Non-Branching Personal Persistence

Given reductionism about people, personal persistence must fundamentally consist in some kind of impersonal continuity relation. Typically, these continuity relations can hold from one to many. And, if they can, the analysis of personal persistence must include a non-branching clause to avoid non-transitive identities or multiple occupancy. It is far from obvious, however, what form this clause should take. This paper argues that previous accounts are inadequate and develops a new proposal.

Reductionism about people is the view that people exist but they're not a fundamental part of the world. The view is perhaps best explained through David Hume's analogy with reductionism about nations. Most of us are reductionists about nations: We believe that nations exist but also that their existing consists in more basic facts, such as the existence of citizens who organize themselves in certain ways on certain territories. So we could, in principle, provide a complete description of the world (and these more basic facts) without asserting that nations exist. In this manner, reductionism about people says that the world could, in principle, be completely described without asserting that people exist.
Given reductionism about people, personal persistence must fundamentally consist in an impersonal continuity relation holding over time. Some standard candidates for this impersonal continuity are di erent kinds of psychological, physical, and phenomenal continuity. Typically, these continuity relations can (at least in principle) branch by holding from one person at one time to two or more people at other times. And, if these relations can branch, the analysis of personal persistence must include a non-branching clause in order to avoid either of two problems, namely, the problem of non-transitive identities and the problem of multiple occupancy. In this paper, I shall explore what form this non-branching clause should take. I shall argue that previous accounts are implausible. But, with the help of some ideas from these accounts, I shall develop a new proposal.
Assuming a perdurance framework, we analyse personal persistence in terms of person-stages at di erent times being I-related, that is, being stages of the same continuant person. A person, on this framework, is an aggregate of person-stages such that (i) each stage in the aggregate is I-related to all stages in the aggregate and (ii) no person-stage that isn't in the aggregate is I-related to all stages in the aggregate.
Person-stages can (but need not) be extended in time, that is, they can be present not only at a single instant but also at each point in an interval of time. This opens up for some ambiguities about temporal order, which we should try to avoid. Let us say that a person-stage x is simultaneous with a person-stage y if and only if there is a time at which x and y are both present. Let us say that a person-stage x is present before a person-stage y if and only if there is a time at which x is present which is earlier than all times at which y is present. And let us say that a person-stage x is present a er a person-stage y if and only if there is a time at which x is present which is later than all times at which y is present.
Given reductionism about people, Relation I is analysed in terms of some basic form of connectedness, such as psychological, physical, or phenomenal connectedness. Psychological accounts take the relevant kind of connectedness between person-stages to be direct psychological connections, such as one stage's having an experience memory (or quasi-memory) of the experiences of the other. Physical accounts take the relevant kind of connectedness between p. ) and Noonan ( , pp.
-)-or that they are ad hoc-see Oderberg ( , p. ), Schechtman ( , p. ), Olson ( , p. ), and Hawley ( ); yet compare Demarest ( , pp. -). Lewis ( b, p. ). On perdurance, persons persist by having stages present at di erent times, with no stage being wholly present at more than one time. The assumption of perdurance won't be crucial for the argument of this paper. An alternative to perdurance is endurance. On endurance, persons persist by being fully present at di erent times; see Lewis ( , p. ) for the endurance/perdurance distinction. While we shall assume a perdurance framework for our discussion, we could translate the proposals from this framework to an endurance framework, replacing person-stages by people and Relation I by personal identity (see Appendix D for an endurance translation of each proposal). This translation would mainly strengthen the arguments for my proposal in so far as they rely on transitivity, because it is more obvious that identity is transitive than that Relation I is transitive. For another advantage, see note .
Lewis ( b, p. ). Lewis ( , p. ). To distinguish cases of ssion or fusion from cases where someone time travels to a time where a younger or older stage of them is also present, we can rely on some kind of personal time rather than external time; see Lewis ( a, p. ). We can then treat the continuities in ssion and fusion cases as branching without thereby treating the continuities in time-travel cases as branching (unless they also feature ssion or fusion).
person-stages to be their sharing a su ciently large portion of their brains. And phenomenal accounts take the relevant kind of connectedness between person-stages to be their sharing the same stream of consciousness. For the purposes of our discussion, we can, for the most part, leave open which one of these kinds of connectedness is the relevant one. Let Relation C be a temporally ordered version of the relevant kind of connectedness: Temporally Ordered Connectedness Person-stage x is C-related to person-stage y (xCy) = df x and y are connected by the right kind of connection and x is either simultaneous with y or present earlier than y.
And let Relation C ′ be a temporally unordered version of this kind of connectedness: Temporally Unordered Connectedness Person-stage x is C ′ -related to person-stage y (xC ′ y) = df x and y are connected by the right kind of connection.
We shall assume that Relation C is re exive and that Relation C ′ is re exive and symmetric over person-stages. Consider, rst, a direct analysis of Relation I as Relation C ′ : ( ) Person-stage x is I-related to person-stage y if and only if xC ′ y.
