Dynamical infomorphism: form of endo-perspective

Dynamical infomorphism: form of endo-perspective

Yukio-Pegio Gunji a,b,*, Tatsuji Takahashi b, Masashi Aono b

a Department of Earth and Planetary Sciences, Faculty of Science, Kobe University, Nada, Kobe 657-8501, Japanb Graduate School of Science and Technology, Kobe University, Nada, Kobe 657-8501, Japan

Accepted 8 March 2004

Communicated by Prof. Y. Aizawa

Abstract

The essential feature of the endo-perspective is examined, and a formal model of the endo-perspective is proposed byintroducing the mixture of intra- and inter-operations. Because such a mixture in its naive realization entails a paradoxwithin a formal system, we weaken the inter-operation in order to allow the formal system to be endowed with thatmixture without a contradiction. The weakened inter-operation is related to the infomorphism proposed by Barwise[Information Flow, The Logic of Distributed Systems, Cambridge Univ. Press, 1997]. The formal model of the endo-perspective is thereby expressed as the dynamical infomorphism driven by that mixture. The endo-perspective is de-scribed as a formal system that includes the outside of the occupied perspective. If such an inclusion is applied to thecommon definition of a set, it entails Russel’s paradox. Retaining the outside can be expressed as the mixture of theintent and the extent of a set together with the mixture of intra-operations within the intent (or the extent) and inter-operations between the intent and the extent. The endo-perspective, therefore, consists of two subsystems corre-sponding to the intent and the extent, respectively, and is defined as a system involving a particular mathematical tool(i.e., infomorphism) that allows for retaining the outside without a contradiction. Within that framework, the mixtureof the intra- and the inter-operation drives the dynamical transition of the system, however, it can be terminated by itscollapse. This collapse can be predicted from the internal logic defined within the system. The model is constructedthrough the verification of ‘‘a weakened paradox’’. Because the definition of the system involves a weakened paradoxonly, it does not always lead to a contradiction, although the collapse of the system corresponds to a contradiction. Thedouble standards can be embedded into the system, the domain with truth-values (the inside) and the domain in whichthe collapse of the logic can occur (the outside).! 2004 Elsevier Ltd. All rights reserved.

1. Introduction

From two streams emerging from theoretical biology and from physics, the perspective that might raise a paradoxwhich has been mostly neglected in the past has been put into the focus of research. Concretely, these are the endo- [2,3]and/or the internal-perspective [4–6], respectively. In theoretical biology, the interaction between intra-cellular andinter-cellular reactions has to be taken into consideration when treating a multi-cellular system as a whole [7]. It isassumed that in a model of partial di!erential equations the time constants corresponding to the intra-cellular reactionsare much faster than those of the inter-cellular ones. If that assumption is abandoned when a multi-cellular system hasto be described one encounters a discrepancy between the identification of a state and the proceeding reaction that hasbeen called vertical scheme by Conrad [8]. The aspect of the vertical scheme is formalized as a self-referential property

*Corresponding author.E-mail address: [email protected] (Y.-P. Gunji).

0960-0779/$ - see front matter ! 2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.chaos.2004.03.001

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[9]. Also Rosen refers to the interaction between operand and operator, although he describes it in a dualistic way[10,11]. In his later work, Rosen regarded complexity as a paradox [12] that is comparable to Conrad’s notion of thevertical scheme. They both conclude that a biological system should be described from a particular perspective which is,on the one hand, destined to be paradoxical, however, on the other hand, does not imply impossibility.

The aforementioned aspect can be generalized as following. Assume an observer who focuses on a system like aparticular cell, for example, in order to describe it. One can define a particular formal system consisting of a cell and itsenvironment. However, even if the environment of the cell has been properly defined one can usually find somethingthat was excluded from the previous definition of the formal system and can change it when being included. What isfound a posteriori might be inconsistent with what has been defined a priori. Such an experience of an observer maylead him to assume an indefinite environment of a cell. As a result, the observer has to describe the indefinite envi-ronment within a formal system. The transition from ‘‘a priori’’ to ‘‘a posteriori’’ is embedded in the notion ofindefiniteness [13,14]. In other words, time in the form of transition results from the formal description involvingindefinite environments.

The notion of an environment is not closed in a consistent description. All attempts to assign indefinite environmentsto a set lead to infinite regression. Indefinite environments can never be consistently described because the very notionof indefiniteness is outside and/or beyond any description which means that one has to give up the Cartesian cut [15].Endo-physics and the theory of internal measurement appear in the field of physics in order to embed an observer’sstance in an object [3,4,6]. Although a formal description is always accompanied with an observer, one does not have todescribe a super-observer (i.e., exo-observer) explicitly because a super-observer is defined in the form of a consistentrelation between local and global descriptions such as the relation between the principle of detailed balance and thestationary solution of a master-equation (e.g., [16]). In endo-physics a local observer is defined by introducing causalvacillations between local causes and global actions that work even in a Hamiltonian system [17]. In the theory ofinternal measurement a local observer at a local site is confronted with the perpetual dis-equilibration in order to makea local motion consistent with the global conservation laws [4,5]. A local observer’s intentions can never be achievedcompletely. Both in endo-physics as well as in the internal measurement description, indefinite environments areintroduced as an empirical law such as a conservation law, and the notion of indefiniteness is described as the dis-crepancies between local and global descriptions. As a result a dynamical change results from the motion to negotiatethe discrepancies. Only in the results of an observation (i.e., by ignoring the dynamical process), an observer can findthe equivalence between local and global descriptions [4].

The only way to formalize the endo-perspective is the following one: (i) Firstly, the local and the global descriptionsof a system are prepared independently, and the system is defined to entail a contradiction if the local description isdirectly mixed with the global one. (ii) Then, the interaction between the local and the global description is introducedwhile avoiding a contradiction. (iii) Eventually, if the interaction does not fail by producing a contradiction and if itallows the transition from the a priori to the a posteriori, then the indefinite environments and/or endo-observer can beembedded. An endo-perspective has to account for the outside of the description although it can never be directlydescribed. To allow a previously consistent relation between the local and the global descriptions to become incon-sistent but thereby ruling out contradictions, is a well-defined way to describe an endo-perspective which is a non-closeddescription. In other words, such a description retains and refers to the outside (i.e., is opened to the outside and/or thefuture).

Local and global descriptions are abstractly found in mathematics. In set theory, an element of a set (the extent) is alocal, and the attribute (the intent) of a set is a global entity. The mixture of them leads to the well-known Russelianparadox [18]. Therefore, in the context of formalizing the endo-perspective, it is necessary to define a system (i.e.,perspective) that in spite of mixing intent and extent never fails. In category theory, the equivalence relation betweenextent and intent can be expressed as the equivalence between two categories through adjunctive functors [19].Therefore, it is possible to formalize a system within category theory that never ends up with a paradox in spite of themixture of intent and extent. We proceed with the following steps: (i) We start with the emergence of Russel’s paradoxin category theory, (ii) We investigate the causal relation of a paradox, and (iii) We avoid a contradiction in spite of theinteraction between the intent and the extent by allowing the formal model to be inconsistent. Eventually this leads toan abstract formal endo-perspective.

If one introduces the mixture of the intent and the extent, one has to weaken some mathematical tools so as to avoida contradiction. One way is to weaken the notion of a functor that can equivalently bridge between the intent (globaldescription) and the extent (local description). Infomorphism [1] is one of the tools that can be used to avoid a paradoxwhich otherwise would result from mixing intent and extent. Infomorphism is, as it is called, a weakened adjunctivefunctor. Infomorphism has originally been proposed by Barwise [20] to describe the communication between twodi!erent perspectives and to sophisticate the situational semantics [21]. Although two perspectives are not isomorphic toeach other, there is a kind of equivalence between them based on a particular map called infomorphism. However, the

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conditions under which two perspectives can communicate by infomorphism as defined by Barwise are too static for ourpurpose. We here relate these two perspectives to the intent and the extent in an abstract sense. In our context, theintent is dynamically mixed up with the extent. The mixture of the intent and the extent drives a dynamical change ofthe infomorphism. Our definition of the mixture ensures that the equivalence between two perspectives (the intent andthe extent in an abstract sense) holds even in the changing infomorphisms between the intent and the extent by avoidinga contradiction. We then discuss the significance of our formal expression of indefinite environments or the perspectivethat accounts for the outside.

2. Endo-perspective as the interaction of intent and extent

In order to introduce the interaction between local and global descriptions in an abstract sense, and to generalize theformal description of an endo-perspective, we first examine the well-known Russelian paradox. In classical set theorybefore the ZF-axiomatic system was introduced, a set had been defined by the equivalence of its intent and its extent.For example, the intent of a set of even numbers is defined by 2x, where x is a natural number, and its extent is definedby f0; 2; 4; . . .g. The definition through the extent is a local expression, whereas the intent corresponds to a globaldefinition. An observer has an overall view of the whole using the intent definition but focuses on a local indivi-dual element when the extent is used instead. The extent is a domain to which the concept of even numbers isapplied, whereas the intent is the attribute of even numbers. The relation between them is expressed as,0; 2; . . . 2 y () 2x. Because the intent of a set is given as an arbitrary expression (i.e., a sequence consisting of symbols)that contains a variable x, it is generally denoted as A!x". The extent of a set, y, is expressed as x 2 y. Then the generalform of a set is

8x9y !x 2 y () A!x"": !1"

This definition can lead to a paradox as explained in the following. Because A!x" is an arbitrary expression involving avariable x, it should hold that x 62 x. Therefore, one obtains 8x9y (x 2 y () x 62 x). Now, since x is an arbitrary objectand y is a particular one, one can substitute x by y. As a result, one obtains

y 2 y () y 62 y: !2"

The intent is inconsistent with the extent. One obtains a paradox that results from (i) the mixture of the intent and theextent whereby the intent is expressed through the extent, and (ii) the unrestrained use of a universal quantifier (i.e., themixture of a part and the whole). In other words, the distinction of local and global descriptions is articulated asthe distinction of the intent and the extent and the distinction of existential (indication of an element) and universal(indication of the wholeness) quantifier. Therefore, the mixture of local and global descriptions can be expressedthrough the two steps mentioned above. To tackle Russel’s paradox in set theory (using the Z-F axiomatic system) theoriginal axiom is replaced by the axiom of specification such that

8a8x9y !x 2 y () A!x" ^ !x 2 a"": !3"

The restriction x 2 a prohibits the mixture of the intent and the extent, which is why we cannot use it in our contextwhere we have to introduce that mixture to account for the endo-perspective.

