Ontological Dependence

First published Thu May 12, 2005; substantive revision Thu May 1, 2025

Ontological dependence is a relation—or, more accurately, afamily of relations—between entities orbeings(onta in Greek, whenceontological). For there arevarious ways in which one being may be said todepend uponone or more other beings, in a sense of “depend” that isdistinctlymetaphysical in character and that may becontrasted, thus, with variouscausal senses of this word.More specifically, a being may be said to depend, in such a sense,upon one or more other beings for itsexistence or for itsidentity. A traditional approach was to analyze all forms ofontological dependence inmodal terms, that is, in terms ofdistinctly metaphysical notions ofpossibility and necessity,but more recently an analysis in terms of the notion ofessence has become popular. A third option is to treat thenotion of ontological dependence as a metaphysicalprimitive,in which case it is not possible to give a further definition of it.This entry will remain neutral about the ultimate basis of ontologicaldependence. At least in some cases, it is possible to use ontologicaldependence as a tool to describe the relationship between entitieswithout taking a stand about its basis, but in other cases this maynot be possible. Notions of ontological dependence are frequentlycalled upon by metaphysicians in their proposed analyses of othermetaphysically important notions, such as the notion ofsubstance or the thought thattropes depend fortheir existence on a substance.

1. Varieties of Ontological Dependence

A crucial notion in metaphysics is that of one entitydependingfor its existence upon another entity—not in a merelycausal sense, but in a deeper, ontological sense (but see Ingthorsson2002: 96 for discussion on the idea that causal relations could beunderstood as one-sided existential dependence; cf. also Bennett 2017:Ch. 4, where it is argued that the line between causal and noncausaldetermination is not clear). The kind of dependence in question mustalso be distinguished from any kind oflogical dependence,because logical relations, strictly speaking, cannot obtain betweenconcrete objects, nor between abstract objects that are notpropositional in nature. We should also distinguish ontologicaldependence, broadly conceived, from what is usually considered astricter type of metaphysical dependence, namelymetaphysicalgrounding. The link between ontological dependence andmetaphysical grounding (or simply “metaphysicaldependence”, as it is sometimes called, see Rosen 2010 and theseparate entry onmetaphysical grounding; see also Trogdon 2013 for discussion) will be discussed below, butthis entry concerns a somewhat broader family of relations ofdependence. The notion of ontologicalindependence is alsoimportant, and it is often used to describe the idea that something isfundamental. This idea as well will be discussed below (butsee also the separate entry onfundamentality).

While much of the early work on ontological dependence was doneprimarily in a setting where dependence was thought to be aninherentlymodal notion (e.g., Simons 1987), the recent trendinhyperintensional metaphysics (see Nolan 2014) has had anenormous impact for discussion surrounding dependence. The thoughthere is that we need more fine-grained tools than mere necessaryequivalence: two statements can apply to the very same entities acrossall possible worlds but not be equivalent. Here, the notion ofessence is often invoked, and many would now regard theultimate basis of dependence (and indeed of modality itself) to beessence (see Fine 1994a, Lowe 1998, Correia 2005, and Koslicki 2012).Finally, it is entirely possible to deny both approaches and regardthe notion of ontological dependence itself as a primitive (seeThomasson 1998: Ch. 2 and Barnes 2012: 879). Some would regard thenotions of metaphysical grounding or fundamentality to be similarlyprimitive (see Fine 2012, Rodriguez-Pereyra 2005, Rosen 2010, Schaffer2009, and Wilson 2014). Primitivism does not mean that the notion ismysterious or obscure, as we can still illuminate it, e.g., bydescribing its instances. But it does mean that the notion isunanalyzable.

In what follows, the discussion will roughly follow the historicaldevelopment, starting with the attempts to give a modal-existentialanalysis of ontological dependence in§2. As we will see in§2.2, these attempts do have some serious shortcomings, even if they can beof use especially in certain restricted contexts. In§3, 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§4, a discussion of the relationship between metaphysical grounding anddependence in§5, finishing with a brief discussion of various other connections andapplications in§6.

To get us started, we may consider a few simplified examples. It isnot uncommon to see the notion of ontological dependence used in arather coarse-grained manner, given that it encompasses a family ofrelations. 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 singletonSocrates.”

While all of the above no doubt express some important type ofdependence relationship, there are also important differences betweenthem. (1) may be understood as stating that a set {x,y,z} could not exist if its members, namelyx,y,z, did not exist. One way to analyzethe type of dependence in question is asrigid existentialdependence, to be clarified in a moment. Another sense ofdependence that could be at work in (1) isidentity-dependence, but we will return to this later on. In(2), we seem to have in mind a more general kind of dependence: therecould not be electricity, now or ever, if there were no electrons. Sothe existence of electricity depends on the existence of a veryspecific kind of particle, the electron. This second type ofdependence is also existential, but to separate it from the rigiddependence in (1), we may call itgeneric existentialdependence. (3) is a famous case borrowed from Fine (1994a,1994b). Singleton Socrates, expressed by “{Socrates}”, isthe set containing Socrates as its sole member. From (1) we see thatthe set itself plausibly depends on Socrates, but since the setnecessarily exists when Socrates does, the dependence between themwould seem to besymmetric. However, (3) captures the veryplausible idea that it is not any part of theessence oridentity of Socrates that it should belong to {Socrates}. So,(3) must refer to some form ofessential dependence (or lackthereof in this case).

We will provide a more detailed analysis of these examples below, butwe can already see that a family of—sometimescompeting—notions is beginning to emerge. However, we shouldformulate each notion somewhat more precisely. The first thing to notein defining ontological dependence is themodal-existentialelement in dependence claims. For instance, we’ve said that asetcannot exist unless its members do. So there is a sensein which the existence of a setnecessitates the existence ofits members. Indeed, it is common to talk about, e.g.,rigidexistential necessitation as synonymous with rigid existentialdependence. Typically, statements of ontological dependence arethought to refer tometaphysical modality (rather than, say,conceptual or logical modality), primarily because they concernmatters that are broader than just conceptual or logical. Forinstance,substances are often considered to be entities thatdonot depend, in a distinctively metaphysical sense, fortheir existence upon anything else (see Lowe 1998: Ch. 6). We can seethis already in Descartes, who asserts that

bysubstance we can understand nothing other than a thingwhich exists in such a way as to depend on no other thing for itsexistence. (The Philosophical Writings of Descartes: vol. I,p. 210)

The idea of ontological dependence goes back further than Descartesthough, at least to Aristotle (see Corkum 2008, Koslicki2013, Peramatzis 2011, and Edelhoff 2020), and it appearsthroughout the history of philosophy. For instance, there is a notionof dependence operative in the philosophy of John Duns Scotus, whouses it in his proof of the existence of God (see the separate entryonJohn Duns Scotus, §2.2 for details), and the Aristotelian tradition is alsoreflected in Aquinas’s work (for a recent analysis arguing thatAquinas was committed to a pluralirty of irreducible dependencerelations, see Paolini Paoletti forthcoming). In the Easterntradition, relevant discussion can be found for instance inHuayan Buddhism (see Kang forthcoming). Later, precursors to thecontemporary work on ontological dependence can be found in the workof Franz Brentano and Edmund Husserl; in particular, Husserl’swork on the dependence between wholes and parts (for discussion, seeSimons 1982 and Johansson 2004: Ch. 9). Husserl’s student RomanIngarden should also be mentioned in this connection. Ingardendistinguished between four different types of dependence, using thelabel of “existential moments”. These were presentedas the following pairs: autonomy – heteronomy; originality– derivativity; self-sufficiency – non-self-sufficiency;and independence – dependence (see De Rizzo 2021).

