Young Socrates is the same person as old Socrates, but, differing in age, they cannot be absolutely identical. Leibnizian substitutivity of identity|one of the two great principles of absolute identity|fails for the terms \young Socrates" and \old Socrates". For substituting the former term for the latter one in the sentence \Old Socrates was sentenced to death" yields a falsehood; young Socrates has not been sentenced to death. Thus identity over time cannot be a species of absolute or Leibnizian identity. It's got to be a species of relative identity, then. The purpose of this paper is to o er an event-ontological clari cation of this notion.
Event-ontology treats natural individuals as events in space and time. Since the spatial dimension of event-ontology is still under construction I have to con ne my treatment to their temporal dimension. All I need for doing so is the notion of a temporal ordering, that is an ordered pair hT; <i, consisting of a non-empty set T and a two-place relation < on T . The non-empty set is comprised of moments or instants of time; and the two-place relation is the earlier/later relation. Globally we shall not require more than that it be a subset of T T ; but locally it may be irre exive, transitive and branching towards the future. Because we permit branching some people might prefer to characterize the elements of T as tempo-modal rather than as temporal. In any case, my main concern is with their point-likeness. My notion of a point is not Euclidean; for Euclidean points do not have parts. Points in my sense do because in my ontology something is a point of a given type i it can be bisected so that none of the two parts is of that type again. So the elements of the set T are required to be points of type T . Hence they may be years or days or milliseconds because none of the parts of these stretches of time are years or days or milliseconds in turn. The point requirement prohibits that material from T may be used to construct temporal entities which can be found among this very material. The upshot of our requirement is a sharp distinction between elements of construction and constructed items.
Starting from our rather thin notion of a temporal ordering, we shall rst introduce the concept of biography. This notion is needed in order to say what it means to age. Aging is kindred to living. For he who is living has lived and therefore must have grown older. As it appears aging or, for that matter, living is an event that combines occurring with having occurred. This combination lies at the heart of an individual's biography. Moreover, natural individuals have options for their future being. This presupposes the availability of alternative future courses of events. But having such options would be futile if we could not live to see these events occur. Hence biographies seem to branch towards the future and to exhibit a kind of hand-likeness.
Roughly speaking, biographies are events which contain temporal stretches of growing duration, each stretch corresponding to a particular age. This intuitive idea can be spelled out by a couple of de nitions starting out from a rigorous de nition of \stretch of time" or, as I prefer to say, \occurrence". For events are sets of occurrences. The event called \sunset", for example, is the totality of the particular episodes which consist in the sun's setting.
De nition 1. \occurrence" v is an occurrence on hT; <i i (a) v T and v 6= ;; (b) for all t; t0 2 v: t < t0 or t = t0 or t0 < t; (c) for all t; t0 2 v: if t < t0 then not t0 < t; (d) for all t; t0; t00 2 v: if t < t0 and t0 < t00 then t < t00; (e) for all t; t00 2 v and t0 2 T : if t < t0 and t0 < t00 then t0 2 v.
A set of occurrences on hT; <i will be called \a (generic) event on hT; <i".
Consider an event that resembles the event of aging in that every occurrence harbours each of its initial segments. Living is such an event; opening a door is another one. And if Aristotle is correct even seeing belongs to this kind of events. He called events exhibiting this temporal structure \energeiai " [1, 6]. I propose to use the adjective \entelic"1 for them which may be de ned as follows:
is (an) entelic (event) on hT; <i i (a) is a (generic) event on hT; <i; (b) for all v 2 and for all occurrences v0 v: if there is no t 2 v such that t < v0 then v0 2 .2 Although each biography has a unique past it is open to several future developments. Hence all occurrences belonging to one and the same biography must have the same past. Moreover, belonging to the same biography, they must have the same origin in time, that is, they must share an initial segment. But their 1 In our days some people classify them as telic [2] and some people as atelic [3]. Since I do not want to stir up the wrong associations in my readers I suggest to use a completely new term which still has an Aristotelian ring about it. 2 The formula \t 2 v0" is short for \for all t0: if t0 2 v0 then t < t0". behaviour vis-a-vis the future can be left undetermined. Therefore biographies present themselves in the form of trees or hands.
is (a) hand-like (event) on hT; <i i (a) is a (generic) event on hT; <i; (b) for all v; v0 2 and t 2 T : t < v i t < v0; (c) for all v; v0 2 : (v \ v0) 6= ;. A (generic) event on hT; <i which is non-empty, entelic, and hand-like will be called \biography on hT; <i".
