## 1. Varieties of Ontological Dependence



A crucial notion in metaphysics is that of one entity *depending
for its existence upon* another entity--not in a merely
causal sense, but in a deeper, ontological sense (but see Ingthorsson
2002: 96 for discussion on the idea that causal relations could be
understood as one-sided existential dependence; cf. also Bennett 2017:
Ch. 4, where it is argued that the line between causal and noncausal
determination is not clear). The kind of dependence in question must
also be distinguished from any kind of *logical* dependence,
because logical relations, strictly speaking, cannot obtain between
concrete objects, nor between abstract objects that are not
propositional in nature. We should also distinguish ontological
dependence, broadly conceived, from what is usually considered a
stricter type of metaphysical dependence, namely *metaphysical
grounding*. The link between ontological dependence and
metaphysical grounding (or simply "metaphysical
dependence", as it is sometimes called, see Rosen 2010 and the
separate entry on
 metaphysical grounding;
 see also Trogdon 2013 for discussion) will be discussed below, but
this entry concerns a somewhat broader family of relations of
dependence. The notion of ontological *in*dependence is also
important, and it is often used to describe the idea that something is
*fundamental*. This idea as well will be discussed below (but
see also the separate entry on
 fundamentality).



While much of the early work on ontological dependence was done
primarily in a setting where dependence was thought to be an
inherently *modal* notion (e.g., Simons 1987), the recent trend
in *hyperintensional* metaphysics (see Nolan 2014) has had an
enormous impact for discussion surrounding dependence. The thought
here is that we need more fine-grained tools than mere necessary
equivalence: two statements can apply to the very same entities across
all possible worlds but not be equivalent. Here, the notion of
*essence* is often invoked, and many would now regard the
ultimate basis of dependence (and indeed of modality itself) to be
essence (see Fine 1994a, Lowe 1998, Correia 2005, and Koslicki 2012).
Finally, it is entirely possible to deny both approaches and regard
the notion of ontological dependence itself as a primitive (see
Thomasson 1998: Ch. 2 and Barnes 2012: 879). Some would regard the
notions of metaphysical grounding or fundamentality to be similarly
primitive (see Fine 2012, Rodriguez-Pereyra 2005, Rosen 2010, Schaffer
2009, and Wilson 2014). Primitivism does not mean that the notion is
mysterious or obscure, as we can still illuminate it, e.g., by
describing its instances. But it does mean that the notion is
unanalyzable.



In what follows, the discussion will roughly follow the historical
development, starting with the attempts to give a modal-existential
analysis of ontological dependence in
 SS2.
 As we will see in
 SS2.2,
 these attempts do have some serious shortcomings, even if they can be
of use especially in certain restricted contexts. In
 SS3,
 we see that there are important cases where a more fine-grained,
*asymmetric* notion of explanatory dependence seems warranted.
We will then proceed to the essentialist analysis of dependence in
 SS4,
 a discussion of the relationship between metaphysical grounding and
dependence in
 SS5,
 finishing with a brief discussion of various other connections and
applications in
 SS6.



To get us started, we may consider a few simplified examples. It is
not uncommon to see the notion of ontological dependence used in a
rather coarse-grained manner, given that it encompasses a family of
relations. For instance, we often see claims such as:



(1)
"Sets ontologically depend on their members."
(2)
"Electricity ontologically depends on electrons."
(3)
"Socrates does not ontologically depend on singleton
Socrates."


While all of the above no doubt express some important type of
dependence relationship, there are also important differences between
them. (1) may be understood as stating that a set {*x*, *y*,
*z*} could not exist if its members, namely *x*, *y*,
*z*, did not exist. One way to analyze the type of dependence in
question is as *rigid existential dependence*, to be clarified
in a moment. Another sense of dependence that could be at work in (1)
is *identity-dependence*, but we will return to this later on.
In (2), we seem to have in mind a more general kind of dependence:
there could not be electricity, now or ever, if there were no
electrons. So the existence of electricity depends on the existence of
a very specific kind of particle, the electron. This second type of
dependence is also existential, but to separate it from the rigid
dependence in (1), we may call it *generic existential
dependence*. (3) is a famous case borrowed from Fine (1994a,
1994b). Singleton Socrates, expressed by "{Socrates}", is
the set containing Socrates as its sole member. From (1) we see that
the set itself plausibly depends on Socrates, but since the set
necessarily exists when Socrates does, the dependence between them
would seem to be *symmetric*. However, (3) captures the very
plausible idea that it is not any part of the *essence* or
*identity* of Socrates that it should belong to {Socrates}. So,
(3) must refer to some form of *essential dependence* (or lack
thereof in this case).



We will provide a more detailed analysis of these examples below, but
we can already see that a family of--sometimes
competing--notions is beginning to emerge. However, we should
formulate each notion somewhat more precisely. The first thing to note
in defining ontological dependence is the *modal-existential*
element in dependence claims. For instance, we've said that a
set *cannot* exist unless its members do. So there is a sense
in which the existence of a set *necessitates* the existence of
its members. Indeed, it is common to talk about, e.g., *rigid
existential necessitation* as synonymous with rigid existential
dependence. Typically, statements of ontological dependence are
thought to refer to *metaphysical modality* (rather than, say,
conceptual or logical modality), primarily because they concern
matters that are broader than just conceptual or logical. For
instance, *substances* are often considered to be entities that
do *not* depend, in a distinctively metaphysical sense, for
their existence upon anything else (see Lowe 1998: Ch. 6). We can see
this already in Descartes, who asserts that



> 
> 
> by *substance* we can understand nothing other than a thing
> which exists in such a way as to depend on no other thing for its
> existence. (*The Philosophical Writings of Descartes*: vol. I,
> p. 210)
> 
> 
> 



The idea of ontological dependence goes back further than Descartes
though, at least to Aristotle (see Corkum 2008, Koslicki 2013, and
Peramatzis 2011), and it appears throughout the history of philosophy.
For instance, there is a notion of dependence operative in the
philosophy of John Duns Scotus, who uses it in his proof of the
existence of God (see the separate entry on
 John Duns Scotus,
 SS2.2 for details). Later, precursors to the contemporary work on
ontological dependence can be found in the work of Franz Brentano and
Edmund Husserl; in particular, Husserl's work on the dependence
between wholes and parts (for discussion, see Simons 1982 and
Johansson 2004: Ch. 9).



Finally, there are also those who are sceptical about the metaphysical
import of notions of ontological dependence. There are several reasons
why one might consider ontological dependence relations to be
unintelligible or unhelpful; for instance, one might think that they
conflate other relations or that they are extraneous additions to
being. However, it is commonly held among those working with
ontological dependence that these dependence relations, being
*formal ontological relations*, are not additional elements of
being (cf. Lowe 2006: Ch. 3). Such relations are typically thought to
be *internal* (see Armstrong 1978: 85, who defines two things
to be internally related if and only if there exist properties of them
which logically necessitate that the relation holds; for further
discussion on internal and external relations, see the separate entry
on
 relations).
 Still, there is at least an ideological cost in some approaches to
ontological dependence and hence some would prefer a
*deflationary* approach. According to this type of approach, we
may be able to account for supposed ontological dependence relations
in terms of a "cluster" of modal and mereological
relations (for a defense of this type of approach, see Kovacs
2018).


## 2. The Modal-Existential Analysis of Dependence



In this section we will focus on the modal-existential analysis of
ontological dependence, which used to be the dominant approach. But
even if this approach has fallen out of fashion, it is important to
see why that is the case and we cannot do that without understanding
the approach. Moreover, there are those who do not wish to commit to
the ontologically heavyweight notion of essence, but who nevertheless
employ the notion of ontological dependence (e.g., trope bundle
theorists, such as Keinanen, Keskinen, & Hakkarainen 2019).
We will discuss some of the problems that this account faces in
 SS2.2
 and return to the essentialist analysis in
 SS4.


### 2.1 The Modal-Existential Analysis



Let us start from
 (1)
 in the previous section; how should this relationship between sets
and their members be defined according to the modal-existential
analysis? An obvious proposal would be to say, quite simply:



(EDR)
*x* dependsR for its existence upon *y*
=df Necessarily, *x* exists only if *y*
exists.