An example of an account of this kind is the memory criterion of personal identity. This simple account is open to Thomas Reid's well known counterexample, namely, his case The Brave O cer: Suppose a brave o cer to have been ogged when a boy at school, for robbing an orchard, to have taken a standard from the enemy in his rst campaign, and to have been made a general in advanced life: Suppose also, which must be admitted to be possible, that when he took the standard, he was conscious of his having been ogged at school, and that when made a general he was conscious of his taking the standard, but had absolutely lost the consciousness of his ogging. That is, as an o cer, a man remembers being a boy and, as a general, the man remembers being an o cer but not being a boy. Suppose that The Boy, The Ofcer, and The General are person-stages from the three periods of the man's life. Then, if the relevant kind of connections are memories, The O cer is connected both to The Boy and to The General but The General isn't connected to The Boy. The pattern of connections in this case can be represented diagrammatically as follows, where the double-headed arrows represent C ′ -relations: The Boy The O cer The General In The Brave O cer, ( ) yields that The O cer is I-related both to The Boy and to The General and that The Boy isn't I-related to The General. Accordingly, we have an instance of multiple occupancy, that is, one person-stage is a stage of two or more people. Given ( ), we have two maximal aggregates of I-related person-stages in this case: one consisting of The Boy and The O cer and one consisting of The O cer and The General. The O cer is a stage of one person who has The Boy as a stage and of one person who has The General as a stage. These people cannot be identical, since The Boy isn't I-related to The General. In order to avoid multiple occupancy in The Brave O cer, we shall-instead of a connectedness relation-rely on a continuity relation of overlapping connectedness. Let Relation R be the relevant kind of temporally ordered continuity (we shall consider temporally unordered continuity later):
Given the re exivity of Relation C, Relation R is re exive and symmetric over person-stages. John Perry and David Lewis both suggest ( ) Person-stage x is I-related to person-stage y if and only if xRy.
We follow Locke's (Essay II.xxvii. -; , p. ) distinction between man and person. A man or a woman in Locke's sense is a human animal rather than a person. Perry ( , pp. -) and Lewis ( b, pp. -). Lewis ( b, p. ) allows, however, that Relation R might have some restrictions on the maximal duration between two R-related person-stages, which can come into play in cases of extreme longevity. Lewis is also indecisive regarding whether to emphasize Relation C ′ as in ( ) or Relation R as in ( ). If the sole objection to ( ) is that it allows multiple occupancy, ( ) should be acceptable to Lewis, since he accepts multiple occupancy. This proposal avoids multiple occupancy in The Brave O cer: In that case, it yields that all person-stages are I-related.
Even so, ( ) still allows for multiple occupancy. It will do so in division cases such as Derek Par t's case My Division. Par t presents My Division as follows: My body is fatally injured, as are the brains of my two brothers. My brain is divided, and each half is successfully transplanted into the body of one of my brothers. Each of the resulting people believes that he is me, seems to remember living my life, has my character, and is in every other way psychologically continuous with me. And he has a body that is very like mine.
The pattern of connections in this case can be represented diagrammatically as follows, where Wholly is the person-stage before the division and Le y and Righty are the two resulting person-stages a erwards:

Wholly
Le y

Righty
In this case, each of ( ) and ( ) yields that Wholly is I-related both to Le y and to Righty while Le y isn't I-related to Righty. Wholly is then a person-stage of two persons: one who has Le y as a stage and one who has Righty as a stage.
To preserve the transitivity of Relation I and avoid multiple occupancy, reductionist accounts of personal persistence typically include a non-branching clause. Par t rst suggested that The criterion might be sketched as follows. "X and Y are the same person if they are psychologically continuous and there is no person who is contemporary with either and psychologically continuous with the other. " -) presents a psychological variant with the same structure. The 'Le y'/'Righty' terminology is due to Strawson ( , p. ). The idea of analysing personal identity in terms of a one-many relation in combination with a non-branching clause dates back to Shoemaker ( , pp. -). Par t ( , p. ). Against Par t, Demarest ( , p. ) argues that it would be better to just analyse personal persistence in terms of non-branching continuity. The trouble, however, is that it is far from clear when a continuity relation has a non-branching form. Par t's proposal is a rst attempt to clarify the notion of a non-branching form, on which we shall try to improve.
In terms of perdurance, we can state this suggestion as follows: ( ) Person-stage x is I-related to person-stage y if and only if xRy and there is no person-stage z such that either (i) xRz and y and z are distinct and simultaneous or (ii) yRz and x and z are distinct and simultaneous.
