
Theory of Levels in Personal Identity V.Moiseyev 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 timeinstants 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 1possible worlds, or simply 1worlds, w^{1}. Every 1world is defined in a moment t such that one can used the following explicit notation for this circumstance: w^{1}(t). Hence we also suppose that a set of different moments of time is defined, for example, set T = [t_{0},t_{K}], 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 W^{1} : T W^{1} is defined here, where W^{1} is set of all 1worlds, W^{1}(t_{i}) is set of all 1worlds at a moment t_{i}, i.e., W^{1}(t_{i}) = {w^{1}(t): t = t_{i}}. Further we shall use the symbol “W^{1}(t_{i})” for set W^{1}(t_{i}). We can accept that W^{1} is union of all sets W^{1}(t_{i}), where t_{i} T. Further we shall denote separate 1worlds as w^{1}_{i}(t), w^{1}_{j}(t), w^{1}_{k}(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 2worlds, or w^{2}. Only one assumption will be accepted here. This is the idea of a map P: W_{P}^{1} W_{P}^{2}, where W_{P}^{1} is set of all 1worlds w^{1} for which at least one 2wold w^{2} exists such that w^{2} = P(w^{1}), W_{P}^{2} is set of all persons as 2worlds. In general case set W_{P}^{2} is a subset of set W^{2}. Therefore, if w^{1}(t)W_{P}^{1}, then P(w^{1}(t)) is a w^{2} and w^{2} is a person. We shall call set W_{P}^{1 }as set of personalmanifested 1worlds, or simply P1worlds. Accordingly, if w^{2} W_{P}^{2}, then w^{2} can be called as P2world. Meaning of Pmapping consists an idea of a link between some P1world w^{1}(t) and some P2world (or worlds) w^{2}. We shall only demand that map P determines at least one P2world for every P1world. Interpretation of Pmapping can be the following. If P(w_{i}^{1}(t)) = w_{j}^{2}, then 1world w_{i}^{1}(t) is an expression of a person (P2world) w_{j}^{2} at the moment t. Separate 2worlds can be marked by different symbols: w^{2}_{i}, w^{2}_{j}, w^{2}_{k}, etc. 3. Level of 3worlds w^{3}_{i} which are states of affairs including all above described structures specified for w^{3}_{i}: 1) itime T^{i} = [t^{i}_{0},t^{i}_{K}] with iorder <^{i} on the pairs of moments of itime, 2) set W^{i1} of all i1worlds w^{i1} defined during itime T^{i}, 3) imap W^{i1} : T^{i} W^{i1} from itime to set of all i1worlds, 4) set W^{i2} of all i2worlds w^{i2} and subset W_{P}^{i2} of all Pi2worlds, 5) imap P^{i}: W_{P}^{i1} W_{P}^{i2} from set of all Pi1worlds to set of all Pi2worlds (hence subset W_{P}^{i1} of set W^{i1} is determined also). Therefore i3world w^{3}_{i} can be represented as follows: w^{3}_{i} = < T^{i}, <^{i}, W^{i1}, W_{P}^{i1}, W^{i1}, W^{i2}, W_{P}^{i2}, P^{i}>. Set of all 3worlds can be marked as W^{3}. Described structure can be denoted as ^ (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 3placed 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 timeinstant 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 carryingout 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 carryingout 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 carryingout of R and at the same time they are used as initial data before the carryingout of the criterion. To solve the paradoxe it is sufficient to understand that R is defined on instanttime 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 timedistributive 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 6^{th} 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 Now we can reformulate DI’ criterion in the following form: two timesingular states S Confusion takes place here through the indistinguishness of distributive and singular temporal predication. In fact we could to write that P(t) = S Such reformulation of DI’criterion provides explicite necessity of distinguishness of two temporal levels in PI theories. These are level of timesingular states, as expressions of persons at the moments of time, and level of persons as timedistributive things. These two levels are represented themselves in PIS as sets of 1worlds and 2worlds accordingly. In such terms DI’criterion is a predicate R(w_{j}^{i1} ^ 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 3world has the following structure w^{3}_{i} = < T^{i}, <^{i}, W^{i1}, W_{P}^{i1}, W^{i1}, W^{i2}, P^{i}>, there must be according expressions for basic parts of the structure of some 3world. Only index “i” must be absent in T_{}, since T_{}, as modal theory, can assert somewhat only in the framework of separate possible 3worlds. Consequently, we shall denote expression for structure X^{i} 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 T^{i}, 2placed predicate symbol < for predicate <^{i}. 2. 