Finally, there are also those who are sceptical about the metaphysicalimport of notions of ontological dependence. There are several reasonswhy one might consider ontological dependence relations to beunintelligible or unhelpful; for instance, one might think that theyconflate other relations or that they are extraneous additions tobeing. However, it is commonly held among those working withontological dependence that these dependence relations, beingformal ontological relations, are not additional elements ofbeing (cf. Lowe 2006: Ch. 3; see also Hakkarainen and Keinänen2023 for an in-depth analysis of formal ontological relations). Suchrelations are typically thought to beinternal (see Armstrong1978: 85, who defines two things to be internally related if and onlyif there exist properties of them which logically necessitate that therelation holds; for further discussion on internal and externalrelations, see the separate entry onrelations). Still, there is at least an ideological cost in some approaches toontological dependence and hence some would prefer adeflationary approach. According to this type of approach, wemay be able to account for supposed ontological dependence relationsin terms of a “cluster” of modal and mereologicalrelations (for a defense of this type of approach, see Kovacs2018).

2. The Modal-Existential Analysis of Dependence

In this section we will focus on the modal-existential analysis ofontological dependence, which used to be the dominant approach. Buteven if this approach has fallen out of fashion, it is important tosee why that is the case and we cannot do that without understandingthe approach. Moreover, there are those who do not wish to commit tothe ontologically heavyweight notion of essence, but who neverthelessemploy the notion of ontological dependence (e.g., trope bundletheorists, such as Keinänen, Keskinen, & Hakkarainen 2019).We will discuss some of the problems that this account faces in§2.2 and return to the essentialist analysis in§4.

2.1 The Modal-Existential Analysis

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

(EDR): x depends_R for its existence upony=_df Necessarily,x exists only ifyexists.

The subscript “R” is used here because it seemsappropriate to call this species of existential dependence of oneobject upon anotherrigid existential dependence. Why“rigid”? Because there is no flexibility here: theexistence of a givenx requires the existence ofthatveryy. It could not be something a little bit likey, something falling roughly in the same category, forinstance; itmust bey. Thedefiniens in(EDR) is equivalent to “Necessarily, ifx exists, theny exists”, so that according to (EDR) the existentialdependence_R ofx upony amounts to thestrict implication ofy’s existence byx’s existence. We have mentioned one example of rigidexistential dependence, namely, sets ontologically depending on theirmembers (more precisely, a set depends rigidly on the very members ithas, i.e., any change in a set’s members will change the setitself). Another, although more controversial, example is a particularperson depending for her existence on her parents, or, more precisely,on the particular sperm and egg that she originates from. This exampleis of course related tothe essentiality of origin (asdiscussed in Kripke 1980).

We can express (EDR) in a more compact manner by resorting to furtherformalization (the following notation is used, e.g., in Correia 2008).We can use the sentential operator “\(\square\)” formetaphysical necessity, the one-place predicate“E” for existence, and the two-place sententialoperator “→” for material implication. Following thisnotation, we can formalize rigid existential dependence as follows:“\(\square(Ex \rightarrow Ey)\)”, which can be read as “xrigidly depends for its existence ony”, oralternatively “x rigidly necessitatesy”. Note that (EDR), somewhat controversially, impliesthat everything depends_R for its existence uponitself. It would, of course, be easy enough to modify(EDR)’sdefiniens to read “y is notidentical withx and, necessarily,x exists only ify exists”, but that would have the disadvantage ofprecluding anything from depending_R for itsexistence upon itself.

We can quite naturally contrastrigid existential dependenceas defined by (EDR) with what might appropriately be callednon-rigid existential dependence, defined as follows:

(EDN): x depends_N for its existence upon theF =_df Necessarily,x exists only if theF exists.

The thought here is that—to use the language of “possibleworlds”—“theF” in any instance of(EDN) might well denotedifferent entities in differentpossible worlds. So, for example, it might be said that a materialobjectx depends_N for its existence uponthematter composingx, even though it might have beencomposed of different matter, because in every possible world in whichx exists the matter composingx in thatworld exists in that world.

On the face of it, (EDR) seems to capture precisely one stronglyintuitive notion of existential dependence; or more accurately, onedirection of that notion. The intuitive idea behind this notion isthat ify does not exist, then the dependent entityx does not exist. For example, when it is said that aparticular event, such as the assassination of Caesar, depends for itsexistence upon Caesar, (EDR) seems to explicate this appropriately interms of the fact that the assassination could not have existed ifCaesar had not existed to be assassinated. Someotherassassination, we may suppose, could have existed at that very timeand place, but forthat very assassination to have existed,Caesar himself had to exist.

However, there are clearly cases in which (EDR) fails to capture theintuitive sense of dependence at hand. Consider a living organism. Aliving organism would appear to depend for its existence upon itsparts, such as cells. But we also know that a living organism maysurvive a change of any of its cells, provided that the change iseffected in a non-disruptive manner. It is true, of course, that suchan organism musthave parts such as cells if it is to exist,butwhich objects those parts are is inessential—andconsequently it is not the case that it depends for its existence, inthe sense defined by (EDR), upon any one of those parts. But it ispossible to defineanother sense of existential dependence inwhich itis true to say that a composite object depends forits existence upon its proper parts; ageneric notion ofexistential dependence, defined as follows:

(EDG): x depends_G for its existence uponFs=_df Necessarily,x exists only if someFexists.

Composite objects are existentially dependent objects in the sense of(EDG), since they require the existence of proper parts (setF as “proper part ofx” in (EDG)). Usingthe previous formal notation, we could express (EDG) as“\(\square(Ex \rightarrow \exists Fy)\)”. Here we haveadded the existential quantifier “∃” as well as thegeneral term “F” to express the thought that“x generically depends for its existence on somethingbeing anF”, or alternatively “xgenerically necessitatesF”. The important differencebetween the rigid and the generic cases is that (EDR) refers to aspecific object whereas (EDG) only requires that at least someFs exist. Another example, mentioned earlier, where (EDG)would seem to capture the correct sense of dependence is(2), “Electricity ontologically depends on electrons”. Moreprecisely, we could say that electricity depends_G for itsexistence upon electrons.

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

(4): “Children ontologically depend on their parents.”

On the face of it, what we mean in (4) is that if parentsxandy had not existed, then their childz could nothave come into existence. This looks like a case of rigid existentialdependence, but it is clear thatoncez has beenborn, her parents can go out of existence without any effect on herown existence. At that point, there is onlypast rigidexistential dependence. For cases such as this, we would requiretemporally 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 temporallyrelativized ontological dependence).

Note, incidentally, thatgeneric existential dependence asdefined by (EDG) above is very close to a pluralized form ofnon-rigid existential dependence, as defined by (EDN). Wecould call thisplural non-rigid existential dependence,definable as follows:

(EDP): x depends_P for its existence upon theFs =_df Necessarily,x exists only if theFs exist.

The similarity between(EDG) and(EDN) is apparent: saying—for example—that a composite objectx depends_G for its existence upon proper parts ofx is very close indeed to saying thatxdepends_P for its existence upon the existence ofthevery kinds of proper parts thatx has.

