The Endurance/Perdurance Distinction

Neil McKinnon

 

It is now usual to say that something persists iff it is located at more than one time.  This neutral term gives us a means of framing the question, how does a thing persist?  One answer is to say that a thing’s persistence involves its perduring.  What is it for a thing to perdure?  Generally, it has been held that perdurance involves persisting in virtue of having temporal, as well as spatial, parts.  And what is it for a thing to endure?  Often, this is put in terms of a thing’s being wholly present at all times at which it exists.  Again, sometimes the endurance/perdurance distinction is put in terms of the difference between strict identity and a looser unity relation sometimes labelled ‘genidentity’.  On this understanding, a persisting thing endures iff for any time at which the persisting thing is located, there is something which is identical to that thing.  A persisting thing perdures iff for any pair of times at which it is located, it has different temporal parts at those times which stand in the genidentity relation to each other.

 

Hopefully the above gives the reader some sort of feeling for what the endurance/perdurance distinction might look like.  Unfortunately, none of the suggestions above seem adequate to capture the distinction.  For instance, the idea that a thing’s endurance can be captured in terms of being its being ‘wholly present at all times at which it is located’ has been shown by Theodore Sider [13] to be problematic.  He says that for an endurantist parthood is an irreducibly temporally relative matter.  Contrast the situation for endurance with the situation for perdurance.  We can state without temporal indexing what parts a perduring thing has.  What parts a perduring thing has, it has simpliciter.  These parts have the further property of being located at various times.  On the other hand, we cannot state what parts an enduring thing has without mentioning the times relative to which it has those parts.

 

But if this is so, what can it mean to say that an enduring thing is wholly present at a time?  The intended idea was, perhaps, to say that for any time at which an enduring thing is located, all of its parts are located at that time.  However, once we admit that, for the endurantist, parthood at a time is irreducibly temporally relative, we realise that there is a blank to fill in: being wholly present at a time is to have all of its parts …when???… located at that time?  Since enduring things don’t have parts simpliciter, a statement like ‘a is a part of enduring thing b’ must always be qualified with a time reference.  The problem is, how can we fill in the blank while charting a course between triviality (all of its parts relative to that time are located at that time) and absurdity (all of its parts relative to some other time are located at that time)? (Sider, [13], p. 209.)

 

What of the idea that we can use strict identity to capture endurance?  Trenton Merricks brings to light a problem with this approach ([6], p. 427).  It is not only the endurantist who holds that for any time at which a persisting thing is located, there is something that is identical to the persisting thing.  The perdurantist also assents to this statement.  For any time at which a part of a perduring thing is located, the perdurantist will say that the perduring thing is located at that time.  Thus, for any perduring thing that has a part at t, the perduring thing is located at t.  Naturally, it follows from this that there is something located at t which is identical to the perduring thing.  How can we alter the ‘strict identity’ attempt to account for endurance so that it does not also subsume perduring things?  Only, it seems, by adding a proviso to the effect that enduring things are wholly present at every time at which they are located.  Thus, the account of endurance now becomes: a persisting thing endures iff for any time at which the persisting thing is wholly located, there is something that is identical to that thing.  In view of the problems associated with using ‘wholly present’ in an account of endurance, we don’t seem to have advanced very far.

 

At this point, it is important to recognise that understanding endurance in terms of enduring things being wholly present at each time is unproblematic when taken in the context of presentism.[1]  This is because presentists do not hold that talk of things having parts at times involves irreducible relationships to times.  For the presentist endurantist, those parts that a thing now has, it has simpliciter.  Yesterday, it may have had different parts simpliciter, and likewise for tomorrow.  It may be possible to introduce the notion of a thing having parts at a time, but only in a derivative sense.  So, for instance, we may be able to think of times as constructions out of propositions, as structured universals, or something like that, and say that x has y as a part at t iff t represents x as having y simpliciter.[2]  Sider’s argument against our understanding a thing’s endurance in terms of its being wholly present whenever it exists is fuelled by the concern that irreducible mereological relationships between things and times might be required by endurance.  Presentism, then, offers us a counterinstance to Sider’s argument.[3]

 

Does this mean that we have rehabilitated the ‘wholly present’ understanding of endurance?  Not yet.  Some people (myself included) believe that endurance is compatible not only with presentism, but also with the tenseless view of time, according to which the past, present and future are all equally real and time’s passing consists in nothing more than the obtaining of temporal precedence relations.  What should endurantists who favour tenseless time say about whether parthood at a time is irreducibly temporally relative?  Those who are convinced mereological essentialists can maintain that all parthood is parthood simpliciter, and can thus remain happy with the ‘wholly present’ understanding of endurance.  Those who think otherwise (and I take it that this includes most people who believe that endurance is consistent with the tenseless theory) are committed to viewing parthood at a time as an irreducibly temporally relative matter, as I will now explain.[4]

 

Consider a cart that loses a wheel at some time during its lifetime.  According to the tenseless theory, all stages of the cart’s history exist, and so the cart’s having four wheels and its having three wheels are facts about the cart that are of equal ontological standing.  But we can’t just say that the cart has four wheels and that it has three wheels, since that would involve a blatant contradiction.  So we must say that the cart has four wheels relative to some times and three wheels relative to others.  In other words, the temporal indexing of parthood cannot be eliminated.

 

Thus, very few people who think that endurance is consistent with the tenseless view of time can be satisfied with the ‘wholly present’ understanding as a general account of endurance.  And given this fact, I intend to proceed by looking for an account of the endurance/perdurance distinction that separates endurance and perdurance with tenseless time in mind.  So, for the time being I will largely ignore complications related to presentism.  However, after the account has been produced, I will give its presentist analogue.  In so doing, I intend to produce a general account of the distinction.[5]

 

How, then, should we proceed?  A useful entry-point involves attending to the notion of a temporal part.  It is useful, first, because the notion of temporal parthood has long been associated with the notion of perdurance.  And as I explain in Section 3, one of the ways that I think we can make out the endurance/perdurance distinction requires the notion of temporal parthood, though perhaps not quite in a way that would be expected.

 

1.1  Temporal Parthood

 

What is a temporal part?  Those who believe in temporal parts take them to be among the parts of persisting things.  As such, they are located at instants and across intervals of time.   Let’s start by focusing on instantaneous temporal parts.

 

Plausibly, x is an instantaneous temporal part located at t of y iff (i) x is a part of y, (ii) x is located only at t, and (iii) x overlaps every spatial part of y that is located at t.

 

This formulation captures the idea that a temporal part of y is made up of all y’s spatial parts at t, and suffices to give us a pretty good idea of what a temporal part is.  The only problem is the restriction to spatial parts.  In order to be more accurate we ought to give a purely mereological definition; we don’t want to automatically rule out the possibility of there being persisting things that have temporal parts without having spatial location.  So we drop the reference to spatial parts:

 

(TP): x is an instantaneous temporal part located at t of y iff (i) x is a part of y, (ii) x is located only at t, and (iii) x overlaps every part of y that is located at t.

