=Paper=
{{Paper
|id=Vol-460/paper-11
|storemode=property
|title=Identity over Time
|pdfUrl=https://ceur-ws.org/Vol-460/paper11.pdf
|volume=Vol-460
}}
==Identity over Time==
<pdf width="1500px">https://ceur-ws.org/Vol-460/paper11.pdf</pdf>
<pre>
                          Identity over Time

                                  Bertram Kienzle

                                 Universität Rostock
                               Institut für Philosophie
                           bertram.kienzle@uni-rostock.de



      Abstract. Young Socrates is the same person as old Socrates, but, dif-
      fering in age, they cannot be absolutely identical. Leibnizian substitutiv-
      ity 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 sen-
      tenced to death” yields a falsehood; young Socrates has not been sen-
      tenced 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 offer an event-ontological clarification of
      this notion.

      Key words: biography, character, event, identity over time, person, per-
      sonal identity, relative identity


1   Introduction

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
confine 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 irreflexive, 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 iff 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.
114     Proceedings of ONTOSE 2009

2     Biographies
Starting from our rather thin notion of a temporal ordering, we shall first in-
troduce 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 definitions starting out from a rigorous
definition 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.
Definition 1. “occurrence”
v is an occurrence on hT, <i iff
(a) v ⊆ T and v 6= ∅;
(b) for all t, t0 ∈ v: t < t0 or t = t0 or t0 < t;
(c) for all t, t0 ∈ v: if t < t0 then not t0 < t;
(d) for all t, t0 , t00 ∈ v: if t < t0 and t0 < t00 then t < t00 ;
(e) for all t, t00 ∈ v and t0 ∈ T : if t < t0 and t0 < t00 then t0 ∈ 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, Θ 6]. I propose
to use the adjective “entelic”1 for them which may be defined as follows:
Definition 2. “entelicity”
 is (an) entelic (event) on hT, <i iff (a)  is a (generic) event on hT, <i; (b) for
all v ∈  and for all occurrences v 0 ⊆ v: if there is no t ∈ v such that t < v 0 then
v 0 ∈ .2
Although each biography has a unique past it is open to several future devel-
opments. 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 ∈ v 0 ” is short for “for all t0 : if t0 ∈ v 0 then t < t0 ”.
                                                 Proceedings of ONTOSE 2009           115

behaviour vis-à-vis the future can be left undetermined. Therefore biographies
present themselves in the form of trees or hands.
Definition 3. “hand-likeness”
 is (a) hand-like (event) on hT, <i iff (a)  is a (generic) event on hT, <i; (b) for
all v, v 0 ∈  and t ∈ T : t < v iff t < v 0 ; (c) for all v, v 0 ∈ : (v ∩ v 0 ) 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:
Theorem
   S        1. If κ is a biography on hT, <i then < has the following properties
on κ:
                    ∀t, t0 ∈ κS(t < t0 → ¬t0 < t);
                                S
(a) asymmetry:
(b) transitivity:   ∀t, t0 , t00 ∈ S κ ((t < t0 ∧ t0 < t00 ) → t < t00 );
(c) fusionlessness: ∀t, t0 , t00S∈ κ ((t < t00 ∧ t0 < t00 ) → (t = t0 ∨ t < t0 ∨ t0 < t));
(d) pentachotomy: ∀t, t0 ∈S κ (t < t0 ∨ t = t0 ∨ t0 < tS∨
                    ∃t00 ∈ κ (t < t00 ∨ t0 < t00 ) ∨ ∃t00 ∈ κ (t00 < t ∨ t00 < t0 )).4
There are different kinds of biographies. Some of them do not only grow in time
but also spread over alternative histories. Some of them are flat like a line and
might therefore be called “linear”. Here is a definition of this notion:
Definition 4. “linear biography”                                         S
κ is a linear biography on hT, <i iff (a) κ is a biography on hT, <i; (b) κ ∈ κ.
If two linear biographies on hT, <i have a non-empty intersection this intersection
is a linear biography on hT, <i.5

