Mereology

Classical mereology concerns the parthood relation, which is stipulated to satisfy the following three properties.Footnote ²

Although improper parthood offers a viable concept of parthood, DDS cannot plausibly be interpreted as a thesis that God lacks parts in this sense, because the reflexivity of improper parthood implies that God has at least himself as a part. Instead, DDS is more plausibly understood as saying that God lacks proper parts – call this the mereological formulation of DDS.

The mereological formulation seems to be a reasonable initial explication of DDS, since simplicity is after all naturally understood as a mereological concept.Footnote ³ But the mereological formulation is potentially ambiguous, because proper parthood can be defined in several ways. On one possible definition, proper parthood is distinct parthood:

Alternatively, proper parthood can be defined as non-symmetric parthood:

The former definition is more typical, but the latter has been considered in the accounts of mereology given by Leonard and Goodman (Reference Leonard and Goodman1940), Simons (Reference Simons1987), Casati and Varzi (Reference Casati and Varzi1999), Cotnoir (Reference Cotnoir2010), and Cotnoir and Bacon (Reference Cotnoir and Bacon2012), among others. How the mereological formulation is further explicated depends on which definition of proper parthood is adopted. If proper parthood is distinct parthood, the mereological formulation of DDS is equivalent to the common understanding of DDS as a thesis identifying God with his parts – call this the identity formulation of DDS.

But if proper parthood is defined as non-symmetric parthood, then explicating proper parthood in the mereological formulation yields the symmetry formulation of DDS.

Under classical mereology, the difference between the identity and symmetry formulations is merely notational. The antisymmetry of improper parthood entails that to be a distinct part is to be a non-symmetric part, and hence that the identity and symmetry formulations of DDS are logically equivalent. So whenever classical mereology is assumed, the usual discussions of DDS as an identity thesis leave open the possibility that DDS can also be understood according to the symmetry formulation.

However, there are independent reasons to consider mereological theories without antisymmetry, and indeed some such theories have been discussed in the literature.Footnote ⁴ There are also intuitive cases, apart from DDS, that appear to violate antisymmetry. It might be thought that constitution involves mutual parthood, such that when a statue is made from a lump of clay, the statue and clay are parts of each other, but the two nevertheless remain distinct (Cotnoir Reference Cotnoir2010, 396–405; Thomson Reference Thomson1998, 149–173). Similarly, Cotnoir and Bacon (Reference Cotnoir and Bacon2012), Cotnoir (Reference Cotnoir2017), and Molto (Reference Molto2018) argue that trinitarianism can be modelled in such a way that the divine persons are parts of each other, in which case the distinctness of these persons would violate antisymmetry.

If antisymmetry is not assumed, then the two definitions of proper parthood diverge: all identical parts are symmetric parts (by reflexivity), but the converse fails when parthood is not antisymmetric. Accordingly, DDS under the symmetric formulation is strictly weaker than under the identity formulation: if God lacks distinct parts, he also lacks non-symmetric parts, but not vice versa. Hence, if proper parthood is defined as non-symmetric parthood, the mereological formulation of DDS is no longer equivalent to the identity formulation, but equivalent instead to the symmetry formulation.

The upshot is that as far as the formal mereological framework goes, the symmetry formulation presents a viable and distinct way of understanding DDS. The viability of the symmetry formulation is given by the fact that it is equivalent to the mereological formulation if parthood is assumed antisymmetric, or if proper parthood is defined as non-symmetric parthood, either of which would be independently plausible. The distinctness of the symmetry formulation from the identity formulation is given by the fact that the two are non-equivalent whenever parthood is not antisymmetric, which is also a reasonable possibility.

To be sure, adopting the symmetry formulation does not require abandoning the identity formulation altogether, nor is it being suggested that antisymmetry should be altogether dropped from our concept of parthood. Given the range of applications of mereology, it is possible that parthood relations between different kinds of entities may vary. In particular, parthood could obey classical mereology in ordinary instances, with violations of antisymmetry limited to atypical cases like constitution or cases involving God. Indeed, it will be seen in section ‘Motivations’ that some of the usual motivations for DDS can be upheld by taking the mereological relations involving God to differ from those between ordinary objects, particularly with respect to antisymmetry. Insofar as parthood behaves antisymmetrically, the identity formulation of DDS is not ruled out by the symmetry formulation, and is in fact equivalent to it.