This proposal yields the desired result in My Division: Wholly, Le y, and Righty are all I-unrelated to each other. Nevertheless, consider the following unbalanced variant of My Division, which is just like My Division except that the man with the le half of the brain lives on longer than the man with the right half:

Wholly
Le y

Righty
Old Le y Like Le y, Old Le y is a person-stage with the le half of the brain. But Old Le y is a later stage than Le y, existing at t a er the man with the right half Brueckner ( , p. ) interprets Par t as relying on a temporally unordered continuity. Par t ( , p. ) doesn't mention a temporal-order requirement in his main de nition of psychological continuity, which is the continuity his ( , p. ) psychological criterion relies on. But Par t ( , p. n; , p. ; , pp. -) makes clear elsewhere that the chain of psychological connections needs to be temporally ordered. One might object that a di erence between our de nition of Relation R and Par t's de nition of psychological continuity is that he ( , p. ; , p. n ) at times seems to take psychological continuity to be transitive. Par t ( , p. n ; , p. ; , pp. -) makes clear, however, that psychological continuity is only supposed to be transitive when considered in one direction in time. Note, however, that, in Par t's terminology, 'Relation R' isn't psychological continuity. In his ( , p. ) terminology, psychological continuity consists in overlapping chains of strong psychological connectedness, whereas his ( , p. ) 'Relation R' is 'psychological connectedness and/or continuity with the right kind of cause. ' As de ned by Par t, 'Relation R' di ers from psychological continuity, because two person-stages can be psychologically connected without being strongly psychologically connected. Unlike psychological continuity and Relation R as we de ne it, Par t's 'Relation R' isn't transitive when considered in one direction in time. To see this, consider a variant of The Brave O cer with the same pattern of connections except that the connections aren't strong. Par t's relation then holds from The Boy to The O cer and from The O cer to The General but not from The Boy to The General (because The Boy and The General are neither psychologically connected nor related by overlapping chains of strong connectedness).
has died. (The C ′ -relation between Wholly and Old Le y won't be crucial for our discussion, because Wholly would still be R-related to Old Le y if this connection were removed.) In My Unbalanced Division, ( ) yields that Wholly is I-related to Old Le y, since they are R-related and neither of them is simultaneous with any other person-stage. Likewise, ( ) yields that Old Le y is I-related to Le y, since they are R-related and-even though Le y is simultaneous with Righty-Old Le y isn't R-related to Righty. But, according to ( ), Wholly isn't I-related to Le y, since Wholly is R-related to Righty and Righty is simultaneous with Le y. So ( ) yields that Wholly is I-related to Old Le y, Old Le y is I-related to Le y, and Wholly isn't I-related to Old Le y. Hence, given ( ), we have a non-transitive Relation I and thus multiple occupancy. Moreover, it's implausible that Wholly is I-related to Old Le y, especially given that Wholly isn't I-related to Le y.
Consider, furthermore, the following variant of The Brave O cer, where The General is connected to The Boy but not to The O cer: The Boy The O cer The General This pattern of connections could occur if the relevant kind of connections are memories and The General has irrevocably lost all memories of the experiences of The O cer but, just like The O cer, The General remembers the experiences of The Boy. This pattern of connections could also be realized on some physical accounts of the relevant kind of connections. Consider a variant of My Division, where the transplant of the right half of the brain is delayed and the man with the le half dies before the transplant of the right half: Wholly Le y

Righty
The idea here is that, even though the right half of the brain exists at t , nothing with the right half quali es as a person-stage while the man with the le half is alive. Furthermore, if the relevant kind of connections could be preserved Grice ( , pp. -). The name comes from Perry ( , p. ).
through travel by teletransportation, the same pattern of connections could be realized in a teletransportation case. Consider a case where I step into a scanner on Earth at t . My body is then scanned and destroyed. My scanned information is beamed both to the Moon and to Mars. My information reaches the Moon rst, where a replica is created at t . Later on, my information reaches Mars, where another replica is created at t . The replica on the Moon, however, has died before t . Let Earthy be the person-stage being scanned on earth, let Moony be a person-stage of the man on the Moon, and let Marsy be a person-stage of the man on Mars: My Asynchronous Replication (C ′ -relations; The Senile General pattern) t t t

Moony
Marsy I mention these variants of The Senile General in order to illustrate di erent ways in which this pattern of connections could arise given di erent views on the relevant kind of connectedness. For the purposes of our discussion, however, the di erences between these cases won't matter much. In The Senile General, ( ) yields that The Boy is I-related both to The O cer and to The General, since The Boy is R-related to The O cer and to The General while there are no distinct and simultaneous person-stages in that case-so the non-branching clause in ( ) doesn't apply. And, given ( ), The O cer isn't I-related to The General, since these stages are not R-related. So, like before, we have a non-transitive Relation I and hence multiple occupancy. This result is problematic, since the motivation for having a non-branching clause is to retain the transitivity of Relation I and avoid multiple occupancy. If non-transitivity and multiple occupancy weren't problematic, we could stick with ( ), which is simpler than ( ), or with ( ), which is simpler still.