Russel’s paradox is generalized by the diagonal argument in a categorical expression [22]. Because the symbol 2 is abinary relation, the extent can be replaced by a map 2 : X # X ! 2 defined such that for !x; y" 2 X # X , 2 !x; y" $ 1 ifx 2 y; 2 !x; y" $ 0 otherwise, where 2 $ f0; 1g. The intent is replaced by choosing a particular map expressed byg : X ! 2X , where 2X $ ff : X ! 2g. (Actually, one can define fy!x" $ 1 if x 2 y; fy!x" $ 0 otherwise, and g!x" $ fy , ifx 2 y; g!x" is defined properly otherwise.) Finally the equivalence between the intent and the extent of a set is expressedas,

X # X ! 2 () X ! 2X : !4"

Note, that the mixture between a set and an element is partly described in this form because a set y is also assumed to bean element of X . From the equivalence (4), in assuming that the map g is onto, one obtains g!y"!x" $ hg!y"!x" for 8x9y,where h : 2 ! 2 is defined by h!0" $ 1 and h!1" $ 0. Substituting y for x entails a fixed point with respect to h such thatg!y"!y" $ hg!y"!y" (see Appendix A and B). The assumption that g is an onto map is a formal expression for the mixtureof a part and the whole, i.e. the unrestrained use of the wholeness. We conclude that the contradiction results from (i)the mixture of the intent and the extent, and (ii) the unrestrained use of the notion of wholeness (i.e., the assumptionthat g is an onto map).

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In category theory (see Appendix A), the equivalence between the intent and the extent is expressed in a more generalform. Starting out from the equivalence, X # X ! 2 () X ! 2X , a ensemble of a set and maps constitutes a categorydefined by a collection of objects and arrows. Between two categories, A and B, a functor F : A ! B can be definedthat preserves the composition of arrows (e.g., maps), F !gf " $ F !g"F !f " as well as the identity map, F !idX " $ idF !X ". Bya functor, a collection of arrows is transformed as a whole. That is why a functor and an arrow can be compared with aset (wholeness) and with an element (part), respectively. In the equivalence expression X # X ! 2 () X ! 2X , onefinds two dual (i.e., adjunctive) functors, namely !%" # X and !%"X (Appendix A). We rewrite !%" # X and !%"X by Fand G, respectively, and the set 2 by Y to obtain

FX ! Y () X ! GY : !5"

In a general category, two functors satisfying the equivalence (5) are called adjunctive functors (Appendix A).The most general form of the equivalence between the intent (global description) and the extent (local description) is

given by Eq. (5). Remember that a paradox, i.e. discrepancies between the intent and the extent, results both from themixture of intent and extent as well as from the restrained use of the universal quantifier (the mixture of part andwhole). The latter can be expressed as the mixture of an arrow and a functor because a functor implies the transfor-mation of arrows as a whole. We show that the mixture of FX ! Y and X ! GY as well as the mixture of an arrow anda functor can entail a paradox, namely the failure of the equivalence (5). To understand this paradox will help us toconstruct the mixture of a functor and an arrow to account for an abstract endo-perspective––against a paradox. Thisin turn raises the question of how one can construct the mixture of a functor (the wholeness) and a map (an element)without contradiction under the condition of the mixture of FX ! Y and X ! GY .

The mixture of FX ! Y and X ! GY is partly expressed by setting Y $ X . The equivalence, FX ! X () X ! GX ,can be interpreted as a one-to-one relation between a chosen arrow g : X ! GX and second arrow, h : FX ! X . Weexpress that one-to-one relationship by a one-to-one and onto map denoted by / : fg : X ! GXg ! fh : FX ! Y gwith /!g" $ h. From the equivalence (5), one obtains the relationship between FX ! Y and X ! GY as

GY FGY ev

g Fg X

F !(g)

X FX

: !6"

The diagram commutes which implies that evFg $ /!g", where g : X ! GY is transformed to Fg : FX ! FGY by afunctor F (Appendix A). Under that condition, we define the mixture between a functor and an arrow.

To invalidate the di!erence between a functor and an arrow can be done by means of a particular definition of thefunctor, such that is application to an arrow, Fg, is given by the composition of arrows. Specifically, we replace theexpression Fg, meaning F is applied to g, by the composition of g and some arrows. Since g is composed of somearrows, we interpret the functor as an arrow, too. This leads to the mixture of a functor and an arrow. A functor isdefined both for an object (e.g., a set) and an arrow (e.g., a map). For an object, A, FA is defined as a particular object.For an arrow f : A ! A0, Ff : FA ! FA0 is defined as a commutative diagram,

sFA A

Ff f

u

FA" A"

!7"

from which follows that Ff $ ufs, where u and s are particular arrows. Since a functor has to preserve compositions, wedefine the composition of arrows S : FA0 ! A0 and u : A0 ! FA0 as idFA. In other words, a category consists of aparticular collection of arrows. That is why for f : A ! A0 and f 00 : A0 ! A00 it holds that

F !f 0"F !f " $ u0f 0s0ufs $ u0f 0idA0fs $ u0!f 0f "s $ F !f 0f " !8"

and for idA : A ! A, F !idA" $ u idAs $ us $ idFA (i.e., identity is preserved). Especially, for g : A ! GA the diagram


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!(g)

FA A

Fg g

u

FGA GA

!9"

is assumed to be commutative with s $ /!g", and Fg $ ug/!g". The latter expression can be recognized as a part of themixture of intent and extent.

It is assumed that FX ! X () X ! GX or alternatively the commutative diagram (6), as well as the mixture of afunctor with an arrow is introduced by a particular functor as defined by diagram (7). Under this condition it is provedthat the equivalence, FX ! X () X ! GX , fails. Substituting Fg $ ug/!g" for the commutative diagram (6), weobtain ev ug/!g" $ /!g" from which follows that

evug $ idA: !10"

Since the arrow ev : FGA ! A is defined by /!idGA", we conclude that ev is uniquely determined for an object A(Appendix A). An arrow g is chosen from a set ff : X ! GXg and g is chosen such that evug 6$ idA. As a result thediagram (6) does not commute which is a contradiction.

Finally, from assuming an equivalence between the intent and the extent in a categorical form (i.e.,FX ! X () X ! GX ) as well as the unrestrained use of the wholeness (i.e., the mixture of a functor and an arrow orthe definition of an application of a functor by the composition of arrows), we conclude that a contradiction is obtainedin the form of a failure of the previously assumed equivalence. To avoid the contradiction, one has to give up either theequivalence or the mixture of a functor and an arrow. Since we here stick with the idea of formalizing an endo-per-spective we cannot abandon the mixture of a functor and an arrow nor the unrestrained use of the notion of wholeness.We also cannot give up the equivalence between the intent (global description) and extent (local description), becausewe want to keep the fact that a global description is empirically consistent with a local one. Of course, the fact that alocal description is empirically equivalent to a global one has to be distinguished from the axiom that demands for theequivalence of local and global descriptions. One has to verify that the empirical fact results from the interactionbetween local and global descriptions. We choose the third way to construct the formal end perspective. Although wemaintain both the equivalence, FX ! X () X ! GX and the mixture of a functor and an arrow to some extent, thenotion of a functor is re-defined.

A functor is originally defined both for an object and an arrow. We weaken the role of a functor while we keep theequivalence given by FX ! X () X ! GX . Even if G and F are rewritten by two maps, f ^ : X ! Y and f _ : Y ! X ,respectively (X and Y are sets), f _!y" ! x () y ! f ^!x" is also kept for all y 2 Y and x 2 X . The pair hf ^; f _i is calledan infomorphism [1]. If a functor is weakened and replaced by a map we can mix an arrow within the intent or withinthe extent (i.e., an arrow) with an arrow pointing from the intent to the extent (i.e., hf ^; f _i). We call the former arrowan intra-action and the latter one an inter-action, respectively. Thus, we are enabled to keep both the equivalencebetween the intent and the extent as well as the unstrained use of wholeness.

The question concerning the formal endo-perspective arises whether one can construct a dynamical model under theframework, of which the equivalence between the intent and extent is expressed as f _!y" ! x () y ! f ^!x", and a mapwithin the extent or the intent is mixed up with a map from the intent to the extent. How does such a dynamical modellook like? If such a model is possible, it embodies the unrestrained use of the notion of wholeness by allowing theempirical consistent relation between the local and the global descriptions. In the context mentioned in the introduc-tion, it involves the notion of indefinite environments. That is why such a model can posses an engine that transform itsown equivalence of the intent and the extent and has time built-in. Our aim on the formal endo-perspective is sum-marized with the following question: How can one define a time development (i.e., a dynamical system) caused by theconfusion of intra- and inter-action within the framework of infomorphisms?