The subscript "R" is used here because it seems
appropriate to call this species of existential dependence of one
object upon another *rigid* existential dependence. Why
"rigid"? Because there is no flexibility here: the
existence of a given *x* requires the existence of *that very
*y**. It could not be something a little bit like *y*,
something falling roughly in the same category, for instance; it
*must* be *y*. The *definiens* in (EDR) is
equivalent to "Necessarily, if *x* exists, then *y*
exists", so that according to (EDR) the existential
dependenceR of *x* upon *y* amounts to the
*strict implication* of *y*'s existence by
*x*'s existence. We have mentioned one example of rigid
existential dependence, namely, sets ontologically depending on their
members (more precisely, a set depends rigidly on the very members it
has, i.e., any change in a set's members will change the set
itself). Another, although more controversial, example is a particular
person depending for her existence on her parents, or, more precisely,
on the particular sperm and egg that she originates from. This example
is of course related to *the essentiality of origin* (as
discussed in Kripke 1980).



We can express (EDR) in a more compact manner by resorting to further
formalization (the following notation is used, e.g., in Correia 2008).
We can use the sentential operator "\(\square\)" for
metaphysical necessity, the one-place predicate "*E*"
for existence, and the two-place sentential operator
"-" for material implication. Following this
notation, we can formalize rigid existential dependence as follows:
"\(\square(Ex \rightarrow Ey)\)", which can be read as
"*x* rigidly depends for its existence on *y*",
or alternatively "*x* rigidly necessitates *y*".
Note that (EDR), somewhat controversially, implies that everything
dependsR for its existence upon *itself*. It would,
of course, be easy enough to modify (EDR)'s *definiens*
to read "*y* is not identical with *x* and,
necessarily, *x* exists only if *y* exists", but that
would have the disadvantage of *precluding* anything from
dependingR for its existence upon itself.



We can quite naturally contrast *rigid* existential dependence
as defined by (EDR) with what might appropriately be called
*non*-rigid existential dependence, defined as follows:



(EDN)
 *x* dependsN for its existence upon the *F*
=df Necessarily, *x* exists only if the *F*
exists.


The thought here is that--to use the language of "possible
worlds"--"the *F*" in any instance of
(EDN) might well denote *different* entities in different
possible worlds. So, for example, it might be said that a material
object *x* dependsN for its existence upon *the
matter composing *x**, even though it might have been
composed of different matter, because in every possible world in which
*x* exists the matter composing **x* in that world*
exists in that world.



On the face of it, (EDR) seems to capture precisely one strongly
intuitive notion of existential dependence; or more accurately, one
direction of that notion. The intuitive idea behind this notion is
that if *y* does not exist, then the dependent entity *x*
does not exist. For example, when it is said that a particular event,
such as the assassination of Caesar, depends for its existence upon
Caesar, (EDR) seems to explicate this appropriately in terms of the
fact that the assassination could not have existed if Caesar had not
existed to be assassinated. Some *other* assassination, we may
suppose, could have existed at that very time and place, but for
*that very assassination* to have existed, Caesar himself had
to exist.



However, there are clearly cases in which (EDR) fails to capture the
intuitive sense of dependence at hand. Consider a living organism. A
living organism would appear to depend for its existence upon its
parts, such as cells. But we also know that a living organism may
survive a change of any of its cells, provided that the change is
effected in a non-disruptive manner. It is true, of course, that such
an organism must *have* parts such as cells if it is to exist,
but *which* objects those parts are is inessential--and
consequently it is not the case that it depends for its existence, in
the sense defined by (EDR), upon any one of those parts. But it is
possible to define *another* sense of existential dependence in
which it *is* true to say that a composite object depends for
its existence upon its proper parts; a *generic* notion of
existential dependence, defined as follows:



(EDG)
 *x* dependsG for its existence upon *F*s
=df Necessarily, *x* exists only if some *F*
exists.


Composite objects are existentially dependent objects in the sense of
(EDG), since they require the existence of proper parts (set *F*
as "proper part of *x*" in (EDG)). Using the previous
formal notation, we could express (EDG) as "\(\square(Ex
\rightarrow \exists Fy)\)". Here we have added the existential
quantifier "[?]" as well as the general term
"*F*" to express the thought that "*x*
generically depends for its existence on something being an
*F*", or alternatively "*x* generically
necessitates *F*". The important difference between the
rigid and the generic cases is that (EDR) refers to a specific object
whereas (EDG) only requires that at least some *F*s exist.
Another example, mentioned earlier, where (EDG) would seem to capture
the correct sense of dependence is
 (2),
 "Electricity ontologically depends on electrons". More
precisely, we could say that electricity dependsG for its
existence upon electrons.



We would now have the tools to formalize most of the mentioned
examples, but note that there are cases where further tools are
required. Consider:



(4)
"Children ontologically depend on their parents."


On the face of it, what we mean in (4) is that if parents *x* and
*y* had not existed, then their child *z* could not have
come into existence. This looks like a case of rigid existential
dependence, but it is clear that *once *z** has been born,
her parents can go out of existence without any effect on her own
existence. At that point, there is only *past* rigid
existential dependence. For cases such as this, we would require
temporally relativized versions of
 (EDN)
 and
 (EDG),
 but we will omit these complications here (see Thomasson 1998:
24-34 and Correia 2005, 2008 for some versions of temporally
relativized ontological dependence).



Note, incidentally, that *generic* existential dependence as
defined by (EDG) above is very close to a pluralized form of
*non-rigid* existential dependence, as defined by (EDN). We
could call this *plural* non-rigid existential dependence,
definable as follows:



(EDP)
 *x* dependsP for its existence upon the *F*s
=df Necessarily, *x* exists only if the *F*s
exist.


The similarity between
 (EDG)
 and
 (EDN)
 is apparent: saying--for example--that a composite object
*x* dependsG for its existence upon proper parts of
*x* is very close indeed to saying that *x*
dependsP for its existence upon the existence of
*the very kinds* of proper parts that *x*
has.


### 2.2 Some Problems for the Modal-Existential Analysis



An important group of problems for the modal-existential analysis of
ontological dependence emerges from essential properties (at least if
they are considered as distinct from modal properties, as argued in
Fine 1994a). What could be a plausible example of an essential
property of an individual object, say, Socrates? Let us assume that
properties are to be conceived of as property *instances*, such
as the particular redness of a certain apple. Of course, an apple can
change its color, so that this is not an example of an
*essential* property of the apple. But what about--in the
case of Socrates--his *humanity* (as it were, his
particular *being human*)? Certainly, if there is such a thing
as the particular humanity of Socrates, he cannot lose it without
ceasing to exist. Perhaps we can deny that Socrates's particular
humanity is anything distinct from *Socrates himself*: after
all, its existence necessarily coincides with his. Properties are
commonly said to depend for their existence upon the entities that
possess them. One might propose to state this in the form of a
principle, with the help of
 (EDR),
 as follows:



(PROP-DEP)
If *x* is a property and *y* is an entity possessing
*x*, then *x* dependsR for its existence upon
*y*.


Now, substituting the definiens of (EDR) into (PROP-DEP) gives us:



(PROP-NEC)
 If *x* is a property and *y* is an entity possessing
*x*, then, necessarily, *x* exists only if *y*
exists.


Note that (PROP-NEC) is not intended to apply to properties understood
as universals, but only to particularized properties (otherwise
variously known as property instances, individual accidents, tropes,
or modes). These are items exactly like the particular redness of a
particular apple mentioned above, conceived of as an object distinct
from the redness of any *other* apple, no matter how well
matched in color to the first. On this interpretation, (PROP-NEC) has
some plausibility, complying as it does with the intuition that
particularized properties cannot "migrate" from one object
to another. (Actually, (PROP-NEC) itself does not quite imply this,
although it does imply that a particularized property cannot migrate
from one object to another when the first object ceases to exist.
Here, the friend of an essentialist analysis of dependence has some
further options, i.e., one could say that it is part of the nature or
essence of a particularized property to be instantiated by the very
object that it is instantiated by. But we will leave this aside for
now.)



