because of this, and because discussing such issues now would interrupt the flow of the paper, I have reserved doing so for the appendix.

Schaffer (

In the monist structure,

What, though, should we look to? Some say the fundamental since monist theories have fewer fundamental things than dualist theories (

Here, the salient facts are the same: in both the non-bottomless and the bottomless cases, we are able to get monist structure from dualist structure by having the

Again though, what should we look to? Independent things! In dualist theories, the largest number of independent things is two. In monist theories, it is one. So according to the independence approach, monisms (bottomless or not) are simpler than dualisms (bottomless or not). This gets the seeming facts about simplicity right. In all this, we have reason to accept the independence approach.

The independence approach identifies simplicity with a kind of unity; a unity amongst the things, taken collectively, a theory posits. Call this ‘ontological unity’.

Ontological unity is a function of the relations that things stand in. Consider Oppenheim and Putnam’s (

Here, we see the same kind of unity in the independence approach to simplicity. Recall the distinction between the width of a theory and the height, where the former is measured in terms of the size of the largest pluralities of independent things a theory posits. Like simplicity, what matters is width when it comes to unity. The grounding structures depicted in §1.1 illustrate this nicely. The first structure is more unified than the second which is more unified than the third. And the most natural and straightforward explanation of this has everything to do with their width. This is also clear in monist and dualist theories. Monisms are more unified than dualisms precisely because they have fewer independent things (they are, after all, monisms).

That the independence approach identifies simplicity with ontological unity yields two nice things. First, it explains why focusing on just the number of fundamental things will not do. Since there can be ontological unity

As seen above, bottomless monist and dualist grounding structures tell against thinking that when it comes to simplicity, fundamental things are what we should be counting. I want to continue to push this line by providing further cases that the independence approach can, but a fundamentality approach cannot, make sense of.

Let us begin by comparing a foundationalist structure which posits one and only one fundamental thing with a mixed structure which posits a fundamental thing and something which has no fundamental ground. Where the ellipsis tells us that it is grounds all the way down, we have

In positing

There are other ways of showing what we just did. For example, suppose we get rid of

There are other ways of denying the existence of a fundamental level. Consider

If to be fundamental is to be ungrounded, then every theory with reflexive or symmetric structure lacks fundamental things. So an approach to simplicity that counts only fundamental things has it that any theory with either of these structures is simpler than a theory with the above foundationalist structure and as simple as a theory with a nihilist one! This is not so for the independence approach. The largest number of independent things in the reflexive structure is one (pluralities of one are, vacuously, pluralities of independent things). The same holds for the symmetric structure since

Perhaps a revision in our notion of fundamentality is called for. Let us say that to be fundamental is to be either ungrounded, or, if grounded, then grounded only in itself. But this helps little: given this notion of fundamentality, any theory with the above symmetric structure still has no fundamental things. But then any theory with this structure is still simpler than a theory with the above foundationalist structure and as simple as a theory with a nihilist one. So let us revise this notion further by saying that for something to be fundamental is for it to be ungrounded, or, if grounded, then grounded only in something that it grounds. This will make each of

Here is a revealing comparison. Where the ellipses tell us that the grounding structure is preserved all the way down, consider the following two structures:

Now, any theory with this linear structure seems simpler than any theory with this criss-crossed structure. After all, for any level

Let me motivate this a bit differently. Notice that for the criss-crossed structure, each thing is so bound up with everything else that to get rid of some is to get rid of all. For example, removing _{3} removes everything below it (grounds necessitate what they ground)._{2}, _{2}, _{1}, _{1}, … would go. But in no longer having a ground, _{3} would also go. So everything would go! Or to go lower down the hierarchy, removing _{2} removes everything below it. But then in no longer having a ground, _{2} would go. But then in longer having a ground, _{3} and _{3} would go. Again, everything would go! In the criss-crossed structure then, nothing stands apart from anything else. All is bound to all. In this respect, the criss-crossed structure and the linear structure are the same: both are highly unified. (Indeed, this unity claim holds for any criss-crossed structure and so holds for an infinitely extended version of the above structure where the grounding structure is not only “bottomless” but also “sideless”.)