2.2 Some Problems for the Modal-Existential Analysis

An important group of problems for the modal-existential analysis ofontological dependence emerges from essential properties (at least ifthey are considered as distinct from modal properties, as argued inFine 1994a). What could be a plausible example of an essentialproperty of an individual object, say, Socrates? Let us assume thatproperties are to be conceived of as propertyinstances, suchas the particular redness of a certain apple. Of course, an apple canchange its color, so that this is not an example of anessential property of the apple. But what about—in thecase of Socrates—hishumanity (as it were, hisparticularbeing human)? Certainly, if there is such a thingas the particular humanity of Socrates, he cannot lose it withoutceasing to exist. Perhaps we can deny that Socrates’s particularhumanity is anything distinct fromSocrates himself: afterall, its existence necessarily coincides with his. Properties arecommonly said to depend for their existence upon the entities thatpossess them. One might propose to state this in the form of aprinciple, with the help of(EDR), as follows:

(PROP-DEP): Ifx is a property andy is an entity possessingx, thenx depends_R for its existence upony.

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

(PROP-NEC): Ifx is a property andy is an entity possessingx, then, necessarily,x exists only ifyexists.

Note that (PROP-NEC) is not intended to apply to properties understoodas universals, but only to particularized properties (otherwisevariously known as property instances, individual accidents, tropes,or modes). These are items exactly like the particular redness of aparticular apple mentioned above, conceived of as an object distinctfrom the redness of anyother apple, no matter how wellmatched in color to the first. On this interpretation, (PROP-NEC) hassome plausibility, complying as it does with the intuition thatparticularized properties cannot “migrate” from one objectto another. (Actually, (PROP-NEC) itself does not quite imply this,although it does imply that a particularized property cannot migratefrom one object to another when the first object ceases to exist.Here, the friend of an essentialist analysis of dependence has somefurther options, i.e., one could say that it is part of the nature oressence of a particularized property to be instantiated by the veryobject that it is instantiated by. But we will leave this aside fornow.)

A line of argument that could be developed on this basis suggests thatif indeed there are such things as essential particularized propertiesthat are “possessed” by composite objects, then they arein fact to beidentified with those objects. But it would bevery problematic, quite generally, to argue that wherever itemsx andy are mutually existentially dependent asdefined by(EDR), they are identical. For instance, consider the relationship betweenSocrates and the temporally extended event or process that washislife. Clearly, in terms of (EDR), Socrates’s lifedepends_R for its existence upon Socrates—but so,plausibly, does his existence upon it: it is, for Socrates, what wemight call aninalienable event. And yet there are thingstrue of the life of Socrates that are not true of him andviceversa (for example, that it was so many years long, and that heweighed so many pounds)—so there is no question of their beingidentical. Even more strikingly, if we assume that numbers arenecessary existents, and hence mutually existentially dependent, thisline of thought would seem to entail that any two numbers areidentical!

But it could be disputed whether Socrates is existentiallydependent_R upon his life—whetherhenecessarily would not have existed ifit had not—for itmay be urged that he might have had or led adifferent life.Now, it is true enough that his life might have beenqualitatively different in many ways, but what is currentlyat issue is whether he might have had anumerically differentlife—and it is hard, perhaps, to see how he could. For if it isaccepted thatlives depend_R for their existenceupon the persons whose lives they are, then, necessarily,x’s life exists only ifx exists. Suppose, forthe sake of argument, that Socrates could have had a numericallydifferent life: then it would still have been a life which could onlyhave been Socrates’s—no one other than Socrates could havehad that “other” life. But then what could underpin thesupposition that it is indeed a life “other” than the lifehe actually had (except qualitatively)? Other possible worlds clearlydo contain lives that do not exist in the actual world, insofar asthey containpeople who do not exist in the actual world: butthere seem to be no strong grounds to suppose that they do so otherthan for that reason. It seems thus that there are certain difficultquestions which the modal-existential analysis may not fully address,at least not in the form that we have presented it until now. We willcontinue to discuss these issues in§3 and§4.

3. Asymmetrical Existential Dependence

Where does the above analysis of existential dependence lead us? Thecase of Socrates’s life demonstrates that(EDR) permits the possibility ofmutual existentialdependence_R between non-identical things. But thisimplication does not seem to be acceptable in every context and indeedsymmetric dependence more generally is a controversial issue (seeBarnes 2018 for discussion). Take, again, the relationship betweenSocrates and his life. According to (EDR), Socrates is quite as muchexistentially dependent_R upon his life as his life is uponhim. And yet there is a strong intuition that, in another and perhapsmore important sense, Socrates’s life is the trulydependent object here, while Socrates is, in some sense, anindependent existent. We might say that Socrates’s lifeexists onlybecause Socrates does, whereas it would beputting the cart before the horse to say that Socrates existsbecause his life does. Now, it appears that the conjunction“because” must beasymmetrical, because itexpresses anexplanatory relationship and explanation isasymmetrical. Plausibly, two distinct states of affairs cannotexplain each other. Theremay, quite conceivably, beself-explanatory states of affairs, so we should only want tourge thatnon-identical states of affairs cannot be mutuallyexplanatory. Technically, this means that we should strictly describeexplanation as an “antisymmetric” rather than as anasymmetric relation. The asymmetry of explanation is, of course,intimately related to the unacceptability ofcirculararguments. It is also closely linked to recent discussionsregardingmetaphysical grounding, to which we will returnlater, in§5.

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

(EDX): x depends_X for its existence upony=_df Necessarily,x exists only becauseyexists.

We use the subscript “X” because, in line with foregoingremarks, it seems appropriate to call this species of ontologicaldependenceeXplanatory existential dependence. (We shall needthe subscript “E” for another use later.) Here it isimportant to note that the presence of the word “only” in(EDX)’sdefiniens should not be understood as implyingthat an objectx may not depend_X for its existenceupon two (or more)different things,y andz. Thus the particularized relation of Mary’s lovingTom—supposing such an object to exist—plausibly existsonly because Mary exists, but plausiblyalso existsonly because Tom exists. Furthermore, we may assume that itisnot an implication of (EDX) that a composite objectdepends_X for its existence upon its proper parts, that is,that it isnot the case that it “exists only becausethey exist”—on the grounds that it could still exist inthe absence ofthose particular parts, provided suitablealternative parts were substituted for them. Thus (EDX) is quiteunlike(EDG) in its implications for part-whole dependence relations, as far ascomposite objects are concerned. For the same reason, we may assumethat (EDX) does not imply that an “Aristotelian” universaldepends_X for its existence upon its particular exemplars.Indeed, we may take it that thedefiniens of (EDX) entailsthedefiniens of(EDR)—although notvice versa, of course—so that the following is aprinciple that one could accept:

(B-NEC): If, necessarily,x exists onlybecausey exists, then, necessarily,x exists onlyify exists.

However, despite these clarifications, it must be conceded that thelocution “x exists only becauseyexists” is hardly very perspicuous, either as to its logicalform or as to its exact meaning. Moreover, precisely because we haveintroduced the conjunction “because” as anexplanatory conjunction, it may be felt that it is notwell-suited to the ontological role now being devised for it (for oneof the many recent attempts to specify the explanatory role of“because”, see deRosset 2013). There are perhaps twosources of worry here: first, that this approach invites a confusionbetween metaphysics and epistemology; and secondly (but relatedly)that contexts governed by the conjunction “because” areopaque (in the technical sense of the term, in which itimplies the non-applicability of Leibniz’s Principle of theIdentity of Indiscernibles).

Some of these worries can perhaps be allayed by resorting to therecent work on metaphysical grounding, which presents a much morerigorous 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 groundingand metaphysical explanation (see §4 of the separate entry onmetaphysical grounding), but only to note that the discussion surrounding explanatorydependence has progressed, in part, independently of the widerconsiderations surrounding ontological dependence. So we shouldperhaps accept that(EDX) as it stands does not really constitute a satisfactory definition ofa species of existential dependence, conceived as an objectivemetaphysical relation between entities, because it is insufficientlyperspicuous. In any case, the fact that (B-NEC) but not its converseholds some appeal indicates that what we should be trying to frame isa perspicuous definition of a species of ontological dependence thatis a relation betweenx andystronger than(entailing but not entailed by) “necessarily,x existsonly ify exists”. This should moreover be (for reasonsdiscussed earlier) anasymmetrical relation—or, moreaccurately, anantisymmetric 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 anticipatedsense, depend ontologically uponitself, controversial asthis may be (it is nevertheless a position that is of at leasthistorical interest, e.g., because of the potential special case ofGod depending on herself).