Par t later put forward the following suggestion: ). In the reprinting, the fourth clause was changed to "it has not taken a 'branching' form. " Nevertheless, in that reprinting, Par t ( , p. ) still claims that, 'On what I call the Psychological Criterion, a future person will be me if he will be Rrelated to me as I am now, and no di erent person will be R-related to me. ' It is hard to make -In terms of perdurance, this suggestion can be interpreted as follows: ( ) Person-stage x is I-related to person-stage y if and only if xRy and there is no person-stage z such that either (i) xRz and not yIz or (ii) yRz and not xIz.
This proposal has been charged with circularity since it analyses Relation I partly in terms of itself. Analysing a relation partly in terms of the same relation needn't be a problem, however; this is a standard feature of recursive de nitions. Rather, the problem with ( ) is incompleteness. Consider, for example, The In addition to incompleteness, there's a further problem with ( ). Consider a version of My Division which is extended before the ssion, highlighting that sense of this claim given Par t's terminology. In his ( , p. ) terminology, 'Relation R' isn't psychological continuity but 'psychological connectedness and/or continuity with the right kind of cause. ' There can be psychological connectedness without psychological continuity, since psychological continuity requires overlapping chains of strong psychological connectedness and there can be psychological connectedness that isn't strong; see note . Consider a case with two person-stages, s at t and s at t , where these stages are psychologically connected but not strongly psychologically connected. Suppose that these stages aren't psychologically connected to any other person-stages. Then, at t , there is only one future person-stage that will be related to s in terms of Par t's 'Relation R' , namely, s . But, according to Par t's psychological criterion, s and s cannot be I-related, because they're not psychologically continuous (as they're not related by overlapping chains of strong connectedness).
Par t's criterion only rules out branching in cases of ssion and not in cases of fusion. My interpretation, however, treats ssion and fusion in the same way. But this di erence won't matter for my objections to ( ), because they only rely on cases of ssion.
Rather than ( ), Brueckner ( , p. n ; , p. ) actually criticizes a temporally unordered variant of ( ), that is, a proposal just like ( ) except that Relation R has been replaced by Relation R ′ (temporally unordered psychological continuity, de ned later). In the same way as ( ), this variant is open to the problem with incompleteness in The Brave O cer-presented later. Moreover, in My Extended Pre-Division (presented later), this variant implausibly rules out that Young Wholly is I-related to Wholly unless Le y is I-related to Righty. For proof by contradiction, assume that Young Wholly is I-related to Wholly and that Le y isn't I-related to Righty. Since the non-branching clause in this variant of ( ) then cannot rule out that Young Wholly is I-related to Wholly, it's not the case that Young Wholly is R ′ -related to Le y while Wholly isn't I-related to Le y. Therefore, since Young Wholly is R ′ -related to Le y, we have that Wholly is I-related to Le y. But, given this variant of ( ), Wholly isn't I-related to Le y, since Wholly is R ′ -related to Righty and we assumed that Le y isn't I-related to Righty. the man who divides lived undivided for an extended period of time before the division: Young Wholly Wholly Le y Righty This is, of course, how My Division is usually understood; the di erence is merely that we highlight Wholly's extended past in the model. (The C ′ -relations between Young Wholly and Le y and between Young Wholly and Righty won't be crucial for our discussion, because Young Wholly would still be R-related both to Le y and to Righty if these connections were removed.) In My Extended Pre-Division, it seems that Young Wholly should be Irelated to Wholly. But this is ruled out by ( ). For proof by contradiction, assume that Young Wholly is I-related to Wholly. Since the non-branching clause in ( ) then cannot rule out that Young Wholly is I-related to Wholly, it is not the case that Young Wholly is R-related to Le y while Wholly isn't I-related to Le y. Therefore, since Young Wholly is R-related to Le y, we have that Wholly is I-related to Le y. According to ( ), Le y isn't I-related to Righty, since Le y isn't R-related to Righty. Therefore, since Wholly is R-related to Righty while Le y isn't I-related to Righty, the non-branching clause in ( ) rules out that Wholly is I-related to Le y. We then have the contradiction that Wholly both is and is not I-related to Le y.
Par t's nal suggestion was the following: This suggestion is better. In fact, it avoids all problems we have discussed so far. And, given ( ), Relation I is an equivalence relation-that is, it is re exive, symmetric, and transitive-over person-stages.
Commenting on Brueckner's ( , p. n ) proposal ( ), Par t ( , pp. -) suggests that it 'may be what we need' given that the continuity is taken to be temporally ordered rather than temporally unordered. Note that the double-headed arrow between Wholly and Old Wholly just represents a direct connection between these stages and not the existence of a further person-stage at t in addition to Le y and Righty. This connection won't matter for our discussion. For our discussion, we may just as well consider a variant without this connection: Wholly Le y

Righty Old Wholly
In My Temporary Division or My Forgetful Temporary Division, ( ) yields that the only I-related person-stages are Wholly and Old Wholly. To see this, note that, for all distinct pairs of person-stages other than the pair of Wholly and Old Wholly, one of Le y and Righty is R-related to one stage in the pair but not to the other. So, for these pairs, the non-branching clause in ( ) rules out that the I-relation holds. Wholly and Old Wholly, however, are R-related to the same person-stages. It seems odd that Wholly and Old Wholly would be I-related, because there is branching in the continuity between them (especially in My Forgetful Temporary Division). We can get around this problem with the following proposal from Sydney Shoemaker: Shoemaker and Swinburne ( , p. ). Shoemaker ( , p. ) hints at a similar proposal.