A line of argument that could be developed on this basis suggests that
if indeed there are such things as essential particularized properties
that are "possessed" by composite objects, then they are
in fact to be *identified* with those objects. But it would be
very problematic, quite generally, to argue that wherever items
*x* and *y* are mutually existentially dependent as defined
by
 (EDR),
 they are identical. For instance, consider the relationship between
Socrates and the temporally extended event or process that was *his
life*. Clearly, in terms of (EDR), Socrates's life
dependsR for its existence upon Socrates--but so,
plausibly, does his existence upon it: it is, for Socrates, what we
might call an *inalienable* event. And yet there are things
true of the life of Socrates that are not true of him and *vice
versa* (for example, that it was so many years long, and that he
weighed so many pounds)--so there is no question of their being
identical. Even more strikingly, if we assume that numbers are
necessary existents, and hence mutually existentially dependent, this
line of thought would seem to entail that any two numbers are
identical!



But it could be disputed whether Socrates is existentially
dependentR upon his life--whether *he*
necessarily would not have existed if *it* had not--for it
may be urged that he might have had or led a *different* life.
Now, it is true enough that his life might have been
*qualitatively* different in many ways, but what is currently
at issue is whether he might have had a *numerically* different
life--and it is hard, perhaps, to see how he could. For if it is
accepted that *lives* dependR for their existence
upon the persons whose lives they are, then, necessarily,
*x*'s life exists only if *x* exists. Suppose, for the
sake of argument, that Socrates could have had a numerically different
life: then it would still have been a life which could only have been
Socrates's--no one other than Socrates could have had that
"other" life. But then what could underpin the supposition
that it is indeed a life "other" than the life he actually
had (except qualitatively)? Other possible worlds clearly do contain
lives that do not exist in the actual world, insofar as they contain
*people* who do not exist in the actual world: but there seem
to be no strong grounds to suppose that they do so other than for that
reason. It seems thus that there are certain difficult questions which
the modal-existential analysis may not fully address, at least not in
the form that we have presented it until now. We will continue to
discuss these issues in
 SS3
 and
 SS4.


## 3. Asymmetrical Existential Dependence



Where does the above analysis of existential dependence lead us? The
case of Socrates's life demonstrates that
 (EDR)
 permits the possibility of *mutual* existential
dependenceR between non-identical things. But this
implication does not seem to be acceptable in every context and indeed
symmetric dependence more generally is a controversial issue (see
Barnes 2018 for discussion). Take, again, the relationship between
Socrates and his life. According to (EDR), Socrates is quite as much
existentially dependentR upon his life as his life is upon
him. And yet there is a strong intuition that, in another and perhaps
more important sense, Socrates's life is the truly
*dependent* object here, while Socrates is, in some sense, an
*in*dependent existent. We might say that Socrates's life
exists only *because* Socrates does, whereas it would be
putting the cart before the horse to say that Socrates exists
*because* his life does. Now, it appears that the conjunction
"because" must be *asymmetrical*, because it
expresses an *explanatory* relationship and explanation is
asymmetrical. Plausibly, two distinct states of affairs cannot
*explain each other*. There *may*, quite conceivably, be
*self*-explanatory states of affairs, so we should only want to
urge that *non-identical* states of affairs cannot be mutually
explanatory. Technically, this means that we should strictly describe
explanation as an "antisymmetric" rather than as an
asymmetric relation. The asymmetry of explanation is, of course,
intimately related to the unacceptability of *circular
arguments*. It is also closely linked to recent discussions
regarding *metaphysical grounding*, to which we will return
later, in
 SS5.



One upshot of all this is that, *for the purpose of defining
ontologically independent existents (if there are any)*, (EDR)
should be replaced, at least to a first approximation, by something
like:



(EDX)
 *x* dependsX for its existence upon *y*
=df Necessarily, *x* exists only because *y*
exists.


We use the subscript "X" because, in line with foregoing
remarks, it seems appropriate to call this species of ontological
dependence *eXplanatory* existential dependence. (We shall need
the subscript "E" for another use later.) Here it is
important to note that the presence of the word "only" in
(EDX)'s *definiens* should not be understood as implying
that an object *x* may not dependX for its existence
upon two (or more) *different* things, *y* and *z*.
Thus the particularized relation of Mary's loving
Tom--supposing such an object to exist--plausibly exists
*only because Mary exists*, but plausibly *also* exists
*only because Tom exists*. Furthermore, we may assume that it
is *not* an implication of (EDX) that a composite object
dependsX for its existence upon its proper parts, that is,
that it is *not* the case that it "exists only because
they exist"--on the grounds that it could still exist in
the absence of *those* particular parts, provided suitable
alternative parts were substituted for them. Thus (EDX) is quite
unlike
 (EDG)
 in its implications for part-whole dependence relations, as far as
composite objects are concerned. For the same reason, we may assume
that (EDX) does not imply that an "Aristotelian" universal
dependsX for its existence upon its particular exemplars.
Indeed, we may take it that the *definiens* of (EDX) entails
the *definiens* of
 (EDR)--although
 not *vice versa*, of course--so that the following is a
principle that one could accept:



(B-NEC)
If, necessarily, *x* exists only *because *y**
exists, then, necessarily, *x* exists only *if *y**
exists.


However, despite these clarifications, it must be conceded that the
locution "*x* exists only because *y* exists" is
hardly very perspicuous, either as to its logical form or as to its
exact meaning. Moreover, precisely because we have introduced the
conjunction "because" as an *explanatory*
conjunction, it may be felt that it is not well-suited to the
ontological role now being devised for it (for one of the many recent
attempts to specify the explanatory role of "because", see
deRosset 2013). There are perhaps two sources of worry here: first,
that this approach invites a confusion between metaphysics and
epistemology; and secondly (but relatedly) that contexts governed by
the conjunction "because" are *opaque* (in the
technical sense of the term, in which it implies the non-applicability
of Leibniz's Principle of the Identity of Indiscernibles).



Some of these worries can perhaps be allayed by resorting to the
recent work on metaphysical grounding, which presents a much more
rigorous analysis of non-causal, metaphysical explanation of this type
(see especially the articles in Correia & Schnieder (eds.) 2012a).
This is not to take any stand on the relationship between grounding
and metaphysical explanation (see SS4 of the separate entry on
 metaphysical grounding),
 but only to note that the discussion surrounding explanatory
dependence has progressed, in part, independently of the wider
considerations surrounding ontological dependence. So we should
perhaps accept that
 (EDX)
 as it stands does not really constitute a satisfactory definition of
a species of existential dependence, conceived as an objective
metaphysical relation between entities, because it is insufficiently
perspicuous. In any case, the fact that (B-NEC) but not its
converse holds some appeal indicates that what we should be trying to
frame is a perspicuous definition of a species of ontological
dependence that is a relation between *x* and *y*
*stronger than* (entailing but not entailed by)
"necessarily, *x* exists only if *y* exists".
This should moreover be (for reasons discussed earlier) an
*asymmetrical* relation--or, more accurately, an
*antisymmetric* relation, that is, a relation \(\Rrel\) such
that if \(x\Rrel y\) and \(y\Rrel x\), then \(x = y\). This is to
allow that in principle an object may, in the anticipated sense,
depend ontologically upon *itself*, controversial as this may
be (it is nevertheless a position that is of at least historical
interest, e.g., because of the potential special case of God depending
on herself).



As we shall see in a moment, a relation that may be of some assistance
in spelling out the relevant sense of dependence is the relation of
*identity-dependence*, to be explained below. But first we
should digress for a moment to note that the *one-sided*
holding of the relation defined by
 (EDR)
 is unlikely to do the job. According to this suggestion, we have:



(EDA)
 *x* dependsA for its existence upon *y*
=df

(i)
 necessarily, *x* exists only if *y* exists and 
(ii)
 it is *not* the case that, necessarily, *y* exists
only if *x* exists.
 


Notice that the relation thus defined is *asymmetric* (rather
than *antisymmetric*): it doesn't permit any object to be
existentially dependent upon *itself*. Indeed, we have chosen
to use the subscript "A" here because the relation in
question may aptly be called *asymmetrical* rigid existential
dependence. One might think that this is at it should be, as on the
face of it the idea of something existentially depending on itself
appears very strange. There is of course an obvious theological
candidate that may violate this condition, namely God, so perhaps we
should not rule out the possibility outright. But there may be
slightly less controversial candidates as well, for it could be
suggested that there are *fundamental* entities that
existentially depend on themselves (rather than on nothing). At any
rate, if anything of the sort is possible, we better not rule it out
by definition, as (EDA) does.



