1. The modal account of essence: simple versus sparse

In his contemporary classic ‘Essence and Modality’, Fine (Reference Fine1994) is widely thought to have refuted the once-dominant view that the essential properties of an individual are simply those properties it must exemplify if it exists. According to Fine, every property that is essential to an individual is one it must exemplify if it exists – but not vice versa. For some properties satisfy what occurs to the right of the following ‘simple’ modal account, but not what occurs to the left:

As counterexamples, Fine points to properties such as (i) is such that there are infinitely many prime numbers, (ii) is distinct from the Eiffel Tower, and (iii) his most famous case, is a member of {Socrates}. Although all of these are properties that Socrates cannot exist without exemplifying, none intuitively concern his ‘nature’ or ‘identity’ (1994, 9). As Fine puts it, citing facts about how Socrates is related to things such as the Eiffel Tower, the prime numbers, and {Socrates} does not help to ‘define’ Socrates, at least not in the sense of specifying ‘what it is’ to be the very individual he is (1994, 2), and so MP does not specify what is sufficient for a property to be essential to him. Due to Fine’s influential critique, literature on the metaphysics of essence has largely moved on, either following him in treating the notion as reductively unanalyzable, or instead by exploring the viability of reductive accounts that either appeal to unfamiliar pieces of ideology or make use of non-standard modal notions (or both).Footnote ²

However, a number of philosophers have suggested that this retreat from the simple modal account has been too drastic, since one could restrict the class of properties eligible for essentiality in some way. For instance, Almog (Reference Almog2003), Gorman (Reference Gorman2005), and Correia (Reference Correia2007) tentatively suggest the possibility of undermining Fine’s counterexamples by distinguishing between properties corresponding to ‘anything that can be said about a thing’ no matter how ‘artificial’, as opposed to those properties that ‘really characterize’ it (as Gorman puts it: 2005, 278–279), restricting MP to the latter class. In a similar vein, Barbara Vetter, after noting that Fine’s putative counterexamples are ‘strange properties’, observes that MP seems to function perfectly well for ‘such more natural candidate properties as being human or being a philosopher’, and wonders whether Fine only establishes ‘the more modest conclusion that some naturalness restriction needs to be imposed on candidate properties for essentiality’ (Reference Vetter2009, 3). The challenge, of course, will be to show that this restriction is not an unprincipled one, does not itself presuppose the notion of essence, and that suitably clarified will classify non-essential yet necessary properties of Socrates such as is a member of {Socrates} as ‘strange’ and ‘artificial’ in the relevant sense while classifying essential properties such as is a human as ‘natural’, ‘real characteristics’ of him – a ‘tricky task’, as Correia concedes (2006, fn. 2).

Although Almog et al. neither develop this response to Fine’s counterexamples in detail nor explicitly endorse it, Cowling (Reference Cowling2013) and Wildman (Reference Wildman2013) have independently proposed appealing to what Lewis (Reference Lewis1983, Reference Lewis1986) famously calls the distinction between the sparse and abundant properties in order to clarify the distinction upon which the response rests. Rather than taking an essential property of a thing to be any property it cannot exist without exemplifying, Cowling and Wildman propose taking an essential property of a thing to be any sparse property it cannot exist without exemplifying:

As Cowling and Wildman both note, non-equivalent conceptions of what it is for a property to be sparse rather than abundant yield non-equivalent variants of SPM.Footnote ⁴ For instance, under Lewis’s conception of the distinction, the sparse properties are all and only the fundamental (or ‘perfectly natural’) properties, which collectively share the following characteristics:

Lewis contrasts these with the abundant properties, which

Call this fundamentalist SPM. One obvious worry with this variant of SPM is that it yields a criterion that seems far too narrow – certainly more narrow than the ‘natural’ properties ‘really characterizing’ a thing that Almog et al. allude to. Presumably, is a human is a biological property, and therefore not a fundamental one; hence fundamentalist SPM would entail that Socrates is not essentially human. Cowling and Wildman note that those worried with this consequence could instead opt for what Schaffer (Reference Schaffer2004) calls the scientific conception of sparsity. As Wildman puts it, the sparse properties under this conception are any ‘involved in the total scientific understanding of the world, regardless of level of occurrence’ (Reference Wildman2013, 766). Call SPM interpreted under this broader conception of sparsity scientific SPM.Footnote ⁵

