The point of departure for our solution is Aristotle's notion of "accidental sameness''. Elsewhere, we have proposed (for the sake of argument, at any rate) that the phenomenon of material constitution be understood in terms of accidental sameness.8 What we here propose is that the unity of the divine Persons also be understood in terms of this relation (or more accurately, in terms of the genus of which it is a species—namely, numerical sameness without identity9). In this section, therefore, we review the way in which appeal to accidental sameness provides a solution to the problem of material constitution and address what we take to be the most natural objections to it.
According to Aristotle, familiar particulars (trees, cats, human beings, etc.) are hylomorphic compounds—things that exist because and just so long as some matter instantiates a certain kind of form. Forms, for Aristotle, are complex organizational properties, and properties are immanent universals (or, as some have it, tropes). The matter of a thing is not itself an individual thing; rather, it is that which combines with a form to make an individual thing.10 Thus, for example, a human being exists just in case some matter instantiates the complex organizational property humanity. Each human being depends for its continued existence on the continued instantiation of humanity by some matter; and each human being is appropriately viewed as a composite whose parts (at one level of decomposition) are just its matter and (its) humanity.11
On Aristotle's view, living organisms are the paradigmatic examples ofmaterial objects. But Aristotle also acknowledges the existence of other hylomorphic
8 See Rea 1998 and Brower 2004b.
9 For reasons that we shall explain below, the label 'accidental sameness' is not appropriate in the context of the Trinity.
10 This claim is negotiable; and, in fact, there are independent (non-Aristotelian) reasons for thinking that ''masses of matter'' must be treated as individuals. (See, e.g., Zimmerman 1995). But the view of matter articulated here seems to comport best with Aristotle's metaphysics and with the solution to the problem of the Trinity that we will propose, and so we will go ahead and endorse it here. Those who think of masses of matter as individuals may be inclined (in Section 3 below) also to think of what we will call ''the divine essence'' as an individual. Were we to endorse this view, we would deny that the divine essence is a fourth Person or a second God (just as we would deny that Socrates's matter is a second man co-located with Socrates). Rather, we would say that the divine essence is one in number with God, a sui generis individual distinct from the Persons and, indeed, nothing other than a substrate for the Persons. We would also deny that there is any sense in which the divine essence is prior to or independent of God.
11 We place 'its' in parentheses to signal our neutrality on the question whether, say, the humanity of Plato is a special kind of trope or a multiply instantiated universal.
compounds. Thus, books, caskets, beds, thresholds, hands, hearts, and various other non-organisms populate his ontology, and (like an organism) each one exists because and only so long as some matter instantiates a particular complex organizational property.12 Indeed, Aristotle even countenances what Gareth Matthews calls "kooky'' objects—objects like 'seated-Socrates', a thing that comes into existence when Socrates sits down and which passes away when Socrates ceases to be seated.13 Seated-Socrates is an 'accidental unity'—a unified thing that exists only by virtue of the instantiation of an accidental (non-essential) property (like seatedness) by a substance (like Socrates). The substance plays the role of matter in this sort of hylomorphic compound (though, of course, unlike matter properly conceived, the substance is a pre-existing individual thing), and the accidental property plays the role of form. Accidental sameness, according to Aristotle, is just the relation that obtains between an accidental unity and its parent substance.14
One might balk at this point on the grounds that Aristotle's accidental unities are just a bit too kooky for serious ontology. We see that Socrates has seated himself; but why believe that in doing so he has brought into existence a new object—seated-Socrates? Indeed, one might think it's clear that we shouldn't believe this. For there is nothing special about seatedness, and so, if we acknowledge the existence of seated-Socrates, we must also acknowledge the existence of a myriad other kooky objects: pale-Socrates, bald-Socrates, barefoot-Socrates, and so on. But surely there are not millions of objects completely overlapping Socrates.