 

This account is easily generalised to cover non-instantaneous temporal parts: x is a temporal part (extending through the temporal interval T) of y iff (i) x is a part of y, (ii) x is located only throughout T, and (iii) for any time in T, x overlaps every part of y that is located at that time.

 

The above account of temporal parthood is Theodore Sider’s ([13], p. 206).  I would take this account to be a satisfactory resting point, but for methodological reasons, Sider chiefly employs a different formulation.  Since I plan to use the version of Sider’s account of temporal parthood that I have just outlined, I ought to defend this decision.  This is my next task.

 

 

1.2  Sider’s Neutral Accounts of Temporal Parthood and Parthood at a Time

 

Recall that Sider thinks (correctly, in my view) that perdurantists ultimately reduce talk about parthood at a time to facts about parthood simpliciter.  However, references to times are not entirely eliminated, but are shifted from qualifying parthood to qualifying the locations of a perduring thing’s parts.[6]  Perdurantists can, if they choose, use the idiom of parthood at a time, since they have a ready way of translating such talk into canonical vocabulary. Sider offers the following translation:

 

Necessarily, x is a part of y at t iff x and y are each located at t, and x’s temporal part located at t is part of y’s temporal part located at t ([13], p. 200).[7]

 

Moreover, Sider also thinks that perdurantists may also employ a temporally indexed notion of temporal parthood, namely, temporal parthood at a time.  He offers us an account of temporal parthood at a time, which is defined in terms of parthood at a time and overlap at a time:

 

x is an (instantaneous) temporal part of y at t iff (i) x is a part of y at t, (ii) x exists at, but only at, t, and (iii) x overlaps at t everything that is part of y at t ([13], p. 205).

 

  We can, if we wish, pass from the neutral to the atemporal reading of temporal parthood by substituting parthood and overlap simpliciter for their temporally indexed relatives.

When speaking of perdurance and parthood, Sider mostly employs temporally indexed notions of parthood, and temporal parthood.  Why do this if perdurance presupposes parthood and temporal parthood simpliciter?  As far as I can see there are two reasons.  Each reason pertains to a different task that Sider sets for himself in  ‘Four Dimensionalism’.  The first task is to see what can be made of the endurance/perdurance distinction.  The second task is to provide an argument against endurance based on considerations about vagueness ([13], pp. 224-9).  He is certainly warranted in employing parthood at a time (without committing to an endurantist or perdurantist reading of this notion) to discharge his argument against endurance.  This argument makes use of the notion of parthood at a time, and so it is of course important that he does not assume at the outset of that argument an understanding of parthood at a time which prejudges the issue in favour of perdurance.

How do things stand with respect to the first task?  Sider offers the following account of perdurance in terms of parthood at a time:

x perdures iff x persists, and for any two disjoint sets of times, T1 and T2 whose union is the timespan of x, there is a y and z such that (i) x and y have the same parts at every time in T1, (ii) x and z have the same parts at every time in T2, (iii) the timespan of y is T1 and the timespan of z is T2 ([13], p. 204).[8]

He then uses temporal parthood at a time to confirm that his account says what perdurantists want to say; he claims that his account of perdurance entails that perduring things have a temporal part at every time at which they exist ([13], pp. 205-6).

Sider’s reason for constructing an account of perdurance in terms of parthood at a time stems from his view that parthood is irreducibly temporally relative for the endurantist but not for the perdurantist.  He writes:

…it is desirable to state opposing views in a neutral language, so that the opponents may agree on the identity of the proposition under dispute.  Moreover, we do not want to hide [perdurance] in the very language we use to raise the question of its truth ([13], p. 202).

He further remarks that endurantists and perdurantists will disagree over how to treat parthood at a time (and hence, temporal parthood), but will agree that the notion is intelligible, and thus, that it affords a neutral framework within which the endurance/perdurance distinction can be framed ([13], p. 202).

Even if it were true that endurance and perdurance have different implications for how parthood is construed, these considerations appear unconvincing.  First, it is not clear that using a neutral view of parthood in our accounts of endurance and perdurance does help opponents to agree about the content of the distinction.  If the worry was that endurantists will assume an endurantist understanding of parthood when characterising perdurance, and that perdurantists will assume a perdurantist understanding of parthood when characterising endurance, then the move to a neutral language does not seem to advance us very far.  This is because any perdurantists, for example, who would have insisted upon mischaracterising endurance by using an inappropriate sense of parthood, will make sure that they always parse the neutral language in their own terms.  Second, by making it explicit in an account of perdurance that parthood is to be treated as parthood simpliciter, we do not thereby assume that parthood is just parthood simpliciter.  So even though parthood simpliciter might appear in our account of perdurance, that does not mean that when we come to discuss whether perdurance is the right view of persistence to hold, we must assume that parthood  is just parthood simpliciter; only at the point when we come to argue for endurance or perdurance does a neutral reading of parthood becomes important.

When stating opposing views, such as perdurance and endurance, we ought to have as much 'up front' in the statement of the views as possible.  Where opposing views have different metaphysical primitives, this ought to be made explicit.  Non-primitive terms may be used in the statements of the views, but only where it is clear that the analysis of the non-primitive terms does not differ between the views in question.  This means that if perdurance were to be understood (partly) in terms of a primitive notion of parthood simpliciter, then there would be no temporal indexing of parthood in the account of perdurance.  Similarly, if endurance were to be understood (partly) in terms of irreducibly temporally relative parthood, then there would be temporal indexing in the account of endurance.

In any case, since Sider's motivation for the neutral reading of parthood at a time was based on endurance and perdurance each embodying different understandings of parthood at a time, and since it appears that endurance is consistent with the correct notion being parthood simpliciter, it might be thought that Sider's motivation has been undercut.  However, I suspect that this realisation ought to have a different effect; it actually introduces a new reason for making some use of a neutral account of parthood at a time.  A neutral reading is not required in the account of perdurance, but it might be useful in an account of endurance that makes use of the notion of parthood.  An endurantist’s underlying view of time and stance on the issue of mereological essentialism may influence whether parthood is held to be fundamentally temporally relative or not.  So a neutral reading of parthood at a time may be useful if we desire to frame an account of endurance that does not smuggle aboard assumptions about time or mereological change.

Where does this leave us with respect to temporal parthood?  If the having of temporal parts is to make a positive contribution to the formulation of the endurance/perdurance distinction it will be in the framing of perdurance.  There, the appropriate notion is temporal parthood simpliciter, since temporal parthood simpliciter is defined in terms of the appropriate notion of parthood for perdurance, namely, parthood simpliciter.  In contrast, any role that temporal parts might have in an account of endurance will be a purely negative role.  That is, endurantists will want to deny that enduring things have temporal parts as temporal parts are construed in the framing of perdurance.  And so, again, the proper understanding of temporal parthood is temporal parthood simpliciter.