As we shall see in a moment, a relation that may be of some assistancein spelling out the relevant sense of dependence is the relation ofidentity-dependence, to be explained below. But first weshould digress for a moment to note that theone-sidedholding of the relation defined by(EDR) is unlikely to do the job. According to this suggestion, we have:

(EDA)

x depends_A for its existence upony=_df

(i): necessarily,x exists only ify exists and
(ii): it isnot the case that, necessarily,y existsonly ifx exists.

Notice that the relation thus defined isasymmetric (ratherthanantisymmetric): it doesn’t permit any object to beexistentially dependent uponitself. Indeed, we have chosento use the subscript “A” here because the relation inquestion may aptly be calledasymmetrical rigid existentialdependence. One might think that this is at it should be, as on theface of it the idea of something existentially depending on itselfappears very strange. There is of course an obvious theologicalcandidate that may violate this condition, namely God, so perhaps weshould not rule out the possibility outright. But there may beslightly less controversial candidates as well, for it could besuggested that there arefundamental entities thatexistentially depend on themselves (rather than on nothing). At anyrate, if anything of the sort is possible, we better not rule it outby definition, as (EDA) does.

Perhaps more importantly, an appeal to (EDA) does not seem to help toresolve the difficulty raised by the example of Socrates’s life.Forneither Socratesnor his life is existentiallydependent_A 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 istrue tosay that Socrates’s life is ontologically dependent uponhim, but notvice versa.

4. Essential Dependence and Identity-Dependence

4.1 Why Essential Dependence?

Until quite recently, it was common to think that ontologicaldependence can be fully characterized in modal-existential terms, aswe have seen above. One obvious reason for this is that if one adoptsthe usual “modalist” analysis ofessence,essential dependence will collapse into a form of modal-existentialdependence (one classic defense of the “modalist” analysisis Marcus 1967). But it seems clear already starting from Aristotlethat there is an alternative way to formulate ontological dependenceif essence is not analyzed in modal terms (as Fine 1994a andespecially those working in the “neo-Aristotelian”tradition would have it). However, this is not to suggest thatAristotle did not rely on the modal-existential notion of ontologicaldependence. Rather, there are reasons to think that Aristotle’sunderstanding of dependence encompassesboth themodal-existential notion and the essentialist notion to be describedbelow (for historical details and further discussion, see Corkum 2008,Koslicki 2013, and Peramatzis 2011).

One motivation for developing anon-modal conception ofontological dependence is that the modal-existential analysis appearsto be too coarse-grained for some cases. We have already discussedsome of these cases, but the most well-known examples have been madefamous by Fine (e.g., 1994b). Consider, for instance, what themodal-existential account entails in the case of necessary existents.Take Socrates and the number 2, for example. Given that numbersnecessarily exist, it is necessarily the case that 2 exists ifSocrates does. But presumably we do not want to say that Socratesdepends upon the number 2, or indeed on most necessary existents thatyou might put in the place of 2. So the modal-existential accountmakes everything depend upon every necessary existent, which seemslike the wrong result.

Admittedly, those who defend a modal-existential analysis ofontological dependence could insist that it applies only tocontingent objects (cf. Simons 1987: 295). Simons makes thistype of qualification by focusing onconcrete entities, henceexcluding necessary existents by definition; he also excludesself-dependence. Simons calls the resulting notion of dependenceweak rigid dependence, but a stronger notion,strongrigid dependence (Simons 1987: 303) is also defined—thelatter is a special case of the former. One example of weak rigiddependence as defined by Simons would be a particular water moleculedepending for its existence on a particular oxygen atom. In the caseof strong rigid dependence, the dependent object cannot be a properpart of the object it depends upon. So objectx is stronglyrigidly dependent on objecty ifx depends for itsexistence ony andy is not a proper part ofx. One example of strong rigid dependence defined thus wouldbe a trope (or mode) depending for its existence on a substance. Inaddition to these rigid notions, Simons defines corresponding notionsof (weak and strong)generic dependence.

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

The modal-existential analysis of ontological dependence can thus bedeveloped further, and it can perhaps overcome some of the problemsthat were pointed out earlier. But there are areas where a morefine-grained notion would seem to be required and there is also ahistorical precedent (e.g., in Aristotle) for such a notion. It shouldbe noted though that much of the contemporary literature in defense ofa non-modal, fine-grained analysis (such as Fine 1994b and Koslicki2012), operates in a “neo-Aristotelian” framework whichtypically assumes some “non-modalist” version ofessentialism. Accordingly, there is an obvious rift between themodal-existential analysis and the essentialist analysis—onethat we cannot fully bridge here. In any case, it is good to keep inmind that the notion of ontological dependence itself, when used as atool to formulate the commitments of a specific theory, does notimmediately force one to make a commitment in this regard, even thoughsome 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 notanalyzable in terms of the modal-existential account—and hence amotivation to pursue a non-modal analysis of dependencealtogether—are cases that do not seem to involve a requirementforexistence at all. Indeed, as we already saw in the caseof(3), “Socrates does not ontologically depend on singletonSocrates”, it seems that something beyond mere existentialindependence is being expressed. Instead, one might say that singletonSocrates does not have anything to do with theidentity oressence of Socrates, and it is this lack ofidentity-dependence between Socrates and {Socrates} that wewish to express with (3). In other words, being a member of singletonSocrates is not anessential property of Socrates. Recallthat a similar issue seemed to arise with regard to the possibility ofcomposite objects havingessential proper parts. We will nowconsider a more systematic method for dealing with cases of dependenceinvolving 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, weought to get a better picture of what it means to say that an objectdepends upon something for itsidentity, that is, we shouldclarify the relation ofidentity-dependence. Note that thenotion of “identity” at play here is not the onesymbolized with the “equals” sign, i.e., “=”.Rather, we mean “identity” in the sense ofwhat athing is, orwhich thing of a certain kind a thing is(compare this with Aristotle’sto ti esti, “thewhat it is”, see §7 of the separate entry onAristotle’s metaphysics). Informally speaking, to say that the identity ofx dependson the identity ofy—or, more briefly, thatxdepends for its identity upony—is to say thatwhich thing of its kindy is fixes (or at leasthelps to fix)which thing of its kindx is. By“fixes” in this context is meantmetaphysicallydetermines. For instance, then, the identity of aset isfixed by the identities of its members, as was hinted already in§1. Likewise, the identity of anassassination is (at leastpartially) fixed by the identity of the person assassinated. Theserelationships of identity-dependence are direct consequences of theidentity-criteria governing the kinds of which the items thus relatedare instances. (For further discussion on identity-criteria andidentity-dependence, see Lowe 1989, 2009: Ch. 2, and 2012.)

One way to attempt to precisify the notion of essence is to note thatwe are here concerned with identity in the sense ofindividuality rather than therelation of identity.In other words, identity-dependence expresses the determination of theindividuality of objects in terms of the individuality of otherobjects. Thus, the identity-dependence of a set upon its members couldbe regarded as a consequence of the fact that the Axiom ofExtensionality functions as the criterion of identity for sets. Theset’s members determine the individuality of the set—theyindividuate the set. Notice, here, that we allow thatx may be said to depend for its identity upony evenin cases in which the identity ofy alone does notsuffice to fix the identity ofx. So, for example, a set withtwo or more members depends for its identity uponeach ofthem, although its identity is onlycompletely fixed by theidentities ofall of them.