Perhaps more importantly, an appeal to (EDA) does not seem to help to
resolve the difficulty raised by the example of Socrates's life.
For *neither* Socrates *nor* his life is existentially
dependentA on the other, since in neither case is clause
(ii) of (EDA) satisfied. But we were looking for a sense of
"ontologically dependent" in which it is *true* to
say that Socrates's life is ontologically dependent upon
*him*, but not *vice versa*.


## 4. Essential Dependence and Identity-Dependence


### 4.1 Why Essential Dependence?



Until quite recently, it was common to think that ontological
dependence can be fully characterized in modal-existential terms, as
we have seen above. One obvious reason for this is that if one adopts
the usual "modalist" analysis of *essence*,
essential dependence will collapse into a form of modal-existential
dependence (one classic defense of the "modalist" analysis
is Marcus 1967). But it seems clear already starting from Aristotle
that there is an alternative way to formulate ontological dependence
if essence is not analyzed in modal terms (as Fine 1994a and
especially those working in the "neo-Aristotelian"
tradition would have it). However, this is not to suggest that
Aristotle did not rely on the modal-existential notion of ontological
dependence. Rather, there are reasons to think that Aristotle's
understanding of dependence encompasses *both* the
modal-existential notion and the essentialist notion to be described
below (for historical details and further discussion, see Corkum 2008,
Koslicki 2013, and Peramatzis 2011).



One motivation for developing a *non-modal* conception of
ontological dependence is that the modal-existential analysis appears
to be too coarse-grained for some cases. We have already discussed
some of these cases, but the most well-known examples have been made
famous by Fine (e.g., 1994b). Consider, for instance, what the
modal-existential account entails in the case of necessary existents.
Take Socrates and the number 2, for example. Given that numbers
necessarily exist, it is necessarily the case that 2 exists if
Socrates does. But presumably we do not want to say that Socrates
depends upon the number 2, or indeed on most necessary existents that
you might put in the place of 2. So the modal-existential account
makes everything depend upon every necessary existent, which seems
like the wrong result.



Admittedly, those who defend a modal-existential analysis of
ontological dependence could insist that it applies only to
*contingent* objects (cf. Simons 1987: 295). Simons makes this
type of qualification by focusing on *concrete* entities, hence
excluding necessary existents by definition; he also excludes
self-dependence. Simons calls the resulting notion of dependence
*weak rigid dependence*, but a stronger notion, *strong
rigid dependence* (Simons 1987: 303) is also defined--the
latter is a special case of the former. One example of weak rigid
dependence as defined by Simons would be a particular water molecule
depending for its existence on a particular oxygen atom. In the case
of strong rigid dependence, the dependent object cannot be a proper
part of the object it depends upon. So object *x* is strongly
rigidly dependent on object *y* if *x* depends for its
existence on *y* and *y* is not a proper part of *x*.
One example of strong rigid dependence defined thus would be a trope
(or mode) depending for its existence on a substance. In addition to
these rigid notions, Simons defines corresponding notions of (weak and
strong) *generic dependence*.



While it is possible to avoid some of the challenges raised for the
modal-existential account with these qualifications, they do
nevertheless call for an alternative, more fine-grained account of
ontological dependence that could also be applied to necessary
existents. As last resort, a proponent of the modal-existential
account who does not wish to buy into essentialist ontology could
simply bite the bullet and insist that every contingent entity
*does* rigidly depend for its existence on necessary existents.
One reason to do so would be the ability to get by with a sparser
battery of formal ontological tools--a consideration motivated by
parsimony.



The modal-existential analysis of ontological dependence can thus be
developed further, and it can perhaps overcome some of the problems
that were pointed out earlier. But there are areas where a more
fine-grained notion would seem to be required and there is also a
historical precedent (e.g., in Aristotle) for such a notion. It should
be noted though that much of the contemporary literature in defense of
a non-modal, fine-grained analysis (such as Fine 1994b and Koslicki
2012), operates in a "neo-Aristotelian" framework which
typically assumes some "non-modalist" version of
essentialism. Accordingly, there is an obvious rift between the
modal-existential analysis and the essentialist analysis--one
that we cannot fully bridge here. In any case, it is good to keep in
mind that the notion of ontological dependence itself, when used as a
tool to formulate the commitments of a specific theory, does not
immediately force one to make a commitment in this regard, even though
some of its applications may entail such a commitment.


### 4.2 Identity-Dependence



A point of entry to the idea that there are notions of dependence not
analyzable in terms of the modal-existential account--and hence a
motivation to pursue a non-modal analysis of dependence
altogether--are cases that do not seem to involve a requirement
for *existence* at all. Indeed, as we already saw in the case
of
 (3),
 "Socrates does not ontologically depend on singleton
Socrates", it seems that something beyond mere existential
independence is being expressed. Instead, one might say that singleton
Socrates does not have anything to do with the *identity* or
*essence of Socrates*, and it is this lack of
*identity-dependence* between Socrates and {Socrates} that we
wish to express with (3). In other words, being a member of singleton
Socrates is not an *essential property* of Socrates. Recall
that a similar issue seemed to arise with regard to the possibility of
composite objects having *essential* proper parts. We will now
consider a more systematic method for dealing with cases of dependence
involving the essences or essential properties of objects.



It is not quite straight-forward to define essential dependence,
although a formal definition will be given below. Before that, we
ought to get a better picture of what it means to say that an object
depends upon something for its *identity*, that is, we should
clarify the relation of *identity-dependence*. Note that the
notion of "identity" at play here is not the one
symbolized with the "equals" sign, i.e., "=".
Rather, we mean "identity" in the sense of *what* a
thing is, or *which* thing of a certain kind a thing is
(compare this with Aristotle's *to ti esti*, "the
what it is", see SS7 of the separate entry on
 Aristotle's metaphysics).
 Informally speaking, to say that the identity of *x* depends on
the identity of *y*--or, more briefly, that *x* depends
for its identity upon *y*--is to say that *which*
thing of its kind *y* is fixes (or at least helps to fix)
*which* thing of its kind *x* is. By "fixes"
in this context is meant *metaphysically determines*. For
instance, then, the identity of a *set* is fixed by the
identities of its members, as was hinted already in
 SS1.
 Likewise, the identity of an *assassination* is (at least
partially) fixed by the identity of the person assassinated. These
relationships of identity-dependence are direct consequences of the
identity-criteria governing the kinds of which the items thus related
are instances. (For further discussion on identity-criteria and
identity-dependence, see Lowe 1989, 2009: Ch. 2, and 2012.)



One way to attempt to precisify the notion of essence is to note that
we are here concerned with identity in the sense of
*individuality* rather than the *relation* of identity.
In other words, identity-dependence expresses the determination of the
individuality of objects in terms of the individuality of other
objects. Thus, the identity-dependence of a set upon its members could
be regarded as a consequence of the fact that the Axiom of
Extensionality functions as the criterion of identity for sets. The
set's members determine the individuality of the set--they
*individuate* the set. Notice, here, that we allow that
*x* may be said to depend for its identity upon *y* even in
cases in which the identity of **y* alone* does not
suffice to fix the identity of *x*. So, for example, a set with
two or more members depends for its identity upon *each* of
them, although its identity is only *completely* fixed by the
identities of *all* of them.