Fair enough; and nothing here depends on our believing in seated-Socrates or his cohorts. But note that, regardless of what we think of seated-Socrates, we (fans of common sense) believe in many things relevantly like seated-Socrates. That is, we believe in things that are very plausibly characterized as hylomorphic compounds whose matter is a familiar material object and whose form is an accidental property. For example, we believe in fists and hands, bronze statues and lumps of bronze, cats and heaps of cat tissue, and so on. Why we should believe all this but not that sitting down is a way of replacing one kind of object (a standing-man) with another (a seated-man) is an interesting and surprisingly difficult question. But never mind that for now. The important point here is that, whether we go along with Aristotle in believing in what he calls accidental unities, the fact is that many of us will be inclined to believe in things relevantly like accidental unities along with other things that are relevantly like the parent substances of accidental unities.
12 See, e.g., Metaphysics H2, 1042b15-25.
13 Matthews 1982, 1992.
14 Topics A7, 103a23-31; Physics A3, 190a17-21, 190b18-22; Metaphysics D6, 1015b16-22, 1016b32—1017a6; Metaphysics D9, 1024b30-1.
This last point is important because the things we have listed as being relevantly like accidental unities and their parent substances are precisely the sorts of things belief in which gives rise to the problem of material constitution. Hence the relevance of Aristotle's doctrine of accidental sameness. Aristotle agrees with common sense in thinking that there is only one material object that fills the region occupied by Socrates when he is seated. Thus, he says that the relation between accidental unities and their parent substances is a variety of numerical sameness. Socrates and seated-Socrates are, as he would put it, one in number but not one in being.15 They are distinct, but they are to be counted as one material object.16 But once one is committed to believing in such a relation, one has a solution to the problem of material constitution ready to hand. Recall that the problem arises whenever it appears that an object a and an object b share all of the same parts and yet have different modal properties. In such cases we are pushed in the direction of denying that the relevant a and b are identical and yet we also want to avoid saying that they are two material objects occupying the same place at the same time. Belief in the relation of accidental sameness solves this problem because it allows us to deny that the relevant a and b are identical without thereby committing us to the claim that a and b are two material objects. Thus, one can continue to believe that (e.g.) there are bronze statues and lumps of bronze, that every region occupied by a bronze statue is occupied by a lump of bronze, that no bronze statue is identical to a lump of bronze (after all, statues and lumps have different persistence conditions), but also that there are never two material objects occupying precisely the same place at the same time. One can believe all this because one can say that bronze statues and their constitutive lumps stand in the relation of accidental sameness: they are one in number but not one in being.
But should we believe in accidental sameness? The fact of the matter is that this sort of solution to the problem of material constitution is probably the single most neglected solution to that problem in the contemporary literature; and it is not hard to see why. Initially it is hard to swallow the idea that there is a variety of numerical sameness that falls short of identity. But, in our view, the most obvious and serious objections are failures, and the bare fact that the doctrine of accidental sameness is counterintuitive is mitigated by the fact that every solution to the problem of material constitution is counterintuitive (a fact which largely explains the problem's lasting philosophical interest). In the remainder of this section, we will address what we take to be the four most serious objections
15 Topics A7, 103a23-31; Metaphysics D6, 1015b16-22, 1016b32-1017a6.
16 And, we might add, the same would hold true for Socrates and his matter, if indeed the matter of a thing were to be understood as an individual distinct from that thing.
against the doctrine of accidental sameness. We will also explain how the relation of accidental sameness differs from two other relations to which it bears some superficial resemblance. In doing all this, we hope to shed further light on the metaphysics of material objects that attends belief in accidental sameness.
First objection: Most contemporary philosophers think that, for any material objects a and b, a and b are to be counted as one if and only if a and b are identical. Indeed, it is fairly standard to define number in terms of identity, as follows:
(1F) there is exactly one F =df9x(Fx & Vy(F_y = y = x)) (2F) there are exactly two Fs =df9x3y(Fx & F_y & x = y & 8z (Fz = y = z V x = z))
But if that is right, then it is hard to see how there could be a relation that does not obey Leibniz's Law but is nevertheless such that objects standing in that relation are to be counted as one.