Another point of clarification is that the symmetry formulation is not intended to be an interpretation of DDS as traditionally presented. DDS is often discussed as an identity thesis by both traditional and contemporary authors, and by both proponents and opponents of the doctrine. So insofar as the identity and symmetry formulations are not equivalent, the latter would deviate from popular presentations of DDS.

Instantiation

DDS is sometimes taken to be a general claim that God lacks proper parts of any kind, with some authors even denying that God instantiates composition by being part of something else (Aquinas Reference Aquinas2005b).Footnote ⁵ This paper will focus just on the implications of DDS for God’s relations to his attributes. One reason for this focus is that simplicity with respect to attributes would set God apart from other entities more distinctly than simplicity with respect to other kinds of parthood, in line with a common motivation for DDS from God’s transcendence (see section ‘Motivations’). While God may lack spatiotemporal parts, this would not be a unique feature of God if there are other kinds of abstracta, such as mathematical entities, which also lack spatiotemporal parts. In contrast, entities other than God are not usually taken to be simple with respect to their properties. Furthermore, some popular objections against DDS concern God’s relation to his attributes (see sections ‘Properties’ and ‘Modality’), so an analysis of this kind of simplicity may shed light on how DDS can be defended against common objections.

When it comes to God’s relation to his attributes, the symmetry formulation raises the question of how the relation between objects and their properties can be modelled as a form of parthood.Footnote ⁶ Furthermore, if the symmetry formulation is to present a genuine alternative to the identity formulation, it should also be plausible that parthood in this sense can violate antisymmetry. And, it has to be plausible that proper parthood can be defined as non-symmetric parthood, if the symmetry formulation is to be viable here. This section will address these issues.

Some authors argue that the instantiation relation can be modelled as parthood.Footnote ⁷ On the mereological bundle theory suggested by Laurie Paul (Reference Paul2002, Reference Paul2006, Reference Paul2012, Reference Paul and Keller2017), for instance, objects are mereological fusions of the properties they bear. Similarly, on a mereological nominalist view of properties, properties are mereological fusions of their instances (Effingham Reference Effingham2020, 160–185; Zemach Reference Zemach1982, 527–537). To lend legitimacy to these suggestions, it might be considered how instantiation resembles more familiar instances of parthood. In ordinary cases, parthood is closely related to connection, in that whenever one thing is a part of another, whatever is connected to the first thing becomes connected also to the second. Since my left hand is part of my body, whatever comes into contact with my left hand thereby comes into contact with my body. Indeed, in some extensions of mereology involving both parthood and connection, parthood is defined by this relation to connection (Clarke Reference Clarke1981, 204–218, Reference Clarke1985, 61–75; Randell et al Reference Randell, Cui and Cohn1992, 165–176).

The notion of connection admits an analogue between properties, namely, as resemblance. It has been suggested that similarity between properties can be quantified in such a way that degrees of dissimilarity form a distance-like metric (Bigelow Reference Bigelow1976, 299–320, Reference Bigelow1977, 459–472; Blumson Reference Blumson2018, 211–239). So just as ordinary objects are connected in the sense of spatial contact if they are close enough with respect to spatial distance, properties (and collections thereof) can be said to be connected in the sense of resemblance if they are close enough with respect to similarity. A reasonable notion of parthood can then be constructed in terms of connection in this sense: one property (collection) is part of another just in case whatever becomes more similar to the first also thereby becomes more similar to the second.Footnote ⁸

When objects are seen as collections of the properties they bear, instantiation becomes a particular case of this kind of parthood. Whenever an object $o$ has a property $p$, whatever bears a property more similar to $p$ also thereby becomes more similar to $o$; hence $p$ is part of $o$ with respect to similarity. Whenever something has a property more similar to that of being a hand, that thing thereby becomes more similar to my left hand; the property of being a hand is thus part of my left hand. It is also possible, at least in principle, for objects to be part of properties with respect to similarity: an object $o$ is part of a property $p$ just in case nothing can become more similar to $o$ without bearing a property that is more similar to $p$.Footnote ⁹ My left hand is not part of the property of being a hand, since my left foot is more similar to my left hand than my right foot is (since the former two but not the latter two share the property of being on the left side of my body), but my feet do not differ in similarity with respect to the property of being a hand. In this way, relations between objects and properties, in both directions, can be legitimately modelled in terms of parthood.