3. Infomorphism viewed from category theory

The empirical equivalence between local and global description is replaced by the equivalence between the intent andextent, and by the equivalence of FX ! X () X ! GX [19]. Defining the endo-perspective requires the mixture oflocal and global descriptions. In our context, that mixture is expressed as (i) the mixture of the intent and extent, and (ii)


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the mixture of a functor F (and G) and an arrow f : FX ! X (and g : X ! GX ). As discussed in Section 2, thosemixtures entail a paradox, which raises the question how one can weaken a particular mathematical tool to define asystem that does not entail a paradox. Our suggestion is to weaken a functor by replacing it with an arrow. Theequivalence of intent and extent is replaced by, f _!y" ! x () y ! f ^!x" for all y 2 Y and x 2 X , where f ^ : X ! Yand f _ : Y ! X are maps between sets, X and Y . Within this framework one can formulate two kinds of maps, namely(i) f _!y" ! x and y ! f ^!x", as well as (ii) f _ and f ^. In our context, the former maps are best called ‘‘intra-actions’’,and the latter ones ‘‘inter-actions’’. If the intra-action is replaced with a binary relation, the proposed frameworkcoincides with the concept of infomorphism [1].

Within the framework of infomorphism, we construct the time development via the mixture of intra- and inter-action, because an infomorphism consists of both a relation expressed as the intra-action and a pair of maps (info-morphism) expressed as the inter-action. In the following, we recapitulate the infomorphism proposed by Barwise[1,21]. The framework of infomorphism is based on a triplet called classification. A classification A!A1;A2;RA" consistsof two sets, A1 called type of A, and A2 called token of A, and a binary relation RA & A1 # A2. This is a model of aperspective of where formal symbols (type) are related to real objects (token). The communication between two per-spectives (classifications) are expressed by a pair of maps, f $ hf ^; f _i with f ^ : A1 ! B1 and f _ : B2 ! A2. Thecondition to communicate between two classifications is defined by the fundamental property such that for all a 2 A1

and b 2 B2,

f _!b"RAa () bRBf ^!a": !11"

A pair of maps, f $ hf ^; f _i, that satisfies the fundamental property given by Eq. (11) is called an infomorphism. Giventwo arbitrary classifications, it is not easy to see whether there is an infomorphism between them, satisfying the fun-damental property of Eq. (11). However, one can implement an algorithm to find an infomorphism on a machine. Fig. 1shows the result for the problem where two classifications with two relations are expressed as 4 · 4 matrices. Although44 · 44 possible pairs of infomorphisms exist, the number of an infomorphism satisfying the fundamental property isfairly small. Frequently there is no infomorphism between two arbitrarily chosen classifications.

First we sketch the infomorphism in terms of category theory, because categorical expressions can reduce theinvolution of expressions, compared to the original definitions in [1]. A category is defined as a collection of objects(A;B; . . .) and arrows between objects (f : A ! B, g : B ! C; . . .), where (i) each object A has an identity arrowidA : A ! A such that for any arrows, f : A ! X , g : Y ! A, f idA $ f and idAg $ g; (ii) if f : A ! B, g : B ! C arearrows, the composition gf : A ! C is also an arrow; (iii) given f : A ! B, g : B ! C and h : C ! D, the associativelaw hgf $ h!gf " $ !hg"f holds.

Fig. 1. The result for the problem to determine an infomorphism, given two classifications expressed as two binary relations, RA andRB. Each relations has four tokens (row) and four types (column), where !type; token" $ !a; a" with aRa is depicted by a filled square(above). Possible sixteen pairs of infomorphism, f ^ and f _, are shown below, where left diagram represents f ^ : typ!A" ! typ!B" (acell with filled square represents !a; f ^!a""" and right one represents f _ : tok!B" ! tok!A" (a cell with filled square represents!b; f _!b""".


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A category of infomorphisms, Inf is defined by a collection of classifications as objects and of infomorphisms asarrows. Because id^ : A1 ! A1 and id_ : A2 ! A2 with id^!a" $ a for all a in A1 and id_!a" $ a for all a in A2 is aninfomorphism, the axiom (i) holds. The axiom (ii) also holds because gf : A ! B ! C with gf $ hg^f ^; f _g_i satisfiesthe fundamental property such that for all a 2 A1 and c 2 C2, f _g_!c"RAa () g_!c"RBf ^!a" () cRAg^f ^!a". Similarly,the axiom (iii) is satisfied as can be seen in the following. Given f : A ! B, g : B ! C , and h : C ! D, for all a 2 A1

and d 2 D2, f _g_h_!d"RAa () g_h_!d"RBf ^!a" () h_!d"RCg^f ^!a" () dRCh^g^f ^!a".It can be proven that a category Inf has a co-limit for each finite subcategory (i.e., a subset of objects and arrows

collected from a category, and that satisfies the axiom of a category). The existence of a co-limit of a subcategory is aroutine verification in a category theory, and that is verified from the existence of a co-product and a co-equalizer [19].In [1], it is proven for a sum of two arbitrary classifications. In terms of category theory, that is called a co-product.Given A!A1;A2;RA" and B!B1;B2;RB" (i.e., two objects), one obtains the following infomorphismsiA : A!A1;A2;RA" ! A' B!A1 ' B1;A2 # B2;RA'B" and iB : B!B1;B2;RB" ! A' B!A1 ' B1;A2 # B2;RA'B" such that

iA iB

A1 A1+B1 B1

RA iA RA+B iB RB

A2 A2#B2 B2

: !12"

The two maps iÂ : A1 ! A1 ' B1 and i^B : B1 ! A1 ' B1 are injection maps, and for all a 2 A1 and b 2 B1,

iÂ!a" $ ha; 0i; i^B!b" $ hb; 1i; A1 ' B1 $ fha; 0ija 2 A1g [ fhb; 1ijb 2 B1g: !13"

Maps i_A : A2 # B2 ! A2 and i_B : A2 # B2 ! B2 are projection maps, and for all ha; bi 2 A2 # B2,

i_A!ha; bi" $ a; i_B!ha; bi" $ b: !14"

and A2 # B2 is a product set of A2 and B2. It is easy to see that if RA'B is defined by

ha; biRA'Bha; 0i : () aRAa; ha; biRA'Bhb; 1i : () bRBb; !15"

so that iA $ hiÂ ; i_Ai and iB $ hi^B ; i_Bi are infomorphisms satisfying the fundamental property such asi_A!ha; bi"RAa () ha; biRA'BiÂ!a" and i_B!ha; bi"RBb () ha; biRA'Bi^B!b".

In terms of category theory, a co-product is defined as following. Assume given two objects A and B. If there exists aunique arrow hf ; gi : A' B ! X with an arbitrary object X and arrows f : A ! X , g : B ! X , that commuteshf ; giiA $ f and hf ; giiB $ g, an object A' B with arrows iA : A ! A' B and iB : B ! A' B is called a co-product.The commutative diagram is drawn by the below left, and in a category Inf, the object A' B!A1 ' B1;A2 # B2;RA'B"with arrows iA : A!A1;A2;RA" ! A' B!A1 ' B1;A2 # B2;RA'B" and iB : B!B1;B2;RB" ! A' B!A1 ' B1;A2 # B2;RA'B"is a co-product, whereby the following right diagram commutes:

X XX

f <f,g> g f <f,g> g

iA iB iA iB

A A+B B AA AA +BB BB

: !16"

In terms of category theory, a co-product is dual to a product (i.e., if all arrows are reversed in the diagram of a product,the latter is replaced with a co-product). It is also known that there exist both products and co-products in a category ofsets (where objects are given by sets). As mentioned before, an object of the category Inf consists of two sets (type andtoken) and a binary relation as well asa map on type and one on token that have opposite directions (i.e., are dual).Therefore, if a co-product is defined on type it is related to a product defined on token by a particular binary relation.Thus, one can obtain a co-product in a category Inf.

Barwise and Seligman [1] do not refer to the existence of co-equalizer, however, it is easy to check the existence of aco-equalizer in category Inf. The same holds for the construction of a co-product. The existence of a co-equalizer in acategory Inf can be verified by connecting the co-equalizer on type and the equalizer on token with a particular relation.The co-equalizer is the dual of the equalizer.


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We define a co-equalizer and an equalizer in a category theory [19] as following. Assume given two objects A and Band two arrows f ; g : A ! B, then, if the following condition is satisfied, an object C with an arrow c : B ! C is calledco-equalizer: (i) cf $ cg, (ii) given an arbitrary object X with an arrow s : B ! X , if sf $ sg, there is a unique arrowk : C ! X such that kc $ s. An equalizer is defined through the same diagram as the co-equalizer, however withopposite arrows. Assume given two objects A and B and two arrows f ; g : B ! A. If the following condition is satisfied,an object E with an arrow e : E ! B is called equalizer: (i) fe $ ge, (ii) given an arbitrary object X with an arrows : X ! B, if fs $ gs, there is a unique arrow k : X ! E such that ek $ s. Diagrammatically co-equalizer and equalizerare drawn as:

f c f e

A B C A B E

g s k g s k

X X

: !17"

Actually, both co-equalizer and equalizer are defined in a category of sets (i.e., its objects are sets). If one definesco-equalizers in type-sets and equalizers in token-sets together with a particular relation between types and tokens, onecan obtain a co-equalizer in a category of Inf. Given two classifications (i.e., two objects), A!A1;A2;RA" andB!B1;B2;RB" and two infomorphisms (i.e., two arrows), f $ hf ^; f _i : A!A1;A2;RA" ! B!B1;B2;RB" andg $ hg^; g_i : A!A1;A2;RA" ! B!B1;B2;RB", it follows that co-equalizers in type-sets and equalizers in token-sets aredefined. In focusing on type-sets with two given sets A1 and B1, and two maps f ^; g^ : A1 ! B1, we define a quotient setC1 $ B1=J where