Obviously enough, a believer in accidental sameness must reject standard definitions like 1F and 2F. But this does not seem to us to be an especially radical move. As is often pointed out, common sense does not always count by identity.17 If you sell a piano, you won't charge for the piano and for the lump of wood, ivory, and metal that constitutes it. As a fan of common sense, you will probably believe that there are pianos and lumps, and that the persistence conditions of pianos differ from the persistence conditions of lumps. Still, for sales purposes, and so for common sense counting purposes, pianos and their constitutive lumps are counted as one material object. One might say that common sense is wrong to count this way. But why go along with that? Even if we grant that 1F and its relatives are strongly intuitive, we must still reckon with the fact that we have strong intuitions that support the following:
(MC) In the region occupied by a bronze statue, there is a statue and there is a lump of bronze; the lump is not identical with the statue (the statue but not the lump would be destroyed if the lump were melted down and recast in the shape of a disc); but only one material object fills that region.
If we did not have intuitions that support MC, there would be no problem of material constitution. But if MC is true, then 1F and its relatives are false, and there seems to be no compelling reason to prefer the latter over the former.
Of course, if rejecting 1F and its relatives were to leave us without any way of defining number, then our move would be radical, and there would be compelling reason to give up MC. But the fact is, rejecting 1F and its relatives does not leave us in any such situation. Indeed, belief in accidental sameness doesn't even preclude us altogether from counting by identity. At worst, it simply requires us to acknowledge a distinction between sortals that permit counting by identity
and sortals that do not. For example, according to the believer in accidental sameness, we do not count material objects by identity. Rather, we count them by numerical sameness (the more general relation of which both accidental sameness and identity are species). Thus:
(1M) there is exactly one material object = df9x (x is a material object & 8y (y is a material object = y is numerically the same as x))
(2M) there are exactly two material objects = df 9x9y (x is a material object &y is a material object and x is not numerically the same asy and 8z (z is a material object = z is numerically the same as x or z is numerically the same as y))
Perhaps the same is true for other familiar sortals. For example: Suppose a lump of bronze that constitutes a bronze statue is nominally, but not essentially, a statue.18 Then the lump and the statue are distinct, and both are statues. But, intuitively, the region occupied by the lump/statue is occupied by only one statue. Thus, given the initial supposition, we should not count statues by identity either. Nevertheless, we can still grant that there are some sortals that do allow us to count by identity. Likely candidates are technical philosophical sortals like 'hylomorphic compound', or maximally general sortals, like 'thing' or 'being'. For such sortals, number terms can be defined in the style of 1F and its relatives. Admittedly, the business of defining number is a bit more complicated for those who believe in accidental sameness (we must recognize at least two diflErent styles of defining number corresponding to two different kinds of sortal terms). The important point, however, is that it is not impossible.
In saying what we have about the categories of hylomorphic compound, thing, and being, we grant that proponents of our Aristotelian solution to the problem of material constitution are committed to a kind of co-locationism. Although cases of material constitution will never, on the view we are proposing, present us with two material objects in the same place at the same time, they will present us with (at least) two hylomorphic compounds or things in the same place at the same time. But we deny that this commitment is problematic. By our lights, it is a conceptual truth that material objects cannot be co-located; but it is not a conceptual truth that hylomorphic compounds (e.g., a statue and a lump, a fist and a hand, etc.) or things (e.g., a material object and an event) cannot be co-located. We take it as an advantage of the Aristotelian solution that it respects these prima facie truths.
Second objection: To say that hylomorphic compounds, or mere things, can be co-located but material objects cannot smacks of pretense. For while it preserves the letter, it does not preserve the spirit of the intuition that material objects cannot be co-located. If counting two material objects in the same place
!8 An object belongs to a kind in the nominal way just in case it displays the superficial features distinctive of members of that kind.
at the same time "reeks of double counting",19 then the same reek must attend the counting of two hylomorphic compounds or two things in the same place at the same time. At best, therefore, the Aristotelian solution is only verbally distinct from the co-locationists solution. For co-locationists and fans of accidental sameness will still have the same metaphysical story to tell about statues and their constitutive lumps—namely, that they are distinct, despite occupying precisely the same region of spacetime—and that metaphysical story is all that matters.
