2. Some Coincidence Cases

3. Four Responses to the Coincidence Cases

I now set out four responses to the coincidence cases, although only the first two responses will be discussed in detail in this paper.

(II) ‘Perdurance’ Theory (Four-Dimensional Worms)

Temporary Coincidence is not Identity, but Permanent Coincidence is Identity. (‘Intermediate Monism’)

The second response is the one typically given by a perdurance theorist about persistence. According to perdurance theory, statues and pieces of clay are four-dimensional space-time worms, which persist by having different temporal parts (or ‘stages’) at different times, and are never wholly present at any one time during their existence. It can happen (often does happen) that a four-dimensional worm that is a statue (i.e., whose temporal parts are united by the ‘statue’ unity relation) shares some, but not all, of its temporal parts with a four-dimensional worm that is a piece of clay (i.e., whose temporal parts are united by the ‘piece-of-clay’ unity relation). In such a case, the 4-D statue-worm and the 4-D piece-of-clay-worm overlap, but are not identical, and we have a case of ‘temporary coincidence’. (See Figure 1.) It can also happen that a 4-D worm that is a statue shares all its temporal parts with a 4-D worm that is a piece of clay. In such a case (‘permanent coincidence’), the 4-D worm that is a statue completely coincides with, and is identical with, the 4-D worm that is a piece of clay. (Again, see Figure 1.)Footnote ⁵

Figure 1.

Thus the perdurance theorist holds that temporary coincidence is not identity, although permanent coincidence is identity. For what I hope are obvious reasons, I shall refer to this position as ‘intermediate monism’.Footnote ⁶

4. Permanent Coincidence, Identity, Leibniz's Law, and Modal Predicates

It is a familiar point that, if permanent coincidence is identity, then we must explain, consistently with Leibniz's Law, the fact that the statue and the piece of clay in Case 3 appear to differ in their modal properties: for example, it seems that the piece of clay could have survived radical reshaping, whereas the statue could not have survived radical reshaping. The only obvious way to achieve this is to treat a modal predicate like ‘could have survived radical reshaping’ as what Harold Noonan has called an ‘Abelardian predicate’ (Noonan Reference Noonan1991, Reference Noonan1993), which stands for a different property when attached to the expression ‘the statue’ from the property that it stands for when attached to the expression ‘the piece of clay’. Perhaps the best known version of such an account is the ‘inconstant’ version of counterpart theory advocated by David Lewis in his (Reference Lewis1971) and (Reference Lewis1986), but there are others (see Noonan Reference Noonan1991, Reference Noonan1993; Gibbard Reference Gibbard1975).

This way of dealing with the Leibniz's Law objection has come under attack by Kit Fine (Reference Fine2003). However, in this paper I shall assume, for the sake of argument, that the ‘Abelardian’ strategy can succeed in disarming the ‘Leibniz's Law’ objection (to the view that permanent coincidence is identity) that is based on an apparent difference between the de re modal properties of the statue and the piece of clay in a permanent coincidence case.

5. Permanent Coincidence, Identity, de dicto Persistence Conditions

If permanent coincidence is identity, then, in Case 3, there is a single entity that is both a piece of clay and a statue. It might seem that this provides the defenders of pluralism with a very simple argument against their opponents, as follows:

Unfortunately, however, the view that permanent coincidence is identity is not so easily refuted. Although the persistence conditions associated with being a statue and those associated with being a piece of clay are, indeed, different, it is not obvious that they are incompatible. Moreover, the defence of the claim that they are compatible does not depend on the adoption of the ‘Abelardian’ account of de re modal predication described in §4.

Consider the following de dicto principles, which we might take to express implications of the persistence conditions associated with being a statue and being a piece of clay.

(S)

1. (i) Necessarily (for all x, if x is a statue, then x does not survive a radical reshaping (even if the reshaping preserves all its matter in one coherent mass)).

2. (ii) Necessarily (for all x, if x is a statue at any time in its existence, then x is a statue at every time in its existence).