I $ fhf ^!a"; g^!a"ija 2 A1g !18"

and J is the smallest equivalence relation such that I & J & B# B, and then J satisfies reflective, symmetric, andtransitive law. A map c^ : B1 ! B1=J is defined by c^!b" $ (b)J , whereby (b)J is an equivalent class such as

(b)J $ fb0 2 B1jbJb0g: !19"

b 62 f ^!A1", then (b)J $ fbg because bJb. From the definitions it follows, that for all a 2 A1, c^!f ^!a"" $(f ^!a")J $ (g^!a")J $ c^!g^!a"", and thus c^f ^ $ c^g^. Supposing that s^f ^ $ s^g^ with s^ : B1 ! X1, one can defines^!b" $ k^!c^!b"" $ k^!(b)J " with k^ : B1=J ! X1. This definition can be verified by checking that an equivalent classdoes not depend on a representative. Taking b and b0 2 B1, b 6$ b0 with b $ f ^!a" and b0 $ g^!a", one obtains

s^!b" $ s^!f ^!a"" $ s^!g^!a"" $ s^!b0" !20"

k^!c^!b"" $ k^!(b)J " $ k^!(b0)J " $ k^!c^!b0"": !21"

Therefore, the definition of k^ can be verified. From the definition and supposing that there is another k^ : B1=J ! X1

such that it satisfies the same condition as k0^ : B1=J ! X1, one obtains that for all (b)J 2 B1=J ,k^!(b)J " $ k^!c^!b"" $ s^!b" $ k0!c^!b"" $ k0^!(b)J ", and then k^ $ k0^. Finally, C1 $ B1=J with a map c^ : B1 ! B1=Jis verified as a co-equalizer.

In focusing on token-sets with two given sets, A2 and B2, as well as two maps f _; g_ : B2 ! A2, we define a token setas

C2 $ fb 2 B2jf _!b" $ g_!b"g !22"

and c_ : C2 ! B2 such that is leads to an inclusion map, c_!b" $ b for all b 2 C2 From this definition we obtainf _!c_!b"" $ f _!b" $ g_!b" $ g_!c_!b"". Supposing that f _s_ $ g_s_ with s_ : X2 ! B2, there uniquely existsk_ : X2 ! C2 such that for x 2 X1s_!x" $ c_!k_!x"". The latter expressions are verified by the following. Suppose thatthere exists another map k0_ : X2 ! C2 satisfying the condition, k_!x" $ c_!k_!x"" $ s_!x" $ c_!k0_!x"" $ k0_!x", whichyields k_ $ k0_.

Finally, both co-equalizers on types and equalizers on tokens can be well-defined, and one can construct co-equalizers in a category of Inf by an object, C!B1=J ;C2;RC" where for (b)J 2 B1=J and b 2 C2, bRC(b)J : () bRBb, andan arrow c $ hc^; c_i : C!B1=J ;C2;RC" ! B!B1;B2;RB". Diagrammatically, this can be depicted as


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f $, g$ c$

A1 B1/J B1

RA f %, g% RC c% RB

A2 C2 B2

: !23"

Therefore, as mentioned above, the existence of a co-product and a co-equalizer entails a co-limit for an arbitrarysubcategory.

Our main purpose is just to use the category Inf as a tool to examine dynamical infomorphism. After defin-ing the dynamical infomorphism, we verify that a new infomorphism is obtained through the mixture of the intra-and the inter-action by using the category Inf. As mentioned, if a classification is regarded as a perspective andthe communication between perspectives is defined as an infomorphism with the fundamental property, then thereis a super-perspective in the universe of classifications. That super-perspective turns out to be the co-limit of Inf.We take the endo-perspective instead of the exo-(super-) perspective. The endo-perspective is not an incompleteperspective in the sense that there is a better and/or more complete perspective as a co-limit in a universe. That iswhy our perspective is not consistent with Barwise’s point of view. Therefore, we define a perspective as a pairof classifications connected with infomorphisms, where a perspective consists of two classifications that are comparedto the intent and the extent. The aspect that the perspective is surrounded by an indefinite environment, is expressedby the dynamic transformation of a perspective via the mixture of binary relations in a classification and an info-morphism.

By using a classification, one can define a logic that is defined in a power set of types. Depending on thebinary relation between types and tokens, there are various logics; some are monotonous and others are not [1]. Toestimate a logic in a dynamic infomorphism, we add some statements to the discussion concerning the logic given byBarwise.

4. Logic of a perspective

Assume given a classification, then the Boolean operations ^ (conjunction), _ (disjunction) and : (negation) can bedefined on this classification [1]. For the sake of convenience, we define the terms ^-, _- and :-closed classification. Aclassification A!A1;A2;RA" is ^-closed (or _-closed, respectively) if and only if a map ^ : Â ! A (_ : _A ! A,respectively) is an infomorphism, where the classification Â!!Â"1; !Â"2;RÂ" is defined by !Â"1 $ }!A1" (i.e.,a power set of A1) and !Â"2 $ A2 and if and only if aRAa for all a 2 H for a given a 2 Â2 and H & A1aRÂH.Analogously, _A!!_A"1; !_A"2;R_A" is defined by !_A"1 $ }!A1" (i.e., power set of A1), !_A"2 $ A2, and for a givena 2 !_A"2 and H & A1; aR_AH, if and only if aRAa for some a 2 H. Especially it holds that both ^ : Â ! A and_ : _A ! A are token identical maps such that ^_!a" $ __!a" $ a for a 2 !Â"2 $ !_A"2 $ A2. The fundamentalproperty is then expressed as follows: For all a 2 !Â"2 $ !_A"2 and H & A1, aRAa () aRÂ ^^ !H"!aRAa () aR_A_^

!H", respectively).Similarly, a classification A!A1;A2;RA" is :-closed if and only if a map : : :A ! A is an infomorphism, where a

classification :A!!:A"1; !:A"2;R:A" is defined by !:A"1 $ A1, !:A"2 $ A2, and for a given a 2 A2 and a 2 A1aR:Aa if andonly if a–RAa.

In order to estimate a logical operation in a dynamical sequence, we check whether a classification is ^-, _- and :-closed. For this purpose, we first have to verify some theorems. A set of tokA!a" in a classification A!A1;A2;RA" isdefined by tokA!a" $ fa 2 A2jaRAag [1]. In using the term tokA!a", it is easy to see that the operations ^, _ and : can beregarded as the three set operations intersection, union and complement, respectively.

Lemma 4.1. Given a classification A!A1;A2;RA", for all ai and aj 2 A1,

(i) tokA!ai" \ tokA!aj" $ tokÂ!fai; ajg" with fai; ajg 2 !Â"1;(ii) tokA!ai" [ tokA!aj" $ tok_A!fai; ajg" with fai; ajg 2 !_A"1.

Proof. It is proved by the following series of equivalence.


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a 2 tokA!ai" \ tokA!aj" () aRAai and aRAaj

() aRÂfai; ajg !from definition RÂ"

() a 2 tokÂ!fai; ajg" !from definition of tok";

a 2 tokA!ai" [ tokA!aj" () aRAai or aRAaj

() aR_Afai; ajg !from definition R_A"

() a 2 tok_A!fai; ajg" !from definition of tok": !

Corollary 4.2. Given a classification A!A1;A2;RA", for any H & A1 denoted by H $ fa1; a2; . . . ; ang, the following equa-tions hold:

(i) tokA!a1" \ tokA!a2" \ * * * \ tokA!an" $ tokÂ!H";(ii) tokA!a1" [ tokA!a2" [ * * * [ tokA!an" $ tokÂ!H".

This can be proven analogously to Lemma 1.

Lemma 4.3. A classification A!A1;A2;RA" is ^-closed if and only if there exists a 2 A1 such that tokÂ!H" $ tokA!a" for allH 2 !Â"1. Similarly, a classification A!A1;A2;RA" is _-closed if and only if there exists a 2 A1 such thattok_A!H" $ tokA!a" for all H 2 !_A"1.

Proof.

A!A1;A2;RA" is ^ -closed () ^ : Â ! A satisfies the fundamental property

() 8a 2 A28H 2 !Â"1!aRÂH () aRA ^^ !H""

() 8H 2 !Â"1;9 a 2 A1!tokÂ!H" $ tokA!a"";

where ^^!H" $ a. The second statement can be proven analogously to the first. h

Theorem 4.4.(i) A!A1;A2;RA" is ^-closed

() For 8ai; aj 2 A1, 9as 2 A1 such that tokA!as" $ tokA!ai" \ tokA!aj" and for [a2A1tokA!a", 9ap 2 A1 such thattokA!ap" $ [a2A1

tok!a".(ii) A!A1;A2;RA" is _-closed

() For 8ai; aj 2 A1, 9as 2 A1 such that tokA!as" $ tokA!ai" [ tokA!aj" and for \a2A1tokA!a", 9ap 2 A1 such thattokA!ap" $ \a2A1tok!a".

Proof. (i) ()): For 8ai; aj 2 A1, from Lemma 4.1,

tokÂ!fai; ajg" $ tokA!ai" \ tokA!aj" () 9as 2 A1!tokA!as" $ tokÂ!fai; ajg" !from Lemma 4:3":

Next we consider the unification [a2A1tok!a". Since A!A1;A2;RA" is ^-closed (assumption), there exists ^!£" 2 A1 suchthat

aRA ^ !£" () aRÂ£

for all a 2 A2. In denoting ap $ ^!£", this means that

tokA!ap" $ tokÂ!£":

It can also be seen that tokÂ!£" $ fa 2 A2jaRÂ£g $ fa 2 A2jaRAa; 8a 2 £g $ A2 $ [a2A1tokA!a", and thentokA!ap" $ [a2A1tok!a".