Another way to specify the notion of essence being used is to resortto Fine’s (1994b, 1995) distinction betweenconstitutive andconsequential essence:

A property belongs to theconstitutive essence of an objectif it is not had in virtue of being a logical consequence of some morebasic essential properties; and a property might be said to belong totheconsequential essence of an object if it is a logicalconsequence 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 partof the constitutive essence of {Socrates}. In contrast, the propertyof containing some member or another is only part of its consequentialessence. It is evident that in cases of essential andidentity-dependence, we are typically interested in the constitutivenotion of essence, because it is presumably also a part of theconsequentialist essence of Socrates that 2 = 2. So, using theconsequentialist notion of essence to define essential andidentity-dependence would quickly lead to problems not unlike the onesfaced by the modal-existential account. Accordingly, in what follows,the various definitions should in the first instance be understood toconcern the constitutive essence. But the problem that remains is thatthe distinction between constitutive and consequential essence itselfis not always easy to make and it is inevitable that there will besome controversial cases (for discussion, see Fine 1994b). Theexamples used to illustrate the various forms of essential dependenceshould thus be taken with a grain of salt, as it is often not entirelyclear what should be included in the constitutive essence of anentity. In any case, provided that this distinction can be made, thefriend of the essentialist analysis of dependence has a clearadvantage, given that they are able to express a broader range ofdependence claims with the help of this distinction.

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

(ID-NEC): Ifx depends for its identity upony, then thereis a functionf such thatx is necessarily identicalwithf(y).

(Note: “f(y)” here may be pronounced“thef ofy”.) For example: if we saythat the identity of a marriage depends on the identities of the twopeople being married, ifx is a marriage andy andz are the two people in question, (ID-NEC) is satisfied inrespect ofx andy in virtue of the fact thatx is necessarily identical withthe marriage ofy withz—so that in this case therequired function is themarriage withz functionfrom persons to events. (We here ignore the complications created bythe fact that, under some legal systems, the same two persons may bemarried to one another more than once.) However, (ID-NEC) turns out tobe unsatisfactory as a definition, because it can be regarded astrivially true, given standard set theory, that anyx andy that may be connected with a function will satisfy theconsequent of (ID-NEC). So we better find a more satisfactorydefinition of identity-dependence.

Evidently, it would not do simply to replace the conditionalconnective in (ID-NEC) with abiconditional connective andthence attempt to turn it into adefinition ofidentity-dependence, unless at the same time one could impose somesuitable restriction on the kind of function involved. One possibilitywould be to exclude any functionf which is not such that itispart of the essence ofx that it is thef ofy. But since it would appear strange to saythat essences quite generally involve anything aboutfunctions, we might be better off with a definition using anappropriate yet to be specified place-holder relation that would dothe same job instead, arriving at the following definition ofidentity-dependence:

(ID): x depends for its identity upony =_dfThere is a relation “R” such that it is part ofthe essence ofx thatx isrelated by R toy.

We can exemplify (ID) by lettingx be {z} andy bez, in which case we have, as is intuitivelycorrect, that {z} depends for its identity uponz,because there is a relation—namely, “being a member of thesingleton set” (also known as theunit setfunction)—such that it is part of the essence of {z}that it is the singleton set ofz. Accordingly, we mayinterpret the relationR more precisely on a case by casebasis.

Building on this example, we can see thatx is necessarilyidentical with the sole member of {x} and hence we can arriveat a principle not unlike (ID-NEC). However, even thoughx isnecessarily identical with the sole member of {x}, it is,once again, plausibly notpart of the essence ofxthatx is the sole member of {x} (cf. Fine 1994a:4–5; see also Fine 1994b for a related discussion of the notionof 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 taskwhich cannot be undertaken here. However, various attempts toconstruct such an account have been made, for instance by Fine (1994a)and Lowe (2008). In this sense, a thing’s essence may be said toconstitute its identity, when one uses the word“identity” in this distinctive manner to speak ofathing’s identity, rather than using it to speak of theidentityrelation. Seen in this light, identity-dependence asdefined by (ID) is simply a species ofessential dependence,that is, a way in which the essence of a certain thing is determinedby a relation in which it stands to another thing.

Although the above account of identity-dependence is not entirelyuncontroversial, there are some interesting applications of it in theliterature. One of these applications concerns a version of OnticStructural Realism (OSR, see Ladyman 1998), which claims that objectsdepend for their identities on the structures to which they belong(ID-OSR) (see French 2010 for one version, and see Lowe 2012 andO’Conaill 2014 for further discussion of OSR anddependence).

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

(ID-EX): Ifx depends for its identity upony, then,necessarily,x exists only ify exists.

And second:

(ID-NEX): Ifx isnot identical withy andx depends for its identity upony, thenydoesnot depend for its identity uponx.

An immediate implication of (ID-EX) in conjunction with(EDR) is that ifx depends for its identity upony, thenx isrigidly existentially dependent upony(although not necessarilyvice versa, of course). These twoprinciples are, it seems, quite plausible. As for (ID-EX): surely,x cannot exist unless everything upon whichx’s identity depends also exists. Thus an assassinationcannot exist unless the person assassinated exists, and a set cannotexist unless its members exist. Indeed, it would seem that we canderive (ID-EX) with the aid of(ID). For suppose thatx depends for its identity upony.Then, by (ID), there is a relationR such thatx isrelated byR toy. Given, however, that the relevantR cannot exist unlessy exists—because arelation can obtain only between entities all of whichexist—it follows thatx likewise cannot existunlessy exists. As for (ID-NEX), this seems to follow fromthe requirement ofnon-circularity which is a condition onthe adequacy of any criterion of identity. For example, given thatunit sets are not to be identified with their members, we cannot sayboth that the identity of a unit set depends upon theidentity of its memberand that the identity of that memberdepends upon the identity of that unit set, for this would engender avicious circle which would seem to depriveboth unit setsand their members of well-defined identity-conditions.

Note that it may be urged, with some plausibility, that every objectx trivially depends for its identity uponitself.And, certainly,(ID) would appear to have this implication, because for any objectx, there is a relation—namely,identity—such that it is part of the essence ofx thatx is the object identical withx.But we must be very careful to distinguish between the claim that anobject depends for its identity upon itself and the claim that anobject depends for its identitysolely upon itself. For evenif the former claim is trivially true of all entities, the latterclaim is certainly not. However, this result may be contested when webear in mind the distinction between constitutive and consequentialessence, because one might very well hold that it is no part of theconstitutive essence of an entity, at least typically, that it shouldbe identical with itself (we will omit a more detailed discussionhere, but see Fine 1994b).

4.3 Essential Dependence

Having defined identity-dependence by(ID) as a species ofessential dependence, we are now in aposition to identify other species of essential dependence, the mostobvious being what may aptly be calledessential(existential)dependence, which can be defined asfollows:

(EDE): x depends_E for its existence upony=_df It is part of the essence ofx thatxexists only ify exists.