2.      Understanding Perdurance: Some Proposals

Now that we have an understanding of temporal parthood, namely, (TP) from Section 1, we can use this notion to give a simple account of endurance/perdurance.  An entity perdures iff it has temporal parts.  And an entity endures iff it persists and has no temporal parts.

This simple account is a good starting point.  Regrettably, however, the component that deals with perdurance is flawed because it does not allow us to distinguish between perduring things and things which are both perdurers and endurers.[9]  We should not, I think, in our formulation of the endurance/perdurance distinction assume that there could not be things that have at least some enduring temporal parts.  And if the tenseless theory of time were right, composition were unrestricted, and ordinary things persisted by enduring, endurer/perdurers might not be far away.  Under these conditions, there would be endurer/perdurers like Cromwell-Disraeli-Blair (the mereological sum of the temporally non-overlapping enduring persons Cromwell, Disraeli and Blair). And unfortunately, the above account lumps perdurers and endurer/perdurers together.

Perhaps we could try instead Theodore Sider’s account of perdurance, set up without the neutral reading of parthood:

x perdures iff x persists, and for any two disjoint sets of times, T1 and T2 whose union is the timespan of x, there is a y and z such that (i) x is the fusion of y and z, (ii) the timespan of y is T1 and the timespan of z is T2, and (iii) xyz ([13], p. 206).[10]

Although Sider’s account makes no mention of temporal parts, he claims that it entails something about temporal parts.  He claims that x’s satisfying the definiens of his account entails that x has a temporal part located at each instant encompassed by x’s timespan.[11]

Note that Sider’s account and this entailment are not equivalent.  In order to arrive at something equivalent to Sider’s account, we would need to augment the entailment with the following assumptions that are built into his account: (a) unrestricted composition, and (b) the assumption that any way of partitioning x’s timespan into two sets reveals two things which are parts of x.  These assumptions ought not to be built into an account of perdurance if at all possible.  Building (a) into an account of perdurance automatically rules out the possibility of perduring things whose only temporal parts are instantaneous ones.  This is because (a) ensures that any pair of instantaneous entities have a merelogical sum. Thus, a persisting thing made up of many instantaneous temporal parts is sure to have many non-instantaneous ones as well. And incorporating (b) automatically rules out the possibility that perduring things have no instantaneous temporal parts, but only smaller and smaller ones tending towards a limit.[12]  Perhaps (a) and (b) register necessary truths.  Yet, it is certainly contentious whether they do.  So maybe the entailment improves on Sider’s own account.  Let’s try it out. Our new account becomes: x perdures iff x has a temporal part located at each instant encompassed by x’s timespan.

This account does not presuppose unrestricted composition, nor does it presuppose that perduring things have instantaneous parts, since it does not assume that if something is located at an instant, then it is not also located at other instants.  But unfortunately, by avoiding a commitment to instantaneous temporal parts, this account, like the simple account that said a thing perdures in virtue of having no temporal parts, is vulnerable to the objection that it fails to distinguish between perdurers and endurer/perdurers.  Endurer/perdurers satisfy this account of perdurance, since it true that an endurer/perdurer has a temporal part located at every instant encompassed by the endurer/perdurer’s timespan.

Where should we go from here?  Perhaps we could seek to revise this account, but I doubt that it would be profitable to do so.  This is because any account of perdurance which entails that perduring things must have temporal parts faces a formidable objection formulated by Trenton Merricks ([6], p. 431).  Here is the objection, slightly rephrased from Merricks’ original formulation.

Consider a world where every cell is a perduring thing with temporal parts.  Next, consider a persisting organism composed entirely of such cells.  Further, assume that the organism has no proper parts aside from these cells and their parts.  Since the organism is composed entirely of cells with temporal parts, we ought to say that the organism perdures.  Unfortunately, the organism itself has no temporal parts.  To see this, note that none of the organism’s parts satisfy the account of temporal parthood we have considered.  Recall that it was a necessary condition of x’s being a temporal part of y located at t that x should overlap every spatial part of y that is located at t.  For any t during the organism’s life, those of its parts that are t-located are temporal parts of the various cells that compose the organism.  But these parts do not compose anything.  So none of the organism’s t-parts satisfy the above necessary condition for temporal parthood.  This means that the organism does not perdure according to an account of perdurance given in terms of temporal parthood.  Yet, the organism is a perduring entity since it is composed only by perduring cells and their parts.  The conclusion to be drawn, then, is that an account of perdurance given in terms of temporal parthood is inadequate.

Merricks does offer those who favour an account of perdurance in terms of temporal parthood an escape route.  He suggests that they might weaken the account of perdurance in the following way: ‘…perduring objects either have temporal parts (i.e. parts that are big enough) at each time/interval at which they exist or have proper parts that have such temporal parts’ ([6], p. 432).  This amendment allows for the classification of the organism as a perdurer, but we can alter Merricks’ example so that the amendment fails.  Why not say that neither the organism nor any of its parts have temporal parts?  Take the organism’s cells, for instance.  In Merricks’ original example, for any t during the organism’s timespan, the organism’s cells have temporal parts located at t, but (he stipulates) these temporal parts fail to compose anything.  Thus, the organism has no temporal parts.  Why not treat the cells in the way that Merricks treats the organism?  Each cell has various t-parts, but (we stipulate) none of the cells’ t-parts compose anything.  Like the organism, each cell is a perduring entity that lacks temporal parts.  And we can say the same thing for any of the parts of the organism, no matter how small.  Once we make this change to Merricks’ example, we can see that it cannot be subdued by weakening the temporal parthood-based account of perdurance.[13]

Given the outlandish nature of Merricks’ organism we might wonder whether it is genuinely possible.  Is there any motivation for thinking so?  Merricks does, in passing, mention a possible motivation for the view that his organism perdures but lacks temporal parts, namely, a distaste for arbitrary undetached parts ([6], p. 431).  However, it is unlikely that the arguments which have been marshalled against the doctrine of arbitrary undetached parts in the philosophical literature provide any support for his organism.[14]

What other motivation could there be? Consider the set of entities, E, whose members compose the organism.  Next, consider R, a subset of E, which differs from E only by excluding those entities that are located only at the organism’s first instant.  And consider S, also a subset of E, which differs from E only by excluding those entities that are located only at the organism’s last instant.[15]  Do the members of R compose something? And do the members of S compose something?  If we answer positively to both questions, we have to say that the organism has (at least) two temporal parts.  And there seems to be a good reason for answering positively.   If R were to compose something, then that thing would be almost indiscernible from the organism itself.  So it is going to be hard to find a believable restriction on composition that would ensure both that the organism exists but that R fails to compose anything.  The same point applies also to S.[16]

 

However, it may well be that even if we have no motivation for believing that the organism is possible, nevertheless we ought, for the purposes of framing the endurance/perdurance distinction, to proceed as though it is possible. For perhaps all that Merricks requires is the mere epistemic possibility of composition being restricted so that the organism doesn’t have temporal parts (his note 22 is suggestive).  If this is right, then we might need something akin to a proof that the organism is impossible before an account of perdurance that rests on the notion of temporal parts could proceed.  But proofs are hard to come by.  There might well be arguments against denying that there are arbitrary undetached parts.  There are indeed arguments for certain views about composition that affirm the existence of arbitrary undetached parts.  For instance, there are arguments for unrestricted mereological composition.[17]  But however strong we think these arguments are, should we really regard any of them as proofs?[18]

 

According to Merricks’ preferred way of understanding perdurance, it turns out that endurance entails presentism ([6], p. 424).  Are we really forced to adopt this position?  Fortunately, (since I prefer to juxtapose endurance with tenseless time), even if we grant that Merricks has refuted the idea that perdurance requires the having of temporal parts, we can still avoid his understanding of the endurance/perdurance distinction.