((): For H 6$ ;, H 2 !Â"1, we denote H $ fa1; a2; . . . ; ang, and then


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tokÂ!H" $ tokÂ!fa1; a2; * * * ; ang"$ !tokA!a1" \ tokA!a2"" \ tokA!a3" \ * * * \ tokA!an" !Corollary 4:2"$ !tokA!as!1"" \ tokA!a3"" \ tokA!a4" \ * * * \ tokA!an" !assumption"

..

.

$ tokA!as!n%1"": !assumption"

In the next step we consider the case H $ £ 2 !Â"1. For all a 2 A2 it holds that aRÂ£. This means thattokÂ!£" $ fa 2 A2jaRÂ£g $ A2 $ [a2A1tok!a". Because of the assumption we have 9!ap" 2 A1 such thattokA!ap" $ [a2A1tok!a", and then 9ap 2 A1 such that tokA!ap" $ tokÂ!£".

As a result, it follows from Lemma 4.3 that A!A1;A2;RA" is ^-closed. (ii) can be verified in the same way as (i). h

Theorem 4.5.A!A1;A2;RA" is :-closed() For 8a 2 A1; 9b 2 A1 such that tokA!b" $ tokA!:a" $ !tokA!a""c,

where Sc is a complement of a set, S.

Proof. A !A1;A2;RA" is :-closed () : : :A ! A satisfies the fundamental property

() 8a 2 A1;8a 2 A2!aRA:a () aR:A!a";

() 8a 2 A1;9b $ :a 2 A1!tokA!b" $ !tokA!a""c $ tok:A!a"":

The final statement holds because tokA!b" $ fa 2 A2jaRA:ag $ fa 2 A2jaR:Aag $ fa 2 A2jaRAagc $ !tokA!a""c. h

In using Theorem 4.4 and/or Theorem 4.5, one can check whether a given classification is ^, _ and/or :-closed ornot. We illustrate such a check and show how to construct ^, _ and/or :-closed classification when an arbitraryclassification is given.

Example 4.6 (Construction of ^-closed classification). Assume given a classification A!A1;A2;RA" where A1 $ fa1;a2; a3g, A2 $ fa1; a2; a3g, RA is defined as

&1 &2 &3

a1

a2

a3

and where the filled squares located at the a-th token and at the a-th type symbolizes aRa whereas the un-filled squaresstand for a–Ra (i.e., a token a is not of type a). In this classification, one obtains

tokA!a1" $ fa1; a2g; tokA!a2" $ fa3g; tokA!a3" $ fa2; a3g:

From Theorem 4.4, ^-closed classification requires

tokA!a1" \ tokA!a2" $ £ 2 ftokA!a"ja 2 A1g;

tokA!a1" \ tokA!a3" $ fa2g 2 ftokA!a"ja 2 A1g;

tokA!a2" \ tokA!a3" $ fa3g 2 ftokA!a"ja 2 A1g;

[a2A1 tok!a" 2 ftokA!a"ja 2 A1g:

In our example, however, £ 62 ftokA!a"ja 2 A1g, fa2g 62 ftokA!a"ja 2 A1g and [a2A1tok!a" 62 ftokA!a"ja 2 A1g whichmeans that A!A1;A2;RA" is not ^-closed.

From the classification A!A1;A2;RA", one can construct a ^-closed classification B!B1;B2;RB" defined byB1 $ A1 [ fajtok!a" $ £; fa2g;A2g $ fa4; a5; a6g, B2 $ A2 and RB $ RA [ RB where for all a 2 B2, a–RBa4 and aRBa6, andfor a2 2 B2, aRBa5 and a 2 B2, a 6$ a2, a–RBa5. Therefore, one obtains


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&1 &2 &3 &4 &5 &6

a1

a2

a3

as a ^-closed classification B!B1;B2;RB". By constructing ^B!!^B"1; !^B"2;R^B" as,

' {&1} {&3} {&5} {&1, &3}{&1, &5} {&1,&2,&3 ,&4,&5}

a1

a2

a3

{&2} {&4} {&1, &4}{&1, &6} {&1,&2,&3 ,&4,&5,&6}

one can check the existence of infomorphism. It is obvious that for all H 2 !^B"1 $ }!B1", there exists a $ ^!H" 2 B1,and then B!B1;B2;RB" is ^-closed.

Example 4.7 (Construction of :-closed classification). Given a classification A!A1;A2;RA" where A1 $ fa1; a2; a3g,A2 $ fa1; a2g, and RA is defined as

&1 &2 &3

a1

a2

one obtains tokA!a1" $ fa1g, tokA!a2" $ fa2g, tokA!a3" $ fa1; a2g. From Theorem 4.5, tokA!:a1" $ !tokA!a1""c $fa2g 2 ftok!a"ja 2 A1g, tokA!:a2" $ !tokA!a2""c $ fa1g 2 ftok!a"ja 2 A1g, tokA!:a3" $ !tokAa3""c $ £ 62 ftok!a"ja 2A1g. It leads that A!A1;A2;RA" is not :-closed.

By constructing a classification such that for all a 2 A1 tokA!:a" 2 ftok!a"ja 2 A1g, one obtains the left relation, B.Clearly, there is an infomorphism from :B (right diagram) to A.

&1 &2 &3 &4 &1 &2 &3 &4

a1 a1

a2 a2

In the subsequent section we define the dynamical infomorphism as a model of the endo-perspective and we estimatethe change of infomorphisms in terms of ^, _ and :-operations.

From these investigations it is easy to see that a binary relation in a classification is a kind of filter by which aparticular topological space is defined. Given a set of tokens as A2, one can define a subset of a power-set of A2. Hence,tokA!a" is expressed as a subset of A2. It turns out that a type a 2 A1 indicates a subset of A2, and a set of types, A1,represents a set of observable subsets of A2. This is why a binary relation can be regarded as a filter. If A1 indicates allpossible subsets of A2, a topological space is defined by a power-set of A2, and if A1 indicates an empty subset of A2, thena topological space is defined by f£;A2g. In the same way, a variety of topological spaces can immediately be defined asa classification.

However, in a dynamical infomorphism the logic is found in a dynamical sequence of classifications. We discuss sucha dynamical logic in a dynamic infomorphism.


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5. Dynamical infomorphism

5.1. Interaction between formal intent and formal extent

In formalizing an endo-perspective, we use a pair of classifications endowed with an infomorphism that satisfies thefundamental property. Thereafter, a relation in the classification is mixed with the infomorphism that is the formalexpression for opening the perspective to the outside. In this definition, it is assumed that an observer is placed at themargin of the perspective as a pair of classifications referring to the outside of his own perspective. The first perspectiveis expressed as F : A ! B with an infomorphism f $ hf ^; f _i, f ^ : A1 ! B1 and f _ : B2 ! A2 where A and B areabbreviations for A!A1;A2;RA" and B!B1;B2;RB", respectively. Classifications A and B are regarded as the intent and theextent, respectively.

As mentioned in Section 2, the mixture of the intent and the extent is defined as the mixture of the intra- (relation, RA

or RB) and inter-actions (f $ hf ^; f _i), because it is based on the mixture of a functor and an arrow. In following thisconsideration, the application of the inter-action to the intra-action is expressed as a composition of intra-actions. Inassuming that the inter-action, i.e. f $ hf ^; f _i, is a (pseudo-) functor, the application of the inter-action to the intra-action (the classification called intent) implies a process of interpretation from the intent to the extent (the otherclassification). The definition of the mixture of inter- and intra-actions, therefore, implies that the process of inter-pretation is expressed as the composition of intra-actions, and the so-interpreted intent cannot coincide with the extentand vice versa. As a result, the mixture of the relation, RA or RB with f $ hf ^; f _i entails the interpreted relations, R0

Aand R0

B. In other words, interpreted pair of classifications A!A1;A2;R0A" and B!B1;B2;R0

B" is obtained, accompanied byf $ hf ^; f _i, whereby, however, this infomorphism is not defined as the connecting device between R0

A and R0B, because

there is no assumption of a super-observer who can overview the whole perspective. An endo-observer only sees a partof the perspective, either the intent or the extent. If he focuses on the intent A!A1;A2;RA", he obtains R0

B by theinterpretation. Once he obtained R0

B and then moves to the extent, he has to constitute a new intent consistent with RB

with respect to f $ hf ^; f _i. As a result, he moves to the intent again obtaining a new one, A!A1;A2;R#A". In an

analogous fashion, starting from the extent B!B1;B2;RB", he eventually obtains B!B1;B2;R#B". Finally, the perspective

becomes a pair of classification, A!A1;A2;R#A" and B!B1;B2;R#

B" and it is not trivial to see whether there is an info-morphism between them. If there exists an infomorphism between them, the transition from a pair A!A1;A2;RA" andB!B1;B2;RB" to a pair A!A1;A2;R#

A" and B!B1;B2;R#B" is regarded as a time transition. In this sense, the time transition

results from the mixture of intra- and inter-action, or from formal expression of indefinite environments surroundingthe perspective.

The dynamical infomorphism is schematically shown in Fig. 2. Because an endo-observer cannot overview hisperspective as a whole, it is easy to imagine that an endo-observer can be assigned to one out of two cases, either to anintent- or to an extent-observer. Because of the incompleteness of observation, an intent-observer cannot directly

RAt

RBt

A1

A2

B1 B2

RB’

RA’

f t

RBt+1

RAt+1

f t+1

MakingImage

Reconstructionof New Relation

Possible?