Note that, whereas—assuming the truth of(ID-NEX)—two distinct entities cannot beidentity-dependent uponeachother, it very plausiblyis possible for each of twodistinct entities to depend_E for itsexistence onthe other. For example, consider a solid sphere. Let us assume for thepurposes of this example that the solid sphere is asubstance, a kind of basic, ontologicallyindependent entity (see§6.3 for further discussion). If we think of the top and bottom“halves” of a solid sphere as beinggeometricallydefined entities whose boundaries are specified by reference to thewhole sphere of which they are equal subdivisions, then it could beargued that it is part of the essence of each such hemisphere that itexists only if the other does. (These “halves”, it shouldbe emphasized, must not be confused with the portions of matter“filling” them at any given time, and hence should not bethought of as “parts” of the sphere of which it ismateriallycomposed.) At the same time, each hemispheredepends for itsidentity upon the whole sphere—onebeing identified as the top half of that sphere and the other as thebottom half—whereas the sphere itself does not likewise dependfor its identity upon either of these halves, given that we haveassumed that it is a substance in its own right. This example isadmittedly controversial: one could insist that the constitutiveessence for each hemisphere involves the whole sphere, but not theother hemisphere. Moreover, one could also focus on theplural orcollective essence of the halves, whichmight help us get around this circularity (on plural/collectiveessence, see Zylstra 2019). Note also that, very plausibly, ifx depends_E for its existence upony, thenx also depends_R for its existence upony:essential existential dependence entailsrigidexistential dependence—but not, of course,viceversa.

We have seen that (ID) and (EDE) correspond torigid forms ofmodal-existential dependence, such as (EDR). But we can also formulateageneric notion of essential dependence, just like we did inthe case of modal-existential dependence. Corresponding to genericexistential dependence (EDG), we can formulategeneric essential(existential) dependence as follows:

(EDEG): x depends_EG for its existence uponFs=_df It is part of the essence ofx thatxexists only if someF exists.

For instance, ifx is a particular water molecule, we may saythat it is a part of the essence of that water molecule that it existsonly if some atoms of hydrogen and oxygen exist. In other words, watermolecules essentially depend for their existence on there being atomsof the kind hydrogen and oxygen. It may be helpful to introduce adistinction betweenindividual andgeneral essencesin this connection. (EDEG) as applied to the case of water certainlyapplies to each and every water molecule rather than just one of them.Accordingly, when we speak of the essence of water in this context, weare not referring to the individual essence of any particular watermolecule, but to the general essence of the natural kindwater. (EDEG) concerns what an entity depends upon for itsgeneral essence (e.g., what makes something count as a watermolecule), whereas (EDE) concerns what an entity depends upon for itsindividual essence (e.g., what makes something the particular watermolecule it is, as opposed to any other). (For further discussion ofthis distinction between individual and general essence, see Lowe2008.)

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

(EDC): An entityx ontologically depends on an entity (orentities),y, just in casey is a constituent (orare constituents) inx’s essence.

This account relies on the notion ofconstitutive essence,which is developed in Fine’s work (1994b, 1995). We have alreadydefined this notion above, but a recap may be useful. As Fine putsit:

we may takex to depend upony ify is aconstituent of a proposition that is true in virtue of the identity ofx or, alternatively, ify is a constituent of anessential property ofx. (Fine 1994b: 275)

It should also be noted that even though the notions of essentialdependence defined above are no doubt more fine-grained than themodal-existential notions, there may be reasons to think that eventhey are not sufficiently fine-grained for all purposes (as argued inKoslicki 2012). To illustrate, consider Fine’s well-knowndiscussion of Socrates and the singleton set that has Socrates as itssole member. To use Fine’s terminology, we could say that it ispart of the constitutive essence of Socrates’s singleton setthat it has Socrates as its sole member, whereas it isnotpart of the constitutive essence of Socrates to be the sole member ofSocrates’s singleton set. But as Koslicki (2012: 195) pointsout, this is really just to say that Socrates’s singleton setontologically depends on Socrates whereas Socrates doesnotontologically depend on Socrates’s singleton set. The relevantnotion of dependence appears to be built into Fine’s notion ofessence and, similarly, the notion of essence assumed by Fine (andmany other “neo-Aristotelians”) is already built into(EDC).

Before we conclude our discussion of essential dependence, a majorissue in the literature concerning ontological dependence should bementioned. For it appears that those who are willing to buy into asuitably fine-grained notion of essence, popular in the“neo-Aristotelian” tradition, will find themodal-existential account of ontological dependence far toocoarse-grained. Yet, those who are not sympathetic to the“neo-Aristotelian” notion of essence but would ratheranalyze essence in terms of modality would insist that themodal-existential analysis is quite sufficient, and indeed thatessential dependence collapses into modal-existential dependence. Therecent 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 ofessential dependence or the “non-modalist” analysis ofessence), but this is indeed only a relatively recent phenomenon. Afurther issue is that there is some disagreement about how therelevant notion of essence is to be constrained amongst those whothink that a more fine-grained analysis than the modal-existentialaccount is needed. For instance, Koslicki (2012: 196 ff.) regardsFine’s propositional notion of essence according to which thereis little or no distinction between essence andrealdefinition as overly restrictive. She identifies the source ofthis restrictive conception of essence, which is also present inLowe’s work, to be the focus on essences asindividuating—this is the type of view regarding essentialdependence, namely identity-dependence, which we have here beenfocusing 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 uniquelyidentify and delineate the entities under consideration at every timeand 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 analyticmetaphysics in the beginning of the twenty-first century, but theroots of the notion go back to Aristotle (for an overview, see Fine2012). Sometimes grounding is understood as “metaphysicalexplanation”. To be more precise, when somex isgrounded in somey, it is usually thought thatyexplainsx, and grounding either is that explanatoryrelationship or ‘backs’ that explanation (there is a lotof debate on these issues, see the separate entry onmetaphysical grounding). Moreover, the status ofy is generally thought to be somehowprior to that ofx—grounding is typicallyunderstood to express priority between things. For instance, we mightsay that the members of a set are prior to the set itself; theexistence of the set is grounded in its members. Or to take a moreconcrete example, the existence of any given composite object isgrounded in the existence of its parts. For instance, we might suggestthat the existence of any given water molecule is grounded in theexistence 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 ontologicaldependence relation between the grounded entity and the groundingentity or entities. In fact, we have already discussed some of theseexamples in terms of ontological dependence above. This naturallyleads one to question whether grounding just is (a variety of)ontological dependence—or whether ontological dependence isgrounding. One motivation for finding a systematic link betweenontological dependence and grounding is that it would be moreparsimonious than having two primitive notions (for discussion, seeCorreia & Schnieder 2012b).

An initial reason to distinguish ontological dependence andmetaphysical 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 ontologicaldependence obtaining in such a way that a given object is dependent onitself. It is of course controversial whether there actually are anysuch entities, but this is not enough to consider irreflexivity as anecessary requirement for ontological dependence. In contrast,metaphysical grounding is usually considered to be necessarilyirreflexive (of course, there are those who think that thisrequirement could be dropped in the case of grounding just like it mayin 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 bothnecessary and sufficient for grounding? This seems unlikely (as arguedin Rydéhn 2021). While there are some instances of groundingwhich are clearly rigid in the same sense as rigid existentialdependence, it is also clear that this connection is not necessary andsufficient. Take for instance the fact [f] and assume that it is fullygrounded 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 isalso plausible that [f] depends for its identity upon [g]. However,other instances of grounding do not require that the grounded factrigidly depends upon its grounds. For example, suppose that [f] is thefact that a particular subject is feeling pain, and that [f] is fullygrounded in a particular neural fact about this subject, [g]. It seemspossible that this subject could have been in pain, so [f] could haveobtained, 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 rigidlydependent upon [g]. This rules out some simple options for linkinggrounding and rigid ontological dependence, but there are arguablyalso problems for linking it with generic varieties of ontologicaldependence (for a more detailed discussion, see Rydéhn2021).