If a temporal ordering hT; <i is to contain biographies it must possess a subset S such that < is asymmetric, transitive, fusionless3 and pentachotomous on S, as is clear from the following theorem:
is a biography on hT; <i then < has the following properties Theorem 1. If on S : (a) asymmetry: 8t; t0 2 S (b) transitivity: 8t; t0; t00 2 S (c) fusionlessness: 8t; t0; t00 2 S (d) pentachotomy: 8t; t0 2 S 9t00 2 S (t < t0 ! :t0 < t); ((t < t0 ^ t0 < t00) ! t < t00); ((t < t00 ^ t0 < t00) ! (t = t0 _ t < t0 _ t0 < t)); (t < t0 _ t = t0 _ t0 < t _ (t < t00 _ t0 < t00) _ 9t00 2 S (t00 < t _ t00 < t0)).4 There are di erent kinds of biographies. Some of them do not only grow in time but also spread over alternative histories. Some of them are at like a line and might therefore be called \linear". Here is a de nition of this notion:
is a linear biography on hT; <i i (a) If two linear biographies on hT; <i have a non-empty intersection this intersection is a linear biography on hT; <i.5 is a biography on hT; <i; (b) S Natural individuals are entities with both a temporal and a spatial dimension. Their temporal dimension is a biography. Their spatial dimension consists in a trajectory through space. To simplify matters I'll reduce the trajectory of a given individual to its birthplace. My reason for this reduction is that the exploration of the spatial dimension of natural individuals is still work in progress.
Since our experience presents time and temporal episodes to us in a linear order we may forget about individuals whose biographies are voluminous like trees or hands and concentrate on individuals with a linear biography. They might aptly be called \l-individuals". Here is a de nition of them: De nition 5. \l-individual" h ; i is a l-individual on hT; <i i (a) the birthplace of h ; i. is a linear biography on hT; <i; (b) is
4 For a proof see [6, theorem 5.1].
5 This holds even for biographies in general. For a proof of the general claim see [6,
lemma 5.2 (
Although a l-individual is set-theoretically included in a single linear history its intuitive counterpart may be found in di erent alternative histories. This brings me to the gist of my argument. Since l-individuals are set-theoretical constructs they possess a set-theoretical identity. It is well known that this identity consists in identity of extension. But this is only one way of construing relative identity; identity of origin is another one. There are spatiotemporal origins and there are material origins. Since I do not consider the material dimension of natural individuals in this paper I want to focus on identity of spatiotemporal origin.
It is characteristic of l-individuals that they bear being divided. Take Mary's little lamb. If we cut o a piece of its skin we'll have two parts, namely one part being a lamb and the other one not. This procedure can be applied to lindividuals in particular as well as to individuals in general. We may even de ne an individual of type T to be something that can be cut into two pieces such that just one of them is of type T , again. If we bisect a biography which is long enough to be divided at all we shall nd out that it is an individual of type biography.6
Take two occurrences of one and the same biography. They must contain at least one initial occurrence in common, since sharing an initial occurrence guarantees a biography's identity over time. For the same reason it guarantees its identity across (possible) worlds or, if you prefer, world histories. Therefore we may consider two set-theoretically di erent biographies to be one and the same biography i their set-theoretical intersection is not empty. The relation of being the same biography is an equivalence relation on the set of biographies.7
Since a l-individual is composed of a biography and a birthplace we may de ne its identity as follows:
h ; i is the same l-individual on hT; <i as h 0; 0i i (a) \ 0 is a linear biography on hT; <i; (b) = 0.