(P)

1. (i) Necessarily (for all x, if x is a piece of clay, then x does survive a radical reshaping in which all its matter is preserved in one coherent mass).

2. (ii) Necessarily (for all x, if x is a piece of clay at any time in its existence, then x is a piece of clay at every time in its existence).

It does not follow, from (S) and (P) alone, that nothing can be both a statue and a piece of clay – as the piece of clay in Case 3 is alleged, by the proponents of intermediate monism, to be. What does follow, from (S) and (P), is that if something is both a piece of clay and a statue, then it is not subjected to a radical reshaping in which all its matter is preserved in one coherent mass. But this is consistent with its being the case that although the statue and the piece of clay in cases of temporary coincidence such as Case 2 are distinct, the statue and the piece of clay in a case of permanent coincidence are identical. In other words, the de dicto persistence conditions represented by (S) and (P) can be jointly satisfied by a single entity, as long as that entity is not subjected to a radical transformation of its shape in which all its matter is retained in one coherent mass.Footnote ⁹

To sum up, then, the intermediate monist can consistently accept that the sortals statue and piece of clay are associated with distinct de dicto persistence conditions such as (S) and (P).Footnote ¹⁰ Indeed, the intermediate monist may even appeal to these very persistence conditions in order to argue that in a temporary coincidence case such as Case 2, the statue and the piece of clay are distinct entities. In addition, since (S) and (P) are de dicto principles, the intermediate monist who accepts them is not in danger of thereby being committed to any difference in the de re modal properties of a statue and a piece of clay. For (S) and (P) are completely silent on the question whether an entity that satisfies the persistence conditions for being a statue, or for being a piece of clay, in one possible world also satisfies those conditions in other possible worlds.Footnote ¹¹

6. Temporary Coincidence and Permanent Coincidence

I shall assume, then, at least for the sake of argument, that the perdurance theorist's intermediate monism is not refuted either by an appeal to Leibniz's Law applied to modal properties (§4), or by a simple appeal to the potentially conflicting persistence conditions associated with the concepts of being a statue and being a piece of clay (§5).

It seems to me, though, that, even when these concessions are granted, intermediate monism is deeply counterintuitive. For it is, I think, deeply counterintuitive to claim that there is distinctness in a same-origin temporary coincidence case (such as Case 2) and yet that there is identity in a permanent coincidence case (such as Case 3). In this section, I shall try to explain and justify this claim.

Here is a simple-minded argument against intermediate monism:Footnote ¹²

1. (1) If same-origin temporary coincidence is not identity, then permanent coincidence is not identity.

2. (2) Same-origin temporary coincidence is not identity.

Therefore:

1. (3) Permanent coincidence is not identity.

Since the intermediate monist accepts Premise (2) of this (obviously valid) simple-minded argument, but rejects its conclusion, the intermediate monist must reject Premise (1). But Premise (1) of the simple-minded argument is, it seems to me, extremely plausible.

7. What's Supposed to be so Bad about Permanent Coincidence Without Identity (PCWI)?

I have tried to establish that Premise (1) of my ‘simple-minded’ argument against the perdurance theorist's intermediate monism is very plausible, and that its plausibility does not rest simply on assumptions that are straightforwardly question-begging against a perdurantist.

However, the endurantist pluralist view obviously faces a challenge of its own. For several writers have suggested that there is something deeply repugnant about the idea of permanent coincidence without identity, and regard it as a conclusion that it is worth some considerable trouble to avoid. In particular, it remains to be seen whether the perdurantist's intermediate monism (with the counterintuitive consequences I have claimed, in §6, that it has) is, on balance, in conflict with intuition to a greater extent than is the endurantist pluralist view.

But what exactly is supposed to be so bad about permanent coincidence without identity (PCWI)? One argument against PCWI is that it is inconsistent with (or at least does not sit comfortably with) a four-dimensionalist metaphysics. But this argument, by itself, is obviously forceful only in combination with other arguments for four-dimensionalism.Footnote ¹³ I shall not attempt to discuss these arguments here.Footnote ¹⁴ Instead, in the sections that follow, I shall consider three further arguments against PCWI. I shall conclude that these arguments are uncompelling – or, at least, that they are uncompelling when employed in an attempt to defend the position of intermediate monism.