Continuing, ignore everything that occurs below _{1} and _{1} in the criss-crossed structure and assume that, for all practical purposes, _{1} and _{1} are fundamental. Given this ignoring, we should no longer think that the criss-crossed structure is as unified as the linear structure. In disregarding what occurs below _{1}, and so in treating _{1} as fundamental, we have no reason to think that removing it would result in any other thing being removed (the same holds for _{1}). But then in disregarding what occurs below _{1}, we have no reason to think that each of the things that we are not disregarding (_{1}, _{1}, and everything above them) is bound up with every other. We see then that for any level

In light of all this, that any theory with the above linear structure is simpler than one with the above criss-crossed structure is no longer so clear.

There is something else we can glean from all this. In response to worries infinitist structures pose for a fundamentality approach to simplicity, Schaffer (

T is simpler than T* iff there is a level

Suppose that the non-fundamental level that _{1} appears on in both the linear and criss-crossed structure is

But this is the wrong result. And the above bi-conditional gives us this result because it does what it should not. In going to counterfactual scenarios where

There are other structures and so other comparisons we can make.

In spite of fundamentality being the wrong thing to focus on when it comes to simplicity, fundamentality and simplicity are related. To see why, assume that every non-fundamental thing is fully grounded in some fundamental things.

Proof: since, if some things are fundamental, then each is independent of the others, it cannot be that the number of fundamental things in T is greater than the largest number of independent things in T (from here on out, ‘in T’ will be dropped).

Suppose, for

Suppose then that they are not all grounded. So some are fundamental and some are grounded.

Now since these independent things are either all grounded or not all grounded, and since both disjuncts lead to a contradiction on the assumption that the largest number of independent things is greater than the number of fundamental things, this assumption must not be true. And from this and that the number of fundamental things cannot be greater than the largest number of independent things, it follows that the number of fundamental things is the same as and the largest number of independent things. Thus, Equivalence.

From Equivalence (and recall, we only get Equivalence by assuming that every non-fundamental thing is fully grounded in some fundamental things), it follows that T has fewer fundamental things than T* if and only if the largest number of independent things T posits is less than the largest number T* posits. So from the independence approach, T has fewer fundamental things than T* if and only if T is simpler than T*. So fundamental things

Where the dashed arrows represent partial grounding, consider these two grounding structures:

A theory with the hierarchical structure on the left is simple. Everything boils down to a single thing. This is not so for a theory with the egalitarian structure on the right. Given it, everything is grounded in no less than everything

It is helpful to state this reason for thinking that one theory is simpler than the other in terms of the notion of a complete minimal basis. Say that _{1}, …, _{n}_{df}._{1}, …, _{n}_{1}, …, _{n}_{1}, …, _{n}_{df}. x_{1}, …, _{n}_{1}, …, _{n}

Now for hierarchically structured theories that have complete minimal bases (and as we have seen, not all do), this reason for thinking that one theory is simpler than another seems right. But this is not always so when it comes to non-hierarchical theories. Here is why.

In a theory with the above egalitarian structure, the ontological demands that

What has happened here? How is it that the complete minimal basis in the egalitarian structure is

There are two things we can take away from this. First, given that nothing counts against the simplicity of a theory with an egalitarian structure any more than anything else, such a theory is no less simple than a theory with the above hierarchical structure. And this is what the independence approach says. Far then from being a problem for such an approach, in the end, this objection from egality serves to confirm it.