But this objection is sound only on the assumption that the properties being a material object, being a hylomorphic compound, and being a thing are on a par with one another. From 'x is a hylomorphic compound & y is a hylomorphic compound & x = y , we rightly infer that x and y are two hylomorphic compounds. And if, somehow, we come to believe that x and y are co-located, we'd have no choice but to conclude that x and y are two distinct hylomorphic compounds sharing the same place at the same time. The reason is that the following seems to be a necessary truth about the property of being a hylomorphic compound:
(H1) x is a hylomorphic compound iff x is a matter-form composite; exactly one hylomorphic compound fills a region R iff some matter instantiates exactly one form; and x is (numerically) the same hylomorphic compound as y iff x is a hylomorphic compound and x = y.
According to the second objection, a parallel principle expresses a necessary truth about the property of being a material object:
(M1) x is a material object iff x is a hylomorphic compound; exactly one material object fills a region R iff exactly one hylomorphic compound fills R; and x is (numerically) the same material object as y iff x is a material object and x = y
Note that Ml is not a mere linguistic principle; it is a substantive claim about the necessary and sufficient conditions for having a material object in a region, having exactly one material object in a region, and having (numerically) the same material object in a region. But M1 is a claim that will be denied by proponents of the Aristotelian solution we have been describing here. As should by now be clear, proponents of that solution will reject Ml in favor of something like M2:
(M2) x is a material object iff x is a hylomorphic compound; exactly one material object fills a region R iff at least one hylomorphic compound fills R; and x is (numerically) the same material object as y iff x andy are hylomorphic compounds sharing the same matter in common.
M2 is equivalent to Ml on the assumption that no two hylomorphic compounds can share the same matter in common; but, short of treating the technical
philosophical category hylomorphic compound as co-extensive with the common-sense category material object, it is hard to see what would motivate that assumption. Thus, there is room for disagreement on the question whether M2 is true or whether M2 is equivalent to M1; and, importantly, accidental-sameness theorists and co-locationists will come down on different sides of those questions. Thus, there is a substantive (as opposed to a merely verbal) disagreement to be had here after all.
Two further points should be made before we move on to the third objection. First, though M2 is specifically a thesis about the property being a material object, the doctrine of accidental sameness makes it plausible to think that similar theses about various other properties will be true. In particular, if one thinks that sortals like 'cat', 'house', 'lump', 'statue', and so on can apply nominally to things that constitute cats, houses, lumps, or statues, then something like M2 is true of most familiar composite object kinds. Second, though it may be tempting to think that the relation of accidental sameness (or of numerical sameness without identity) is nothing other than the relation of sharing exactly the same matter, as we see it, this isn't quite correct. On our view (though probably not on Aristotle's), the relation of numerical sameness without identity can hold between immaterial objects, so long as the relevant immaterial objects are plausibly thought of on analogy with hylomorphic compounds. Thus, it is inappropriate to say (as might so far seem natural to say) that the relation of numerical sameness without identity is nothing other than the relation of material constitution. Rather, what is appropriate to say is that material constitution is a species of numerical sameness without identity.
Third objection: The principles for counting that we have just described (i.e., H1 and M2) are apparently inconsistent with the doctrine of accidental sameness. To see why, consider the following argument. Let Athena be a particular bronze statue; let Lump be the lump of bronze that constitutes it. Let R be the region filled by Athena and Lump. Then:
(1) Athena is identical with the material object in R whose matter is arranged statuewise.
(2) Lump is identical with the material object in R whose matter is arranged lumpwise.
(3) The material object whose matter is arranged statuewise is identical with the material object whose matter is arranged lumpwise.
(4) Therefore, Athena is identical with Lump (contrary to the doctrine of accidental sameness).