( ) Person-stage x is I-related to person-stage y if and only if xRy and there
is no person-stage z such that (i) z is not present before each of x and y is present, (ii) z is not present a er each of x and y is present, (iv) zRy, and (v) there are two distinct and simultaneous person-stages u and v such that zC ′ u and zC ′ v.
In My Temporary Division and My Forgetful Temporary Division, ( ) yields that all person-stages are I-unrelated to each other. According to ( ), Le y and Righty are not I-related, since they're not R-related. And neither Wholly nor Old Wholly is I-related to any person-stage given ( ), since they are both C ′ -related to the distinct and simultaneous Le y and Righty. Note, however, that this entails that Wholly and Old Wholly are not I-related to themselves. So ( ) violates the re exivity of the Relation I. Obviously, each person-stage should be I-related to itself. In My Extended Pre-Division, ( ) yields-just like ( )-the implausible result that Young Wholly isn't I-related to Wholly. Young Wholly's being Irelated to Wholly is ruled out by the non-branching clause in ( ), since Young Wholly is R-related to Wholly while Wholly is R-related to itself and is C ′related both to Le y and to Righty.
In The Senile General, the non-branching clause in ( ) doesn't apply, because there are no distinct and simultaneous person-stages in that case. So ( ) yields that The Boy is I-related both to The O cer and to The General (since The Boy is R-related to them) and that The O cer isn't I-related to The General (since they're not R-related). Hence we get a non-transitive Relation I and thus multiple-occupancy, which defeats the purpose of having a non-branching clause.
Finally, consider a variant of My Extended Pre-Division where Righty isn't connected to Wholly while Le y isn't connected to Young Wholly:

Young Wholly Wholly
Le y Righty -Suppose, for example, that the relevant kind of connections are memories and that the brain transplants cause irrevocable losses of memory: Le y remembers Wholly but can't remember Young Wholly; Righty remembers Young Wholly but can't remember Wholly. And suppose that the transplants happen between t and t ; so, at t , the brain halves are still united in Wholly. In My Forgetful Division, ( ) yields that Young Wholly is I-related both to Le y and to Righty, even though there is clearly branching here given that Le y and Righty are distinct and simultaneous person-stages. Clause (v) in ( ) doesn't hold in this case, because no person-stage is C ′ -related to each of two simultaneous person-stages.
Anthony Brueckner maintains that, in The Senile General, The O cer should be I-related to The General. To get this result, Brueckner argues that Relation I needs to be analysed in terms of temporally unordered continuity, rather than the temporally ordered variety. Let Relation R ′ be the temporally unordered variant of Relation R: Temporally Unordered Continuity Person-stage x is R ′ -related to person-stage y (xR ′ y) = df xC ′ y or there are person-stages z , z , . . . , z n such that xC ′ z , z C ′ z , . . . , z n− C ′ z n , z n C ′ y.
Given the re exivity and symmetry of Relation C ′ , Relation R ′ is re exive, symmetric, and transitive over person-stages.
Brueckner considers, but does not defend, the following temporally unordered variant of ( ): ( ) Person-stage x is I-related to person-stage y if and only if xR ′ y and there is no person-stage z such that either (i) xR ′ z and not yR ′ z or (ii) yR ′ z and not xR ′ z.
The main problem with ( ) is that, since Relation R ′ is already transitive, the non-branching clause in ( ) doesn't rule out anything: ( ) is equivalent to an account that identi es Relation I with Relation R ′ . So, in My Division, ( ) yields that all person-stages are I-related. Hence we have that Le y is I-related to Righty even though they are distinct and simultaneous person-stages, which seems wrong. Brueckner also considers a temporally unordered variant of ( ): Le y and Righty are distinct and simultaneous not only in external time but also in personal time; see note .
( ) Person-stage x is I-related to person-stage y if and only if xR ′ y and there is no person-stage z such that either (i) xR ′ z and y and z are distinct and simultaneous or (ii) yR ′ z and x and z are distinct and simultaneous.