Second, notice that no matter how large we increase the complete minimal basis in an egalitarian structured theory (four, five, six, … aleph null, …), nothing in such a basis would count against the simplicity of this theory any more than anything else. So increasing the size of this basis does not result in a less simple theory. This is important since it shows us where the problem really lies. The problem is not with the independence approach. It is not with

Suppose that theory T posits ten fundamental and no grounded things and that theory T* posits nine fundamental and 1,000 grounded things. Now, if this is the only difference between them, then according to Fiddaman & Rodriguez-Pereyra, “[T] is the better theory, since [T*] is unnecessarily profligate” (

Fiddaman & Rodriguez-Pereyra think that T* is unnecessarily profligate on account of positing more things than T without a corresponding advantage. But this is not a good reason for thinking that T* is

Now, that T* should be rejected on account of its positing grounded, and so superfluous, things is an extreme claim since it amounts to a ban on grounded things: according to this claim, any theory with grounded things should be rejected in favor of a theory just like it

What bearing does this have on Fiddaman & Rodriguez-Pereyra’s insisting that T is simpler than T*? As just seen, that T* posits more things than a theory that posits just its fundamental things is no mark against it. But then, where T** is gotten from T* by eliminating the latter’s grounded things, we should accept

T* and T** are co-simple.

Now since T posits ten fundamental things, T** nine, and since neither posits grounded things, it is uncontroversial that

T** is simpler than T.

And from this and that T* and T** are co-simple, it follows that

T* is simpler than T,

contradicting Fiddaman & Rodriguez-Pereyra’s judgement. Since that T** is simpler than T is uncontroversial, if they want to maintain their claim that T* is objectionably profligate, they need to show that T* and T** are not co-simple. In short, they need to take the extreme route and argue that theories with grounded things should be rejected in favor of theories without them.

Though this paper’s concern is not over whether preference should be given to simpler theories, there is a way of justifying such a preference that tells against the independence approach.

Here is an attractive idea: simpler theories are more likely to be true because they are better supported by the data. Huemer elaborates on this when he says that “a simple theory can accommodate fewer possible sets of observations than a complex theory can … [so the] realization of its predictions is consequently more impressive than the realization of the relatively weak predictions of the complex theory” (

P(T|E) = [(P(E|T) × P(T)] / P(E).

Consider a complex theory T_{c}_{s}_{s}

if [T_{s}_{1} and E_{2}, while [T_{c}_{1}, E_{2}, E_{3} and E_{4} (where the E_{i}_{1}|[T_{s}_{1}|[T_{c}_{s}_{3} or E_{4}, but if E_{1} or E_{2} is observed, [T_{s}_{c}

Assuming then that the prior probabilities of T_{s}_{c}_{s}_{c}_{s}_{c}

This seems all well and good. But Baron & Tallant (

What should we think of this argument? Put to the side the controversial claim that a preference for simpler theories can be justified in something like the above manner.

Here is another worry. In order for us to infer that one theory is more probable than another on the basis of their likelihoods, we have to assume that their priors are the same. But why make such an assumption in the present context? Baron & Tallant answer

when we are at the point of choosing between theories using theoretical virtues, … we already know that the theories at issue do not come apart in any of the normal ways, and so something extra is needed to select between them. If our priors were not equal between the theories, then the theories would probably come apart in a standard way, and so considerations of parsimony would be less likely to weigh in. (609)

On the contrary, one would have thought that when it comes to choosing between theories on the basis of the theoretical virtues, such theories predict, and predict equally well, the evidence; the ‘all else being equal’ clause seems to rule out a difference in the likelihoods. As Sober says (

The debate between nominalism and platonism provides us with a nice example._{m}_{m}_{m}_{m}

I have so far assumed that the largest number of independent things a theory posits is finite. Given this, comparisons of simplicity can proceed based on the largest number of independent things theories have. But what happens when the theories being compared each posit an infinity of independent things?

Notice what this calls for: an account of what makes it that one plurality is smaller than another that works for any theory, and so works for theories that posit an infinite number of independent things. For convenience’s sake, let us, for now, restrict ourselves to theories each of whose things is independent of any other thing. And let us assume that the number of things in each of these theories is of the same infinite size. Given this, distinguish between a pair of theories each of which has an infinite number of independent things that the other does not and a pair of theories where this is false. That is, distinguish between a pair of theories each of which

Assume that theory T_{a}_{c}_{a}_{c}_{a}_{a}_{+}’. Now T_{a}_{a}_{+} since everything in T_{a}_{a}_{+} but not vice versa. But T_{c}_{a}_{+} (the reason for thinking this is the same as the reason for thinking that T_{a}_{c}_{a}_{c}

T_{a}_{c}

Since T_{a}_{a}_{+}, it follows from T_{a}_{c}

T_{c}_{a}_{+}.