 

 

3.  Drawing the Distinction

 

To secure an adequate understanding of endurance/perdurance, we need very little in addition to the material we have thus far covered.  On the supposition that Merricks’ organism is possible, an interesting fact emerges that we can use to underwrite an account of the endurance/perdurance distinction.  If an enduring thing is located at a certain time, then necessarily there is a set whose members compose that thing at that time.  But in the light of Merricks’ organism, there is no corresponding fact that obtains for perduring things.  His organism perdures, yet for any time, its parts located at that time compose nothing.[19]  We can use this disanalogy to frame a general account of endurance/perdurance.  Here is the distinction:

 

(E) x endures iff (a) x persists, (b) x has no temporal parts, and (c) for any time at which x is located, there is a set whose members compose x at that time.[20],[21]

 

(P) x perdures iff (a) there is a y and z such that y and z are parts of x, y and z have temporal location, and there is no t such that y and z are both located at t, and (b) x has no temporal parts that endure.

 

(EP) x endures/perdures iff x has at least one enduring temporal part.[22],[23]

 

A straightforward account of endurance/perdurance put in terms of perduring things having temporal parts and enduring things lacking them falls foul of Merricks’ example.  But by defining endurance only partly in terms of lacking temporal parts, and then proceeding to define perdurance and perdurance/endurance, we can avoid this pitfall.  The above account classifies Merricks’ organism as a perduring thing.  It does not count as an endurer since it violates (c) of (E).  And it does not count as an endurer/perdurer since it has no temporal parts, and therefore, no temporal parts that satisfy (E).

 

In addition, (P) has further advantages over the temporal parts-based accounts of perdurance I discussed in Section 2.  For instance, it does not presuppose unrestricted mereological composition.  It is also neutral on the issue of whether perduring things with temporal parts must have instantaneous temporal parts.  Even granting that the times quantified over in (P) are instants, it does not follow that perduring things have instantaneous temporal parts.  This is because there are ways of construing instants as constructions of non-instantaneous things.  For example, we could think of instants as the limits of sequences of perduring things such that each thing in the sequence temporally encloses each thing further along in the series.  Or we could think of instants as sets of temporally overlapping things.[24]  Neither of these alternatives requires perdurers to have instantaneous parts.

 

To close this section I would like to discuss an interesting objection to (P) suggested to me by Theodore Sider.  The objection involves us imagining a certain variant of mereological essentialism which allows that entities can gain or lose parts.  But if x is a part of y, then it is essential to x that it is a part of y, and thus x is a part of y throughout x’s entire timespan.

 

Now suppose that I am an endurer and that a certain enduring electron is a part of me only during 1980.  And suppose that another enduring electron is a part of me only during 1990.   This looks like a counterexample to (P).  (P) uses the notion of parthood simpliciter, and the problem case has been set up so as to accommodate this.  Since it is essential to both electrons that they are parts of me, there seems to be no reason not to say that both are parts of me simplicitier.  And since the electrons do not overlap temporally, I satisfy (a) of (P).  As an endurer, I also satisfy (b) of (P).  Therefore, (P) mistakenly classifies me as a perdurer.

 

Notice something disquieting about Sider’s objection.  Since I am an endurer and I gain and lose parts (e.g. the 1980 electron), the relevant reading of ‘has as a part’ is ‘has as a part at t’.  I do not have parts simpliciter.  Both of the electrons are part of me simpliciter, and yet I have neither of them as parts simpliciter.  This is quite odd, but is it incoherent?   I suspect that it is.

 

One good reason for thinking it is incoherent is the thought that ‘x is a part of y’ and ‘y has x as a part’ are just different ways of saying that a certain relationship holds between x and y; they denote the same fact.  And if that is so, then x can’t be a part of y simpliciter without y having x as a part simpliciter.  Yet, even if this is wrong, and ‘is a part of’ and ‘has as a part’ denote different relations, it’s hard to imagine how these relations could come apart so that x could be a part of y simpliciter without y’s having x as a part simpliciter. 

 

Even so, it’s worth remembering that I am bound by my own dialectical constraints to give any benefit of the doubt to Sider here.  Certainly, Sider’s example, although unusual, seems otherwise coherently describable.  If I am wrong in claiming that it is incoherent, perhaps what is needed is just some sort of connection to be made between x’s standing in the dyadic is a part of relation to y and y’s only standing in the triadic has as a part at relation to x.  We might say something like this, for example: if x is a part of y, then (y has x as a part, or for any t at which x is located, y has x as a part at t).

 

However, even if we accept the coherence of Sider’s example, (P) merely requires a small amendment.  We supplement it with (c): for any x, x is a part of y iff y has x as a part.  Without the benefit of Sider’s example, (c) would appear entirely trivial.  But if Sider’s example is coherent then we learn that endurance is compatible with its falsity.  But perdurance is not compatible with its falsity.  Perdurance requires both is a part of and has as a part to be dyadic relations.

4.  Presentism

Having shown how to frame an account of endurance/perdurance in the context of the tenseless theory of time, I turn now to the task of providing presentist analogues.[25]

I begin by providing a presentist account of temporal parthood.  All tensed locutions and all quantifiers are to be construed as irreducibly tensed:

x is an instantaneous temporal part of y iff

(i)   x is a part of y.

 

(iia) If x exists then Ř(it was the case that x exists) and Ř(it will be the case that x exists).

(iib) If x existed exactly z minutes ago then x does not exist, Ř(it will be the case that x exists), and there is no nz such that x existed n minutes ago.[26]

(iic) If x will exist in exactly z minutes then x does not exist, Ř(it was the case that x exists), and there is no nz such that x will exist in n minutes.

 

(iiia) If x exists then x overlaps every part of y that exists.

(iiib) If x existed exactly z minutes ago then x overlapped exactly z minutes ago every part of y that existed exactly z minutes ago.