According to clause (a) and 0 must share at least one occurrence. Of course, this occurrence must include the origins both of and of 0. Therefore our de nition of l-individual identity realizes the idea of identity as identity of origin.
Let's condense these remarks into another theorem: Theorem 2. L-individual identity on hT; <i is an equivalence relation on the set of all l-individuals on hT; <i.8 5
Persons are l-individuals who are endowed with a capability to choose. Typical specimens of persons are adult human beings. They are able to compare the
value of objects, properties, relations, states of a airs, events, actions, reasons, feelings, and so on, and so forth. They are able to evaluate both themselves and one another. They are able to estimate the quality of their lives and to choose among shorter or longer periods thereof. They are able to judge episodes in the past and to distinguish between good and bad actions to be done in the future. I propose to call their capability to compare, to evaluate, to estimate, to choose, to judge, to distinguish, etc. \character". It would be a gross mistake to conclude from this sketchy explanation that my understanding of the word \character" deviates considerably from its usual meaning. Usually, character is taken to be \the particular nature which distinguishes human beings from one another" [5, s. v. \Charakter "]. That is exactly what I have in mind.
People say things like \She is a good mother", \Your reply was good", \The new knife cuts well", \It's not good that you did not show up yesterday", thereby informing us about the results of their comparisons, evaluations, estimations, choices, judgments, and distinctions. As these examples suggest, and are meant to suggest, people can be characterized by what they choose to be good. This fact motivates my conception of character as something to do with choosing what is taken to be good. Let me emphasize that I am far from restricting a person's choices to what is socially, let alone morally, good; I want to include everything that she considers to be a good alternative in one respect or another. As I understand the adjective \good", its uses by default express the user's individual choices. So they unveil her particular nature and make it publicly accessible to everybody.
These considerations suggest that the study of the adjective \good" plays an important methodical role in our enquiry into the nature of character. Let's have a closer look at it, then, and consider the statement \Good apples are apples". Of course, that's true. Good apples do form a part of the class of apples. Extrapolating from this fact, we may describe what the adjective \good" stands for as the device which assigns to a given totality a uniquely determined particular choice thereof. Nobody would want to deny that good apples are a choice of apples, to be sure; but why should this choice be unique? Why identify the classes of apples which are good with respect to taste and which are good with respect to look? Obviously, the results of our choices depend on the respect of comparison. But that is not enough. We have to take into account the moment of choice as well. Thus we arrive at the conclusion that \good" stands for a threeplace choice function g which maps a moment t, a respect r and a comparison class9 X onto a subset of X.
The fact that g is a choice function can be rendered by the following postulate: g(t; r; X)
X :
(
If X
Y and g(t; r; X) 6= ; then g(t; r; Y ) 6= ; :
If x; y 2 X then (g(t; r; X) \ fx; yg)
If x 2 g(t; r; fx; y; zg) then g(t; r; fx; yg)
g(t; r; fx; yg) :
g(t; r; fx; y; zg) :
(
x 2 g(t; r; fx; yg) and y 2= g(t; r; fx; yg) .
At t and with respect to r, x and y are equally good :=
x 2 g(t; r; fx; yg) i y 2 g(t; r; fx; yg) .
Our next theorem shows that our four choice postulates are su cient for
establishing the semantic connection between \good" and its comparatives:
Theorem 3. Every function which satis es postulates (
g is the character of h ; i on M i h ; i is a l-individual on hT; <i; (b) M S ;
(c) g is a (possibly partial) choice function whose temporal arguments are in M
and which satis es postulates (
The character of h ; i on S will be called \the character of h ; i".