3    Natural Individuals
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 definition of them:
Definition 5. “l-individual”
hκ, γi is a l-individual on hT, <i iff (a) κ is a linear biography on hT, <i; (b) γ is
the birthplace of hκ, γi.
3
  Sometimes called “left linear” or “linear in the past”.
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 (2)].
116     Proceedings of ONTOSE 2009

4     L-individuals and their Identity
Although a l-individual is set-theoretically included in a single linear history its
intuitive counterpart may be found in different 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 off 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 l-
individuals in particular as well as to individuals in general. We may even define
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 find 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 different biographies to be one and the
same biography iff 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
define its identity as follows:
Definition 6. “l-individual identity”
hκ, γi is the same l-individual on hT, <i as hκ0 , γ 0 i iff (a) κ∩κ0 is a linear biography
on hT, <i; (b) γ = γ 0 .
According to clause (a) κ and κ0 must share at least one occurrence. Of course,
this occurrence must include the origins both of κ and of κ0 . Therefore our
definition 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     Characters, Persons and their Identity
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
6
  For a proof see [6, lemma 5.3].
7
  The proof of this claim is an immediate corollary to theorem 5.3 in [6].
8
  For a proof see [6, theorem 5.6].
                                                Proceedings of ONTOSE 2009         117

value of objects, properties, relations, states of affairs, 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 three-
place 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 .                                 (1)

It is pretty clear that there must be some further postulates which regiment the
behaviour of our choice function; and it is equally clear that they will restrict
the chooser’s predilections and aversions. But which restrictions are tolerable?
Since there is a semantical connection between the positive “good” on the one
9
    This technical term for the class to choose from was coined by R. M. Hare in [4].
118       Proceedings of ONTOSE 2009

hand and the comparatives “better” and “equally good” on the other we should
accept only postulates that contribute to establishing this connection. Thus we
eventually arrive at the following postulates:

              If X ⊆ Y and g(t, r, X) 6= ∅ then g(t, r, Y ) 6= ∅ .                     (2)
              If x, y ∈ X then (g(t, r, X) ∩ {x, y}) ⊆ g(t, r, {x, y}) .               (3)
              If x ∈ g(t, r, {x, y, z}) then g(t, r, {x, y}) ⊆ g(t, r, {x, y, z}) .    (4)

Let’s define the comparatives “better” and “equally good” as follows:
         At t and with respect to r, x is better than y :=
                                     x ∈ g(t, r, {x, y}) and y ∈ / g(t, r, {x, y}) .
         At t and with respect to r, x and y are equally good :=
                                     x ∈ g(t, r, {x, y}) iff y ∈ g(t, r, {x, y}) .

Our next theorem shows that our four choice postulates are sufficient for estab-
lishing the semantic connection between “good” and its comparatives:
Theorem 3. Every function which satisfies postulates (1)–(4) induces on any
given set S an ordering h better, equally good i with the following properties:
(a) better is transitive on S;
(b) equally good is an equivalence relation on S;
(c) any two elements of S stand in just one of these two relations.10
Obviously, sometimes our choices are identical, sometimes they are different, and
sometimes there is no choice at all. The last possibility is due to the fact that
we do not always choose. Sometimes we are asleep, and sometimes we are not
choosing because we are busy with something else. In order to account for the
possibility of not choosing we are well advised to model the notion of character
with partial choice functions. Thus we arrive at the following definition:

Definition 7. “character”                                                        S
g is the character of hκ, γi on M iff hκ, γi is a l-individual on hT, <i; (b) M ⊆ κ;
(c) g is a (possibly partial) choice function whose temporal arguments are in M
and which satisfies postulates (1)–(4).
                              S
The character of hκ, γi on κ will be called “the character of hκ, γi”.
    Unfortunately we cannot define persons as l-individuals endowed with a char-
acter because this definition does not make personal identity into an equivalence
relation. For consider the following line diagram and suppose that each of the
three persons π1 , π2 and π3 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 π1 , π2 and π3 share the occurrence that reaches from the beginning of the
figure to the first branching point the intersections of their biographies are not
empty. Therefore and because π1 , π2 and π3 have the same birthplace they are
one and the same l-individual. In particular, π3 is the same l-individual as π1
10
     For a proof see [6, theorem 5.16].
                                                 Proceedings of ONTOSE 2009          119