(iiic) If x will exist in exactly z minutes then x will overlap in exactly z minutes every part of y that will exist in exactly z minutes.

 

Note that conditions (iib), (iic), (iiib) and (iiic) are included so as to not prejudge the question of whether a thing can have non-existent temporal parts; if x is to count as a non-existent temporal part of y, then it must satisfy these conditions.[27]  Indeed, it should be obvious from the above account that things can have temporal parts according to presentism only if non-existents can stand in relations.  For instance, suppose that Evad has temporal parts.  In that case, he has a temporal part, p, which is not present, and therefore, which does not exist. And given that p is one of Evad’s temporal parts, it stands in a mereological relationship to Evad.

 

The next question is what sort of presentist analogue could be given of non-instantaneous temporal parts?  Constraints of length preclude my setting out an account here, but I expect that some sort of construction could be made out of instantaneous temporal parts.  However, thinking about the analogue of non-instantaneous parts, and indeed, the analogue of a temporally extended whole, raises an interesting issue.  Suppose we do grant that non-existents can bear properties and stand in relations.  Even under these conditions, I see what might be a problem for thinking that presentism is compatible with persisting things having temporal parts.  Suppose, as I imagine many presentists do, that time is necessarily presentist; any temporal world is a presentist world.  If that is so, then it is not possible for, say, Igor Stravinsky qua aggregate of temporal parts, to exist.  Igor Stravinsky qua aggregate of temporal parts is a logically impossible entity.

 

It is one thing to say that non-existents can bear properties and stand in relations, but perhaps it is quite another to say that impossible entities can bear properties and stand in relations.  If there are people who maintain that it is quite another thing to say this, then those people still retain a connection of sorts between existence and the bearing of properties/relations: only possible existents can bear properties/relations.  Any presentist of this variety who also thinks that time is necessarily presentist should also think that persisting things do not have temporal parts.

 

Having given the presentist account of temporal parthood[28], let us see what analogues can be provided for (E), (P) and (EP).

 

The presentist analogue of (E) follows.  Again, the analogue is designed not to prejudge the issue of whether a thing might have non-existent temporal parts.  And again, all tensed locutions and all quantifiers are to be read as being irreducibly tensed.  Tense operators, quantifiers and connectives are symbolised where I feel that doing so enhances clarity and readability.  Finally, let us introduce a useful definition.  Let 'A' represent the property of being composed by the members of some set (but no set in particular).

 

(E) x endures iff:

 

(a) x exists & ( P[x exists] Ú F[x exists] )

(b) x has no temporal parts

(c) Ax & n( Pn[x exists] PnAx ) & n( Fn[x exists] FnAx )

 

Note that this account, via (a), entails that a thing can be an endurer only if it exists.  This is certainly right for the presentist who does not believe that non-existents can have properties or stand in relations.  I also suspect that the presentist who does think that non-existents can have properties or stand in relations ought to say that endurance is a property which is existence-entailing although a non-existent could have the property of being such that it was an endurer, or the property of being such that it will be an endurer.  However, those who think that being an endurer is not existence-entailing ought to augment (a) so that it reads as follows:

 

( x exists & (P[x exists] Ú F[x exists]) ) Ú P(x exists and P[x exists]) Ú F(x exists and F[x exists])

 

Next is the analogue of (P).

 

(P) x perdures iff:

 

(a)  some existing y is a part of x and some z is a part of x and does not exist but did or will exist.

(b)  x has no temporal parts that endure.

 

If it is held that x can perdure without having any existing parts, then (a) should read:

 

(a) some y and z are such that y and z are parts of x, and ( y exists and z does not exist, or P(y exists and z does not exist), or F(y exists and z does not exist) ).

 

There are two things to note about this understanding of perdurance.  First, presentists who deny that non-existents can bear properties and stand in relations will read ‘some y’ and ‘some z’ in (a) as existential quantifications, and will thus regard (a) as necessarily false.  The second thing to note is that any presentists who hold that non-existents can bear properties and stand in relations and also think that being an endurer is an existence-entailing property but that temporal parthood is not, should augment (b) so that it reads as follows:

 

(b) x has no temporal parts that endure and x has no temporal parts that did endure and x has no temporal parts that will endure.

 

This alteration is made so that nothing counts as a perdurer if it has a non-existent temporal part that was, or will be, an endurer.

 

(EP) x endures/perdures iff:

 

(a)  x has a part that exists & (x has a part that did exist Ú x has a part that will exist).

(b)  x has, had, or will have, at least one enduring temporal part.

 

 

5.  Statues, Lumps and Presentism

 

To round off the discussion of the endurance/perdurance distinction, I would like to discuss an argument from Sider against understanding endurance in terms of lacking temporal parts.  If his argument succeeds, it also spells trouble for my preferred account of endurance for the following reasons. First, it is a necessary condition of endurance on my account that enduring things lack temporal parts.  And second, Sider argues that some supporters of endurance might want to say that there could be enduring things with a temporal part, so such a circumstance ought not to be ruled out by definition.

 

Sider writes:

 

…imagine a lump of clay that gets made into a statue-shape for only an instant (by a god, say).  It seems to me that some [endurantists] might want to say that in that instant, a statue comes into being, but immediately goes out of existence.  After all, many [endurantists] say that when a lump of clay becomes statue-shaped for some extended period of time and then gets squashed, a statue comes into being for that period of time; the instantaneous statue would be a limiting case ([13], p. 211).

 

He then proceeds to argue that the lump of clay satisfies the conditions for having a temporal part.  He runs the example through his neutral account of temporal parthood and argues that the instantaneous statue satisfies conditions (i)-(iii) for being a temporal part of the lump:

 

‘…a temporal part of the lump at t is anything that (i) is part of the lump at t, (ii) exists only at t, and (iii) overlaps at t everything that is a part of the lump at t.’ ([13], p. 211.)

 

Recall that I argued in Section 1.1 that the reading of temporal parthood pertinent to the endurance/perdurance distinction is temporal parthood simpliciter.  Does this provide me with an easy way to avoid Sider’s argument?  Unfortunately, it does not.  Remember the account of temporal parthood I gave in a presentist setting.  That was an account of temporal parthood simpliciter.  And Sider’s problem case can easily be placed in the context of presentism.  If it succeeds, it shows that a (presentist) enduring thing could have a temporal part simpliciter.  And this would be an unwelcome result for my presentist account of endurance.  Replace Sider’s temporally relativised account of temporal parthood with clauses (i), (iia) and (iiia) of my presentist account of instantaneous temporal parthood.  If the statue satisfies these clauses, then it would seem that the statue counts as a temporal part simpliciter of the lump.  Here are the clauses:

 

(i) The statue is a part of the lump.

(iia) If the statue exists then it is not the case that it did exist and it is not the case that it will exist.

(iiia) If the statue exists then it overlaps every part of the lump that exists.