Another way to specify the notion of essence being used is to resort
to Fine's (1994b, 1995) distinction between
*constitutive* and *consequential* essence:



> 
> 
> A property belongs to the *constitutive* essence of an object
> if it is not had in virtue of being a logical consequence of some more
> basic essential properties; and a property might be said to belong to
> the *consequential* essence of an object if it is a logical
> consequence of properties that belong to the constitutive essence.
> (Fine 1994b: 276)
> 
> 
> 



As Fine then goes on to note, the property of singleton Socrates,
namely, containing Socrates as a sole member, will presumably be part
of the constitutive essence of {Socrates}. In contrast, the property
of containing some member or another is only part of its consequential
essence. It is evident that in cases of essential and
identity-dependence, we are typically interested in the constitutive
notion of essence, because it is presumably also a part of the
consequentialist essence of Socrates that 2 = 2. So, using the
consequentialist notion of essence to define essential and
identity-dependence would quickly lead to problems not unlike the ones
faced by the modal-existential account. Accordingly, in what follows,
the various definitions should in the first instance be understood to
concern the constitutive essence. But the problem that remains is that
the distinction between constitutive and consequential essence itself
is not always easy to make and it is inevitable that there will be
some controversial cases (for discussion, see Fine 1994b). The
examples used to illustrate the various forms of essential dependence
should thus be taken with a grain of salt, as it is often not entirely
clear what should be included in the constitutive essence of an
entity. In any case, provided that this distinction can be made, the
friend of the essentialist analysis of dependence has a clear
advantage, given that they are able to express a broader range of
dependence claims with the help of this distinction.



Now, although we have not yet presented a formal definition of
identity-dependence, it might well seem that a consequence of any such
definition should be the following principle:



(ID-NEC)
 If *x* depends for its identity upon *y*, then there is
a function *f* such that *x* is necessarily identical with
*f*(*y*).


(Note: "*f*(*y*)" here may be pronounced
"the *f* of *y*".) For example: if we say that
the identity of a marriage depends on the identities of the two people
being married, if *x* is a marriage and *y* and *z* are
the two people in question, (ID-NEC) is satisfied in respect of
*x* and *y* in virtue of the fact that *x* is
necessarily identical with *the marriage of *y* with
*z**--so that in this case the required function is the
*marriage with *z** function from persons to events. (We
here ignore the complications created by the fact that, under some
legal systems, the same two persons may be married to one another more
than once.) However, (ID-NEC) turns out to be unsatisfactory as a
definition, because it can be regarded as trivially true, given
standard set theory, that any *x* and *y* that may be
connected with a function will satisfy the consequent of (ID-NEC). So
we better find a more satisfactory *definition* of
identity-dependence.



Evidently, it would not do simply to replace the conditional
connective in (ID-NEC) with a *bi*conditional connective and
thence attempt to turn it into a *definition* of
identity-dependence, unless at the same time one could impose some
suitable restriction on the kind of function involved. One possibility
would be to exclude any function *f* which is not such that it is
*part of the essence of *x** that it is the *f* of
*y*. But since it would appear strange to say that essences quite
generally involve anything about *functions*, we might be
better off with a definition using an appropriate yet to be specified
place-holder relation that would do the same job instead, arriving at
the following definition of identity-dependence:



(ID)
 *x* depends for its identity upon *y* =df
There is a relation "*R*" such that it is part of the
essence of *x* that *x* is *related by R* to
*y*.


We can exemplify (ID) by letting *x* be {*z*} and *y*
be *z*, in which case we have, as is intuitively correct, that
{*z*} depends for its identity upon *z*, because there is a
relation--namely, "being a member of the singleton
set" (also known as the *unit set* function)--such
that it is part of the essence of {*z*} that it is the singleton
set of *z*. Accordingly, we may interpret the relation *R*
more precisely on a case by case basis.



Building on this example, we can see that *x* is necessarily
identical with the sole member of {*x*} and hence we can arrive
at a principle not unlike (ID-NEC). However, even though *x* is
necessarily identical with the sole member of {*x*}, it is, once
again, plausibly not *part of the essence of *x** that
*x* is the sole member of {*x*} (cf. Fine 1994a: 4-5;
see also Fine 1994b for a related discussion of the notion of essence
and essential dependence).



But, of course, for the foregoing strategy to work and thus for (ID)
to be fully vindicated, a perspicuous account of the notion of
"essence" would be required--and that is a large task
which cannot be undertaken here. However, various attempts to
construct such an account have been made, for instance by Fine (1994a)
and Lowe (2008). In this sense, a thing's essence may be said to
*constitute its identity*, when one uses the word
"identity" in this distinctive manner to speak of *a
thing's* identity, rather than using it to speak of the
identity *relation*. Seen in this light, identity-dependence as
defined by (ID) is simply a species of *essential* dependence,
that is, a way in which the essence of a certain thing is determined
by a relation in which it stands to another thing.



Although the above account of identity-dependence is not entirely
uncontroversial, there are some interesting applications of it in the
literature. One of these applications concerns a version of Ontic
Structural Realism (OSR, see Ladyman 1998), which claims that objects
depend for their identities on the structures to which they belong
(ID-OSR) (see French 2010 for one version, and see Lowe 2012 and
O'Conaill 2014 for further discussion of OSR and
dependence).



At this point, two principles concerning identity-dependence, as
defined by
 (ID),
 may be proposed. First:



(ID-EX)
 If *x* depends for its identity upon *y*, then,
necessarily, *x* exists only if *y* exists.


And second:



(ID-NEX)
 If *x* is *not* identical with *y* and *x*
depends for its identity upon *y*, then *y* does
*not* depend for its identity upon *x*.


An immediate implication of (ID-EX) in conjunction with
 (EDR)
 is that if *x* depends for its identity upon *y*, then
*x* is *rigidly* existentially dependent upon *y*
(although not necessarily *vice versa*, of course). These two
principles are, it seems, quite plausible. As for (ID-EX): surely,
*x* cannot exist unless everything upon which *x*'s
identity depends also exists. Thus an assassination cannot exist
unless the person assassinated exists, and a set cannot exist unless
its members exist. Indeed, it would seem that we can derive (ID-EX)
with the aid of
 (ID).
 For suppose that *x* depends for its identity upon *y*.
Then, by (ID), there is a relation *R* such that *x* is
related by *R* to *y*. Given, however, that the relevant
*R* cannot exist unless *y* exists--because a relation
can obtain only between entities all of which *exist*--it
follows that *x* likewise cannot exist unless *y* exists. As
for (ID-NEX), this seems to follow from the requirement of
*non-circularity* which is a condition on the adequacy of any
criterion of identity. For example, given that unit sets are not to be
identified with their members, we cannot say *both* that the
identity of a unit set depends upon the identity of its member
*and* that the identity of that member depends upon the
identity of that unit set, for this would engender a vicious circle
which would seem to deprive *both* unit sets *and* their
members of well-defined identity-conditions.



Note that it may be urged, with some plausibility, that every object
*x* trivially depends for its identity upon *itself*. And,
certainly,
 (ID)
 would appear to have this implication, because for any object
*x*, there is a relation--namely,
*identity*--such that it is part of the essence of
*x* that *x* is the object identical with *x*. But we
must be very careful to distinguish between the claim that an object
depends for its identity upon itself and the claim that an object
depends for its identity *solely* upon itself. For even if the
former claim is trivially true of all entities, the latter claim is
certainly not. However, this result may be contested when we bear in
mind the distinction between constitutive and consequential essence,
because one might very well hold that it is no part of the
constitutive essence of an entity, at least typically, that it should
be identical with itself (we will omit a more detailed discussion
here, but see Fine 1994b).


### 4.3 Essential Dependence



Having defined identity-dependence by
 (ID)
 as a species of *essential* dependence, we are now in a
position to identify other species of essential dependence, the most
obvious being what may aptly be called *essential*
(*existential*) *dependence*, which can be defined as
follows:



(EDE)
 *x* dependsE for its existence upon *y*
=df It is part of the essence of *x* that *x*
exists only if *y* exists.


Note that, whereas--assuming the truth of
 (ID-NEX)--two
 distinct entities cannot be *identity*-dependent upon *each
other*, it very plausibly *is* possible for each of two
distinct entities to dependE for its *existence* on
the other. For example, consider a solid sphere. Let us assume for the
purposes of this example that the solid sphere is a
*substance*, a kind of basic, ontologically
*independent* entity (see
 SS6.3
 for further discussion). If we think of the top and bottom
"halves" of a solid sphere as being *geometrically*
defined entities whose boundaries are specified by reference to the
whole sphere of which they are equal subdivisions, then it could be
argued that it is part of the essence of each such hemisphere that it
exists only if the other does. (These "halves", it should
be emphasized, must not be confused with the portions of matter
"filling" them at any given time, and hence should not be
thought of as "parts" of the sphere of which it is
materially *composed*.) At the same time, each hemisphere
depends for its *identity* upon the whole sphere--one
being identified as the top half of that sphere and the other as the
bottom half--whereas the sphere itself does not likewise depend
for its identity upon either of these halves, given that we have
assumed that it is a substance in its own right. This example is
admittedly controversial: one could insist that the constitutive
essence for each hemisphere involves the whole sphere, but not the
other hemisphere. Moreover, one could also focus on the
*plural* or *collective essence* of the halves, which
might help us get around this circularity (on plural/collective
essence, see Zylstra 2019). Note also that, very plausibly, if
*x* dependsE for its existence upon *y*, then
*x* also dependsR for its existence upon *y*:
*essential* existential dependence entails *rigid*
existential dependence--but not, of course, *vice
versa*.