But as we have just seen, it is not. So T_{a}_{c}

Let us turn to pairs of theories where it is false that each unshares an infinity of independent things with the other. So, either each theory has a mere finite (possibly zero) number of things that the other does not or only one does. (Since we are dealing with theories that have an infinity of independent things, it must be that these theories share an infinity of such things.) Let us represent these ways by means of the following Venn diagrams.

Now in order to make comparisons of simplicity, ignore those things that these theories share and focus only on the unshared things. Looking at the left-hand diagram, suppose that the number of these things in one theory is

Here then, when it comes to theories that have an infinite number of independent things, a basis involving finite numbers has been established on which to make judgments of simplicity. In order to know which theory is simpler, all we have to do is look at the number of their

Because a basis has been established on which judgments of simplicity can be made for theories which posit an infinity of independent things, we can start to give a general account of what makes it that one plurality of independent things is smaller than another. Let us begin by no longer assuming that both theories have an infinity of independent things. This yields the following Venn diagrams (note that these diagrams are consistent with both theories having, and only having, things that the other does not).

Making comparisons of simplicity here proceeds in the same manner as before. Again, ignore the shared things, focus on the unshared things, and make comparisons of simplicity on the basis of the number of these unshared things.

We have so far restricted ourselves to theories each of whose things is independent of any other thing. Doing so made it easy to see the basic idea, which is to ignore the shared things, focus on the unshared things, and make comparisons of simplicity on the basis of the number of these unshared things. But we need to expand on this idea by looking at scenarios where this restriction is not in place.

Where the ellipses tell us that there are an infinite number of

Now, in comparing the simplicity of T_{1} and T_{2}, we want to see what ontological costs, if any, each makes that the other does not. That is, we want to see what unshared things, in each theory, we should be looking at and count these things. But how should we go about doing this? Here is a way we should not:

Choose _{1}, and _{2}, ignore the things these pluralities share, focus on the unshared things, and make a comparison of simplicity based on the number of these unshared things.

Why not? Because it yields inconsistent results. For example, _{1}, _{2}, _{3}, … in T_{1} unshares an infinite number of things with _{1}, _{2}, _{3}, … in T_{2}, and vice-versa, whereas _{1}, _{2}, _{3}, … in T_{1} and _{1}, _{2}, _{3}, … in T_{2} unshare nothing. So, given the first pair of pluralities, T_{1} and T_{2} are simplicity incommensurable but, given the second pair, they are not (being instead co-simple).

How then should we go about comparing the simplicity of T_{1} and T_{2}? First, take any of those largest maximal pluralities of independent things in T_{1} that overlap the _{2}._{1}, _{2}, _{3}, … is one of these pluralities, and since it overlaps the most with the largest maximal plurality of independent things in T_{2} (_{1}, _{2}, _{3}, …), then it is a plurality we can take._{1}, _{2}, _{3}, … in T_{1} (nothing) are the relevant unshared things.

Do the same thing with T_{2}. Take any of those largest maximal pluralities of independent things in T_{2} that overlap the most with some largest maximal plurality of independent things in T_{1}. Since _{1}, _{2}, _{3}, … is the only largest maximal plurality of independent things in T_{2}, and since it overlaps the most with _{1}, _{2}, _{3}, …, which is one of the largest maximal pluralities of independent things in T_{1}, then it is the plurality we should take. Next, ignore the things these pluralities share. What remains in _{1}, _{2}, _{3}, … in T_{2} (nothing) are the relevant unshared things.