 

The statue surely satisfies (iia).  Sider argues that the statue satisfies (i) by appealing to a temporally relativised principle from the Leonard/Goodman Calculus of Individuals.  However, to suit the backdrop of presentism I have removed the temporal relativisation:

 

If x and y exist, but x is not a part of y, then x has some part that does not overlap y ([13], p. 212).

 

This looks convincing, but perhaps this appearance is deceptive.  Those who are inclined to believe that there could be a lump of clay and an instantaneous statue both occupying exactly the same spatial region at the same time would describe this case by saying that the statue is constituted by, but not identical with, the lump.  Whether the statue would count as a part of the lump on this view is controversial, and I suspect that quite a few endurantists who embrace the constitution/identity distinction would not agree that the statue is a part of the lump.  Consider Frederick Doepke, for instance.  He would maintain that while each part of the lump is a part of the statue, the converse does not apply:

 

Consider you and the collection of atoms of which you are now composed.  Appealing to intuition, I suggest that your heart is a part of you but not a part of this collection of atoms.  Similarly, Theseus’ ship, but not the wood of the ship, is composed of boards.  Though every part of the collection of atoms is a part of you and every part of the wood is a part of the ship, you and the ship have ‘additional parts’ not shared by the collection of atoms and the wood ([1], p. 51).

 

Doepke speaks of composition rather than constitution here, but we can ignore this difference; he holds that being composed by x entails being constituted by x.[29]  Likewise, the thought would be that the statue has various parts that the lump lacks, such as a nose, eyes and legs, if it is a statue of a human being.  The fact that we normally ascribe certain properties like mass and colour to both statues and lumps of clay is said to indicate that a clay statue shares various parts with its constituting lump of clay.  Both have the same mass and colour because, for instance, they both have the same micro-parts. ([1], pp. 51-2).  But whereas it might be quite appropriate to ascribe beauty to a certain statue, it is not usual to ascribe beauty to its constituting lump of clay.  The beauty of a statue depicting a person consists in the way certain of its parts inter-relate.  The salient parts here are things like noses, eyes and legs.  And the reason that beauty is not properly ascribable to the constituting lump is that the lump has no nose, eyes, legs, etc.

 

If this is all to the point, then Sider is wrong to think that he has shown that the statue is a part of the lump, since the statue has parts that the lump lacks.  Unfortunately, I am not completely convinced that Doepke is right.  There is at least one upholder of the constitution/identity distinction who holds that the statue and the lump would have exactly the same parts, namely, Lynne Rudder Baker.  She thinks that not only do things ‘borrow’ the properties and parts of their constitutors, but that constitutors also ‘borrow’ the properties and parts of the things that they constitute.[30]  I am not particularly convinced by the example she gives to motivate the ‘downward’ sharing of properties (p.48).  But nor am I thoroughly convinced that ‘downward’ sharing of properties and parts is inconsistent with the core view that the lump constitutes, but is not identical with, the statue.  I admit the epistemic possibility of such sharing.  So, for the purposes of testing my account of endurance/perdurance, I should, by my own lights, go on as if Sider has demonstrated that the statue is a part of the lump.

 

So, allowing that the case of the statue and the lump satisfies (i) and (iia) of the presentist account of temporal parthood, does it also satisfy (iiia)?  It is easy to see that it does.  Regardless of whether the lump 'borrows' parts from the statue, it is true that the statue overlaps everything which is a part of the lump.

 

Fortunately, I think that even if we allow that the case of the statue and the lump shows what he says it shows, nothing follows to endanger my account of the endurance/perdurance distinction.  To see this, we need to ask what sort of part the statue is.  

 

Clearly, the statue is not a proper part of the lump since the lump has no parts which do not overlap the statue.  So it must be an improper part of the statue.  To see this, note that the statue counts as a part of the lump of clay according to Goodman’s definition of improper parthood:

 

x is an improper part of y iff: for any z, z overlaps x iff z overlaps y ([2], p. 35).

 

 

Though the statue and lump are not identical, they nevertheless qualify as improper parts of each other.[31]

 

Now, recall that I argued earlier for an account of endurance according to which it is a necessary condition of a thing’s enduring that it has no temporal parts.  Sider's counterexample is the case of the statue's being an improper temporal part of the lump.  Is my understanding of endurance under threat?  I don’t believe so.  For one thing, I am suspicious of the notion of an improper temporal part.  ‘Temporal’ is a modifier of ‘part’.  I suspect that one of its functions is to exclude improper parts.   So, if a thing has a temporal part simpliciter then it has another temporal part simpliciter.  In support of this, notice how strange it seems to suppose that an entity which does not persist, but is located at only one instant, has a temporal part. If this is right, then what we ought to do is modify our account of temporal parthood while leaving the account of the endurance/perdurance distinction alone.  Thus, we amend clause (i) in both the presentist and tenseless accounts of temporal parthood to read, ‘x is a proper part of y’.  So, for example, the tenseless account of (instantaneous) temporal parthood ends up looking like this:

 

x is a temporal part of y which is located at t iff (i) x is a proper part of y, (ii) x is located only at t, and (iii) x overlaps everything located at t which is part of y.

 

Having said all of this, I am perfectly willing to concede that others might not share my distate for improper temporal parthood.  As I will now argue, the adequacy of my account of endurance does not depend at all on the illegitimacy of the notion of improper temporal parthood.

 

Even if improper temporal parthood is a legitimate notion, I think that there is a good reason for saying that it is a mistake to allow improper temporal parts to play a role in the endurance/perdurance distinction.  Whether or not a thing has an improper temporal part does not dictate any conclusion about the manner of its persistence.  An enduring thing can have one, and perduring things and endurer/perdurers always have one.  So we shouldn’t let them play any role in formulation of the endurance/perdurance distinction.  Thus, we reformulate the endurance/perdurance distinction so that it explicitly excludes temporal parts which are improper parts.  So, for every occurrence of 'temporal part/s' in (E), (P), (EP) and their presentist analogues, we substitute 'proper temporal part/s'.

 

Conclusion

 

Recent work has disturbed hitherto entrenched readings of the endurance/perdurance distinction.  Theodore Sider has pointed out that understanding endurance in terms of being wholly present is problematic.  And Trenton Merricks has highlighted difficulties with understanding perdurance in terms of having temporal parts.  I hope to have provided a more robust account of the distinction.[32]

 

Monash University

 

References

 

1. Doepke, Frederick, ‘Spatially Coinciding Objects’, Ratio, 24 (1982), pp. 45-60.

2. Goodman, Nelson, The Structure of Appearance (3rd edition) (Dordrecht: D. Reidel, 1977).

3. Hinchliff, Mark, ‘The Puzzle of Change’, Philosophical Perspectives, 10 (1996), pp. 119-36.

4. Lewis, David, On the Plurality of Worlds, (Blackwell, 1986).

5. McKinnon, Neil, ‘The Hybrid Theory of Time’, Philosophical Papers, 28 (1999), pp. 37-53.

6. Merricks, Trenton, ‘Persistence, Parts, and Presentism’, Noűs, 33 (1999), pp. 421-38.

7. Rea, Michael C., ‘The Problem of Material Constitution’, Philosophical Review, 104 (1995), pp. 525-52.

8. Rea, Michael C., ‘In Defense of Mereological Universalism’, Philosophy and Phenomenological Research, 58 (1998), pp. 347-60.