A problem with ( ) is that it yields an implausible result in My Unbalanced Division, namely, that Wholly and Old Le y are the only person-stages that are I-related to each other. Given ( ), Wholly and Old Le y are I-related, since they are R ′ -related and neither of them is simultaneous with another person-stage. The non-branching clause in ( ) rules out that Le y or Righty is I-related to any person-stage, since Le y and Righty are distinct, simultaneous, and R ′ -related to each other. This also entails that Relation I isn't re exive given ( ), because it entails that Le y isn't I-related to Le y and that Righty isn't I-related to Righty. Harold W. Noonan amends ( ) as follows: ( ) Person-stage x is I-related to person-stage y if and only if xR ′ y and there are no distinct and simultaneous person-stages u and v such that either (i) uR ′ x, uR ′ y, and vR ′ x or (ii) uR ′ x, uR ′ y, and vR ′ y.
Yet, in My Unbalanced Division, ( ) does not yield the desired result that Le y is I-related to Old Le y. And, given ( ), Relation I still fails to be re exive, because ( ) yields that no person-stage in My Unbalanced Division is I-related to itself. To see this, note that all person-stages are R ′ -related in My Unbalanced Division. Hence the non-branching clause in ( ) yields that no person-stages are I-related, since each person-stage is R ′ -related to the distinct and simultaneous Le y and Righty. Consider once more My Unbalanced Division as diagrammed earlier together with an alternative diagrammatic representation of the case-where, unlike in the earlier diagrams, the double-headed arrows this time represent R ′ -relations rather than C ′ -relations: Since there is a temporally unordered chain of C ′ -relations linking all personstages in My Unbalanced Division, we get that all person-stages in My Unbalanced Division are R ′ -related. In terms of Relation R ′ , there is just us much branching between Wholly at t and the person-stages at t as there is between the person-stages at t and Old Le y at t . Rather than a tree-structured pattern of relationships, we have a collection of person-stages all of which are R ′ -related to all the others. Plausibly, Le y and Old Le y belong to the same branch while Righty and Old Le y do not. The trouble is that there is no way of accounting for this in terms of any sort of branching of Relation R ′ , because, in terms of Relation R ′ , Old Le y's relations to Le y are symmetrical with Old Le y's relations to Righty. Hence any approach, like ( ) or ( ), that tries to analyse non-branching personal identity just in terms of constructions out of Relation R ′ , will fail. What Noonan has in mind, however, is probably a variant of ( ) with temporally ordered, rather than unordered, continuity: Yi ( , pp. -) explores a series of analyses of xIy as xR ′ y in conjunction with a non-branching clause that isn't expressible in terms of Relation R ′ . His three preferred proposals all have the following form: Person-stage x is I-related to person-stage y if and only if (i) xR ′ y ('xRy' in Yi's notation), (ii) xRy ('xC * y or yC * x' in Yi's notation), and (iii) there are no distinct and simultaneous person-stages u and v such that . . .
First of all, requirement (i) is super uous given (ii). But the main problem is that proposals of this form do not preserve the transitivity of Relation I. To see this, consider The Senile General. In The Senile General, (iii) cannot fail to hold, since there are no distinct and simultaneous person-stages in that case. Since The O cer is R-related (and thus R ′ -related) to The Boy, we have that The O cer is I-related to The Boy. And, since The Boy is R-related (and thus R ′ -related) to The General, we have that The Boy is I-related to The General. But, since The O cer is not R-related to The General, (ii) rules out that The O cer is I-related to The General. Hence Relation I is non-transitive given proposals of this form.
( ) Person-stage x is I-related to person-stage y if and only if xRy and there are no distinct and simultaneous person-stages u and v such that either (i) uRx, uRy, and vRx or (ii) uRx, uRy, and vRy.
This proposal yields the desired result in My Unbalanced Division that Le y is I-related to Old Le y. Even so, ( ) yields the wrong result in My Extended Pre-Division. The non-branching clause in ( ) rules out that Young Wholly is I-related to Wholly. This is because Le y and Righty are distinct and simultaneous and Le y is R-related both to Young Wholly and to Wholly while Righty is R-related to Wholly. And ( ) yields that Young Wholly and Wholly are not I-related to themselves, because they are R-related to the distinct and simultaneous Le y and Righty. Hence ( ) violates the re exivity of Relation I.
Moreover, in The Senile General, ( ) yields that The Boy is I-related both to The O cer and to The General while The O cer is not I-related to The General. This is because there are no distinct and simultaneous person-stages in that case; so all person-stages are I-related except The O cer and The General. The O cer and The General are not I-related, since they're not R-related. Hence, as with ( ) and ( ), we get a non-transitive Relation I and thus multiple occupancy.
We have seen that the previously proposed non-branching clauses don't work. But there is, I shall argue, a better approach. I propose Given ( ), Relation I is an equivalence relation over person-stages. For a proof of re exivity, see Appendix A. For a proof of symmetry, see Appendix B. And, for a proof of transitivity, see Appendix C.