1world expressions: constants W^{1} and W_{P}^{1} for sets W^{i1} and W_{P}^{i1} accordingly, 1placed functional symbol W^{1} for mapping W^{i1}. 3. 2world expressions: constant W^{2} for set W^{i2}, 1placed functional symbol P for mapping P^{i}. Besides, theory T_{} must include: 4. 1placed predicate symbol Ql_{} for the expression of some version of quality part of DI’criterion R_{}. 5. 2placed 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 2placed 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_{}. Lxytt’(xW^{1}(t) yW^{1}(t’) t Theory T_{}, as modal one, must be interpreted on frames (R) R(w_{i}^{3},w_{j}^{3}) x(x W^{j2}), i.e., if possible 3world w_{j}^{3} is accessible for possible 3world w_{i}^{3}, then there exists a person in w_{j}^{3} (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. ^ 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 3world w^{i3} is opened where antecedent of PA_{} is true and consequent is false. Let us see the following set A(R_{}) = {w^{i3}: xytt’(xW^{i1}(t) yW^{i1}(t’) t This is the set of all 3worlds where a criterion R_{} of DI holds. We shall denote the set A(R_{}) as ^ of criterion R_{}. So falsification of theory T_{} connects with the appearance of 3world w_{}^{i3} which does not belong to A(R_{}). If w_{}^{i3} belongs to the region of accessibility of some possible 3world, then axiom PA_{} is not true and theory T_{} can be falsificated. As a rule we do not deal with all possible 3worlds 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 W_{C}^{3}(t) be a revealed by our cognition to moment t part (subset) of set of all 3worlds W^{3}. Therefore theory T_{} is interpreted on W_{C}^{3}(t) at the moment t, not on W^{3}. If W_{C}^{3}(t) A(R_{}), then theory T_{} is considered as true. Otherwise, if (W_{C}^{3}(t) A(R_{})) theory T_{} can be falsificated (if scientists, of course, recognize contradiction as a such). As a rule set W_{C}^{3}(t) is constantly increasing with the increasing of time t. We can suppose that the more time t the more set W_{C}^{3}(t) approximates set W^{3}. ^ 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_{}) = W^{3} 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 W_{C}^{3}(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_{}) = W^{3}) 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 W_{C}^{3}(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 W^{i2} is null set and mappings P^{i} are absent also. Then theory of Personal Identity is impossible. ^ 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 W^{i2} is not empty and mappings P^{i} exist (against Parfitt). 2. For any predicate R_{} it is true that (A(R_{}) = W^{3}) and, possibly, borders of A(R_{}) can not be defined completely. This thesis connects Level Theory with Simple View. 3. Not W^{3}, as a such, but only W_{C}^{3}(t) is important for the human cognition at a moment t. We always use conditional tvalidity of T_{} relatively W_{C}^{3}(t). Here we can accept the following conventions: 1) W_{C}^{3}(t) A(R_{}) iff there are no falsificators, as 3worlds, for T_{} at the moment t, 2) W_{C}^{3}(t) A(R_{}) iff there are verificators, as 3worlds, 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 W_{C}^{3}(t). We can define A(R_{}) only in the framework of W_{C}^{3}(t). Such definability can be called as quasidefinability. 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 {R_{i}}, where N is finite or infinite, such that for every i we have A(R_{i}) A(R_{i+1}), i.e., every previous criterion depends upon succeeding one, and A(R_{N}) = W^{3}. Therefore Level Theory strives for Composed View at limit criterion R_{N}. However it is possible that N is infinite and we never can deal with R_{N} as empirical criterion (aspect of Simple View). 6. At every moment t there exists a DI’criterion R_{} such that W_{C}^{3}(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 ^ 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:  ^ 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 w^{3}_{i} when Ql_{}(w_{j}^{i1} 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 “firstpersoninformation”, 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. 
Theory of Levels in Personal Identity  Language levels and their basic units  
Документы 1. /Eric Lewis  The Stoics on Identity and Individuation.pdf  Personal identification  
Personal info  How to write a personal letter  
A theory of the a priori  Судовая роль (crew list) Серия, номер паспорта моряка или иного документа, удостоверяющего личность (Nature and number of identity document (seaman’s...  
Документы 1. /Siegmund M. GSM and Personal Communications Handbook 1998.pdf  Документы 1. /theory.txt 