Now, what matters when it comes to making comparisons of simplicity is the number of things that remain, and so the number of relevant unshared things. Since nothing remains in _{1}, _{2}, _{3}, … in T_{1} and nothing remains in _{1}, _{2}, _{3}, … in T_{2}, then T_{1} and T_{2} are co-simple, which is the result we want (or so it seems to me).

Let us consider a slightly more complicated case. Consider the following two theories, neither of which share anything at the fundamental level (the first has the odd-numbered _{3} includes something (_{1}) that T_{4} does not but not vice-versa:

Following the example set by our last case, take any of those largest maximal pluralities of independent things in T_{3} that overlap the most with some largest maximal plurality of independent things in T_{4}. Since _{1}, _{2}, _{3}, … is one of these pluralities, and since it overlaps the most with _{2}, _{3}, _{4}, … in T_{4}, then it is a plurality we are free to take._{1}, _{2}, _{3}, … in T_{3} (_{1}) are the relevant unshared things.

Do the same thing with T_{4}. Take any of those largest maximal pluralities of independent things in T_{4} that overlap the most with some largest maximal plurality of independent things in T_{1}. Since _{2}, _{3}, _{4}, … is one of these pluralities, and since it overlaps the most with _{1}, _{2}, _{3}, … in T_{3}, then it is a plurality we are free to take. Next, ignore the things these pluralities share. What remains in _{2}, _{3}, _{4}, … in T_{4} (nothing) are the relevant unshared things.

Looking at the number of things that remain, and so at the number of relevant unshared things, since _{1} is what remains in _{1}, _{2}, _{3}, … in T_{3} and nothing remains in _{2}, _{3}, _{4}, … in T_{4}, then it is T_{4} that is the simpler theory, demanding less of the world than T_{3}.

We brought out the basic idea by working with theories each of whose things is independent of any other thing. We have expanded on this idea by applying it to theories where some things are not independent of others. It is now time to turn all of this into an expression of the independence approach. Take then any of those largest maximal pluralities of independent things in T, _{T}, that overlap the most with some largest maximal plurality of independent things in T*, and ignore the things that are shared between these pluralities._{T}, if anything, are the relevant unshared things in T. Do the same thing for T*, taking any of those largest maximal pluralities of independent things in T*, _{T*}, that overlap the most with some largest maximal plurality of independent things in T and ignore the things that are shared between these pluralities. What remains in _{T*}, if anything, are the relevant unshared things in T*. Focusing then on these pluralities, if the number of unshared things in _{T} is _{T*} is at least _{T} is smaller than _{T*}. We can now give a fully general and perspicuous expression of the independence approach:

_{df}._{T} is less than the number of unshared things in _{T*}.

Notice that, given this expression, if the number of unshared things in _{T} is less than the number of unshared things in _{T*}, then it must be that either both numbers are finite, one is finite and the other is infinite, or one is a smaller infinity than the other.

Notice also that if the above tells us what it is for T to be simpler than T*, then in order for T and T* to be co-simple, it must be that the number of unshared things in _{T} and the number of unshared things in _{T*} is finite. This condition on co-simplicity should not come as a surprise. We proved earlier, when working with theories each of whose things is independent of the others, that if the number of unshared things in _{T} is the same as the number of unshared things in _{T*}, then if this number is infinite, T and T* are simplicity incommensurable. And it follows from this that if T and T* are co-simple, and so not simplicity incommensurable, then the number of unshared things in _{T} and the number of unshared things in _{T*} is finite.

Before closing, I want to show that this completed expression of the independence approach is equivalent to our initial, finitist, expression when we assume that the largest number of independent things a theory has is finite. That is, given this assumption, we can prove the following:

_{T} is less than the number of unshared things in _{T*} ↔ the number of things in _{T} is less than the number of things in _{T*}.