The crucial premise, of course, is premise 3; and premise 3 seems to follow directly from a proposition that is entailed by the facts of the example in conjunction with our remarks about counting—namely, that there is exactly one object in R whose matter is arranged both statuewise and lumpwise.
On reflection, however, it is easy to see that this objection is a nonstarter. For premise 3 follows only if the doctrine of accidental sameness is false. Numerical sameness, according to Aristotle, does not entail identity. Thus, if his view is correct, it does not follow from the fact that there is exactly one material object in R whose matter is arranged both statuewise and lumpwise that the object whose matter is arranged lumpwise is identical with the object whose matter is arranged statuewise. Simply to assume otherwise, then, is to beg the question. One might insist that the assumption is nevertheless highly intuitive, and therefore legitimate. But, again, the right response here is that every solution to the problem of material constitution is such that its denial is highly intuitive. That is why we have a problem. Successfully rejecting a solution requires showing that the intuitive cost is higher with the objectionable solution than with some other solution; but, with respect to the doctrine of accidental sameness, this has not yet been done.
Fourth objection: We say that there is one (and only one) material object that fills a region just in case the region is filled by matter unified in any object-constituting way. So consider a region R that is filled by matter arranged both lumpwise and statuewise. What is the object in R? What are its essential properties? If there is exactly one object in R, these two questions should have straightforward answers. But they do not (at least not so long as we continue to say that there is a statue and a lump in R). Thus, there is reason to doubt that there could really be exactly one object in R.
This is probably the most serious objection of the lot. But there is a perfectly sensible reply: To the first question, the correct answer is that the object is both a statue and a lump; to the second question there is no correct answer.20 If the doctrine of accidental sameness is true, a statue and its constitutive lump are numerically the same object. This fact seems sufficient to entitle believers in accidental sameness to say that the object in R 'is' both a statue and a lump, so long as they don't take this to imply either that the statue is identical to the lump or that some statue or lump exemplifies contradictory essential properties. But if this view is right, how could there be any correct answer to the question ''What are its essential properties?'' absent further information about whether the word 'it' is supposed to refer to the statue or the lump? The pronoun is ambiguous, as is the noun ('the object in R') to which it refers.21 Thus, we would need to disambiguate before answering the question. Does this imply that there are two material objects in R? It might appear to because we are accustomed to finding
20 We assume that 'object' in the context here means 'material object'.
21 Here is why 'the object in R' is ambiguous. There aren't two material objects in R; and the material object in R isn't a third thing in addition to Athena and Lump. Thus, 'Athena = the material object in R' and 'Lump = the material object in R' must both express truths. But they can't both express truths unless either Lump = Athena (which the doctrine of accidental sameness denies) or 'the material object in R' is ambiguous.
ambiguity only in cases where a noun or pronoun refers to two objects rather than one. But if the doctrine of accidental sameness is true, we should also expect to find such ambiguity in cases of accidental sameness. Thus, to infer from the fact of pronoun ambiguity the conclusion that there must be two objects in R is simply to beg the question against the doctrine of accidental sameness.
So much for objections. Now, in closing this section, we would like to make it clear how accidental sameness differs from two apparently similar relations.
Those who have followed the recent literature on material constitution will know that, like us, Lynne Baker has spoken of a relation that stands "between identity and separate existence'' (2000: 29) and that this relation is (on her view) to be identified with the relation of material constitution. On hearing this characterization, one might naturally think that what Baker has in mind is something very much like accidental sameness. In fact, however, the similarity between accidental sameness and Baker-style constitution ends with the characterization just quoted. Baker's definition of constitution is somewhat complicated; but for present purposes we needn't go into the details. Suffice it to say that, according to Baker, the relation of material constitution is neither symmetric nor transitive whereas accidental sameness is both symmetric and transitive. (At least, it is synchronically transitive.) Lacking the same formal properties, the two relations could not possibly be the same.22
One might also naturally wonder whether what we call 'numerical sameness without identity isn't just good old-fashioned relative identity under a different name. Different views have been advertised in the literature under the label 'relative identity'. But one doctrine that virtually all of these views (and certainly all that deserve the label) share in common is the following:
(R1) States of affairs of the following sort are possible: x is an F, y is an F, x is a G, y is a G, x is the same F as y, but x is not the same G as y. This is a claim that we will endorse too; and, like those who endorse the Relative-Identity solution to the problem of the Trinity, it is a truth we rely on in order to show that T1—T3 are consistent with one another. It is for this reason, and this reason alone, that we say that our solution may fruitfully be thought of as a version of the Relative Identity strategy. Despite our commitment to R1, it would be a mistake to suppose that we endorse a doctrine of relative identity. Our solution to the problem of the Trinity is therefore importantly different from the Relative-Identity solution in its purest form.23
22 Baker's definition appears in both Baker 1999 and Baker 2000. For critical discussion, see Pereboom 2002, Rea 2002, Sider 2002, and Zimmerman 2002.