If grounding cannot be easily analyzed with the help of rigid orgeneric dependence, then we might instead suggest that if groundingwere to be understood as a type of ontological dependence, it would besome sort ofexplanatory dependence. The idea that whateverdoes the grounding also somehowexplains what is beinggrounded is a crucial part of the notion’s appeal. Relations ofontological dependence often seem to have a similar type ofexplanatory role, but the link to explanation may appear weaker: eventhough the existence of water depends on the existence of hydrogen andoxygen, it does not seem to be the case that the existence of hydrogenand oxygenexplain the existence of water. Rather, whatexplains the existence of water is the ability of hydrogen and oxygenatoms to form molecules (even though this is rather simplified). Sothere are at least some difficulties in accounting for all relationsof 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 tooliberal notion, for we may regard a number of loosely connected thingsexplanatory in some very loose sense. For instance, we might say thatthe fact that Smith murdered Jones is explained by certain events inSmith’s childhood, but a more direct explanation might beSmith’s desire to rob Jones. Certain events in Smith’schildhood may help usunderstand why Smith has murderousdesires, but it’s not clear that they serve to ground the factthat Smith murdered Jones (unless we can establish transitivitybetween these).

One suggestion that may help to make grounding more precise would beto focus onpriority. To reiterate an example used above, ifthe members of a set are prior to the set itself, then the existenceof the set is grounded in its members. So the grounding entities areprior to—or morefundamental than—the groundedentities. Metaphors abound, but a typical way to express the idea isto say thatx is fundamental or ontologically independent inthis sense if and only if nothing groundsx (Schaffer 2009:373). This is another sense in which grounding would seem to comeapart from certain types of ontological dependence, as a purelymodal-existential understanding of dependence in the lines of(EDR) (if one accepts them), without any claim to priority, is alsopossible. This is of course the sense which we defined above in termsof necessitation: even ifx rigidly necessitatesythat does not entail thaty must be ontologically prior tox. At any rate, it would be odd to say that parents aremore fundamental than their children, even if there is a(temporally relativized) rigid existential dependency between parentsand their children.

All this would appear to suggest that it isn’t straightforwardto define grounding simply as a variety of ontological dependence, orthe other way around. In any case, even if we could define one notionin terms of the other, it seems that there are aspects of ontologicaldependence that are not captured by all accounts of grounding as wellas aspects of grounding that are not captured by all accounts ofontological dependence. The situation may of course change dependingon one’s preferred account of dependence and grounding. In fact,a proponent of the modal-existential analysis of dependence mightargue for a more unitary approach (as Casey 2022 hasattempted). On this line of thought, instances ofontological dependence could be analysed as being a subset ofgrounding instances; specifically, those instances of groundingthat involve something’s existence being grounded areinstances of ontological dependence. The hope is that one mightattempt to avoid some of the challenges that the modal-existentialanalysis of dependence encounters, opting for a grounding-basedinstead of an essence-based account of ontological dependence (assuggested in Casey 2022). The suggestion is interesting, but itssuccess will also depend on the specifics of the notion of groundingbeing employed. For instance, there is the apparent differenceresulting from the fact that grounds are generally considered tonecessitate what they ground, while ontological dependence does notentail necessitation (to the same direction). A possible wayaround this is to treat instances of ontological dependence as onlypartial grounds (Casey 2022: 13).

Accordingly, for the time being, it is advisable to keep the notionsapart, especially since there are some further, formal differencesthat need to be taken into account. In particular, grounding is mostcommonly understood as astrict partial order (see Raven2013), which entails irreflexivity, transitivity, and asymmetry. Ifthis is correct, grounding could only capture a very specific varietyof ontological dependence, as we have seen that there are varieties ofontological dependence that violate these formal features. Some havealso argued that grounding itself violates all or some of these formalfeatures, so it is not entirely uncontroversial that grounding trulyis a strict partial ordering (for discussion, see Jenkins 2011,Schaffer 2012, and Tahko 2013). Given this, the exact link betweengrounding and ontological dependence remains open, subject to furtherspecification of the formal features of ground.

6. Applications and Other Related Notions

In addition to the previous discussion regarding metaphysicalgrounding, there are numerous applications of ontological dependenceas well as closely related notions that we could discuss. It will notbe possible to do justice to all of them, but we will mention a few ofthe most important ones (many of these and further applications arediscussed in Hoeltje, Schnieder, and Steinberg (eds.) 2013).

6.1 Supervenience

The notion of supervenience is covered in more detail in the separateentry onsupervenience (see especially §3.5) but a brief mention is in order here, assupervenience has sometimes been conceived as a type of dependence.One reason for this is that when we assert thatA supervenesonB, we might also say thatA-propertiesontologically depend uponB-properties. Take the typicalexample of the beauty of a work of art and the physical manifestationof that work of art. It would seem that if you wish to change theaesthetic properties of an artwork, you will also have to manipulateits physical manifestation. As a first pass, it looks as if thedependence at work here is generic existential dependence as definedby(EDG), so the aestheticA-properties depend_G for theirexistence upon some physicalB-properties. However,supervenience in general is not so easily analyzed, as at least on oneusual conception, supervenience is not irreflexive and hence notasymmetric (see Steinberg 2013). In other words, we can at least saythat supervenience, as opposed to grounding, is not a strict partialorder and hence not a relation of ontological priority. There arevarious ways to further specify the formal features of supervenience,and indeed to argue that it should not be considered as a form ofdependence, but this is not a task that we can undertake here.

6.2 Truthmaking

The connection between ontological dependence and truthmaking is of amore general type. On many construals, the core of truthmaker theoryis considered to be the idea that truth depends—orsupervenes—onbeing (Bigelow 1988: 133; seealso Armstrong 2004, Schaffer 2010a, and Liggins 2012). Recently, somehave resisted this idea (e.g., Merricks 2007), but it does remainpopular among truthmaker theorists (for discussion, see the separateentry ontruthmakers). Moreover, the connection between truthmaking and ontologicaldependence has been a part of truthmaker theory from the verybeginning; Mulligan, Simons, and Smith (1984: 294) rely onHusserl’s work on ontological dependence in their introductionof the idea of truthmaking (for further discussion of Husserl’swork regarding dependence, see Simons 1982 and Johansson 2004: Ch. 9).More precisely, it seems there must be some way in which the world isin virtue of which true propositions are true, but the world itself isnot symmetrically dependent on truth; there is anasymmetricdependence relation between truth and being (accordingly, thisdependence needs to be expressed with an asymmetric notion ofdependence, see§3 above).

However, popular though it is, this view is not entirelyuncontroversial. For instance, MacBride (2014) argues that there is noneed to appeal to truthmaking to explain this type of asymmetricdependence between truth and being. In fact, there are reasons tothink that truthmakingcannot explain the asymmetry, at leastnot on the basis of the idea that truth supervenes on being. Part ofthe confusion here seems to surround the relevant notion ofsupervenience. On Armstrong’s (2004: 8) construal, the relevantnotion of supervenience is symmetric: truth supervenes on being, butbeing also supervenes on truth. So it appears that the requiredasymmetry cannot arise from supervenience itself (see alsoRodriguez-Pereyra 2005). While these problems must be settledelsewhere, it suffices to say thatsome variety ofontological dependence is no doubt at work in truthmakertheory—there is an on-going debate as to how truthmaking shouldbe specified. One recent suggestion that may have some mileage is thatthe relevant asymmetry could be recovered if truthmaking is understoodastruth-grounding, that is, some entityx groundsthe truth of some propositionp ifp is true invirtue of the existence ofx (see Tahko 2013 fordiscussion).