Proof: assume that the number of unshared things in _{T} is _{T*} is _{T} is among those largest maximal pluralities of independent things in T that overlaps the most with some largest maximal plurality of independent things in T*, and _{T*}). Now, it cannot be that the number of shared things in _{T} is greater than the number of shared things in _{T*}. For if it were, then there would be some largest maximal plurality of independent things in T* that overlaps more with some largest maximal plurality of independent things in T than does _{T*}. But by assumption, there is not. By identical reasoning, it cannot be that the number of shared things in _{T*} is greater than the number of shared things in _{T}. So the number of shared things in _{T} is the number of shared things in _{T*}. But then, since the number of unshared things in _{T} is less than the number of unshared things in _{T*}, the number of things in _{T} is less than the number of things in _{T*}.

Going in the other direction, assume that the number of things in _{T} is less than the number of things in _{T*}. Since, as just seen, the number of shared things in _{T} is the number of shared things in _{T*}, then if the number of things in _{T} is less than the number of things in _{T*}, the number of unshared things in _{T} is less than the number of unshared things in _{T*}. Thus, Equivalence*.

So, given our completed expression, in cases where the largest number of independent things is finite, the simplicity of a theory boils down to the largest number of independent things a theory has. And this, of course, is the result we want.

The independence approach to simplicity is an attractive one. In appealing only to grounding, it is cheap. In making simplicity a matter of unity, it is conservative. In getting the facts right in various grounding scenarios, it is flexible. And in yielding surprising results in non-standard grounding structures (criss-crossed and egalitarian ones), it is illuminating.

In this paper’s approach to simplicity, independence takes center stage. But

_{df}. x

Note the difference between independence and partial independence. Unlike the former’s _{2} is partially independent, but not independent, of _{1}.

_{1}, …, _{n}_{1}, …, _{n}

In being none of _{1}, …, _{n}_{1}, …, _{n}

Part to Full has an important consequence. Suppose that each of _{1}, …, _{n}_{1}, …, _{n}_{1}, …, _{n}_{1}, …, _{n}_{1}, …, _{n}

This result is important. It shows us that adding something that is partially independent of some independent things results in a larger plurality of independent things. So adding to a theory something that is partially independent of the things in that theory results in a larger plurality of independent things. Given this paper’s approach to simplicity, it follows that adding partially independent things to a theory is tantamount to decreasing the simplicity of that theory.

This has epistemological import. Knowing that something is partially independent of some things requires knowing less than knowing that something is independent of those things (this is because knowing that something is not grounded in some things requires knowing less than knowing that something is not partially grounded in those things). All else being equal then, knowledge of partial independence is easier to have than knowledge of independence. So, given Part to Full, we can know the harder by means of the easier. Here then, the notion of partial independence proves useful when it comes to the epistemology of simplicity.

_{df}. x

This is to give a “broad” account of being nothing over and above. It is not just that grounded things are nothing over and above their grounds (

When it comes to simplicity, this account of being nothing over and above yields the right results. Given that the stem partially grounds the apple, it is nothing over and above the apple. And so it counts no more against the simplicity of a theory than the apple. And this is right given the independence approach to simplicity. Given that the stem partially grounds the apple, the independent things required by the stem are at most a proper plurality of the independent things required by the apple. But then from the independence approach to simplicity, the stem counts no more against the simplicity of a theory than does the apple. The opposite does not hold. The apple is not nothing over and above the stem. It is very much over and above it. And so it should be that the apple is partially independent of the stem. And it is! It is not the stem, does not partially ground the stem, is not grounded in the stem, and is not grounded in any of the stem’s partial grounds. Given this and Part to Full, it follows that the apple requires a larger plurality of independent things than does the stem (which, intuitively, it does). But then from the independence approach to simplicity, the apple counts more against the simplicity of a theory than does the stem. This is exactly as it should be.

Turning now to independence, it allows us to make sense of a strong notion of being something over and above some things. Here is the thought:

_{df}. x

This captures an intuitive notion. As seen above, the apple is something over and above the stem. It is not the stem, does not partially ground the stem, is not grounded in the stem, and is not grounded in any of the stem’s partial grounds. But it is partially grounded in the stem. And so, in spite of being something over and above the stem, it is not

Given our distinction between independence and partial independence, we have three key notions: being nothing over and above some things, being something over and above some things, and being strongly something over and above some things. Of course, with respect to some things, whatever stands in the first relation to these things cannot stand in the second and third. Whatever stands in the second relation to these things need not stand in the third. But whatever stands in the third relation to these things can and must stand in the second.