Fig. 2. Schematic diagram of the dynamical infomorphism. Given two classifications RtA and Rt

B. with jtyp!A"j $ jtyp!B"j $jtok!A"j $ jtok!B"j, and an infomorphism, f ^ t and f _ t, the images of two classifications are constructed, and they are represented byR0A and R0

B. At the next step, Rt'1A is constructed so as to keep the fundamental property with Rt

B, f^ t and f _ t (i.e., for all a 2 typ!A" and

b 2 tok!B", f _ t!b"Rt'1A a () bRt

Bf^t!a"". Rt'1

B is analogously constructed to satisfy that for all a 2 typ!A" and b 2 tok!B",f _t!b"Rt

Aa () bRt'1B f ^ t!a"". Finally the new infomorphism, f ^ t'1 and f _ t'1 are searched. Only if a particular condition is satisfied,

there exist f ^ t'1 and f _ t'1.


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observe the extent. He constructs the image of the extent that is B!B1;B2;R0B". In order to make his observation of the

intent consistent with the image of the extent, he re-constructs the intent leading to A!A1;A2;R#A". In the same fashion,

the extent-observer constructs the image of the intent and via this image he reconstructs the extent, B!B1;B2;R#B".

Therefore, an endo-observer can be interpreted as a system that consists of two non-consistent observers. Thatinconsistency triggers the mixture of the intra- and inter-actions leading to the time transition process.

The question arises whether the time transition is possible within the proposed scheme of dynamical infomorphism.To answer that question we prepare some propositions. First, the following proposition is verified that allows to checkwhether A!A1;A2;R#

A" is obtained from given B!B1;B2;R0B" and f $ hf ^; f _i.

Proposition 5.1. Assume given a classification B!B1;B2;RB" and f $ hf ^; f _i with f ^ : A1 ! B1 and f _ : B2 ! A2. If andonly if the condition that

8a 2 A1;8b1 6$ b2; b1; b2 2 B2!f _!b1" $ f _!b2"" ) !b1RBf ^!a" and b2RBf ^!a"" or !b1–RBf ^!a" and b2–RBf ^!a""

is satisfied, there exists RA & A1 # A2 such that for 8a0 2 A1, 8b 2 B2!bRBf ^!a0" () f _!b"RAa0".

Proof((): From the assumption, for 8a 2 A1,

8b 2 B2 !bRBf ^!a" () f _!b"RBa". For b1 6$ b2; b1; b2 2 B2,

b1–RBf ^!a" and b2RBf ^!a" () f _!b1"–RBa and f _!b2"RBa;

!or; b1RBf ^!a" and b2–RBf ^!a" () f _!b1"RBa and f _!b2–RBa"":

The assumption f _!b1" $ f _!b2" yields f _!b1"–RBa and f _!b1"RBa which is a contradiction. Therefore, we concludef _!b1" 6$ f _!b2". Finally, we verify the contraposition.()): If one defines RA & A1 # A2 so that for 8a0 2 A1,

8b 2 B2!bRBf ^!a0" () f _!b"RAa0), one obtains the following fourpossible cases

b1RBf ^!a" and b2RBf ^!a" () f _!b1"RAa and f _!b2"RAa;

b1RBf ^!a" and b2–RBf ^!a" () f _!b1"RAa and f _!b2"–RAa;

b1–RBf ^!a" and b2RBf ^!a" () f _!b1"–RAa and f _!b2"RAa;

b1–RBf ^!a" and b2–RBf ^!a" () f _!b1"–RAa and f _!b2"–RAa:

Because f _!b1" 6$ f _!b2", all cases are verified. h

One can implement the procedure to find RA & A1 # A2 satisfying the fundamental property for 8a0 2 A1,8b 2 B2!bRBf ^!a0" () f _!b"RAa0" given a classification B!B1;B2;RB" and f $ hf ^; f _i with f ^ : A1 ! B1 andf _ : B2 ! A2. As mentioned in Proposition 5.1, there is, in general, no such an infomorphism. Fig. 3 shows an examplefor the construction of RA.

Now we define a dynamical infomorphism. To this end, firstly the image of the extent viewed from the intent-ob-server and the image of the intent viewed from the extent-observer is defined in a formal way.

Definition 5.2. Given two classifications, A!A1;A2;RA" and B!B1;B2;RB", and f t $ hf ^; f _i with f ^ : A1 ! B1 andf _ : B2 ! A2 satisfying the fundamental property, A’s image of B is defined as B0!B1;B2;R0

B" with

R0B $ fhb; bij9a 2 A1!bRBf ^!a"; aRAa; kA!a" $ b"; b 2 B1; b 2 B2g;

where kA : A2 ! B2 is a bijective map. Analogously, B’s image of A is defined as A0!A1;A2;R0A" with

R0A $ fha; aij9b 2 B2!f _!b"RAa; bRBb; kB!b" $ a"; a 2 A1; a 2 A2g;

where kB : B1 ! A2 is a bijective map.Note that kA!a"R0

Bb is defined if there exists an a 2 A1 such that bRBf ^!a", aRAa and kA!a" $ b. This in turn meansthat there exists an a0 2 A1 such that b–RBf ^!a0", aRAa0 and kA!a" $ b. Therefore, b–RBkA!a" does not always imply thatb–RBf ^!a0", aRAa0 and kA!a" $ b. It implies that bRBf ^!a", a–RAa and kA!a" $ b which is very important when consideringan infomorphism between A0!A1;A2;R0

A" and B0!B1;B2;R0B".


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Our next step is the definition of the reconstruction of the intent (respectively extent) via the image of the extent(respectively intent). From this transformation we derive the possibility of a time transition.

Definition 5.3. Given two classifications, At!A1;A2;RA" and Bt!B1;B2;RB", and f t $ hf t^; f t_i with f t^ : A1 ! B1 andf t_ : B2 ! A2 satisfying the fundamental property, then the time development of classifications is defined by a sequencefrom At!A1;A2;RA" and Bt!B1;B2;RB" to At'1!A1;A2;R#

A" and Bt'1!B1;B2;R#B", where R#

A is a relation between A1 and A2,of which f t $ hf t^; f t_i satisfies the fundamental property between R#

A and R0B (At’s image of Bt), and R#

B is a relationbetween B1 and B2 of which f t $ hf t^; f t_i satisfies the fundamental property between R#

B and R0A (Bt’s image of At). If

there is an infomorphism between At'1!A1;A2;R#A" and Bt'1!B1;B2;R#

B", it is denoted by and f t'1 $ hf t'1^; f t'1_i.

We verify that there exists at least one infomorphism between !A1;A2;R#A" and !B1;B2;R#

B" (i.e., newly obtained intentand extent) if R0

A and R0B satisfy a particular condition. First it is verified to reconstruct a classification !A1;A2;R#

A" giventhe image of extent or !B1;B2;R0

B" and f t $ hf t^; f t_i.

Lemma 5.4. In the time development of classifications, R#A (respectively, R#

B) can be always made so as to set f t $ hf t^; f t_isatisfy the fundamental property between R#

A and R0B (respectively, R#

B and R0A). In terms of category Inf , there exists an

arrow from !A1;A2;R#A" to !B1;B2;R0

B" (respectively, from !A1;A2;R0A" to !B1;B2;R#

B").

In other words, given f t $ hf t^; f t_i and R0B (At’s image of Bt), there exists a relation R#

A & A1 # A2 satisfying that for8a 2 A1,

8b 2 B2!bR0Bf

^!a0" () f _!b"R#Aa

0".

Proof. The statement will be proven by the application of the contraposition of Proposition 5.1. It is supposed thatf _!b1" $ f _!b2" for 8b1 6$ b2, b1; b2 2 B2. It is also supposed that

b1R0Bf

^!a" and b2–R0Bf

^!a":

The former term means that there exists a0 2 A1 such that

b1RBf ^!a00" and a–RAa0 !from the definition of R0B":

Because kA is surjective, for f ^!a" 2 B1 there exists a 2 A2 such that kA!a" $ f ^!a". The second term means that there isno a00 2 A1 such that

b2RBf ^!a00" and a–RAa00 !from the definition of R0B":

Fig. 3. Two examples for the construction of RB, given classifications RA with jtyp!A"j $ jtyp!B"j $ jtok!A"j $ jtok!B"j, and an in-fomorphism, f ^ : typ!A" ! typ!B" (depicted by a solid arrow) and f _ : tok!B" ! tok!A" (depicted by a broken arrow). In thematrices representing f ^ and f _, a square with square represents !a; f ^!a"" (left) and !b; f _!b"" (right). In the matrices representingconstructed relations, a filled square represents bRBb that is uniquely determined, a square involving a small square represents that it isproved both bRBb and b–RBb, a square with · represents that it is proved neither bRBb nor b–RBb. The first example shows the failure ofconstruction, and the second one shows the success of construction. As far as RA is constructed by the Definition 5.2, RB is alwaysconstructed, given RA, f ^ : typ!A" ! typ!B" and f _ : tok!B" ! tok!A" (see Lemma 5.4).


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On the other hand,

b1RBf ^!a0" and a–RAa0 () f _!b1"RBa0 and a–RAa0 !from the fundamental property"() f _!b2"RBa0 and a–RAa0 !from f _!b1" $ f _!b2""() b2RBf ^!a0" and a–RAa0:

That is a contradiction. Therefore, one obtains that f _!b1" 6$ f _!b2". From the Proposition 5.1, there existsR#A & A1 # A2 satisfying that for 8a0 2 A1,

8b 2 B2!bR0Bf

^!a0" () f _!b"R#Aa

0". h

We can also verify there exists an infomorphism between the image of extent and the image of intent under aparticular condition.

Proposition 5.5. In the time development of classifications, there exists an infomorphism between !A1;A2;R0A" and

!B1;B2;R0B" if bR0

Bb and aR0Aa satisfy the condition such as

b–R0Bb () 8a 2 A1!b–RBf ^!a"; aRAa; b $ kA!a"";

a–R0Aa () 8b 2 B1!f _!b"–RAa; bRBb; a $ kB!b"";

where R0A is Bt’s image of At and R0

B is At’s image of Bt. In terms of category Inf, there exists an arrow from !A1;A2;R0A" to

!B1;B2;R0B".