Cowling and Wildman claim that relational properties such as being a member of {Socrates} are not sparse under either conception, and thus conclude that the relational properties Fine cites against MP pose no difficulty for SPM (cf. Cowling Reference Cowling2013, 259; Wildman Reference Wildman2013, 767 and 770–774). So far, so good – but the problem now is that the account overgeneralizes, ruling out cases in which it seems to be essential to a thing how it is related to other particular things (such as its being essential to {Socrates} to have Socrates as a member), and cases in which it seems to be essential to a thing how it is related to some or other things of a particular kind (such as its being essential to Socrates that he originated from some gametes, though none in particular, to borrow a case from Wildman Reference Wildman2013, 773). Rather than ruling out all such essentialist judgments wholesale, Wildman treats them not as ascribing an abundant relational property to a thing, but rather as ascribing a sparse relation between that thing and others (in particular or of a particular kind). Wildman thus proposes the following supplement to SPM, which takes a thing to be essentially thus-and-so related just in case (i) the relation at issue is a sparse one and (ii) the thing in question cannot exist without also being thus-and-so related (cf. 772–773):

1. a. It is essential to x that it bears Ψ to y=_def. (i) necessarily, if x exists, then Ψ holds of the ordered pair <x, y> and (ii) Ψ is a sparse relation.

2. b. It is essential to x that it bears Ψ to something that has Φ=_def. (i) necessarily, if x exists, then there is some y that has Φ and is such that Ψ holds of the ordered pair <x, y> and (ii) Ψ is a sparse relation.Footnote ⁶

By appealing to SRM, one can accommodate instances of essence involving relations that SPM overlooks. Following Wildman, and purely for the sake of argument, assume that the relation has ___ as member described in set theory and the relation originates from described in biology are both sparse relations, at least when sparsity is conceived of scientifically (2013, 776, and 773, respectively).Footnote ⁷ And following Wildman, and again purely for the sake of argument, also assume that {Socrates} cannot exist unless it has Socrates as a member, and assume that Socrates cannot exist unless he originated from some gametes (although none in particular).Footnote ⁸ Then given these assumptions, one can capture the intuitive essentialist judgment that {Socrates} essentially have Socrates as a member (understood as ascribing the relation has ___ as a member rather than the relational property has Socrates as a member) as well as the intuitive essentialist judgment that Socrates originated from some gametes (understood as ascribing the relation originated from to some gametes rather than the relational property originated from some gametes).

Following Wildman, call the conjunction of SPM and SRM sparse modalism (specifying whether it is the fundamentalist or scientific strand when the context warrants). Sparse modalism summarized, I will spend the remainder of this paper arguing that it is no improvement over the simple modal account that it is intended to replace – and moreover, in at least one respect it does worse. Sparse modalism is not an improvement over MP because it is subject to variants of the counterexamples Fine used to refute MP. And sparse modalism does worse than MP because in addition to failing to specify what is sufficient for a property to be had essentially, unlike MP it also fails to specify what is necessary, since it cannot accommodate paradigmatic instances of essence involving abundant properties and relations.

2. Sparse modalism does not specify what is sufficient for essentiality

Here is a representative illustration of the problem. Recall that to capture the basic essentialist judgment that {Socrates} essentially has Socrates as a member, the sparse modalist must claim both that the relation has ___ as a member is a sparse one and that {Socrates} cannot exist unless it has Socrates as a member. But now let Φ_prime be the conjunctive (and presumably abundant) property:

Since {Socrates} cannot exist without having Socrates as a member, and cannot exist unless there are infinitely many prime numbers – trivially, since they necessarily exist – it follows that {Socrates} cannot exist unless it has something as a member that is both Socrates and is such that there are infinitely many prime numbers. So given condition (b) of SRM, it follows that it is essential to {Socrates} that it have something as a member that is Socrates and is such that there are infinitely many prime numbers. The problem, of course, is that this is precisely the kind of counterexample that Fine originally raised for simple modalism, and which sparse modalism was supposed to avoid. Citing facts about how {Socrates} is related to the prime numbers no more helps define {Socrates} or helps specify what it is to be the very thing it is than it does Socrates. SRM, just like MP, entails otherwise. Hence SRM is no advance over MP (at least along the dimension that the sparse modalist intends it to be).