For a recent defense of this approach, see Baron & Tallant (

See

As the reader can see, on pain of entailing that non-existent things are fundamental, negation takes narrow scope in ‘

For a modal way, see Armstrong (

_{1},…, _{m}_{1}, …, _{n}_{df}._{1}, …, _{m}_{1}, …, _{n}

Note that pluralities of one are pluralities of independent things. For any

What does largeness amount to here? Though I like to talk in terms of pluralities rather than sets, it will perhaps do here to put it set-theoretically: largest in the sense of having the greatest cardinality.

And so, for co-simplicity, T and T* are co-simple ↔_{df}. X_{T} is the same size as _{T*}.

This is neutral over whether it is quantitative or qualitative simplicity that is at issue. For example, if one wants to focus on qualitative simplicity, then the number of things in _{T} amounts to the number of types in _{T}, where one is free to understand types as they see fit: properties, predicates, sets or pluralities of things, or what have you (notice though that if types are pluralities, then the independence approach requires that one makes sense of pluralities of pluralities, and so of super-pluralities).

There is a brief snag. Since the approach requires quantifying over pluralities of independent things, it would seem that we cannot infer from it that theories according to which there is nothing are simpler than ones according to which there is something. There are a few ways to respond. The one I prefer quantifies over the degenerate ‘empty plurality’ and has it that the largest plurality of independent things in a theory that posits nothing is this plurality (thanks to Jonathan Schaffer for this suggestion). Now taken at face value, this involves quantifying over zero things and so involves a

_{1}, …, _{n}_{df}._{1}, …, _{n}_{1}, …, _{n}_{df}._{i}_{j}_{1}, …, _{n}_{i}x_{j}_{j}x_{i}_{i}_{j}

For an outline of such a view, see van Inwagen (

Appealing to bottomless cases allows us to avoid the following response to monist and dualist cases: what makes monisms preferable to dualisms is not that the former are simpler than the latter, but that the former leave fewer things ungrounded (

This notion can and should be contrasted with other notions of unity. Prominent here is epistemological or pragmatic unity, which often has to do with definability, derivability, and explanation, which are frequently understood in semantic or logical terms. For a classic account of this kind of unity in science, see Nagel (

They only require that there must be several levels and that the number of levels must be finite. With respect to the

But of course, we can understand it grounding-theoretically. Especially if wholes are grounded in their parts. Indeed, the independence approach to simplicity applies nicely to Oppenheim and Putnam’s picture. The branches of science are unified

Or at least, grounds

Of course, any such “criss-crossed” theory has a complexity that any such “linear” theory does not. But this complexity is found in its grounding structure taken as a whole. It is not found in its ontology. As the above pictures make clear, criss-crossed structures are not as neat and graceful as linear structures (lines are neater than criss-crosses). And so “linear” theories are more elegant than “criss-crossed” ones. That’s the sense, if any, in which a theory which has the above criss-crossed structure is less simple than a theory which has the linear one.

Some of these are, like the criss-crossed structure, revealing. For one such structure, see §3.1.

This is what Dixon (

They cannot all be fundamental since we are assuming that the number of them is greater than the number of fundamental things.

I thank Jonathan Schaffer for raising this objection.

The unity test we employed earlier with respect to criss-crossed structure can be applied here. Once

For a similar verdict, see Baron & Tallant (

There are two ways for something to be superfluous: the superfluous can be superfluous in virtue of failing to do any work (so they are idle) or in virtue of doing work, but not doing new work (so they overdetermine). In T*, the 1,000 grounded things are superfluous not because they fail to do work, but because the work they do is not new. For an excellent paper on this and related matters, see Barnes (

Their target is not this approach. It is Schaffer’s (

Swinburne (

In fact, the likelihoods can be used to show that a theory’s positing more things than another can favor accepting it if in so doing, it says less about what does not exist. Suppose that there are only three possible things A, B, and C. Further suppose that according to T**, only A and B exist and that according to T***, only A exists. Now an experiment is performed, and the result is that C does not exist. Assuming that both theories have the same priors, the theory with more things comes out as more probable: the probability of performing the experiment and it showing us that C does not exist is 1 given T** but 1/2 given T***.