The possibility of parthood relations in both directions opens the further possibility of mutual parthood between distinct property bundles. If God is related to his omniscience in such a way that whatever becomes more similar to God also thereby has a property that is more similar to God’s omniscience, and conversely, then God and his omniscience would be parts of each other with respect to similarity. Of course, one way for this to be the case is for God to be identical to his omniscience. But it is also possible, at least in principle, for this symmetric parthood relation to hold while God remains numerically distinct from his omniscience. The possibility of such cases would imply that parthood with respect to similarity is not generally antisymmetric.

Parthood in this sense is also compatible with a definition of proper parthood as non-symmetric parthood. On Paul’s mereological bundle theory, the proper parthood relation is required to satisfy irreflexivity, asymmetry, transitivity, and supplementation (Paul Reference Paul2002, 578–596, Reference Paul2006, 623–659, Reference Paul and Keller2017). If proper parthood is defined as non-symmetric parthood with respect to similarity, it would immediately be irreflexive, asymmetric, and transitive. Proper parthood also satisfies supplementation: if a property (bundle) ${p_1}$ is not part of another ${p_2}$ with respect to similarity, there is a property $p$ in respect of which something can be more similar to ${p_1}$ without becoming more similar to ${p_2}$ – $p$ would then be disjoint from ${p_2}$. Since my left hand is not part of the property of being a hand, it has properties that are disjoint from the property of being a hand, such as the property of being on the left side of my body.

The upshot is that the relations between objects and their properties can be modelled mereologically in such a way that DDS on the symmetry formulation is viable as a thesis about the relation between God and his attributes.

Motivations

Although DDS under the symmetry formulation is weaker than under the identity formulation, this section will argue that the symmetry formulation retains four commonly held motivations for DDS, namely those coming from traditionally held doctrines regarding God’s aseity, transcendence, immutability, and unity. The doctrine of aseity says that God metaphysically depends on nothing other than himself, the doctrine of transcendence says that God is unlike everything else, the doctrine of immutability says that God’s intrinsic nature cannot change, and the doctrine of unity says that there are no divisions in God’s nature. DDS has traditionally been motivated by these four doctrines. For example:

Among contemporary authors, Bennett (Reference Bennett1969) considers transcendence and immutability, Dolezal (Reference Dolezal2017) considers all four doctrines as motivations, Leftow (Reference Leftow1990) considers aseity and immutability, Rogers and Rogers (Reference Rogers and Rogers1996), Brower (Reference Brower, Flint and Rea2008), and Fowler (Reference Fowler and Jonathan2013) consider aseity, and Platter (Reference Platter2020) and Platter (Reference Platter, Priest and Zolghadr2021) consider transcendence.Footnote ¹⁰

Consider first God’s aseity. Metaphysical dependence relations are typically thought to track parthood relations, such that if one thing is part of another, the second thing depends on the first (Casati and Varzi Reference Casati and Varzi1999; Fine Reference Fine, Smith and Smith1995, 463–485).Footnote ¹¹ Aseity hence appears to require that God have no parts other than himself. However, it is implausible that improper parthood entails dependence, because dependence is usually thought to be irreflexive (Correia Reference Correia2010, 251–279; Fine Reference Fine, Correia and Schnieder2012, 37–80; Raven Reference Raven2013, 191–199), while improper parthood is reflexive. Even if there are reflexive instances of dependence (Jenkins Reference Jenkins2011, 267–276), it would seem too strong to hold that everything depends on itself. Instead, the parthood-dependence relation is more plausibly understood as an entailment from proper parthood (in some sense) to dependence. Moreover, instances in which distinct things are parts of each other would be at odds with the usual understanding of dependence as asymmetric,Footnote ¹² so it would square better with our usual understanding for dependence to follow from non-symmetric parthood, rather than distinct parthood more generally. And if dependence tracks non-symmetric parthood, then the doctrine of aseity requires only that God lack non-symmetric parts, as the symmetry formulation posits.