We have seen that (ID) and (EDE) correspond to *rigid* forms of
modal-existential dependence, such as (EDR). But we can also formulate
a *generic* notion of essential dependence, just like we did in
the case of modal-existential dependence. Corresponding to generic
existential dependence (EDG), we can formulate *generic essential
(existential) dependence* as follows:



(EDEG) 
*x* dependsEG for its existence upon *F*s
=df It is part of the essence of *x* that *x*
exists only if some *F* exists.


For instance, if *x* is a particular water molecule, we may say
that it is a part of the essence of that water molecule that it exists
only if some atoms of hydrogen and oxygen exist. In other words, water
molecules essentially depend for their existence on there being atoms
of the kind hydrogen and oxygen. It may be helpful to introduce a
distinction between *individual* and *general* essences
in this connection. (EDEG) as applied to the case of water certainly
applies to each and every water molecule rather than just one of them.
Accordingly, when we speak of the essence of water in this context, we
are not referring to the individual essence of any particular water
molecule, but to the general essence of the natural kind
*water*. (EDEG) concerns what an entity depends upon for its
general essence (e.g., what makes something count as a water
molecule), whereas (EDE) concerns what an entity depends upon for its
individual essence (e.g., what makes something the particular water
molecule it is, as opposed to any other). (For further discussion of
this distinction between individual and general essence, see Lowe
2008.)



We can find slightly different formulations of essential dependence in
the literature. For instance, on Kathrin Koslicki's (2012: 190)
re-construal of Fine's essentialist account, we get:



(EDC)
 An entity *x* ontologically depends on an entity (or
entities), *y*, just in case *y* is a constituent (or are
constituents) in *x*'s essence.


This account relies on the notion of *constitutive essence*,
which is developed in Fine's work (1994b, 1995). We have already
defined this notion above, but a recap may be useful. As Fine puts
it:



> 
> 
> we may take *x* to depend upon *y* if *y* is a
> constituent of a proposition that is true in virtue of the identity of
> *x* or, alternatively, if *y* is a constituent of an
> essential property of *x*. (Fine 1994b: 275)
> 
> 
> 



It should also be noted that even though the notions of essential
dependence defined above are no doubt more fine-grained than the
modal-existential notions, there may be reasons to think that even
they are not sufficiently fine-grained for all purposes (as argued in
Koslicki 2012). To illustrate, consider Fine's well-known
discussion of Socrates and the singleton set that has Socrates as its
sole member. To use Fine's terminology, we could say that it is
part of the constitutive essence of Socrates's singleton set
that it has Socrates as its sole member, whereas it is *not*
part of the constitutive essence of Socrates to be the sole member of
Socrates's singleton set. But as Koslicki (2012: 195) points
out, this is really just to say that Socrates's singleton set
ontologically depends on Socrates whereas Socrates does *not*
ontologically depend on Socrates's singleton set. The relevant
notion of dependence appears to be built into Fine's notion of
essence and, similarly, the notion of essence assumed by Fine (and
many other "neo-Aristotelians") is already built into
(EDC).



Before we conclude our discussion of essential dependence, a major
issue in the literature concerning ontological dependence should be
mentioned. For it appears that those who are willing to buy into a
suitably fine-grained notion of essence, popular in the
"neo-Aristotelian" tradition, will find the
modal-existential account of ontological dependence far too
coarse-grained. Yet, those who are not sympathetic to the
"neo-Aristotelian" notion of essence but would rather
analyze essence in terms of modality would insist that the
modal-existential analysis is quite sufficient, and indeed that
essential dependence collapses into modal-existential dependence. The
recent literature has perhaps been dominated by the
"neo-Aristotelian" line (although one does not, of course,
need to be "neo-Aristotelian" to accept the notion of
essential dependence or the "non-modalist" analysis of
essence), but this is indeed only a relatively recent phenomenon. A
further issue is that there is some disagreement about how the
relevant notion of essence is to be constrained amongst those who
think that a more fine-grained analysis than the modal-existential
account is needed. For instance, Koslicki (2012: 196 ff.) regards
Fine's propositional notion of essence according to which there
is little or no distinction between essence and *real
definition* as overly restrictive. She identifies the source of
this restrictive conception of essence, which is also present in
Lowe's work, to be the focus on essences as
individuating--this is the type of view regarding essential
dependence, namely identity-dependence, which we have here been
focusing on. On Koslicki's alternative picture, essences



> 
> 
> must do more than individuate the entities whose essences they are;
> and real definitions must do more than state conditions which uniquely
> identify and delineate the entities under consideration at every time
> and in every world in which they exist. (Koslicki 2012: 200, fn.
> 13)
> 
> 
> 


## 5. Ontological Dependence and Metaphysical Grounding



The notion of "ground" stormed into contemporary analytic
metaphysics in the beginning of the twenty-first century, but the
roots of the notion go back to Aristotle (for an overview, see Fine
2012). Sometimes grounding is understood as "metaphysical
explanation". To be more precise, when some *x* is grounded
in some *y*, it is usually thought that *y* explains
*x*, and grounding either is that explanatory relationship or
'backs' that explanation (there is a lot of debate on
these issues, see the separate entry on
 metaphysical grounding).
 Moreover, the status of *y* is generally thought to be somehow
*prior* to that of *x*--grounding is typically
understood to express priority between things. For instance, we might
say that the members of a set are prior to the set itself; the
existence of the set is grounded in its members. Or to take a more
concrete example, the existence of any given composite object is
grounded in the existence of its parts. For instance, we might suggest
that the existence of any given water molecule is grounded in the
existence of hydrogen and oxygen atoms. Somewhat more controversially,
we might also say that mental states are grounded in physical states.
In each of the mentioned examples there appears to be an ontological
dependence relation between the grounded entity and the grounding
entity or entities. In fact, we have already discussed some of these
examples in terms of ontological dependence above. This naturally
leads one to question whether grounding just is (a variety of)
ontological dependence--or whether ontological dependence is
grounding. One motivation for finding a systematic link between
ontological dependence and grounding is that it would be more
parsimonious than having two primitive notions (for discussion, see
Correia & Schnieder 2012b).



An initial reason to distinguish ontological dependence and
metaphysical grounding is that the latter is a much stricter notion.
Consider the idea that ontological dependence could be reflexive:
there is nothing in theory that rules out a relation of ontological
dependence obtaining in such a way that a given object is dependent on
itself. It is of course controversial whether there actually are any
such entities, but this is not enough to consider irreflexivity as a
necessary requirement for ontological dependence. In contrast,
metaphysical grounding is usually considered to be necessarily
irreflexive (of course, there are those who think that this
requirement could be dropped in the case of grounding just like it may
in the case of ontological dependence; see, e.g., Jenkins 2011).



A more general issue concerns the modal implications of each notion.
In particular, is there a form of ontological dependence that is both
necessary and sufficient for grounding? This seems unlikely (as argued
in Rydehn forthcoming). While there are some instances of
grounding which are clearly rigid in the same sense as rigid
existential dependence, it is also clear that this connection is not
necessary and sufficient. Take for instance the fact [f] and assume
that it is fully grounded in [g]. This entails that [f] cannot be
grounded except by [g]. An example would be the case of the set of
singleton Socrates, whose existence is rigidly grounded in the fact
that Socrates exists. In such cases, [f] rigidly existentially depends
upon [g], and it is also plausible that [f] depends for its identity
upon [g]. However, other instances of grounding do not require that
the grounded fact rigidly depends upon its grounds. For example,
suppose that [f] is the fact that a particular subject is feeling
pain, and that [f] is fully grounded in a particular neural fact about
this subject, [g]. It seems possible that this subject could have been
in pain, so [f] could have obtained, even if [g] did not obtain. This
would be the case if [f] had been grounded in a different neural fact,
[h]. In this case, [f] cannot depend for its identity upon [g], since
[f] is not rigidly dependent upon [g]. This rules out some simple
options for linking grounding and rigid ontological dependence, but
there are arguably also problems for linking it with generic varieties
of ontological dependence (for a more detailed discussion, see
Rydehn forthcoming).