See Sider (

See Jeffreys (

Here, I assume that appeals to simplicity in metaphysics are appropriate. For some who think they are not, see Huemer (

The appeal to sentences is important given certain brands of nominalism. If one is a Quinean about ontological commitment, then it can be that ‘2 × 3 = 6’ is true so long as the proposition it expresses is one that does not involve quantifying over numbers.

Sober (

I thank a referee for showing me the need to say more.

What does overlapping the most amount to? Suppose we have the natural numbers, the natural numbers minus the number one, and the natural numbers minus the number one and the number two. Each of these pluralities share the same number of things (they each share an infinite number of numbers). But the first plurality overlaps more with the second plurality than it does with the third. We can say then that, where _{df}. X

There are an infinite number of pluralities that are among those largest maximal pluralities of independent things in T_{1}. In addition to _{1}, _{2}, _{3}, … and _{1}, _{2}, _{3}, …, we have _{1}, _{2}, _{3}, … and _{1}, _{2}, _{3}, … and _{1}, _{2}, _{3}, … etc. But, with the exception of the first plurality, none of these overlaps the most with the largest maximal plurality of independent things in T_{2} (_{1}, _{2}, _{3}, …), since each unshares at least one thing with such a plurality.

_{1}, _{2}, _{3}, … is another one of these pluralities, overlapping just as much with _{1}, _{2}, _{3}, … in T_{4}.

This assumes that for T, there is some largest maximal plurality of independent things (thanks to a referee for pointing this out). But this seems right. Put differently, for any theory, there is always a theory with a larger number of independent things. But then for any theory, and so for T, there is some largest, maximal, plurality of independent things. Or if there is not, this needs to be shown.

There are other notions of partial independence, some of which are stronger than others. The weakest says that

There are also formal differences. In the third structure, _{2} is partially independent of _{2}, but not vice-versa, and is partially independent of _{1}, _{1} when taken individually, but not when taken collectively. So partial independence is not symmetric and does not satisfy its version of Collection.

Given Part to Full and that independence is stronger than partial independence, we get: ∃_{1}, …, _{n}_{1}, …, _{n}

There are models of grounding where Part to Full fails. Suppose that _{1}, …, _{n}_{1}, …, _{n}

Proof: if _{1}, …, _{n}_{1}, …, _{n}_{1}, …, _{n}_{1}, …, _{n}_{1}, …, _{n}

For some prominent appeals to this notion, see Lewis (

And so _{df}. x

Of course, something can be nothing over and above some things and cost less than them. Take the stem, skin, and core of the apple. Taken collectively, the stem is nothing over and above these things. But it does cost less than them. Why? Because the skin and core are something over and above it.

But what about the claim that things are nothing over and above those things whose partial grounds ground them? Given that things are nothing over and above their grounds and that partial grounds are nothing over and above what they partially ground (as has just been argued), it follows from the transitivity of being nothing over and above that things are nothing over and above those things whose partial grounds ground them.

For helpful comments, I thank audiences at the 2019 Illinois Philosophical Association, the 2019 Central States Philosophical Association, and the 2020 Eastern American Philosophical Association. For those who commented on drafts of this paper, I thank Chad Carmichael, Judith Crane, Louis deRosset, Aaron Griffith, Dan Korman, Bradley Rettler, and Kristin Whaley. Special thanks to Jonathan Schaffer for his very helpful comments on two early drafts and to a referee whose comments greatly improved the last section of this paper. Finally, I thank my wife, Amy Saenz, for her love and support.

_{P}