9. Routley, Richard, Exploring Meinong’s Jungle and Beyond, (Canberra: Departmental Monograph 3, Philosophy Department, Research School of Social Sciences, Australian National University, 1980).

10. Rudder Baker, Lynne, Persons and Their Bodies, (Cambridge, 2000).

11. Russell, Bertrand, Our Knowledge of the External World, (Allen & Unwin, 1926).

12. Salmon, Nathan, ‘Nonexistence’, Noűs, 32 (1998), pp. 277-319.

13. Sider, Theodore, ‘Four-Dimensionalism’, Philosophical Review, 106 (1997), pp. 197-231.

14. Simons, Peter, Parts: A Study in Ontology, (Clarendon:Oxford, 1987).

15. van Inwagen, Peter, ‘The Doctrine of Arbitrary Undetached Parts’, Pacific Philosophical Quarterly, 62 (1981), pp. 123-37.

16. Zimmerman, Dean, ‘Persistence and Presentism’, Philosophical Papers, 25 (1996), pp. 115-26.

 

 


[1] The core thesis of presentism is that there exist no temporal entities which are past or future.  To illustrate this claim, anyone currently reading this paper exists, whereas no one who is dead exists and no one who has yet to be born exists.

[2] Note the analogy with actualism about modality here.  According to actualists, a certain piece of glass may have been spherical simpliciter, although it is planar simpliciter.  Talk of having properties relative to different possible worlds can be introduced, but only in a derivative sense: x has property y relative to w iff w represents x as having y simpliciter.

[3] Some might suspect that if the ‘wholly present’ account of endurance works in the context of presentism, it will work for other tensed views of time.  The thought is that it is not presentism which is doing the relevant work here, but ‘taking tense seriously’.  Non-presentist tensed views make no distinction in terms of existence between the past and the present.  Some of these views also ascribe equal reality to the future.  These views can be said to ‘take tense seriously’ because they hold that the passage of time involves more than just temporal subsequence relations.   For instance, some of these views interpret passage in terms of things and events changing with respect to monadic properties of pastness, presentness and futurity.  I suspect that variations of Sider’s problem for the ‘wholly present’ account of endurance are going to arise for these kinds of views.  If I am right, it is the existence of multiple times along with their contents/constituents which creates the trouble.  Consider the case of the cart discussed on p. 5 of this paper.  The pressure to relativise parthood to times comes from the facts that the cart’s being three wheeled and its being four-wheeled both exist and seem on the face of things, inconsistent.  The problem does not go away just by noting that the cart’s being four-wheeled has some additional property (e.g. monadic pastness) that the cart’s being three-wheeled lacks.  Thanks to an anonymous referee for suggesting that I discuss this issue.

[4] This is a slight oversimplification.  Just because you (qua endurantist tenseless theorist) think that parthood at a time is irreducibly temporally relative, you need not think that it must be irreducibly temporally relative.  Perhaps you admit the possibility of worlds where there are enduring things and it is an accidental property of each of these things that they do not undergo mereological change.  Or perhaps you don’t believe in mereological essentialism as an all-embracing doctrine, yet you countenance the possibility of worlds containing only enduring things that have all of their parts essentially.  So, perhaps you might say that parthood at a time is only accidentally temporally irreducible.  Thus, you might say that parthood is actually a triadic relation, although it could have been a dyadic one.  Or, if you think that an n­-adic property or relation is necessarily n-adic, then what you could say is this: parthood is a triadic relation, and there is another relation, parthood*, which is very much like parthood, but is dyadic.  And the second relation, not the first, is the one that obtains in the worlds we just considered.  The accommodation of this view would lead to some complications in the formulation of endurance, but no real difficulties.  Specifically, it would require room to be made not only for the property of enduring (which involves parthood), but also for a similar property, endurance* (which involves parthood*).   So, for the sake of a simpler exposition, I will continue as if parthood for the endurantist tenseless theorist can be either dyadic or triadic.

[5] Though see note 22 for some qualifications.

[6] In this way, the perdurantist account of parthood at a time differs from the presentist account, where all reference to times is eliminated.

[7] The translation looks a little overcomplicated.  Why not the following?  Necessarily, x is a part of y at t iff x is a part of y and x is located at t.  The simpler understanding may also have two further advantages.  First, it is consistent with the example (discussed in Section 2) of Merricks’ organism, since it makes no reference to temporal parts.  Second, in order to preserve full generality by covering cases of things that are located at only one instant, Sider’s translation must rely on the notion of improper temporal parthood being a legitmate one.  I harbour some reservations about the legitimacy of this notion (see Section 3).

[8] Sider actually gives a global account of perdurance, that is, an account of what it would be for the world to be a world of perdurers (called the Thesis of Temporal Locality).  However, I have localised his account so that it focuses on what it is for a particular thing to be a perdurer.

[9] The component that deals with endurance is also flawed because it counts Merricks’ organism (discussed later in this section) as an endurer.

[10] Clause (iii) is implicit in Sider’s account.

[11] See his note 15 in [13], where he explains why he thinks that this entailment holds (though see also his note 14 for some qualifications).

[12] See Zimmerman [16], pp. 122-4 for more on this.  Admittedly, Sider at one point allows that some people might want to modify his account of perdurance so that it does not require (b); see [13], p. 226.

[13] Here is another means of upholding the spirit of Merricks’ example in a way that is immune to the weakening manoeuvre.  Suppose that the organism is composed exclusively by mereological atoms, each of which is located at only one instant (and place).  The totality of the organism’s ­mereological atoms compose the organism but there is no subclass of that totality whose members compose anything.

[14] The principal argument against arbitrary undetached parts can be found in van Inwagen [15]. Consider the following circumstance.  Jerry has two legs (let’s not commit on the question of whether his legs would count as arbitrary parts, though Inwagen would say that they do).  If Jerry has arbitrary undetached parts, then Jerry has an arbitrary undetached part which consists of the rest of Jerry minus his legs (Jerry minus).  In a fit of pique at his 215 centimetre frame, Jerry hacks off his legs with a scimitar.  Focus on the moment at which his second essay in dismemberment hits the floor. Jerry is on his way down, too, but exactly what is Jerry?  It appears that Jerry is now identical with Jerry-minus.  Since identities are necessary, how can it be that Jerry had Jerry-minus as a proper part and is now identical with Jerry-minus?  Various answers to this puzzle have been suggested, for example, that identities are not necessary but are in fact temporary; that Jerry is constituted by, but not identical with, his parts; that identity is relative to a sortal.  (For more complete inventories of answers that have been suggested, see Rea [7] and Simons [14], pp. 117-21.  Inwagen’s answer is that we should not believe in arbitrary undetached parts.  There never was a Jerry-minus, and so the puzzle disappears.

