Theory of Levels in Personal Identity icon

Theory of Levels in Personal Identity

НазваниеTheory of Levels in Personal Identity
Дата конвертации14.09.2012
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Theory of Levels in Personal Identity


We would like to offer some more strict description of a structure for the demonstration of basic positions in Personal Identity (PI) problem.

From our point of view there exist three main levels of PI structures. These are the follows:

1. Level of time-instants expressions of persons. These can be physical body, or its parts, for example brain, states of consiousness, etc., when all these demonstrations of persons exist during one moment of time. We shall denote all these instantaneous expressions of persons as 1-possible worlds, or simply 1-worlds, w1. Every 1-world is defined in a moment t such that one can used the following explicit notation for this circumstance: w1(t). Hence we also suppose that a set of different moments of time is defined, for example, set T = [t0,tK], where relation of order “<” is defined on the pairs of time moments, and t < t’ means “moment t is before of moment t’ ”. Therefore a map W1 : T  W1 is defined here, where W1 is set of all 1-worlds, W1(ti) is set of all 1-worlds at a moment ti, i.e., W1(ti) = {w1(t): t = ti}. Further we shall use the symbol “W1(ti)” for set W1(ti). We can accept that W1 is union of all sets W1(ti), where ti  T. Further we shall denote separate 1-worlds as w1i(t), w1j(t), w1k(t’), etc.

2. Level of persons as such. We shall not discuss here concrete nature of Person. Our assumption consists only one assertion: there exist such things as persons, irrespective of their concrete nature. Otherwise we do not see the possibility to construct theory of Personal Identity. We shall denote separate persons as 2-worlds, or w2. Only one assumption will be accepted here. This is the idea of a map P: WP1  WP2, where WP1 is set of all 1-worlds w1 for which at least one 2-wold w2 exists such that w2 = P(w1), WP2 is set of all persons as 2-worlds. In general case set WP2 is a sub-set of set W2. Therefore, if w1(t)WP1, then P(w1(t)) is a w2 and w2 is a person. We shall call set WP1 as set of personal-manifested 1-worlds, or simply P1-worlds. Accordingly, if w2  WP2, then w2 can be called as P2-world.
Meaning of P-mapping consists an idea of a link between some P1-world w1(t) and some P2-world (or worlds) w2. We shall only demand that map P determines at least one P2-world for every P1-world. Interpretation of P-mapping can be the following. If P(wi1(t)) = wj2, then 1-world wi1(t) is an expression of a person (P2-world) wj2 at the moment t. Separate 2-worlds can be marked by different symbols: w2i, w2j, w2k, etc.

3. Level of 3-worlds w3i which are states of affairs including all above described structures specified for w3i: 1) i-time Ti = [ti0,tiK] with i-order <i on the pairs of moments of i-time, 2) set Wi1 of all i1-worlds wi1 defined during i-time Ti, 3) i-map Wi1 : Ti  Wi1 from i-time to set of all i1-worlds, 4) set Wi2 of all i2-worlds wi2 and sub-set WPi2 of all Pi2-worlds, 5) i-map Pi: WPi1  WPi2 from set of all Pi1-worlds to set of all Pi2-worlds (hence subset WPi1 of set Wi1 is determined also). Therefore i3-world w3i can be represented as follows: w3i = < Ti, <i, Wi1, WPi1, Wi1, Wi2, WPi2, Pi>. Set of all 3-worlds can be marked as W3.

Described structure can be denoted as ^ Personal Identity Structure (PI Structure, or PIS) – see also fig.1.

Our nearest aim here is to show that PIS can be fruitful for understanding of many problems of present version of PI. We shall try to demonstrate this on some examples below.