The idea behind the non-branching clause in ( ) is to characterize the intuitive idea of there being two person-stages u and v in di erent branches in the continuity between x and y. Clauses (i) and (ii) make sure that the relevant branching doesn't occur before or a er both x and y; so we avoid the problem ( ), ( ), and ( ) had with ruling out that, in My Extended Pre-Division, Young Wholly and Wholly are I-related. Clauses (iii) and (iv) make sure that u and v are part of the relevant continuities to or from x and y respectively. Finally, clause (v) makes sure that u and v belong to di erent branches in the sense that there is a person-stage that is continuous with one of them but not with the other. Clause (v) is similar to the non-branching clause in ( ), but, unlike the non-branching clause in ( ), it doesn't require that u and v are identical with x and y respectively; so it rules out, in My Temporary Division or My Forgetful Temporary Division, that Wholly is I-related to Old Wholly.
Note also that, ( ) doesn't rely on the identity or distinctness of personstages, which (  One might be worried about the result in The Senile General. First, one might think that The O cer should be I-related to The General. If so, one might be tempted to rely on Relation R ′ rather than Relation R; but, as I argued earlier, relying on temporally unordered continuity doesn't t with some plausible ideas about branching in cases like My Unbalanced Division. If there is a possible case structured like The Senile General such that it seems that the stage corresponding to The O cer should be I-related to the stage corresponding to The General, then a more promising approach is to revise the criteria for what counts as the relevant kind of connectedness so that these stages will be connected. Brueckner ( , pp. -) rejects ( ) due to this worry about circularity. Others, such as Noonan ( , pp. -), do not share Bruckner's worry. See, however, Brueckner and Buford ( , pp. -) for a reply. This kind of reliance on the identity or distinctness of person-stages is also a problem for Yi's ( , pp. -) proposals; see note . If we replace our perdurance framework with an endurance framework, these claims about person-stages being distinct becomes claims about persons being distinct-which raises the worry about circularity when they appear in an account of personal identity. See the endurance translations ( *), ( *), ( *), ( *), ( *), and ( *) in Appendix D. Brueckner ( , p. ). This is the approach I defend in Gustafsson ( ). I defend the view that the relevant kind of connectedness is phenomenal connectedness and, accordingly, that the relevant kind of continuity is phenomenal continuity, that is, the relation of sharing the same stream of consciousness. In addition, I defend the view that phenomenal connections can hold over temporal gaps such as periods of dreamless sleep. Regarding The Senile General, it seems to me that nothing in the standard memory-loss story for this case rules out that there is temporally ordered phenomenal continuity holding between The O cer and The General. In the rare Second, one might think that The Boy should be I-related to The O cer. And one might think that whether The Boy at t is I-related to The O cer at t shouldn't depend on what person-stages The General at t (a er both t and t ) is connected to. If The General had a connection to The O cer like in The Brave O cer, then The Boy would be I-related to The O cer. For similar reasons, one might think that, in My Forgetful Division, Young Wholly should be I-related to Wholly. As long as the connections between person-stages are structured like they are in these cases and we rely on temporally ordered continuity, it is hard to deny that the continuity branches between t and t . If this dependence on the future is implausible, there is a more promising way to avoid such dependence than to rely on temporally unordered continuity, namely, to restrict the relevant kinds of connectedness so that it only holds between person-stages without any temporal gaps between them. Given this restriction, the connection in The Senile General between The Boy and The General would be invalidated, and then ( ) would yield that The Boy is I-related to The O cer. Similarly, in My Forgetful Division, this restriction would invalidate the connection between Young Wholly and Righty, and then ( ) would yield that Young Wholly is I-related to Wholly.

A. Re exivity
To see that Relation I is re exive over person-stages given ( ), assume that a is a person-stage. Note rst that, since R is re exive, we have aRa. Replacing x by a and y by a in ( ), any person-stages u and v that are going to make clauses (i)-(iv) true, must be simultaneous with a. But then-given uRa, vRa, and that u, v, and a are simultaneous-we have that u and v must be R-related to the same person-stages, because, if a person-stage is related by temporally ordered continuity to one of u and v, this continuity could be extended to the other of u and v without breaking the temporal order. But, if u and v are R-related to cases where phenomenal connections really are structured in The Senile General pattern, it seems, I think, that the result of ( ) is plausible (that is, the result that The Boy, The O cer, and The General are all I-unrelated to each other), for in that case the stream of consciousness of The Boy would split into two: one stream including the experiences of The O cer but not those of The General and one stream including the experiences of The General but not those of The O cer.
Yi ( , p. ; , pp. -). Yi discusses a case similar to My Forgetful Fission with a psychological kind of connectedness (it's unclear whether Young Wholly is C ′ -related to Le y in his case) where Young Wholly is scanned at t and Righty is a replica created from this scan at t . Yi's complaint is that, if Young Wholly isn't I-related to Wholly, then a mere scan terminates the person of which Young Wholly is a stage. But it's not the mere scan that is to blame; it's the scan combined with the creation of Righty from this scan that rules out Young Wholly being I-related to Wholly.