If grounding cannot be easily analyzed with the help of rigid or
generic dependence, then we might instead suggest that if grounding
were to be understood as a type of ontological dependence, it would be
some sort of *explanatory dependence*. The idea that whatever
does the grounding also somehow *explains* what is being
grounded is a crucial part of the notion's appeal. Relations of
ontological dependence often seem to have a similar type of
explanatory role, but the link to explanation may appear weaker: even
though the existence of water depends on the existence of hydrogen and
oxygen, it does not seem to be the case that the existence of hydrogen
and oxygen *explain* the existence of water. Rather, what
explains the existence of water is the ability of hydrogen and oxygen
atoms to form molecules (even though this is rather simplified). So
there are at least some difficulties in accounting for all relations
of ontological dependence in terms of grounding relations.



We need something stricter than just "an explanatory role"
to identify grounding--otherwise we would end up with a much too
liberal notion, for we may regard a number of loosely connected things
explanatory in some very loose sense. For instance, we might say that
the fact that Smith murdered Jones is explained by certain events in
Smith's childhood, but a more direct explanation might be
Smith's desire to rob Jones. Certain events in Smith's
childhood may help us *understand* why Smith has murderous
desires, but it's not clear that they serve to ground the fact
that Smith murdered Jones (unless we can establish transitivity
between these).



One suggestion that may help to make grounding more precise would be
to focus on *priority*. To reiterate an example used above, if
the members of a set are prior to the set itself, then the existence
of the set is grounded in its members. So the grounding entities are
prior to--or more *fundamental* than--the grounded
entities. Metaphors abound, but a typical way to express the idea is
to say that *x* is fundamental or ontologically independent in
this sense if and only if nothing grounds *x* (Schaffer 2009:
373). This is another sense in which grounding would seem to come
apart from certain types of ontological dependence, as a purely
modal-existential understanding of dependence in the lines of
 (EDR)
 (if one accepts them), without any claim to priority, is also
possible. This is of course the sense which we defined above in terms
of necessitation: even if *x* rigidly necessitates *y* that
does not entail that *y* must be ontologically prior to *x*.
At any rate, it would be odd to say that parents are *more
fundamental* than their children, even if there is a (temporally
relativized) rigid existential dependency between parents and their
children.



All this would appear to suggest that it isn't straightforward
to define grounding simply as a variety of ontological dependence, or
the other way around. In any case, even if we could define one notion
in terms of the other, it seems that there are aspects of ontological
dependence that are not captured by all accounts of grounding as well
as aspects of grounding that are not captured by all accounts of
ontological dependence. The situation may of course change depending
on one's preferred account of dependence and grounding.
Accordingly, for the time being, it is advisable to keep the notions
apart, especially since there are some further, formal differences
that need to be taken into account. In particular, grounding is most
commonly understood as a *strict partial order* (see Raven
2013), which entails irreflexivity, transitivity, and asymmetry. If
this is correct, grounding could only capture a very specific variety
of ontological dependence, as we have seen that there are varieties of
ontological dependence that violate these formal features. Some have
also argued that grounding itself violates all or some of these formal
features, so it is not entirely uncontroversial that grounding truly
is a strict partial ordering (for discussion, see Jenkins 2011,
Schaffer 2012, and Tahko 2013). Given this, the exact link between
grounding and ontological dependence remains open, subject to further
specification of the formal features of ground.


## 6. Applications and Other Related Notions



In addition to the previous discussion regarding metaphysical
grounding, there are numerous applications of ontological dependence
as well as closely related notions that we could discuss. It will not
be possible to do justice to all of them, but we will mention a few of
the most important ones (many of these and further applications are
discussed in Hoeltje, Schnieder, and Steinberg (eds.) 2013).


### 6.1 Supervenience



The notion of supervenience is covered in more detail in the separate
entry on
 supervenience
 (see especially SS3.5) but a brief mention is in order here, as
supervenience has sometimes been conceived as a type of dependence.
One reason for this is that when we assert that *A* supervenes on
*B*, we might also say that *A*-properties ontologically
depend upon *B*-properties. Take the typical example of the
beauty of a work of art and the physical manifestation of that work of
art. It would seem that if you wish to change the aesthetic properties
of an artwork, you will also have to manipulate its physical
manifestation. As a first pass, it looks as if the dependence at work
here is generic existential dependence as defined by
 (EDG),
 so the aesthetic *A*-properties dependG for their
existence upon some physical *B*-properties. However,
supervenience in general is not so easily analyzed, as at least on one
usual conception, supervenience is not irreflexive and hence not
asymmetric (see Steinberg 2013). In other words, we can at least say
that supervenience, as opposed to grounding, is not a strict partial
order and hence not a relation of ontological priority. There are
various ways to further specify the formal features of supervenience,
and indeed to argue that it should not be considered as a form of
dependence, but this is not a task that we can undertake here.


### 6.2 Truthmaking



The connection between ontological dependence and truthmaking is of a
more general type. On many construals, the core of truthmaker theory
is considered to be the idea that truth depends--or
*supervenes*--on *being* (Bigelow 1988: 133; see
also Armstrong 2004, Schaffer 2010a, and Liggins 2012). Recently, some
have resisted this idea (e.g., Merricks 2007), but it does remain
popular among truthmaker theorists (for discussion, see the separate
entry on
 truthmakers).
 Moreover, the connection between truthmaking and ontological
dependence has been a part of truthmaker theory from the very
beginning; Mulligan, Simons, and Smith (1984: 294) rely on
Husserl's work on ontological dependence in their introduction
of the idea of truthmaking (for further discussion of Husserl's
work regarding dependence, see Simons 1982 and Johansson 2004: Ch. 9).
More precisely, it seems there must be some way in which the world is
in virtue of which true propositions are true, but the world itself is
not symmetrically dependent on truth; there is an *asymmetric*
dependence relation between truth and being (accordingly, this
dependence needs to be expressed with an asymmetric notion of
dependence, see
 SS3
 above).



However, popular though it is, this view is not entirely
uncontroversial. For instance, MacBride (2014) argues that there is no
need to appeal to truthmaking to explain this type of asymmetric
dependence between truth and being. In fact, there are reasons to
think that truthmaking *cannot* explain the asymmetry, at least
not on the basis of the idea that truth supervenes on being. Part of
the confusion here seems to surround the relevant notion of
supervenience. On Armstrong's (2004: 8) construal, the relevant
notion of supervenience is symmetric: truth supervenes on being, but
being also supervenes on truth. So it appears that the required
asymmetry cannot arise from supervenience itself (see also
Rodriguez-Pereyra 2005). While these problems must be settled
elsewhere, it suffices to say that *some* variety of
ontological dependence is no doubt at work in truthmaker
theory--there is an on-going debate as to how truthmaking should
be specified. One recent suggestion that may have some mileage is that
the relevant asymmetry could be recovered if truthmaking is understood
as *truth-grounding*, that is, some entity *x* grounds the
truth of some proposition *p* if *p* is true in virtue of
the existence of *x* (see Tahko 2013 for discussion).