DDS is also thought to uphold God’s transcendence by making God unique in the way he relates to his attributes. The symmetry formulation maintains this motivation, because ordinary objects are not parts of their properties. Namely, objects with plural properties are usually not related to a particular property in such a way that nothing can be more similar to the object without bearing a property more similar to that particular one. Instead, ordinary cases are more like that of my left hand, which bears a plurality of properties in such a way that something can be more similar to my left hand without bearing a property more similar to that of being a hand. Moreover, objects do not usually have properties that are all parts of each other. My left hand has at least two properties: that of being on the left side of my body and that of being a hand. But something can be more similar to the former without being more similar to the latter, so the former is not part of the latter. Hence, the symmetry formulation makes God unique by positing that God is part of all his attributes, which are also all parts of each other.

Third, under the symmetry formulation of DDS, God’s immutability can be maintained by taking some parthood relations between God and his attributes to be necessary. This view of parthood may be independently motivated: if instantiation is modelled as parthood, then for any object at all to have necessary properties requires that some parthood relations be necessary. And if God is taken to bear mutual and necessary parthood relations to the properties that constitute his intrinsic nature, God would have these properties necessarily and hence unchangingly, which yields immutability.

Under the symmetric formulation, God’s attributes may not be numerically identical to each other, but the posited symmetric parthood relations between them nevertheless share some important features of identity. For one, symmetric parthood is an equivalence relation. Also, the transitivity of parthood implies that whenever two things are parts of each other, they have all the same parts and are parts of all the same things. That is, just as numerically identical objects are indiscernible with respect to their properties, objects that are parts of each other are indiscernible with respect to mereology. In particular, when objects are parts of each other with respect to similarity, they are also indiscernible in that whatever becomes more similar to one object also thereby becomes more similar to the other. Hence, under the symmetry formulation of DDS, God’s nature can be said to admit no discernible divisions, even if his attributes are not numerically identical to each other.

To be sure, DDS may have other motivations, such as possible considerations involving God’s sovereignty, timelessness, or goodness (Feser Reference Feser2010; Stump Reference Stump2023, 780–795). But the four motivations considered in this section are perhaps the most common ones, and insofar as they are representative, the discussion above shows that the symmetric formulation retains the motivations of DDS.

Properties

Some popular objections have been raised against DDS on the grounds that we intuitively expect distinctions between God and his attributes. First, if God is his omniscience, then God is a property (or property instance). Besides being unintuitive, this might imply that God’s omniscience is self-exemplifying (Miller Reference Miller1994, 474–477), and make it difficult to explain how God can interact with the world if properties (or property instances) are causally inert (Hasker Reference Hasker2016, 699–725; Plantinga Reference Plantinga1980; Swinburne Reference Swinburne1994). Second, if God is simultaneously his omniscience and his omnipotence, the transitivity of identity seems to imply that God’s omniscience is his omnipotence; but the two seem intuitively to be distinct (Morreall Reference Morreall1978; Moreland and Craig Reference Moreland and Craig2003; Baddorf Reference Baddorf2016, 67–70; Plantinga Reference Plantinga1980). Third, the identification of God with his properties raises the question of how things can share in some but not all of God’s attributes. Since God is good, someone resembles God insofar as they are good. But if resemblance is understood as property sharing, then DDS seems to imply that someone who resembles God by being good also possesses all of God’s attributes, and perhaps even God himself (Morris Reference Morris1985; Hughes Reference Hughes1989, 299–318).