8. Common-sense Intuitions About Temporary Coincidence

Many writers have suggested that there is something at least prima facie counterintuitive, and repugnant to common sense, about the idea that the statue and the piece of clay are two even in a temporary coincidence case. Now, of course, anyone who believes, as the standard perdurance theorist does, that temporary coincidence is not identity must deny that the facts of the matter in temporary coincidence cases are consistent with these alleged common-sense intuitions. However, it has been suggested that although perdurance theorists (with their intermediate monism) cannot accept that these ‘monistic’ common-sense intuitions speak the truth about temporary coincidence cases, the perdurance theorist can nevertheless do a better job of accommodating these monistic intuitions than the (pluralist) endurance theorist can.

Here are some examples of alleged common-sense intuitions that (allegedly) support the thesis that even temporarily coincident objects are identical.

Competition for Space

We think of physical objects as things that have to compete for space. So it is, at least prima facie, odd to say that they can coincide. (Cf. Sider Reference Sider2001: 154ff.)

It is sometimes suggested that the perdurantist's model helps to accommodate these monistic common-sense intuitions because it makes temporarily coincident objects ‘no more mysterious than (spatially) overlapping roads’ (e.g., Sider Reference Sider2001: 152Footnote ¹⁵). And it is also sometimes suggested that the perdurantist's ‘overlapping roads’ model mitigates the (alleged) counterintuitiveness of the claim that temporarily coincident objects are not identical because of the following fact: although, according to the perdurantist, the temporarily coincident objects are distinct four-dimensional worms, the perdurantist can provide a single object – namely, the shared temporal segment – that both wholly occupies and is wholly confined to the region of overlap in any case of temporary coincidence (e.g., Sider Reference Sider2001: 156).

I am unconvinced by these claims. I reject the claim that the perdurantist's overlapping worms are no more mysterious than spatially overlapping roads because of the problem concerning the future discussed in §6 above (to which nothing in the spatial analogy corresponds). In addition, I am sceptical about the relevance of the fact that the perdurantist can supply a single object that is wholly contained within the region of overlap in a temporary coincidence case. For example, in any temporary coincidence case concerning a statue and a piece of clay that overlap only for the middle portion of their histories (as illustrated by Figure 2), this ‘single object’ will have to be a four-dimensional worm that is neither a statue nor a piece of clay, and has ‘persistence conditions’ that do not match those of any of the persisting things that common sense recognizes. In the case of a statue and a piece of clay that overlap only for the middle portion of their histories, the ‘single object’ that is the shared temporal segment must ‘come into existence’ at the start of the time of the coincidence and ‘go out of existence’ at the end of the time of the coincidence. But common sense recognizes no persisting objects that come into existence and go out of existence in this fashion. How, then, can an appeal to such an object legitimately be invoked to accommodate an alleged common-sense intuition that there is just one object present in a case of temporary coincidence?Footnote ¹⁶

Figure 2.

9. Sortal Properties, Modal Properties, and Intrinsic Similarity Principles

It is at least questionable (I have just claimed) whether perdurance theorists, with their intermediate monism, can successfully exploit the alleged common-sense intuitions that (even temporary) coincidence is identity in order to get the upper hand over the endurantist pluralist. But there are further arguments which seem to be arguments directly against the pluralist's view that permanent coincidence is not identity.

The pluralist holds that the statue and the piece of clay are distinct entities, even in a permanent coincidence case such as Case 3. Clearly, this invites the question: what is it, in Case 3, that makes them two, given that they come into existence simultaneously, share all their matter throughout their existence, and go out of existence simultaneously?