1. Problem of Diachronic Identity

It seems to us that many authors formulate now problem of diachronic identity (DI) in PI in the following not accessible form (see e.g. []). Criterion of DI is formulated for one person P at different moments t and t’, where t’ > t. If R is predicate expressing the criterion, then one can write here: R(P(t),P(t’)), i.e., predicate R is 3-placed one: R(P,t,t’). We suppose it is not right. Really, main task, in our opinion, of DI’ criterion is to show that two time-instant states, for example S and S’, belong to the domain of R. And we are not able to know, before the R holding, whether these states are two expressions of one person, or not. Therefore we have the following conclusion here: only carrying-out of criterion of DI gives representation of S and S’ as two forms of expression of one person P, i.e., as P(t) and P(t’) accordingly, hence before the carrying-out of the criterion we do not have such representation. So if we write DI criterion in the form R(P(t),P(t’)), then we have here the following paradox: P(t) and P(t’) are results of carrying-out of R and at the same time they are used as initial data before the carrying-out of the criterion. To solve the paradoxe it is sufficient to understand that R is defined on instant-time states S and S’ about which we do not yet know whether they belong to one person or not. Misunderstanding of this simple circumstance lies, in our opinion, in two possible kinds of temporal determination. First kind of temporal determination is one of function f(t), where t is a moment of time. Here function f and its separate value f(t) are not identical. Function f is set of pairs (t, f(t)), not separate f(t). Such a kind of temporal predication takes place for time-distributive things, i.e., for objects as function f. Further, second kind of temporal predication is one when we speak that, for example, event S “Explosion of atomic bomb in Hirosima” was at 6th of August in 1945 (moment t). Here moment t is included into event S, this is a part of whole definition of S, and event S is not determined at other moments of time. This second kind of temporal predication can be called as “singular temporal predication”, whereas first kind as “distributive temporal predication”. Event S also can be designated as S(t), but here we have another case of temporal predication of time t to thing S. To distinguish these two kinds of temporal predication we shall use brackets of different form: round brackets f(t) for the case of distributive temporal predication, and corner brackets S for the case of singular temporal predication. Time-instant events S can be called as time-singular events. For instance 1-worlds, as different time-instant expressions of persons, are examples of time-singular states, and we should write wii1, not wii1(t), here. On the other hand, 2-worlds, as separate persons, are presupposed by DI as time-distributive states, and we can leave here round brackets wki2(t), where wki2(t) is set {wi1 : Pi(wi1) = wki2} – set of all i1-worlds at the moment t, which are placed to i2-world wki2 by i-mapping Pi.

Now we can reformulate DI’ criterion in the following form: two time-singular states S and S’, where t’ > t, belong to one person P iff a predicate R is carryied out on S and S’.

Confusion takes place here through the indistinguishness of distributive and singular temporal predication. In fact we could to write that P(t) = S and P(t’) = S’ if R was already carried out, but we can not do it before this. Hence we always can write R(S, S’) but we can write R(P(t),P(t’)) only after carrying-out of R(S, S’). Therefore notation R(P(t),P(t’)) is not universal one for the DI’problem. Moreover it always presupposes more earlier and universal notation R(S, S’). In accordance with the last, DI’criterion is 4-placed predicate R(S,S’,t’t’) determined on two time-singular events and two moments of time. Or, if we suppose that moment t is part of definition of time-singular state S, then we can assert that predicate R is 2-placed one determined on two singular events S and S’. It is not 3-placed predicate in any case.

Such reformulation of DI’criterion provides explicite necessity of distinguishness of two temporal levels in PI theories. These are level of time-singular states, as expressions of persons at the moments of time, and level of persons as time-distributive things. These two levels are represented themselves in PIS as sets of 1-worlds and 2-worlds accordingly. In such terms DI’criterion is a predicate R(wji1, wki1) defined on sets domiR = {(wji1, wni1) : t’>t}. This predicate does not put 1-worlds to some 2-world. It only determines whether some 2-world wki2 exists such that Pi(wji1) = wki2 and Pi(wni1) = wki2. In such formulation explicit reference to a concrete 2-world can be excluded on the whole, since DI’criterion is turn out to be equivalent to equality Pi(wji1) = Pi(wni1), where i1-worlds wji1 and wni1 must be P1-worlds. From this point of view DI’criterion can be subdivided on two parts: 1) quality part of DI’criterion connected with the criterion of 1-world as P1-world. This criterion is one of belonging wki1  WPi1, i.e., it must define when instant-time expressions are expressions of Personality, not another time-distributive thing. 2) relation part of DI’criterion as criterion of equality Pi(wji1) = Pi(wni1), i.e., this part must define when two instant-time states are two expressions of one time-distributive thing.

^ 2. Theory of PI in terms of modal logic

Let T be a theory of PI. We shall discuss some features of T from the point of view of PIS here.

First of all one need to note that T must include expressions for basic parts of PIS. As far as 3-world has the following structure w3i = < Ti, <i, Wi1, WPi1, Wi1, Wi2, Pi>, there must be according expressions for basic parts of the structure of some 3-world. Only index “i” must be absent in T, since T, as modal theory, can assert somewhat only in the framework of separate possible 3-worlds. Consequently, we shall denote expression for structure Xi by symbol X without “i”. Therefore we receive the following primary symbols in T:

1. Temporal expressions: constant T for set of all moments of time Ti, 2-placed predicate symbol < for predicate <i.

2. 1-world expressions: constants W1 and WP1 for sets Wi1 and WPi1 accordingly, 1-placed functional symbol W1 for mapping Wi1.

3. 2-world expressions: constant W2 for set Wi2, 1-placed functional symbol P for mapping Pi.