In response, some defenders of DDS argue that God can be a property (instance) without the purported unintuitive consequences following (Mann Reference Mann1982, 451–471, Reference Mann1986, 343–353; Schärtl Reference Schärtl2018, 23–60; Vallicella Reference Vallicella1992, 508–525, Reference Vallicella1994, 478–481; Wolterstorff Reference Wolterstorff and Clark1992, 133–149), while some deny that God shares his attributes at all (Miller Reference Miller1996; Vallicella Reference Vallicella1999, 614–616). Others give an account of property attribution on which DDS, even under the identity formulation, does not identify God with his properties (Bergmann and Brower Reference Bergmann, Brower and Zimmerman2005, 357–386; Brower Reference Brower, Flint and Rea2008, 105–128; Fowler Reference Fowler and Jonathan2013, 114–138; Oppy Reference Oppy2003, 10–22; Pruss Reference Pruss and Kvanvig2008, 150–167). Yet other authors propose alternative formulations of DDS (Burns Reference Burns1989, 271–293; Immink Reference Immink1987; Frame Reference Frame2002; Leftow Reference Leftow2006, 365–380; Swinburne Reference Swinburne1994).

This section will argue that the symmetry formulation can address all three difficulties. It is not the goal here to argue that this line of defence is preferable to extant alternatives, but it may be noted as a strength of the symmetry formulation that it remains equivalent to the mereological formulation under a range of common mereological frameworks, as noted in section ‘Mereology’. In contrast, some alternative accounts of DDS explicate simplicity in non-mereological terms. Gerrit Immink (Reference Immink1987) and Robert Burns (Reference Burns1989), for instance, formulate DDS as a thesis positing harmony among God’s attributes. Since this formulation uses an alternative understanding of ‘simplicity’ that deviates from its usual mereological definition, it sheds little light on the plausibility of DDS when interpreted at face value as a thesis about God’s mereology. That is, whether God’s attributes exist in harmony says little about whether God has proper parts. On the other hand, the usual equivalence between the symmetry and mereological formulations suggests that the defensibility of one implies also the defensibility of the other.

The first two difficulties are immediately avoided under the symmetry formulation. DDS under the symmetry formulation allows for God and his attributes to be numerically distinct from each other, and hence does not entail that God is a property (instance), or that God’s attributes are all identical. To be sure, DDS requires that God and his attributes all be parts of each other, but this is hardly an unacceptable implication. It does not seem implausible that something becomes more similar to God insofar as it bears a property more similar to God’s omniscience, or that something becomes more similar to God’s omniscience insofar as it is more similar to God’s omnipotence.

The difficulty regarding property sharing is less easily resolved. Since God and his attributes are parts of God’s goodness under the symmetry formulation, the transitivity of parthood implies that someone who resembles God by having his goodness as a part also has God and his other attributes as parts. Even if this does not mean that someone instantiates all of God’s attributes (since not all instances of parthood represent instantiation – see n. 9), the implication is itself unintuitive. Nevertheless, this implication can be avoided by adopting an alternative, weaker, account of resemblance, on which two things resemble each other just in case they each have one in a pair of sufficiently similar properties. That is, in terms of the interpretation suggested in section ‘Instantiation’, resemblance can be taken to consist in connection with respect to similarity, rather than overlap. Then, for someone to resemble God in terms of goodness requires not that they have God’s goodness itself, but only that they have a sufficiently similar property like (say) the property of imperfect goodness. And since God and his attributes are not part of this latter property, even on the symmetry formulation of DDS, the implication no longer follows that someone who resembles God in goodness has God and his attributes as parts.

The suggested account of resemblance also addresses the property sharing difficulty under the identity formulation, albeit with stronger commitments regarding degrees of resemblance. If God is his goodness, then something resembles God to exactly the same degree as it resembles God’s goodness. The symmetry formulation is committed only to a weaker (and perhaps more plausible) claim: something resembles God more as it bears a property that is more similar to God’s goodness, and vice versa.

Modality

Another group of objections to DDS falls under the broad heading of modal collapse arguments, according to which DDS runs into difficulties with various kinds of contingency. On Tomaszewski’s (Reference Tomaszewski2019, 275–284) formulation, these arguments proceed as follows.Footnote ¹³ If God’s existence is necessary, as is usually held, DDS seems to entail that every aspect of God, being identical to him, is also necessary. But if all of God’s acts are necessary, then God cannot act differently from how he actually does, which implies that God lacks free will. Moreover, God’s omniscience entails that for any true proposition $p$, God knows that $p$. If God’s knowledge that $p$ is necessary, it follows from the factivity of knowledge that is necessary, and hence that there are no contingently true propositions. Objections along these lines have been put forth by authors such as Hughes (Reference Hughes1989), Wolterstorff (Reference Wolterstorff and Clark1992), Moreland and Craig (Reference Moreland and Craig2003), Hasker (Reference Hasker2016), Leftow (Reference Leftow2016), Mullins (Reference Mullins2016), Mullins (Reference Mullins2022), and Waldrop (Reference Waldrop2022).