One response that the pluralist might give is that what makes the piece of clay and the statue distinct in the permanent coincidence case (Case 3) is that they have different modal properties. For example, the piece of clay could have survived radical reshaping, although the statue could not have survived radical reshaping. However, to appeal to such a modal difference as what grounds the difference between the entities clearly flies in the face of the following principle, which Harold Noonan (Reference Noonan1993: 133) has labelled ‘(CII)’ (an acronym for ‘Constitution is Identity’):

Now, the pluralist could, of course, take issue with principle (CII), and simply reject it.Footnote ¹⁷ But, as far as I can see, the pluralist need not do so. For instead of claiming that what makes the piece of clay and the statue two in a permanent coincidence case (such as my Case 3) is fundamentally a modal difference between them, why can't the pluralist say that what makes them two is, rather, a fundamental sortal difference between them: the statue is a statue, and not a piece of clay, and the piece of clay is a piece of clay, and not a statue?

But if the distinctness is, fundamentally, a distinctness that holds in virtue of a difference in sortal properties, then this way of distinguishing between the statue and the piece of clay is not in conflict with Noonan's principle (CII) – unless, of course, sortal properties are themselves ‘modal, dispositional or counterfactual properties’. But why should we suppose that they are? No doubt, the sortal properties have modal, dispositional, or counterfactual implications. But obviously it cannot be maintained that every property that has such implications is itself a modal, dispositional, or counterfactual property.

So far, then, I have suggested that, despite what appears to be sometimes assumed,Footnote ¹⁸ the principle (CII) can be accepted by the defender of the (pluralist) view that permanent coincidence is not identity. If so, then the principle does not capture what is at issue in the debate between pluralists and their opponents (and the label ‘Constitution is Identity’ for the principle must be a misnomer).Footnote ¹⁹

Let us try a different tack. In his (1992), Mark Johnston admits that the following principle is plausible (at least if we ignore the fact that being a statue may require having the right kind of origin):Footnote ²⁰

Johnston argues, in effect, that if (8) is accepted, it provides a reductio ad absurdum of the claim that Statue and Piece are not identical in a permanent coincidence case such as my Case 3. For, he suggests, in Case 3 Piece is intrinsically exactly like Statue, and Statue is a paradigm statue; so, according to (8), Piece is also a statue. But if Piece is a statue in Case 3, this has intolerable consequences for the view that Piece and Statue are numerically distinct. For if Piece is a statue, and Statue is a statue, and Piece and Statue are two, then the case involves two statues, which is emphatically not the verdict that the pluralist wanted.Footnote ²²

Johnston argues, however, that reflections concerning Peter Unger's ‘Problem of the Many’ show that (8), in spite of its initial plausibility, is false. He claims that it should be replaced by a revised principle ((8^′)Footnote ²³) that does not imply that if Piece is intrinsically exactly like Statue, then Piece is a statue. Harold Noonan (Reference Noonan1993), in his response to Johnston, argues that Johnston's revision of (8) is unnecessary, and that the Problem of the Many can be accommodated by a different revision to (8) (namely, (8*)), one that (in contrast to Johnston's (8^′)) is like the original principle (8) in providing a reductio ad absurdum of the (pluralist's) ‘non-identity’ claim about the permanent coincidence case.Footnote ²⁴ And if Noonan is right about this, we could, it seems, safely conduct our discussion in terms of Johnston's original principle (8) after all, bearing in mind that we may, if we like, take it to be implicitly qualified as Noonan suggests (and hence read as (8*)).Footnote ²⁵

I do not want to try to adjudicate this debate between Noonan and Johnston over the implications of the Problem of the Many for intrinsic similarity principles such as (8). Instead, I want to make the following point. I do not see why the defender of the non-identity verdict on the permanent coincidence case (that is, the pluralist) should accept that the ‘intrinsic similarity’ principle (8) does imply that the statue and the piece of clay in the permanent coincidence case are identical. Surely one should accept this only if one accepts that a sortal property such as being a statue or being a piece of clay is not an intrinsic property? But why should the pluralist concede that this is so?