An immediate response would be to claim that has ___ as a member of is not a sparse relation after all. In fact, although Wildman explicitly claims that it is (Reference Wildman2013, 777), his method of handling one of Fine’s key counterexamples to simple modalism within the sparse modalist framework directly entails the non-sparsity of this relation. According to Wildman, SRM is able to accommodate the basic essentialist judgment that it is not essential to Socrates to be distinct from the Eiffel Tower because is distinct from is an internal relation, and Wildman claims that internal relations are not sparse (2013 773–774). Yet similarly, since is a member of is an internal relation in the sense that Wildman has in mind – a non-empty set and its members cannot exist without the latter being members of the former – by Wildman’s own reasoning it would also follow that is a member of is not a sparse relation either. As well, the non-sparsity of has ___ as a member is also suggested by Cowling, who tentatively suggests the more general claim that ‘abstract objects are distinguished from concrete objects precisely because they lack natures’, insofar as ‘they bear no sparse properties’ (2013, fn. 30). One would think that if this were true at all, it would also be true of the relations they bear, in particular set-theoretic relations such as has ____ as a member. Thus, anyone attracted to Cowling tentative thesis about what distinguishes abstract from concrete objects will face pressure to reject that has ___ as a member is a sparse relation.

There are two replies to make to this response. The first is that to deny that non-empty sets essentially have certain individuals as members (either because one keeps SRM but rejects that has ___ as a member is a sparse relation, or because – even more radically – one rejects SRM or that abstract objects have essential natures) seems far too great a departure from the Finean conception of essence under consideration, and that the sparse modalist aims to capture.Footnote ⁹ If the sparse modalist is unable to capture the basic essentialist intuition that it be essential to {Socrates} to have Socrates as a member, then the sparse modalist seems unsuccessful at stating what is necessary for essentiality. (I will return to this issue in Section 3.)

The second reply is that one can replicate the problem with any individual whatsoever, so long as it cannot exist without bearing some sparse relation to something else (or perhaps itself). For instance, take two other basic essentialist judgments concerning something that is clearly not an abstract object: namely, that any particular water molecule essentially have an oxygen atom as a part.Footnote ¹⁰ Let be the conjunctive (and presumably abundant) property:

Supposing that has ___ as a part is a sparse relation, one can reason as before that SRM entails, incorrectly, that it is essential to any particular water molecule that it have as a part something that is an oxygen molecule and is such that there are infinitely many prime numbers. Now of course, once again one might resist by claiming that has ___ as a part is not a sparse relation. Even if we ignore the fact that many maintain otherwise, Footnote ¹¹ the basic problem still remains: one can replicate the same reasoning with any of the many other cases in which it seems essential to a concrete thing that it bear a given relation Ψ to something that has a given property Φ. (I provide further examples in Section 3.) For in any such case, the relevant thing also cannot exist unless it bears Ψ to something that has the conjunctive property is Φ and is such that there are infinitely many prime numbers, and so condition (b) of SRM will incorrectly embed the prime numbers into its essence, just as the simple modal account does.

A more promising method of response would be to somehow qualify SRM – for instance, by restricting the generic condition Φ that occurs within it so that properties such as Φ_PRIME and are excluded. That would block the line of argument presented just above. But the only obvious modifications – I can think of three – are all non-starters. First, one might respond that Φ be restricted to properties that are logical complexes of sparse properties and relations. To see why this will not do, first note that the sparse modalist should at least let is an oxygen atom and is a human be sparse properties (for if not, then the proposed modification would fail to count it as essential to any particular water molecule that it has an oxygen atom as a part, and would fail to count it as essential to Socrates that he be human). Let Φ_HUMAN be the disjunctive property is either an oxygen molecule or a human. Since Φ_HUMAN is a logical complex of sparse properties, the proposed modification would entail that it is essential to any particular water molecule that it has either an oxygen atom or a human as a part. But presumably, just as there is nothing in the essential nature of this water molecule that connects it to the prime numbers (to paraphrase Fine’s turn of phrase: 1994, 5), nothing in its essential nature connects it to humans either. Yet the proposed modification to SRM entails otherwise.