23 Elsewhere we distinguish between pure and impure versions of the Relative Identity strategy (see Rea 2003). Impure versions endorse R1 without endorsing a doctrine of relative identity; pure versions endorse R1 in conjunction with either R2 or R3 below. Our solution is thus an impure version of the Relative Identity solution.
How is it possible to accept R1 while at the same time rejecting relative identity? The answer, as we see it, is that identity is truly relative only if one of the following claims is true:
(R2) Statements of the form 'x = y are incomplete and therefore ill-formed. A proper identity statement has the form 'x is the same F as y. (R3) Sortal-relative identity statements are more fundamental than absolute identity statements.24
R2 is famously associated with P. T. Geach (1967, 1969, and 1973), whereas R3 is defended by, among others, Nicholas Griffin (1977).25 Views according to which classical identity exists and is no less fundamental than other sameness relations are simply not views according to which identity is relative. Perhaps, on those views, there are multiple sameness relations; and perhaps some of those relations are both sortal-relative and such that R1 is true of them. But so long as classical identity exists and is in no way derivative upon or less fundamental than they are, there seems to be no reason whatsoever to think of other ''sameness'' relations as identity relations. Thus, on views that reject both R2 and R3, there seems to be no reason for thinking that identity is relative.
The difference between accidental sameness and relative identity is important, especially in the present context, because it highlights the fact that there is more than one way to make sense of sameness without identity. It is for this reason that endorsing R1 apart from R2 or R3 won't suffice all by itself to solve the problem of the Trinity. As we have argued elsewhere (Rea 2003), absent an appropriate supplemental story about the metaphysics underlying relative-identity relations, endorsing R1 apart from R2 or R3 leaves one, at best, with an incomplete solution to the problem of the Trinity and, at worst, with an heretical solution.26 We think that the doctrine of accidental sameness provides the right sort of supplemental story, and that the solution it yields (in conjunction with R1) is both complete and orthodox.
We suspect, moreover, that failure to distinguish different ways of making sense of sameness without identity is partly responsible for the attraction that the Relative-Identity solution holds for many. As is well known, respected Christian philosophers and theologians—such as Augustine, Anselm, and Aquinas— habitually speak of the Trinity in ways that require the introduction of a form of sameness that fails Leibniz Law. But this way of speaking, it is often assumed,
24 To say that sortal-relative identity statements are more fundamental than absolute identity statements is, at least in part, to say that absolute identity statements are to be analyzed or defined in terms of more primitive sortal-relative identity statements, rather than the other way around. See Rea 2003 for further discussion of views that endorse R3.
25 See also Routley & Griffin 1979.
26 This is, roughly, the problem that we think Peter van Inwagen's solution to the problem of the Trinity faces. (Cf. Rea 2003.)
can only be explained in terms of relative identity.27 In light of what has just been said, however, we can see that this assumption is false. Sameness without identity does not imply relative identity, and hence any appeal to such sameness either to determine the views of actual historical figures or to provide authoritative support for a (pure) Relative-Identity solution is wholly misguided. Relative identity does provide one way of explaining (numerical) sameness without identity, but it does not provide the only way of explaining it.
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