Proof. We recall that At!A1;A2;RA" and Bt!B1;B2;RB", and an infomorphism f t $ hf t^; f t_i are given. We define twomaps, g^ : A1 ! B1 and g_ : B2 ! A2 such that

g^!a" $ kA!a" : () aRAa; g_!b" $ kB!b" : () bRBb:

From the definition of R0A $ fha; aij9b 2 B2!f _!b"RAa; bRB; kB!b" $ a"; a 2 A1; a 2 A2g, one can find the following se-

quence of equivalence relation:

g_!b"R0Aa () kB!b"R0

Aa; bRBb () f _!b"RAa; bRBb () bRBf ^!a":

Because there exists a 2 A2 with aRAa, both aRAa and bRBf ^!a" entailaRAa; bRBf ^!a" () bR0

BkA!a" () bR0Bg

^!a":

The condition concerning about b–R0Bb and a–R0

Aa is necessary to keep the contraposition of the equivalenceg_!b"R0

Aa () bR0Bg

^!a". From that condition, we yield,

g_!b"–R0Aa () f _!b"–RAa; bRBb; g_!b" $ kB!b"

() b–RBf ^!a"; aRAa () b–R0BkA!a" () b–R0

Bg^!a": !

The following statement also shows the existence of an infomorphism between the image of the extent and the imageof the intent, where the direction of an infomorphism is opposite to the infomorphism shown in Proposition 5.5.

Lemma 5.6. In the time development of classifications, there exists an infomorphism from !B1;B2;R0B" to !A1;A2;R0

A" ifbR0

Bb and aR0Aa satisfy the following conditions

b–R0Bb () 8a 2 A1!b–RBf ^!a"; aRAa; b $ kA!a"";

a–R0Aa () 8b 2 B1!f _!b"–RAa; bRBb; a $ kB!b"";

where R0A is Bt’s image of At and R0

B is At’s image of Bt. In terms of category Inf, there exists an arrow from !B1;B2;R0B" to

!A1;A2;R0A".

Proof. We define two maps, h^ : B1 ! A1 and h_ : A2 ! B2 such that

h^!kA!a"" $ a; aRAa and h_!kB!b"" $ b; bRBb:

Because kA and kB are bijective, that definition can be verified. We denote that kA!a0" $ b; kB!b0" $ a: Supposing thataR0

Ah^!b", one obtains the following sequence of equivalent relation:

aR0Ah

^!b" () aR0Ah

^!kA!a0"" !because kA!a0" $ b"() f _!b"RAh^!kA!a0"" and bRBb

0


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(because of the following diagram)

RA' RA'

p q

RA RB RA RB

h$ B2 b

B1 A1 A2 f % B2 h$((A (a')) f %(b) )'

B

A2 B()')=a

((

where arrows without symbols represent projections (denoted as pr), and p!ha; a0i" $ ha; ai with aRAa, andq!ha; a0i" $ hk%1

B !a0"; bi with bRBk%1B !a0". Therefore, R0

A ! RA ! A2 and R0A ! RB ! B2 ! A2 are commutative such as

prp!ha; a0i" $ pr!ha; ai" $ a and f _prq!ha; a0i" $ f _pr!hk%1B !a0"; bi" $ f _!b" $ a".

() bRBf ^!h^!kA!a0""" and a0RAh^!kA!a0"" !because of f having the fundamental property

and of the definition of h^"() bR0

BkA!a0"

(because of the following diagram)

RB' RB'

s r

RB RA RB RA

h% A1

A2 B2 B1 f $ A2 b f $(h$((A (a' ))) a'

(A

B1 (A(a' )= )

h$((A (a' ))

where arrows without symbols represent projections (denoted as pr), and s!hb; b0i" $ hb; bi with bRBb, andr!hb; b0i" $ ha; k%1

A !b"i with k%1A !b"RAa. Therefore, R0

B ! RB ! B1 and R0B ! RA ! A1 ! B1 are commutative such as

pr s!hb; b0i" $ pr!hb; bi" $ b, and f ^pr r!hb; b0i" $ f ^pr!ha; k%1A !b"i" $ f ^!a" $ b".

() h_!kB!b0""R0BkA!a

0" !because bRBb0; h_!kB!b0"" $ b"

() h_!a"R0Bb !because kA!a0" $ b; kB!b0" $ a": !

As depicted in Fig. 4, there exists an infomorphism between At'1!A1;A2;R#A" and Bt'1!B1;B2;R#

B" under a particularcondition as shown in Lemma 5.6. That infomorphism is made by the composition of f t $ hf t^; f t_i and an info-morphism between the image of the intent and the image of the extent. h

Theorem 5.7. If R0A and R0

B satisfy the condition mentioned in Lemma 5.6 for the time development of classifications, thereexists an infomorphism between At'1!A1;A2;R#

A" and Bt'1!B1;B2;R#B", given At!A1;A2;RA", Bt!B1;B2;RB" and an info-

morphism, f t $ hf t^; f t_i with f t^ : A1 ! B1 and f t_ : B2 ! A2. In terms of category Inf, there exists an arrow fromAt'1!A1;A2;R#

A" to Bt'1!B1;B2;R#B".

Proof. In terms of category Inf, there are arrows At'1!A1;A2;R#A" ! !B1;B2;R0

B", !B1;B2;R0B" ! !A1;A2;R0

A" and!A1;A2;R0

A" ! Bt'1!B1;B2;R#B" from Lemmas 5.4 and 5.6. Because a composition of arrows is an arrow, there exists an

arrow, At'1!A1;A2;R#A" ! Bt'1!B1;B2;R#

B", which is an infomorphism. h

Finally, we verify that there can be a new infomorphism for newly obtained intent and extent although the endo-perspective is perpetually transformed. This implies that time emerges from the mixture of intra- and inter-action of aperspective or from an indefinite environment surrounding the perspective.


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Dynamic infomorphism is an expression of a perspective that is opened to an indefinite environment. This is asignificant result because now an infomorphism can exist in spite of the existence of an indefinite environment. Assumethat an indefinite environment is expressed as a particular noise like a thermal perturbation. Then two classifications,the intent and extent, as well as an infomorphism are always perturbed by that noise. We regard this as a particularexpression of an opened perspective. However, such a model of an opened perspective never works well. If a relation ina classification (the intent or extent) is randomly flipped, e.g. from aRAa to a–RAa (or from a–RAa to aRAa), it is verydi"cult to find an infomorphism satisfying the fundamental property. Given two relations, infomorphisms is rarelyfound. That is why perturbation does not serve as a model of indefinite environments.

In the subsequent section we determine the logic of a sequence of dynamical transitions. Because the time transitionof a dynamical infomorphism is an expression for the endo-perspective, logic has to be defined not in a single classi-fication but rather in a sequence of classifications (i.e., intent or extent). It turns out that the logic of the extent (the oneclassification) is likely to be inconsistent with the logic of the intent (the other classification). In this sense, the rule ofinference is destined to be a local one.

5.2. Logic in a dynamical sequence

As far as the endo-perspective is expressed through the time transition that results from the mixture of intra- andinter-actions, logic does not exist in a single classification. Since classifications (i.e., intent and extent) are perpetuallychanged, the feature of classification, namely the ^-, _- and :-closed-ness, are determined through a sequence ofclassifications. In order to determine the logic even in a sequence of classifications, a sub-logic has to be defined asfollows:

Definition 5.8. A classification, A!A1;A2;RA", has a ^- (respectively, _-, :-) closed sub-logic if and only if there exists asubset Sub!A2" & A2 such that Sub!A"!A1; Sub!A2";RA \ !A1 # Sub!A2""" is ^- (respectively, _-, :-) closed.

Example 5.9. Given a classification as the following,

Fig. 4. An example of the time development of infomorphism. In this case, there exists an infomorphism between At'1!A1;A2;Rt'1A " and

Bt'1!B1;B2;Rt'1B " under a particular condition as shown in Lemma 5.6, given At!A1;A2;Rt

A", Bt!B1;B2;Rt

B", f ^ : typ!A" ! typ!B" andf _ : tok!B" ! tok!A". In the matrix of f ^ : typ!A" ! typ!B", a cell with filled square represents !a; f ^!a"" and the matrix off _ : tok!B" ! tok!A", a cell with filled square represents !b; f _!b"". In each relation table, !type; token" $ !a; a" with aRa is depictedby a filled square.


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one can define Sub!A2" $ fa1; a2g and Sub!A"!A1; Sub!A2";RA \ !A1 # Sub!A2""", where RA \ !A1 # Sub!A2"" $fha1; a2i; ha2; a2ig. This means that there exits a subset, Sub!A2" such that Sub!A"!A1; Sub!A2";RA \ !A1 # Sub!A2""" is^-, _-, and :-closed logic. In fact, tokSub!A"!a1" $ fa1; a2g and tokSub!A"!a2" $ ø. Therefore, a set, ftoksub!A2"!a1";tokSub!A2"!a2"g is \-closed, [-closed and complement-closed (see Theorems 4.4 and 4.5).