### 6.3 Substance



One important application of ontological dependence is the analysis of
the notion of "substance"--a basic or fundamental
entity which possesses at least some degree of ontological independence
(for discussion on the various uses of "substance" in
philosophy, see the separate entry on
 substance).
 If the idea of ontological independence is somehow associated with
"substance", then it seems that some account of what the
relevant dependence amounts to will be required. In fact, those
familiar with Lowe 2005 [2010] (an earlier version of the present
entry), will recall that the discussion was entirely focused on
finding the most plausible account of ontological dependence for the
purposes of analyzing the notion of substance (this discussion
originates from Lowe 1998: Ch. 6). This also ties in with a more
general application of ontological dependence, namely, the study of
formal ontological relations obtaining between ontological categories,
of which substance is a prime example (see also Hoffman &
Rosenkrantz 1994, Schaffer 2009, and Nolan 2011). So how should we
define "substance" given the various notions of
ontological dependence that we have at our disposal? It will be
recalled that one particularly powerful notion of ontological
dependence is the notion of identity-dependence, which we arrived at
when looking for an asymmetrical--or, at least, an
*anti*symmetrical--relationship of ontological dependence.
It seems that if substances are considered as ontologically
*independent* entities, then an asymmetric relation is
required. One possibility for defining substance, which relies on
 (EDR)
 is the following:



(SUB-1)
 *x* is a substance =df *x* is a particular
and there is no particular *y* such that *y* is not
identical with *x* and *x* dependsR for its
existence upon *y*.


(SUB-1) could be seen to follow the precedent of Aristotle,
who--in the *Categories*--admitted only
*particulars* as "primary" substances, while
allowing some universals (the species and genera of primary
substances) the status of "secondary" substances (see
Aristotle, *Categories*: Ch. 2). However, (SUB-1), because it
relies on
 (EDR),
 will be unsatisfactory for those who are convinced that the
modal-existential analysis of ontological dependence is not
sufficiently fine-grained; it will face the type of problems that we
have discussed above in
 SS2.2.
 We know already that it won't do to replace the notion of
existential dependence employed in (SUB-1) by appealing instead to the
relation of *essential* existential dependence, as defined by
 (EDE),
 because the latter is neither an asymmetrical nor an antisymmetrical
relation. Indeed, this is one lesson of the example of the
sphere's two hemispheres discussed immediately after the
definition for (EDE) was given. For, whereas the *whole sphere*
might well be taken to qualify as a substance, neither of its
"halves" plausibly can, because they lack the requisite
kind of ontological independence. But, while (SUB-1) reinterpreted in
terms of (EDE) does indeed have the implication that neither of the
hemispheres is a substance--because each dependsE for
its existence on the other and hence on a particular distinct from
itself--it also has the unwanted implication that the *whole
sphere* is not a substance for the same reason, because it is
arguably a part of the essence of the sphere as a whole that it exists
only if each of its hemispheres exists. However, an obvious remedy is
at hand. We can simply replace the appeal to any species of
*existential* dependence in a definition on the pattern of
(SUB-1) by an appeal to the relation of *identity*-dependence,
as defined by
 (ID),
 to give:



(SUB-2)
 *x* is a substance =df *x* is a particular
and there is no particular *y* such that *y* is not
identical with *x* and *x* depends for its identity upon
*y*.


Composite substances appear to comply with (SUB-2): for, plausibly,
although they possess proper parts, they do not depend for their
*identity* upon those parts, since *which* objects those
parts are does not help to determine of *which* substances they
are parts (the same objects being capable of becoming parts of many
different substances). Moreover, substances quite generally do not
depend for their identity upon their (accidental) particularized
properties, if such exist, nor upon the events in which they
participate, nor upon the places they occupy, nor upon other
substances. The particularized properties (i.e., tropes or modes) of
substances and the events in which substances participate--that
is, items such as the particular redness of this apple and the
assassination of Caesar--would appear to depend for their
identity upon those substances, which precludes the reverse
relationship from obtaining, on pain of circularity. (However, see
Keinanen & Hakkarainen 2014 for discussion; in particular,
they propose a strategy for identifying tropes that does not entail
circularity in the individuation of tropes. See also Moltmann 2019,
which discusses the problem of spatial location related to tropes and
other entities.) As for places, although a physical substance must
indeed occupy *some* place, *which* place it occupies
does not determine *which* substance it is, since substances
may exchange places.



One motivation for rejecting the appeal to
 (EDR)
 in the analysis of substance is that it permits two different
entities to be existentially dependentR *upon one
another*--entities such as Socrates and Socrates's
life. As we have seen,
 (ID)
 precludes any analogous symmetry where *identity*-dependence
is concerned: indeed, it delivers the intuitively correct verdict that
it is *Socrates's life* that is, in this sense,
ontologically dependent upon Socrates, rather than *vice
versa*. For Socrates's life is an extended event or process
in which he participates--and *which* person Socrates is
partially determines *which* event this is, but not *vice
versa*. Or, to put it another way--one which should by now be
familiar--it is *part of the essence* of Socrates's
life that it is *the life of Socrates*, but it is *not*
part of the essence of Socrates that he is the person who lived that
life. Of course, we can still acknowledge that the relation defined by
 (EDR)
 does hold mutually between Socrates and his life and we can still
call this relationship a type of *existential
dependence*--namely, "rigid" existential
dependence. Similarly, we can recognize as other species of
existential dependence the "generic" existential
dependence defined by
 (EDG),
 the "asymmetrical" rigid existential dependence defined
by
 (EDA)
 and the "essential" existential dependence defined by
 (EDE).



The key point is simply that, in the sense of "ontological
dependence" which fits quite naturally with the
"neo-Aristotelian" account of substances as
*ontologically independent* objects, the relationships defined
by
 (EDR),
 (EDG),
 (EDA),
 and
 (EDE)
 will *not* serve the purpose, whereas that defined by
 (ID)
*will*. On this view, the sense in which a substance is an
entity which does not depend "ontologically" upon anything
other than itself is exactly the sense in which it does not depend for
its *identity* upon anything else. This still leaves many
interesting questions concerning ontological (in)dependence
unanswered, notably the question of whether there is a *fundamental
level* or *layer* of reality, consisting of one or more
entities upon which all other existing entities depend ontologically
in one way or another (for discussion, see Lowe 1998: 154-73;
Schaffer 2003; Cameron 2008; and Paseau 2010).


### 6.4 Fundamentality



It seems plausible that at least in some cases where one entity is
more *fundamental* than another one, it is because the less
fundamental is ontologically dependent on the more fundamental. Much
of the recent discussion involving ontological dependence has focused
exactly on the question of fundamentality and especially the question
mentioned above, namely whether there is something that is entirely
fundamental, ontologically independent--is there an ontological
"bottom level"? Hence, there are two senses in which a
clearer understanding of ontological dependence may help: we can
specify in what sense fundamental entities are ontologically
independent as well as how other entities may be ontologically
dependent on the fundamental entities, thus gaining further
understanding of the supposed "hierarchy" that terminates
in the bottom level.



The bottom level is usually thought to be at the smaller end of the
spectrum: the *atomistic* view suggests that certain subatomic
particles are fundamental. But this does not mean that the fundamental
level must necessarily be at the bottom--the fundamental end
could also be at the top, i.e., the universe as a whole could be
considered fundamental--a substance in its own right (see
Schaffer 2010b). Each of these views is a type of *metaphysical
foundationalism*, which suggests that chains of ontological
dependence must come to an end, at least at the dependent end. In
other words, metaphysical foundationalism asserts that chains of
asymmetric ontological dependence must terminate, they must be
*well-founded*. The idea is typically formulated in
mereological terms, as the reference to atomism suggests: there is an
asymmetric ontological dependence relation from one end of the
mereological scale to the other. These questions also have
implications for the debate between *pluralism* and
*monism* (one form of such monism is defended in Schaffer 2010b
and Tallant 2015 discusses the notion of ontological dependence in
connection to Schaffer's monism; see also Trogdon 2009 and the
separate entry on
 monism).



However, there have been some speculative suggestions according to
which chains of ontological dependence could go on *ad
infinitum*, and hence violate the requirement of well-foundedness
(for discussion, see Bliss 2013, Morganti 2014, and Tahko 2014). The
resulting view would appear to be some sort of *metaphysical
infinitism* as opposed to metaphysical foundationalism. One
version of such a view, if combined with the idea that the direction
of dependence is towards the smaller, is a "gunky"
ontology, whereby matter is infinitely divisible "gunk"
and objects or matter have no smallest parts. But as interesting as
these views and the discussion surrounding fundamentality are,
discussing them in detail here would take us too far from our original
topic (for further discussion, see the separate entry on
 fundamentality).