6.3 Substance

One important application of ontological dependence is the analysis ofthe notion of “substance”—a basic or fundamentalentity which possesses at least some degree of ontologicalindependence (for discussion on the various uses of“substance” in philosophy, see the separate entry onsubstance). If the idea of ontological independence is somehow associated with“substance”, then it seems that some account of what therelevant dependence amounts to will be required. In fact, thosefamiliar with Lowe 2005 [2010] (an earlier version of the presententry), will recall that the discussion was entirely focused onfinding the most plausible account of ontological dependence for thepurposes of analyzing the notion of substance (this discussionoriginates from Lowe 1998: Ch. 6). This also ties in with a moregeneral application of ontological dependence, namely, the study offormal 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 wedefine “substance” given the various notions ofontological dependence that we have at our disposal? It will berecalled that one particularly powerful notion of ontologicaldependence is the notion of identity-dependence, which we arrived atwhen looking for an asymmetrical—or, at least, anantisymmetrical—relationship of ontological dependence.It seems that if substances are considered as ontologicallyindependent entities, then an asymmetric relation isrequired. One possibility for defining substance, which relies on(EDR) is the following:

(SUB-1): x is a substance =_dfx is aparticular and there is no particulary such thatyis not identical withx andx depends_Rfor its existence upony.

(SUB-1) could be seen to follow the precedent of Aristotle,who—in theCategories—admitted onlyparticulars as “primary” substances, whileallowing some universals (the species and genera of primarysubstances) the status of “secondary” substances (seeAristotle,Categories: Ch. 2). However, (SUB-1), because itrelies on(EDR), will be unsatisfactory for those who are convinced that themodal-existential analysis of ontological dependence is notsufficiently fine-grained; it will face the type of problems that wehave discussed above in§2.2. We know already that it won’t do to replace the notion ofexistential dependence employed in (SUB-1) by appealing instead to therelation ofessential existential dependence, as defined by(EDE), because the latter is neither an asymmetrical nor an antisymmetricalrelation. Indeed, this is one lesson of the example of thesphere’s two hemispheres discussed immediately after thedefinition for (EDE) was given. For, whereas thewhole spheremight well be taken to qualify as a substance, neither of its“halves” plausibly can, because they lack the requisitekind of ontological independence. But, while (SUB-1) reinterpreted interms of (EDE) does indeed have the implication that neither of thehemispheres is a substance—because each depends_E forits existence on the other and hence on a particular distinct fromitself—it also has the unwanted implication that thewholesphere is not a substance for the same reason, because it isarguably a part of the essence of the sphere as a whole that it existsonly if each of its hemispheres exists. However, an obvious remedy isat hand. We can simply replace the appeal to any species ofexistential dependence in a definition on the pattern of(SUB-1) by an appeal to the relation ofidentity-dependence,as defined by(ID), to give:

(SUB-2): x is a substance =_dfx is aparticular and there is no particulary such thatyis not identical withx andx depends for itsidentity upony.

Composite substances appear to comply with (SUB-2): for, plausibly,although they possess proper parts, they do not depend for theiridentity upon those parts, sincewhich objects thoseparts are does not help to determine ofwhich substances theyare parts (the same objects being capable of becoming parts of manydifferent substances). Moreover, substances quite generally do notdepend for their identity upon their (accidental) particularizedproperties, if such exist, nor upon the events in which theyparticipate, nor upon the places they occupy, nor upon othersubstances. The particularized properties (i.e., tropes or modes) ofsubstances and the events in which substances participate—thatis, items such as the particular redness of this apple and theassassination of Caesar—would appear to depend for theiridentity upon those substances, which precludes the reverserelationship from obtaining, on pain of circularity. (However, seeKeinänen & Hakkarainen 2014 for discussion; in particular,they propose a strategy for identifying tropes that does not entailcircularity in the individuation of tropes. See also Moltmann 2019 andforthcoming, which discuss the problem of spatial locationrelated to tropes and other entities.) As for places, although aphysical substance must indeed occupysome place,which place it occupies does not determinewhichsubstance 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 differententities to be existentially dependent_Rupon oneanother—entities such as Socrates and Socrates’slife. As we have seen,(ID) precludes any analogous symmetry whereidentity-dependenceis concerned: indeed, it delivers the intuitively correct verdict thatit isSocrates’s life that is, in this sense,ontologically dependent upon Socrates, rather thanviceversa. For Socrates’s life is an extended event or processin which he participates—andwhich person Socrates ispartially determineswhich event this is, but notviceversa. Or, to put it another way—one which should by now befamiliar—it ispart of the essence of Socrates’slife that it isthe life of Socrates, but it isnotpart of the essence of Socrates that he is the person who lived thatlife. Of course, we can still acknowledge that the relation defined by(EDR) does hold mutually between Socrates and his life and we can stillcall this relationship a type ofexistentialdependence—namely, “rigid” existentialdependence. Similarly, we can recognize as other species ofexistential dependence the “generic” existentialdependence defined by(EDG), the “asymmetrical” rigid existential dependence definedby(EDA) and the “essential” existential dependence defined by(EDE).

The key point is simply that, in the sense of “ontologicaldependence” which fits quite naturally with the“neo-Aristotelian” account of substances asontologically independent objects, the relationships definedby(EDR),(EDG),(EDA), and(EDE) willnot serve the purpose, whereas that defined by(ID)will. On this view, the sense in which a substance is anentity which does not depend “ontologically” upon anythingother than itself is exactly the sense in which it does not depend foritsidentity upon anything else. This still leaves manyinteresting questions concerning ontological (in)dependenceunanswered, notably the question of whether there is afundamentallevel orlayer of reality, consisting of one or moreentities upon which all other existing entities depend ontologicallyin 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 ismorefundamental than another one, it is because the lessfundamental is ontologically dependent on the more fundamental. Muchof the recent discussion involving ontological dependence has focusedexactly on the question of fundamentality and especially the questionmentioned above, namely whether there is something that is entirelyfundamental, ontologically independent—is there an ontological“bottom level”? Hence, there are two senses in which aclearer understanding of ontological dependence may help: we canspecify in what sense fundamental entities are ontologicallyindependent as well as how other entities may be ontologicallydependent on the fundamental entities, thus gaining furtherunderstanding of the supposed “hierarchy” that terminatesin the bottom level.

The bottom level is usually thought to be at the smaller end of thespectrum: theatomistic view suggests that certain subatomicparticles are fundamental. But this does not mean that the fundamentallevel must necessarily be at the bottom—the fundamental endcould also be at the top, i.e., the universe as a whole could beconsidered fundamental—a substance in its own right (seeSchaffer 2010b). Each of these views is a type ofmetaphysicalfoundationalism, which suggests that chains of ontologicaldependence must come to an end, at least at the dependent end. Inother words, metaphysical foundationalism asserts that chains ofasymmetric ontological dependence must terminate, they must bewell-founded. The idea is typically formulated inmereological terms, as the reference to atomism suggests: there is anasymmetric ontological dependence relation from one end of themereological scale to the other. These questions also haveimplications for the debate betweenpluralism andmonism (one form of such monism is defended in Schaffer 2010band Tallant 2015 discusses the notion of ontological dependence inconnection to Schaffer’s monism; see also Trogdon 2009 and theseparate entry onmonism).

However, there have been some speculative suggestions according towhich chains of ontological dependence could go onadinfinitum, and hence violate the requirement of well-foundedness(for discussion, see Bliss 2013, Morganti 2014, and Tahko 2014). Theresulting view would appear to be some sort ofmetaphysicalinfinitism as opposed to metaphysical foundationalism. Oneversion of such a view, if combined with the idea that the directionof 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 asthese views and the discussion surrounding fundamentality are,discussing them in detail here would take us too far from our originaltopic (for further discussion, see the separate entry onfundamentality).

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