Now, I admit that I don't have an account of what it is for a property to be intrinsic that would show that sortal properties, as conceived of by the pluralist, are intrinsic properties. However, I don't see that the onus is on me to provide such an account. Surely, if monists appeal to intrinsic similarity principles such as (8) in order to argue that sortal properties must be grounded in other properties (in a way that rules out a ‘bare difference’ in sortal properties), then the onus is upon them to explain why a sortal difference does not, in and of itself, count as an intrinsic difference.Footnote ²⁶

At the end of the day, then, the crucial issue seems to be, not whether there can be no sortal difference without an intrinsic difference (a principle that can be accepted as trivially true by the pluralist if sortal properties are themselves intrinsic properties), but, rather, whether there can be no sortal difference without some other intrinsic difference. In other words, the crucial issue seems to be, not whether (8) is acceptable, but whether some principle along the lines of (8A) is correct:

For it seems that, even if my complaint about the assumption embodied in the use of (8) (and Noonan's (8*)) against the pluralist – the assumption that sortal properties are not intrinsic properties – is a justified complaint, (8A) would avoid that objection, but would still yield the verdict, unacceptable to the pluralist, that the piece of clay in Case 3 (permanent coincidence) is a statue. The pluralist must, it seems, admit that, in Case 3, Statue is a paradigm statue, and, also that, setting aside the sortal difference between them, Piece and Statue in Case 3 are intrinsically exactly alike. Thus, it follows, if (8A) is accepted, that in Case 3 Piece is a statue, and hence (unless the case involves two statues) identical with Statue, contrary to what the pluralist maintains.

However, unlike (8) (and (8*)), (8A) seems to me to have an air of the ad hoc and question-begging in the context of the debate between the pluralist and the monist. But there is a further consideration. Consider yet another ‘intrinsic similarity principle’, (8B):

Just as (8A) reflects the idea that there is no sortal difference without a non-sortal intrinsic difference (and hence that permanently coincident objects must belong to exactly the same sortal kinds), so (8B) encapsulates the idea that there is no sortal difference between entities that are alike in their intrinsic non-sortal properties at the beginning of their existence (and hence that same-origin coincident objects must belong to exactly the same sortal kinds).

The pluralist must, of course, reject (8B), as well as rejecting (8A), for the pluralist holds that neither permanent coincidence nor temporary coincidence is identity. But what about the intermediate monist? Intermediate monists, although they may (and almost certainly will) accept (8A), must, of course, reject (8B), on pain of taking on board the unwanted conclusion that Statue and Piece are identical in a same-origin temporary coincidence case.

But now I ask: is (8B) really any less plausible than (8A)? It seems to me that it is not. But if (8B) is at least as plausible as (8A), this makes precarious the intermediate monist's appeal to intrinsic similarity principles such as (8A) in support of the view that permanent coincidence is identity.

10. Pluralism and Ontological Profligacy

The final objection to the pluralist view that I shall consider also comes from Harold Noonan. Noonan (Reference Noonan1993, §4) argues that the pluralist who defends the view that there is permanent coincidence without identity in cases such as the statue and the piece of clay, on the grounds that the statue and the piece of clay differ in their modal properties, will find it hard to resist a commitment to an indefinite number of cases of ‘permanent coincidence without identity’ where the allegedly coincident but distinct entities do not (unlike the statue and the piece of clay) belong to distinct ontological categories. For example, what about the case of the permanently coincident snowball and snowdiscball, where the concept of a snowdiscball is like that of a snowball, except that the persistence conditions for snowdiscballs differ from those of snowballs in that they allow (indeed, require) a snowdiscball to persist when the snow of which it is composed is flattened into a disc shape (1993: 145)? Footnote ²⁷ Evidently, if this example is accepted, it provides a model for the generation of a myriad of examples of permanently coincident physical objects which a pluralist who takes a difference in modal properties to be sufficient for numerical distinctness must regard as distinct. But a commitment to this multiplicity of permanently coincident but distinct physical objects represents, according to Noonan, an unacceptable ‘degree of ontological inflation’ (1993: 145).Footnote ²⁸

In the previous section, I have argued that the pluralist need not appeal to a distinction in modal properties as the fundamental ground for a numerical distinction between the statue and the piece of clay in a permanent coincidence case, but may instead appeal to a fundamental sortal difference between them. However, Noonan's argument concerning ontological profligacy remains pertinent. For if the pluralist appeals to a sortal difference between a piece of clay and a statue as that which underlies the apparent difference between their modal properties, should the pluralist not also appeal to a sortal difference between a snowball and a snowdiscball as what underlies the apparent difference between their modal properties? And does this not lead, as Noonan suggests, to an unacceptable degree of ontological inflation?