As mentioned in note , this is not the approach I favour.
the same person-stages, then u and v would not make clause (v) true. Thus, replacing x by a and y by a in ( ), no person-stages u and v can make clauses (i)-(v) all true. So we have aRa and that the non-branching clause in ( ) doesn't apply. Hence ( ) entails aIa. Since we have derived aIa with the help of ( ) without having assumed anything about a other than that a is a person-stage, we have that Relation I is re exive over person-stages given ( ).

B. Symmetry
To see that Relation I is symmetric over person-stages given ( ), assume, for proof by contradiction, that a and b are person-stages such that a is I-related to b but b isn't I-related to a. From aIb, we have, given ( ), aRb. Then, since Relation R is symmetric, we have bRa. Since bRa even though b isn't I-related to a, there must be some person-stages u and v such that, replacing x by b and y by a in ( ), clauses (i)-(v) are all true. From clauses (iii) and (iv), we then have uRb and vRa. From uRb and bRa, we have, since Relation R is transitive, uRa. Likewise, from vRa and aRb, we have vRb. Given uRa and vRb, we have that, replacing x by a and y by b in ( ), clauses (i)-(v) are all true. To see this, note that, given uRa and vRb, clauses (iii) and (iv) are true, and the other clauses are equivalent a er the swap of a and b in these clauses. Hence we have that, replacing x by a and y by b in ( ), the non-branching clause applies. Hence a isn't I-related to b, which contradicts our assumption that a is I-related to b. Since we have derived a contradiction from the assumption that a is I-related to b and b isn't I-related to a, we have that Relation I is symmetric over person-stages given ( ).

C. Transitivity
To see that Relation I is transitive over person-stages given ( ), note rst that, if person-stages x and y are I-related, then, replacing u by x and v by y in ( ), we have that clauses (i)-(iv) hold. And then clause (v) cannot hold, because, if it did, the whole non-branching clause would apply and rule out xIy. So, if we have xIy, we also have that there is no person-stage z such that z is R-related to one of x and y but not to the other. Hence we have the rst observation that, if two person-stages are I-related, every person-stage that is R-related to one of them is also R-related to the other. Assume that a, b, and c are person-stages such that aIb and bIc. From aIb and the rst observation, we have that a and b are R-related to the same person-stages. And, from bIc and the rst observation, we have that b and c are R-related to the same person-stages. Hence we have the second observation that a and c are R-related to the same person-stages.
Let us say that a person-stage is present in the a-c interval if and only if that stage is neither present before each of a and c nor present a er each of a and c. Assume that person-stage d is present in the a-c interval and that d is R-related to at least one of a and c. And assume that person-stage e is R-related to d. Since d is R-related to at least one of a and c, we have, by the second observation, both aRd and cRd. Since we have aRd, cRd, and that d is present in the a-c interval, we have one of two cases: rst, that d is simultaneous with at least one of a and c and, second, that d is present a er one of a and c, present before the other, and not simultaneous with either. We shall consider these cases in turn.
Consider the rst case: that d is simultaneous with at least one of a and c. Let a simultaneous a/c stage be a stage that is simultaneous with d and identical to either a or c. The temporally ordered continuity from e to d can, without breaking the temporal order of this continuity, be extended to any person-stage that is both R-related to d and simultaneous with d. Hence, in this case, we have that e is R-related to a simultaneous a/c stage. Since e is then R-related to at least one of a and c, we have, by the second observation, both eRa and eRc. For any person-stage, if there is temporally ordered continuity from that person-stage to a simultaneous a/c stage, then this continuity can be extended, without breaking the temporal order, to the simultaneous d. Therefore, by the second observation, we also have that d is R-related to all person-stages that a and c are R-related to. Now, consider the second case: that d is present a er one of a and c, present before the other, and not simultaneous with either. If the temporally ordered continuity from one of a and c to d approaches d from one direction in time, then the temporally ordered continuity from the other of a and c to d approaches d from the other direction in time. The temporally ordered continuity from e to d can therefore be extended to at least one of a and c without breaking the temporal order of this continuity. Hence e is R-related to at least one of a and c. Then, by the second observation, we have eRa and eRc. As mentioned, the temporally ordered continuities from a to d and from c to d approach d temporally from di erent directions. Therefore, by the second observation, we then have that, for any person-stage that is R-related to a or c, there is temporally ordered continuity from at least one of a and c such that this continuity can be extended, without breaking the temporal order, to d. Therefore, by the second observation, we also have that d is R-related to all person-stages that a and c are R-related to.
Hence, in either case, e is R-related both to a and to c. And each personstage that is R-related to one of a and c is R-related to d. Since we haven't assumed anything about d and e other than that d is present in the a-c interval and R-related to e and to at least one of a and c, we have the third observation that all person-stages present in the a-c interval which are R-related to one of a and c are R-related to the same person-stages.
Since Relation R is re exive, we have aRa. So, by the second observation,