Besides, theory T must include:

4. 1-placed predicate symbol Ql for the expression of some version of quality part of DI’criterion R.

5. 2-placed predicate symbol Rl for the expression of some version of relation part of DI’criterion R.

Then theory T will use, at least, some part of an axiomatic Theory of Sets S, where 2-placed predicate symbols , possibly, = (equality), and ordinary set axioms are accepted. At last modal symbols L (operator of necessity) and M (operator of possibility) must be used in T. It is clear that some axioms must be accepted in T to describe PIS.

Between nonlogical and not set theoretical axioms of T the following PI axiom must be:

PA. Lxytt’(xW1(t)  yW1(t’)  t(x)Ql(y)  Rl(x,y)  [P(x) = P(y)])

Theory T, as modal one, must be interpreted on frames 3,R>, where R is a relation of accessibility on 3-worlds. One of the important questions here is what is the relation R ? In our oponion, R may be defined by different conditions but, at least, the following conditional must hold here:

(R) R(wi3,wj3)  x(x  Wj2),

i.e., if possible 3-world wj3 is accessible for possible 3-world wi3, then there exists a person in wj3 (it should be noted the assertion (R) is expressed in metalanguage relatively theory T). Really this sense of relation of accessibility is presupposed by the intention of any theory of PI to find DI’criterion in any possible world, by which the sense of “necessity” links with all such worlds where persons could be.

^ 3. About some Epistemological problems in PI

There can be a situation when some theory T may not be true. It connects with the appearance of some situation when axiom PA does not hold. As a rule here some possible 3-world wi3 is opened where antecedent of PA is true and consequent is false.

Let us see the following set

A(R) = {wi3: xytt’(xWi1(t)  yWi1(t’)  t(x)Ql(y)  Rl(x,y)  [Pi(x) = Pi(y)])}

This is the set of all 3-worlds where a criterion R of DI holds. We shall denote the set A(R) as ^ Region of Adequateness of criterion R.

So falsification of theory T connects with the appearance of 3-world wi3 which does not belong to A(R). If wi3 belongs to the region of accessibility of some possible 3-world, then axiom PA is not true and theory T can be falsificated. As a rule we do not deal with all possible 3-worlds but only with a such part which is revealed by our cognition (we do not suppose here that not revealed things exist outside of our consiousness, but, at least, outside of revealed part of the consiousness). Let WC3(t) be a revealed by our cognition to moment t part (subset) of set of all 3-worlds W3. Therefore theory T is interpreted on WC3(t) at the moment t, not on W3. If WC3(t)  A(R), then theory T is considered as true. Otherwise, if (WC3(t)  A(R)) theory T can be falsificated (if scientists, of course, recognize contradiction as a such). As a rule set WC3(t) is constantly increasing with the increasing of time t. We can suppose that the more time t the more set WC3(t) approximates set W3.

^ 4. Basic positions in PI from the point of view of PIS

Here we shall try to represent basic contemporary approaches in PI in the terms of PIS. It seems to us (see also []) that all spectrum of contemporary conceptions in PI can be divided into following three basic positions:

1. Composed View in PI. Representatives of this approach assert that DI’criterion for PI can be formulated in explicit form, i.e., there exists some predicate R such that A(R) = W3 and R can be adecuately expressed in a theory T. In particular, it entails that theory T can not be falsificated at any moment t of time, or WC3(t)  A(R) for any moment t. This approach is divided into two main positions: 1) Physical Composed View, where quality part Ql of DI’criterion is understood as some physical expression (-criterion, or simply ) of Person, 2) Psychological Composed View, where quality part Ql of DI’criterion is understood as some psychological expression (-criterion, or simply ) of Person.

2. Simple View in PI. This approach asserts that we, as human beings with limited cognition, are not able to fomulate DI’criterion in explicit form, i.e., for any predicate R it is true that (A(R) = W3) or/and R can not be adecuately expressed in any theory T. Hence only God knows PIS, we always deals only with its part in the framework of WC3(t). We also can not determine full borders of A(R) for any predicate R.

3. Parfit View in PI. Derek Parfit, as it is known, stands on the position that there are no such things as Persons, i.e., every set Wi2 is null set and mappings Pi are absent also. Then theory of Personal Identity is impossible.

^ 5. Level Theory in PI

Now we would like to offer some principles of a new approach in PI which can be called as Level Theory. From our point of view it could be a possibility of compromise for two basic approaches, Composed and Simple Views, in PI. Parfitt position is, in our opinion, rejection of possibility to solve PI problem.

Main theses of Level Theory are the following:

1. Persons exist, i.e., sets Wi2 is not empty and mappings Pi exist (against Parfitt).

2. For any predicate R it is true that (A(R) = W3) and, possibly, borders of A(R) can not be defined completely. This thesis connects Level Theory with Simple View.