Unfortunately we cannot de ne persons as l-individuals endowed with a character because this de nition does not make personal identity into an equivalence relation. For consider the following line diagram and suppose that each of the three persons 1, 2 and 3 reaches from its beginning to the very end of the line that is tagged with her name and that all of them have the same birthplace. Since 1, 2 and 3 share the occurrence that reaches from the beginning of the gure to the rst branching point the intersections of their biographies are not empty. Therefore and because 1, 2 and 3 have the same birthplace they are one and the same l-individual. In particular, 3 is the same l-individual as 1 10 For a proof see [6, theorem 5.16]. PPs PPPPPPPP t1
PPP 1 2 3 and 2 notwithstanding the fact that 3 does not occur at t2. Otherwise it were not one and the same person who, at t1, has three options.
But being the same l-individual is not su cient for being the same person. Since the character of a person is de ned on her biography, 1, 2 and 3 may choose di erent things from the same class and in the same respect at t1; and 1 and 2 may choose di erent things from the same class and in the same respect at t2. If so our three l-individuals have di erent characters and therefore di erent personal identities. But this would permit a person's possible future past a given moment to in uence her choice at this very moment. If we want to preclude this pathological kind of backward in uence we should see to it that a person's choice at a given moment does not depend on her possible futures. Let's call persons whose characters have this property \common persons". This notion may be de ned as follows: De nition 8. \common person" h ; i is a common person on hT; <i i (a) h ; i is a l-individual on hT; <i; (b) there is a g such that g is the character of h ; i on S and for all lindividuals h 0; 0i, characters h, moments t, respects r and classes X of comparison holds: if (i) h 0; 0i is the same l-individual on hT; <i as h ; i, (ii) h is the character of h 0; 0i on S 0, (iii) t 2 (S \ S 0), and (iv) ht; r; Xi 2 dom(g) then g(t; r; X) = h(t; r; X).
Given this notion of a common person we de ne the pertaining identity relation as follows:
h ; i is the same (common) person on hT; <i as h 0; 0i i (a) h ; i is the same l-individual on hT; <i as h 0; 0i; (b) there are g and g0 such that g is the character of h ; i on S , g0 is the character of h 0; 0i on S 0, and g \ g0 is the character of h \ 0; i on S( \ 0).11 That this de nition induces an equivalence relation is the message of our nal theorem: Theorem 4. Personal identity on hT; <i is an equivalence relation on the set of all common persons on hT; <i.12 11 Clause (a) implies that = 0 so that it doesn't matter whether we take h \ 0; i or h \ 0; 0i as the bearer of the character g \ g0. 12 For a proof see [6, theorem 5.17]. As may be gathered from the very last conjunct of de nition 9, identity of character is a matter of origin, too. Therefore personal identity itself, being composed of l-individual identity and identity of character, is a kind of identity of origin. 6
The goal of my paper was to clarify the notion of identity over time. Because of time restrictions, I con ned myself to elucidating the notion of personal identity which is the most interesting kind of identity over time in philosophy. If we treat young and old Socrates as common persons we can claim both that young Socrates 6= old Socrates and that nevertheless young Socrates is the same person as old Socrates without contradicting ourselves.
Where else can we apply our results? Being myself a philosopher, I must forewarn you that my answer might need some professional rephrasing before it may appear plausible to you. People in the software business tell me that simulation may be an appropriate eld of application. The objects of a simulation fare like persons in my sense. After instantiation, which is comparable to a person's birth, the objects live on without a corporeal component. Only some medium or other is needed in order to store their properties. Now, a person possesses a history, called \biography", and the capability to choose from alternatives, called \character". Since her character may change over time she is able to learn and so are the objects of a simulated social system.
Another eld of application is software engineering. Any use of software leaves some traces in the form of adjustments, installations of extensions, service packs, patches etc. Normally, people tend to forget that every software has its own history. Taking its history to be a biography, the software could be treated as a kind of l-individual. This treatment might contribute to solve problems that arise in software identi cation.
Acknowledgments. I want to thank Reinhold Kienzle, Herzogenaurach, as well as the anonymous referees for their comments. The presentation of this paper was made possible through a travel allowance by Exzellenzforderinitiave Mecklenburg-Vorpommern.