In response, some defenders of DDS propose alternative accounts on which God is related to his attributes not by identity, but by a contingent relation (Oppy Reference Oppy2003, 10–22; Pruss Reference Pruss and Kvanvig2008, 150–167), or a relation that allows for contingency in God’s attributes even if he is himself necessary (O’connor Reference O’connor1999; Huneeus Reference Huneeus2024, 407–408; Pruss Reference Pruss and Kvanvig2008, 150–167). Other authors maintain that God is identical to his attributes, but argue that their necessity does not follow from God’s (Grant Reference Grant2012, 254–274; Mann Reference Mann and Mann2015, 44–56; Nemes Reference Nemes, Da Silveira and Tat2020, 101–119; Tomaszewski Reference Tomaszewski2019, 275–284) or that their necessity is nevertheless compatible with contingency in our world (Rogers Reference Rogers and Szatkowski2024, 375–390; Schmid Reference Schmid2022, 3–22).

The symmetry formulation offers a possible way in which God’s relation to his attributes can be contingent under DDS, namely, by taking God to be related to some of his attributes by contingent (mutual) parthood relations. The account of God’s immutability suggested in section ‘Motivations’ stipulates that some parthood relations between God and his attributes are necessary, but it does not require that this be the case for all of God’s attributes, only those that constitute his intrinsic nature. Immutability thus allows that God may be involved in both necessary and contingent parthood relations. Indeed, if instantiation is modelled as parthood, ordinary objects with both necessary and contingent properties will be involved in both necessary and contingent parthood relations. The combination of different modalities in God’s mereology is thus not unusual.

The suggested account of immutability may even be compatible with contingency in God’s intrinsic properties if finer distinctions are drawn between the kinds of modality involved. The doctrine of immutability requires only that God’s intrinsic properties be unchanging, that is, that these properties be temporally necessary. But the doctrine of immutability, taken strictly at face value, is compatible with these properties being metaphysically contingent. To be sure, such a distinction would call for an account of how something can have metaphysically contingent intrinsic properties, but if this potential issue can be resolved, the idea that God could bear temporally necessary and metaphysically contingent parthood relations to some of his properties would not itself be unusual. Ordinary objects may have metaphysically contingent properties that, in actuality, remain unchanged while they exist – most buildings in our world do not change their location for as long as they stand, but their locations are not metaphysically necessary.

This account leaves the defender of DDS with two ways of explaining contingency in God’s will and knowledge. In actuality, God is willing to create the world, so God and his willingness to create are parts of each other. If God’s willingness to create is not taken to be an aspect of his intrinsic nature, these parthood relations may simply be taken to be contingent, without the need for distinctions between the kinds of modality involved. Then, there is a possible world in which God exists (by necessity), but does not have a willingness to create and hence does not, which implies that God’s actual creative act is free. Alternatively, if God’s willingness to create is taken to be an aspect of his intrinsic (and hence unchanging) nature, such a possible world can still be taken to exist if God’s willingness to create is seen as one of his temporally necessary but metaphysically contingent attributes.Footnote ¹⁴ God’s actual creative act hence remains free in this case.

Similarly, in actuality, God knows as an implication of omniscience that snow is white, such that God, his omniscience, and his knowledge that snow is white are all parts of each other. The parthood relations between God and his omniscience can be taken to be necessary (since omniscience is in God’s intrinsic nature), while those involving God’s knowledge that snow is white are taken to be (metaphysically) contingent. Then, there could be a possible world in which God does not have the knowledge that snow is white, but God remains omniscient because snow is not white. That snow is white is thus a contingent truth in actuality.