One reaction to such examples, on behalf of the pluralist, would be to claim that a concept such as that of a snowdiscball is illegitimate. If it is, then clearly it can generate no genuine additional entities via cases of coincidence, and hence no ontological inflation.

However, I shall not attempt to defend this ‘ontologically illiberal’ pluralist response to Noonan.Footnote ²⁹ Rather, I shall argue that even if the relevant entities (snowdiscballs and the like) are accepted as legitimate, Noonan's charge of ontological inflation has little, if any, force. Or, at least, it lacks force if it is intended, as Noonan appears to intend it, as an argument in favour of intermediate monism rather than pluralism.

This is perhaps most easily seen if we assume that our (‘ontologically liberal’) pluralist is (as pluralists typically are) an endurance theorist, and that our intermediate monist is (as intermediate monists typically are) a perdurance theorist.Footnote ³⁰ If the pluralist accepts the multiplicity of coincident entities envisaged in Noonan's objection, this may seem to be ontologically profligate. But if it is, I think that the intermediate monist perdurantist is in no position to criticize the endurantist on these grounds. For the perdurantist who admits the legitimacy of concepts such as that of a snowdiscball must recognize a multiplicity of four-dimensional worms corresponding to these concepts. Hence, when we compare the ontological commitments of the ‘ontologically liberal’ endurantist pluralist and those of the perdurantist intermediate monist with respect to such entities, the only respect in which the perdurantist can claim to be ontologically more parsimonious seems to be that the perdurantist refuses to acknowledge, in addition to a multiplicity of entities that are temporarily coincident but not identical, a set of further entities that are permanently coincident but not identical. And I do not see that this represents a significant gain in ontological economy, if indeed it is a genuine gain at all.Footnote ³¹ Hence, I submit, it represents a very weak ground for preferring the perdurance theory, with its intermediate monism, to the pluralist endurance theory that holds that there can be permanent coincidence without identity.

To put the point another way, if there is ontological profligacy or inflation here, it consists chiefly in admitting, into one's ontology, entities such as snowdiscballs in addition to snowballs. The perdurantist intermediate monist who admits the legitimacy of the concept of a snowdiscball (as most perdurantist do), and admits that it picks out genuine entities, must also admit that sometimes these entities are entities additional to snowballs, for in some cases these entities will coincide only temporarily with snowballs, in which case (according to intermediate monism) they must be distinct from them. But once this is conceded, why is it supposed to represent a significant gain in ontological economy to say that, in the cases where the coincidence is permanent, we can identify these entities with the corresponding entities of more familiar kinds (such as snowballs)? I submit that any sacrifice of ontological economy here that is required by the pluralist view is, pace Noonan (Reference Noonan1993: 145), a trivial price to pay for any advantages that the pluralist view may have over that of the intermediate monist.

The most that Noonan's ‘ontological inflation’ objection can show, I think, is this. If the pluralist who distinguishes the statue and the piece of clay in a permanent coincidence case relies, in making that distinction, on the claim that the statue and the piece of clay belong to significantly different ontological categories (such as those of physical object and piece of matter), then that pluralist cannot consistently also rely on the claim that a difference in modal properties or sortal properties is sufficient for a difference in identity. For if Noonan's examples are successful, they show that the pluralist must recognize cases of the permanent coincidence of entities that differ in their sortal properties and modal properties even though those entities belong to the same ontological category.Footnote ³² However, although this may be an important point, and although it may constitute a valid ad hominem objection to Johnston and some other pluralists, I do not see that it tends to undermine the case for preferring pluralism to intermediate monism.Footnote ³³