Second, one might respond that not only the relation Ψ but also the generic condition Φ be sparse in order for a thing to be essentially related by Ψ to something that is Φ. Although this proposal excludes such properties as Φ_PRIME, , and Φ_HUMAN, it also excludes a number of paradigmatic cases of essential relatedness. For instance, it would entail that it is not essential to the set {Socrates, Plato} that it have as a member something that is either Socrates or Plato, and not essential to any particular water molecule that it has as a part something that is either an oxygen atom or a hydrogen atom as a part, given that disjunctive properties such as being either Socrates or Plato and being either an oxygen atom or a hydrogen atom are paradigmatically abundant under both the fundamentalist and scientific conceptions of sparsity. (I provide further examples of this sort in Section 3.) Hence, this modification to SRM will not do either.

A final, even sharper departure from SRM goes as follows. Recall that SRM states that if it is essential to x to bear Ψ to some y that is Φ, then Ψ must be a sparse relation. The proposed modification instead requires that the conjunctive relation λxy(Φy & Ψxy) be a sparse relation. An added theoretical benefit of the proposed modification of SRM is that its logical form is more in line with the sparse modalist’s treatment of monadic properties, SPM.Footnote ¹² Yet this modification does no better than the previous one: if being identical to either Socrates or Plato is not a sparse property, for instance, then the result of conjunctively combining it with a relation (even a sparse relation) would presumably fail to be sparse too. Hence just like the previous modification, the one under consideration produces false negatives.Footnote ¹³

3. Sparse modalism does not specify what is necessary for essentiality

Thus far, I have only argued that sparse modalism fails to adequately capture what is sufficient for essentiality. To see why it also fails to adequately capture what is necessary, it suffices to show that there are cases in which an individual has one or more abundant properties essentially, or stands in one or more abundant relationships – and such cases seem commonplace. Take, for instance, human artifacts such as the Eiffel Tower, which essentially exemplifies various abundant properties (e.g., being a tower Footnote ¹⁴) and essentially stand in abundant relations (e.g., the relation was designed and constructed to perform such-and-such function by, which it bears to some engineer or other, or perhaps to Gustave Eiffel in particular).Footnote ¹⁵ Similarly goes for entities such as smiles (the essential nature of which include facts about faces), holes (the essential nature of which include facts about perforated surfaces), tropes (the essential nature of which include facts about the particular things they ‘inhere’ in), and events (the essential nature of which include facts about the objects, properties and times that ‘participate’ in them), among others.Footnote ¹⁶ Clearly, properties and relations such as these are neither fundamental nor are they involved in our total scientific understanding of the world, and therefore sparse under neither the fundamentalist nor scientific conceptions of sparsity, respectively. And yet these are commonly treated as basic essentialist judgments, which the sparse modalist aims to capture, if any are.

A tempting response floated by Cowling is to appeal to some kind of deflationism about human artifacts (2013, fn. 37), which might extend to smiles, holes, tropes, events and so on. Yet even a deflationary conception of such things – at least one that is not outright and implausibly eliminativist rather than deflationist – would take them, even if conceived of in a deflationary way, to have at least some abundant properties essentially, if only category-level properties such as is human artifact, or is a deflationary object. Even a deflationary conception of human artifacts attempts to state what the Eiffel Tower is in terms of such properties and relations, and therefore such properties should still be included in the Eiffel Tower’s essential nature under the Finean conception of essence at issue, their status as abundant notwithstanding.

Even setting this difficulty aside, the response under consideration would force one to take a deflationist attitude towards many other things that have abundant properties essentially and essentially stand in abundant relations. This includes, but is not limited to, such things as propositions (since it is essential to the proposition 〈the Eiffel Tower is a tower〉 that it is true iff the Eiffel Tower is a tower) and properties (since it is essential to the property is a tower that it is an abundant property, and that something has it iff that thing is a tower). These are plainly not incidental features of properties and propositions: they pertain to the very nature of these things that any adequate philosophical account of what they are must accommodate and illuminate. Part of what it is to be the property being a tower is that it is an abundant property, for example, and part of what characteristically differentiates it from other abundant properties is that a thing exemplifies it just in case it is a tower. A thing would not be the very property is a tower is if it were either not a property or not exemplified just when something is a tower. It would be ad hoc to simply reject these cases: essentialist-framed judgments about such cases as these are nearly as strong and prevalent as those the sparse modalist wants to preserve. Sparse modalism would lose much of its motivation – to capture the notion of essence that Fine and others are interested in wholly in terms of facts about modality and sparsity, but without rejecting basic essentialist intuitions en masse – if it were unable to capture them.Footnote ¹⁷