                                           t2 
                                                      π1
                                            s P
                                s 
                                                 PP
                              PP
                                  PP                     π2
                               t1    PP
                                             PP
                                               PP
                                                         π3

                         Fig. 1. Identity and possible futures


and π2 notwithstanding the fact that π3 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 sufficient for being the same person.
Since the character of a person is defined on her biography, π1 , π2 and π3 may
choose different things from the same class and in the same respect at t1 ; and
π1 and π2 may choose different things from the same class and in the same
respect at t2 . If so our three l-individuals have different characters and therefore
different personal identities. But this would permit a person’s possible future
past a given moment to influence her choice at this very moment. If we want to
preclude this pathological kind of backward influence 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 defined as follows:
Definition 8. “common person”
hκ, γi is a common person on hT, <i iff (a) hκ, γi is a l-individual
                                                                   S         on hT, <i;
(b) there is a g such that g is the character of hκ, γi on κ and for all l-
individuals hκ0 , γ 0 i, characters h, moments t, respects r and classes X of com-
parison holds: if (i) hκ0 , γ 0 iSis the same l-individual
                                                      S on hT, <i as hκ, γi, (ii) h is
the character of hκ0 , γ 0 i on κ0 , (iii) t ∈ ( κ ∩ κ0 ), and (iv) ht, r, Xi ∈ dom(g)
                                                S
then g(t, r, X) = h(t, r, X).
Given this notion of a common person we define the pertaining identity relation
as follows:
Definition 9. “personal identity”
hκ, γi is the same (common) person on hT, <i as hκ0 , γ 0 i iff (a) hκ, γi is the
                                  0   0
same l-individual on hT,
                       S <i 0as hκ , γ i; (b) there are g andSg0 such that g is the
character of hκ, γi on κ, Sg is the character of hκ , γ 0 i on κ0 , and g ∩ g0 is the
                                                     0

character of hκ ∩ κ , γi on (κ ∩ κ0 ).11
                    0


That this definition induces an equivalence relation is the message of our final
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 γ = γ 0 so that it doesn’t matter whether we take hκ ∩ κ0 , γi
   or hκ ∩ κ0 , γ 0 i as the bearer of the character g ∩ g0 .
12
   For a proof see [6, theorem 5.17].
120     Proceedings of ONTOSE 2009

As may be gathered from the very last conjunct of definition 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     Concluding Remarks
The goal of my paper was to clarify the notion of identity over time. Because of
time restrictions, I confined 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 fore-
warn 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 field 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 “char-
acter”. Since her character may change over time she is able to learn and so are
the objects of a simulated social system.
    Another field 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 identification.

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 Exzellenzförderinitiave
Mecklenburg-Vorpommern.


References
[1] Aristotle: Metaphysica. W. Jaeger (ed.) Clarendon, Oxford (1957).
[2] Comrie, Bernard: Aspect. Cambridge UP, Cambridge (1976).
[3] Galton, Antony: The Logic of Aspect. Clarendon, Oxford (1984).
[4] Hare Richard M.: The Language of Morals. Clarendon, Oxford (1952).
[5] Historisches Wörterbuch der Philosophie, vol. 1. Ritter, J., Gründer, K., Gabriel,
   G. (eds.) Schwabe, Basel (1971, 2 2005).
[6] Kienzle, Bertram: Die Bestimmung des Janus. Mohr Siebeck, Tübingen (2007).
[7] Müller, Thomas: Arthur Priors Zeitlogik. Mentis, Paderborn (2002).

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