3. Not W3, as a such, but only WC3(t) is important for the human cognition at a moment t. We always use conditional t-validity of T relatively WC3(t). Here we can accept the following conventions: 1) WC3(t)  A(R) iff there are no falsificators, as 3-worlds, for T at the moment t, 2) WC3(t)  A(R)   iff there are verificators, as 3-worlds, for T at the moment t. Also R can be explicitely formulated in the framework of a theory T. This thesis connects Level Theory with Composed View but only in the framework of WC3(t). We can define A(R) only in the framework of WC3(t). Such definability can be called as quasi-definability.

4. We shall say that DI’criterion R depends upon DI’criterion R iff A(R)  A(R). We shall accept that problem of relation between criterions R and R can be solved through theoretical way. For example, contemporary position of representatives of -criterion is such that -criterion can be DI’criterion only when -criterion realises through itself -criterion, i.e., brain is important for PI only when it brings psychological information. It means that -criterion depends upon -criterion, and this relation can be expressed by pure theoretical meanings.

5. There exist sequences of DI’criterions {Ri}, where N is finite or infinite, such that for every i we have A(Ri)  A(Ri+1), i.e., every previous criterion depends upon succeeding one, and A(RN) = W3. Therefore Level Theory strives for Composed View at limit criterion RN. However it is possible that N is infinite and we never can deal with RN as empirical criterion (aspect of Simple View).

6. At every moment t there exists a DI’criterion R such that WC3(t)  A(R). In particular, it means that even if we does not know such DI’criterion at moment t, it does exist and we should find it. In our opinion, all two kinds of DI’criterions have falsificators today and, as well we came to -criterion after appearance of falsificators to -criterion, as now, after Williams falsificators for -criterion, we should to find some new criterion.

7. Every theory T is valid on such frames , where M  A(R), and false oterwise. Therefore any theory T of PI presupposes own frames in relation to which the theory is valid. Theories are valid in relation to own frames and not valid otherwise. Therefore we should find not so much one true theory as regions of adequateness for every theory (at least in the framework of concept of quasi-definability).

^ 6. To the problem of Third Criterion

One of the important problems of contemporary PI theories, in our opinion, is the problem of Third Criterion of PI after -criterion and -criterion. We shall denote it as 2-criterion. 2-criterion must be in that relation to -criterion such as -criterion to -criterion. This relation can be expressed in the following graphic metaphor:

- ^ PI Proportion

It means that -criterion must depend upon 2-criterion, i.e., A()  A(2) and, at least, some falsificators for -criterion must become verificators for 2-criterion. Falsificators for -criterion are Williams arguments of -duplication. Theory of the best candidate is an effort to answer on these falsificators. But this theory has own falsificators. Therefore all -theories have falsificators now. Most falsificators for -criterion and -criterion include basic idea of possibility of duplication, i.e., such situation w3i when Ql(wji1)Ql(wki1)  Rl(wji1, wki1)  (P(wji1) = P(wki1)).

Let us see how -theories solved such contradictions of -criterion. Then, using PI Proportion, we could to try to use this decision for solving of contradictions of -criterion.

Let the situation of duplication of brain be done when two persons with only one hemisphere of the primary brain exist. Then, using only -criterion, we are not able to solve where the original person is. In such situation -theory proposes to pay attention to not simply substance of brain but an -information bearer of which the brain could be. This approach can be more effective only under the condition that, at least, some falsificators of -criterion could be verificators of -criterion. Really, after the -criterion introducing, we can solve some situations of physical duplication of the brain, when one part of the brain has not possibility to keep full -information of the person. This scheme of decision we can use for the formulation of 2-criterion relatively already -duplications.

Let us imagine that -information has importance for PI not in itself but in connection with some “2-factor” which could be or could not be present in an -information. One of the simple decisions here is in the interpretation of 2-factor as “first-person-information”, i.e., as information in the position of first person “I”. We shall call such information as 2-information. As above, with -duplication, we should pay attention to not simply duplication of -information, but we should to solve the following problems here: 1) whether the original -information, before -duplication, was 2-information? 2) whether two versions of -information, after -duplication, became two versions of 2-information, or not? Like with -duplication there can be the situation when, at least, some falsificators for -criterion could be verificators for 2-criterion.

Therefore search of new PI criterions, particularly 2-criterion, is one of the important aims of PI theory. But Level Theory does not necessarily assert that 2-criterion will answer on all questions and the situation of 2-duplication will not be impossible. It only asserts that 2-cruterion has more large region of adequateness and, particularly, it is more difficult to duplicate 2-information than -information, because of some situations of -duplication are ones of absence of 2-duplication.


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