Moreover, problems multiply further when one queries whether facts about identity and distinctness are included in the essences of things.Footnote ¹⁸ Recall from Section 3 that Wildman’s sparse modalist denies that internal relations in general, and the relation of distinctness in particular, are sparse relations: otherwise SRM wrongly entails that it is essential to Socrates to be distinct from {Socrates}. But the identity relation is internal too; thus applying this strategy across the board would preclude Socrates from being essentially identical to himself.Footnote ¹⁹ Even if identity is allowed to be a sparse relation despite also being internal, pressure remains to allow distinctness be one too. As Fine notes, we make basic essentialist judgments not only about the individual things, but also about pluralities of things – including, but not limited to, the judgment that Socrates and {Socrates} are essentially distinct (c.f. 1995, 54). The most natural way to modify SRM is to say that this holds just in case distinctness is a sparse relation, one that Socrates and {Socrates} cannot both exist without exemplifying together. Although this allows the sparse modalist to say that it is essential to Socrates and {Socrates} together to be distinct, the sparse modalist is once again committed to saying that it is essential to Socrates alone that he is distinct from {Socrates}. The sparsity of distinctness allows her to capture the first basic essentialist judgment, but only by forcing her to flout the second.

I provided a number of apparent cases of essence in abundance. But appearances could be deceiving, of course. Can the sparse modalist offer any general reason for believing that sparsity and essentiality go hand in hand, contrary to our intuitive judgments in the cases above? In fact, Wildman offers such an argument (2013, 764). According to Wildman, an essential property P of a individual, o must be ‘metaphysically significant’ to o, in the sense that P ‘plays a role in determining what o is’ (2013, 764). Wildman’s claim is of course undeniable, given that Fine stipulates that by ‘essential properties’ he just means to refer to the properties of an individual that play this individuative role (cf. 1994, 2). Rather, what I take issue with is Wildman’s further claim that if the essential properties of an individual play this role, it follows that P must be a sparse property. According to Wildman, if P is metaphysically significant, then the following must hold:

I have the following two objections to Wildman’s argument.

First, it is only under the narrow, Lewis-inspired fundamentalist conception of sparsity that a given sparse property is required in order to characterize things completely and without redundancy. But it is only if the sparse modalist opts for the broader, Schaffer-inspired scientific conception of sparsity that she is able to capture basic essentialist judgments such as that Socrates essentially be human. Since Wildman’s argument presupposes the narrower conception of sparsity that precludes judgments such as these, it makes matters worse by diverging even further from the Finean conception of essence at issue.

Second, even if an abundant property plays a role in determining what some individual is, it does not follow that one must cite that property in order to characterize things completely and without redundancy. For this abundant property could just play a redundant role in determining what some individual is. Assume, as Wildman does, that the total distribution of sparse properties and relations fully determines how all things are: it leaves no individual uncharacterized. Then it would seem to follow that this total distribution also fully determines what the Eiffel Tower is in particular – namely, a tower. Thus even though the abundant property is a tower must play a role in determining what the Eiffel Tower in order to be essential to it, one need only cite the sparse properties and relations in order to characterize things completely and without redundancy, since these fully determine what it is – namely, a tower. Since Wildman’s argument relies on invalidly inferring otherwise, it provides no good reason to reject what I have maintained in this section: that the essential properties and relations exemplified by things can also be abundant.

4. Conclusion

Sparse modalism does not succeed in improving on the simple modal account of essence refuted by Fine. Just like the simple modal account, sparse modalism fails to specify what is sufficient for essentiality, and for precisely the same reason: both modal accounts bring irrelevant things and kinds into the essences of individuals. However, unlike the modal account, sparse modalism also fails to specify what is necessary for essentiality: individuals can have abundant properties essentially, and can essentially stand in abundant relationships. Lewis noticed many important tasks that the distinction between sparse and abundant properties can be put to in metaphysical theorizing. Resuscitating the simple modal account of essence – at least in the manner that I have considered at length in this article – does not appear to be one of them.Footnote ²⁰