However, the view that Jerry is a perduring thing with temporal parts actually equips us with another answer to the puzzle.  This answer blames the problematic nature of the puzzle on the assumption of endurance, not on the necessity of identity, nor on arbitrary undetached parts, nor on anything else.  Jerry has a two-legged temporal part located just before the amputation and a legless temporal part located afterwards.  These temporal parts are mereologically disjoint, and are thus non-identical.  Moreover, since Jerry is a sum of his temporal parts, there is no pressure to think of him as being identical with two distinct things.   Far from falling prey to Inwagen’s argument, the temporal parts doctrine actually blunts the argument against arbitrary undetached parts.  To think that considerations surrounding arbitrary undetached parts could motivate a perdurer to accept that many perduring things, such as Merricks’ organism, do not have temporal parts, we should need some further reason for thinking that there are problems with the putative arbitrary undetached parts of putative temporal parts.

[15] Here, I am ignoring complications connected with vagueness.

[16] The only sort of restriction that comes to mind is one that would make composition an extrinsic matter.  Consider R.   The sort of restriction I have in mind dictates that the members of R would have composed something, namely, an organism of the same kind as the organism of which the members of R are parts, were it not for the parts that the organism has at its first instant.  And the reason that the members of the set whose members compose the organism do actually compose something is that there isn’t, for instance, something which is an organism but has the members of that set as proper parts.

[17] See Lewis [4], pp. 212-3, Rea [8] and Sider [13], Section 3.1.

[18] I have a fair measure of sympathy for this line of thought.  Indeed, I assume for the rest of the paper that Merricks is right on this point.  But I think this issue is difficult to evaluate, partly because if we are too liberal about what counts as epistemically possible, it may well become impossible to provide a useful account of anything much at all.

[19] Nothing controversial is assumed here about composition, such as whether composition and identity are distinct.

[20] A neutral account of composition is assumed in (c) of (E).  It is neutral between composition simpliciter, which is relevant for the sorts of worlds I discuss in note 2, and an irreducible notion of composition at a time, which is relevant for other worlds at which time is tenseless.

[21] Armed with the notion of temporal parthood, we could provide an alternative to (E) by resurrecting the strict identity approach.  x endures iff x persists, has no temporal parts, and for any t and t1 in xs timespan:

 

(a)   there is a y such that y is located at t and x=y, and

(b)   there is a z such that z is located at t1 and x=z.

 

This account is actually very close to (E).  It incorporates both (a) and (b) of (E), and clearly entails (c).  Notice that there is no way of resurrecting the genidentity characterisation of perdurance, since this involved characterising perdurance in terms of the genidentity relation’s holding between a thing’s temporal parts.  As such, any account of perdurance based on genidentity would be vulnerable to Merricks’ problem case.

[22] The condition that x persists is omitted from (P) and (EP), since it is implicit in (a) of (P) and in (EP).

[23] Note that although I have used temporal parthood in formulating endurance/perdurance, the distinction can be made without any reference to temporal parthood.  Consider:

 

(E*) x endures iff (a) x persists, (b) Ř$yz( y and z are temporally located parts of x, and "t[ y is located at t iff z is not located at t ] ), and (c) for any time at which x is located, there is a set whose members compose x at that time.

 

(P*) x perdures iff (a) x persists, and (b) "t( if x is located at t then $yz [ y and z are parts of x, y is located at t, and z is not located at t ] ).

 

(EP*) x endures/perdures iff (a) x persists, (b) there is an interval of time such that when the domain of t is restricted to that interval, x satisfies (b) and (c) of (E*), and (c) there is an interval of time such that when the domain of t is restricted to that interval, x satisfies (b) of (P*).

 

Note that eschewing temporal parts also allows us to define a broader notion of endurer/perdurers than that which features in (EP) and (EP*).  Endurer/perdurers not only have parts simpliciter, but they also irreducibly have parts at times.  This fact would appear to allow the possibility of persisting entities that are even more bizarre than the endurer/perdurers outlined in (EP) and (EP*).  Such an entity would not have clearly delineated phases at which it endures, followed by clearly delineated phases at which it perdures.  Instead, it would be a kind of a disorderly mish-mash of enduring and perduring parts.  So, for instance, one of its perduring parts simpliciter might be located throughout interval I, while it might also have enduring parts relative to every time in I.  This sort of entity is certainly exceedingly strange.  It may well be impossible, in fact.  However, I do not see that it is obviously impossible.  Thus, a broader notion of endurer/perdurers appears below:

 

(EP**) x endures/perdures iff (a) x persists, (b) x has a part simpliciter which perdures, and (c) there is an interval in x’s timespan such that x has a part throughout that interval which endures.

 

Here, we also need to modify (P*) so that nothing which satisfies (EP**) also satisfies (P*).  To do this, we add an extra condition to the effect that perduring things do not have an enduring parts.

[24] See Russell [11], pp. 123-7, for further details about these alternatives.

[25] Even though I provide accounts of endurance/perdurance in the idioms of  tenseless time and presentism, I cannot really claim to have provided a thoroughly general account of endurance/perdurance.  Though the tenseless theory and presentism seem to me to be the leading candidates when it comes to dealing with the metaphysics of the passage of time, there are other views.  These other views I like to think of as hybrids of presentism and the tenseless view, since they bear some similarities to each of these.  For example, they hold there is more to passage than just the holding of temporal relations, but that at least some non-present entities exist.  Most of these views I consider to be incoherent (though not quite for the usual reasons - see my [5]) so I have ignored them here.  But, space permitting, analogues for these views could also be given.

[26] The domain of variables n and z is the domain of real numbers, so any finitely long time unit can be used in the formulation.

[27] Among those who are sympathetic to presentism and who also believe that non-existents can have properties and stand in relations, are Hinchliff [3], Routley [9], and Salmon [12].

[28] Subject to the expectation that, as I suggest a few paragraphs earlier, a presentist account of non-instantaneous temporal parthood can be fashioned from instantaneous temporal parthood.

[29] And in any case, the putative differences between composition and constitution are not relevant here.  See Doepke [1], pp. 54-5 and Simons [14], p. 238 for discussions of the composition/constitution distinction.

[30] See, for instance, Rudder Baker [10], pp. 181-2.

[31] It is true that improper parthood is often thought to imply identity (or, for an arch-nominalist like Goodman, to be identity), but only by those who do not accept the constitution/identity distinction.

[32] Thanks to John Bigelow, Sam Butchart, Cathy Legg, two anonymous referees and an associate editor of this journal for comments and discussions.  Special thanks to Ted Sider, who provided extensive and helpful written comments on an earlier version of the paper.