Fig. 5 shows a sequence of classifications generated from a dynamical infomorphism. In that sequence, first threepairs of classifications have ^-, _- and :-closed sub-logic. Specifically, the first pair, Sub!A2" $ fa1; a4g andSub!B2" $ fa2; a4g are expressed as

and both parts of the pair have ^-, _- and :-closed sub-logic. In the second and third pairs of classifications, Sub!A2"and Sub!B2" are expressed as isomorphic to either fhak ; a1i; hak ; a2i; has; a3i; has; a4ig or fhak ; a1i; has; a1ig such as

and it is obvious that they are also ^-, _- and :-closed sub-logic. In the fourth pair, the one classification has only ^-and _ sub-logics, whereas the other has a :-closed sub-logic as well as a ^-, _-closed sub-logic. The consistency betweentwo classifications is lost in the fifth pair, where at the next step there no longer exists an infomorphism between the twogenerated classifications. It is easy to see that there is no Sub!A2" closed with respect to ^-, _- and :-operations. That isjust a conjectural remark, however, the collapse of a sub-logic can be relevant for the failure of infomorphism.

As aforementioned, logic defined in a single classification is only virtually verified by means of presenting anexample. Since a pair of classifications is transformed step by step, the rule of inference is also changed. Therefore, thenotion of logic should be also weakened and be defined as a sequence of sub-logics. In the example mentioned here, theweakened logic consisting of various sub-logics is ^-, _- and :-closed as far as the endo-perspective is kept alive.Roughly speaking, one might say that it has a ^-, _- and :-closed weakened logic. The problem of an exact definition ofthe weak logic remains for the future.

The dynamical infomorphism can proceed in changing classifications that can be connected with each other by aninfomorphism and that can be terminated by the failure of the fundamental property (i.e., the equivalence between theintent and the extent). A failure of the fundamental property implies the failure of the system. Therefore, we mightdefine the death of a system in a formal way. If the sub-logic is relevant for the failure of a system, it might imply that

Fig. 5. A sequence of classifications generated from a dynamical infomorphism. In each relation table, !type; token" $ !a; a" with aRais depicted by a filled square. Sub-logic generated by a relation table is changed as time proceeds. This time development is collapsedafterward.


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the behavior of a system is re-entered and re-formed in a system what we regard as a possible formal expression of‘‘recognition’’. The relationship between the behavior of a dynamical infomorphism and a logic defined in a systemmight be a way to formalize the process of perception and/or cognition.

6. Discussion and conclusion

We stick with the formalization of the endo-perspective in order to describe biological systems and/or life formally.Biological systems involve two di!erent standards, for example, intra-cellular reaction and inter-cellular reaction thatentail a contradiction by mixing them up [8]. Such an aspect is analogous to the endo-perspective from which one canboth recognize objects and act towards the open environments. As far as an endo-observer lives in a universe, actionscannot be separated from recognition or observation even if they can be distinguished from each other. An endo-observer enables both mixture and distinction of actions and recognition. Why is the endo-perspective analogous to abiological system? That question results from a perverted view onto the relationship between an object and anobservation. In a self-reflection, one cannot separate actions from recognition. One is aware of both ones own cognitiveperspective and simultaneously of the universe in which this perspective acts. This is the meaning of saying a person hasconsciousness. In other words, as far as somebody is conscious he must be an endo-observer. When an endo-observerfinds an object that has an analogous structure as his endo-perspective, he calls and treats this object as a living bio-logical system. That is why an endo-perspective is destined to be analogous to a biological system and to a model ofconsciousness. Life cannot be understood without an endo-perspective. Life is not a property originated in a particularobject.

We use a basic framework for the formalization of an endo-perspective in which that perspective cannot be con-sistently defined in a direct way. The endo-perspective is expressed as a formal model by retaining the outside of themodel. That is a formal expression of the opened environment and/or of referring to the outside. How can one con-stitute such a model accompanied with retention? We focus on a formal structure of a paradox resulting from themixture of the intent and the extent as well as an unrestrained use of the notion of wholeness. We examine Russel’sparadox and the diagonal argument that is an essential structure of the latter. In that paradox both explicit and implicitformal expressions are indicated, where the implicit expression means the whole universe. In a formal expression, on theone hand, there are some explicit expressions described by symbols, and on the other hand, there is no substitution forthe whole universe. The former one can be compared with a consistent perspective and the latter one with an openenvironment in which the perspective can be operated and allows to react. Both the whole universe in a formalexpression and the open environment surrounding the perspective cannot be directly indicated, while one can refer to itby a symbol (e.g., 8) and conclude that it is possible to indicate the wholeness (or the open environments). Although theterms wholeness, universe or open environments make sense, they cannot be replaced by an indication. That is whyindicating an implicit expression entails a paradox [23,24].

If one attempts to formalize the endo-perspective, one has to adopt both distinction and mixture of the indicatedexpressions (e.g., a particular perspective) and the expressions just referred to (e.g., the open environment). A formalexpression allowing that mixture is a possible formal model with retention of the outside of the perspective. In anyformal model, the mixture of an indicated expression and a referred expression is prohibited. Russel’s results from thatmixture. The next question arises namely how one can build a formal model that allows that mixture without a con-tradiction. For this purpose, some mathematical tools have to be weakened.

In starting from Russel’s paradox, we replace the mixture of the intent and the extent by the mixture of two cat-egories, and we additionally replace the mixture of indicated and referred expressions by the mixture of an arrow (anoperation in a category) and a functor (an operation between categories). This replacement is not just an analogy and itis verified that those mixtures entail a paradox or a collapse of the equivalence between intent and extent (or twocategories). Therefore, our question can be replaced by the following. What kind of mathematical item has to beweakened to allow those mixtures without leading to a contradiction? We weaken the concept of a functor so that itcorresponds to an infomorphism proposed by Barwise.

The essence of our approach is not to be seen in the introduction an infomorphism to construct a static consistentmodel that consists of an intent and an extent, but rather to build a model of the endo-perspective allowing the mixtureof an arrow and a functor. Introducing an infomorphism is a way to allow that mixture. In that framework, the formalmodel of the endo-perspective is summarized as follows:

(i) The endo-perspective consists of two classifications (two categories), called the intent- and extent-observer. Eachclassification is defined as a triplet consisting of two sets (objects) and a relation (an arrow) between the two sets. (ii)The relationship between two classifications is defined by an infomorphism satisfying the fundamental property thatensures the equivalence between intent and extent. (iii) The equivalence relation between intent and extent is regarded


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just as an illusion. This model is still not a model for the endo-perspective unless the retention of that equivalence isformalized. This equivalence is re-estimated by an endo-observer, i.e. by himself, and then an infomorphism is appliedboth to the intent and the extent. (iv) Because the application of an infomorphism to a classification is defined by thecomposition of two relations that originate from two classifications given by an intent and an extent (that is a formalexpression of the mixture of an arrow and a functor), it turns out that the re-made intent and the re-made extent do notcoincide with the intent and extent given before. The new pair of intent and extent is a newly generated one. (v) Onlyunder a particular condition there is a new infomorphism between the new intent and the new extent. In other words, ifthere exists an infomorphism, the endo-perspective can proceed and/or generate time without entailing a failure. (vi) Ineach classification, one can define a logic and a sub-logic. The failure of a system can be predicted by the structure ofsuch an internal logic.

Various models concerning the brain and consciousness have been proposed, generally consisting of two parts (e.g.,[25–28]). The one part is regarded as the internal self, and the other is regarded as the internal environment. In thosemodels, the internal self perpetually repeats the simulation expressed as the dynamic interaction between the internalself and the internal environment [27]. These models can reveal many aspects of the behavior of brains and/or con-sciousness, but it is hard to confirm whether they reveal the failure of a system or not. The interaction between two partsis defined but there is no retention of the verification of the definition in its own right. That is why there is no possibilityfor a failure and a death of the system itself.

Can the formal model featuring both failure and death be constructed in a heuristic way? Instead of constructing amodel in a heuristic way, we start out from a logical contradiction resulting from the mixture of intent and extent. Thereason why we start with a paradox is, (i) the endo-perspective is paradoxical in its own right, and (ii) the endo-observerhas a finite life. If the logic is weakened to allow the mixture of intra-action (a model of the explicit expressionsindicated directly) and inter-action (a model of the implicit expression just referred to), the form involving that mixturedoes not always reveal a contradiction but is terminated by a contradiction. Under such an expectation we formalize aweakened paradox that leads to a system that can advance time but is terminated by a contradiction. That is a system-death.

In the fields of complex systems research, phenomena are reduced to interactions, especially based on nonlineardynamics [29–31]. Although those models can mimic various phenomena, the perspective might be exo-like. Given aparticular manifold, time development of a system is expressed as a trajectory. That is why the behavior of an object isdescribed in a particular consistent universe. As a result there is little possibility for an expression of a failure of asystem. If a system with many degrees of freedom is formally expressed as the interaction of chaotic dynamics, themanifold is not trivial and it is impossible to compute or trace the trajectory with infinite precision because the behaviorof the system are computed by using a digital computer [32–34]. Therefore, one may say that indefiniteness between realvalues and finite binary sequences can be embedded in a system. However, one cannot see at the same time the rela-tionship between computable trajectories with finite precision and incomputable ‘‘real’’ trajectories, because there is noe!ective way to trace real trajectories with finite precision (e.g., [35,36]). Trajectories with finite precision, therefore,cannot be embedded into or interpreted by a manifold defined with real number values. If such an embedding were notpossible, the evaluation of the tracing of a real trajectory with finite precision turns out to be impossible. Otherwise, onecan say neither that it is possible nor that it is impossible, because it is di"cult to describe the relationship between adescribed trajectory of finite precision and a real trajectory.

Of course, the non-linear dynamics with many degrees of freedom might reveal indefiniteness between real trajec-tories and observed trajectories with finite precisions. In that case, a system could reveal the endo-perspective as aformal expression, and some tools such as chaotic itinerancy [37] and homeochaos [38] could be a formal expression foran endo-observer with finite life. Time development is attracted to a particular attractor and is stationary stable byhomeochaos and can then leave the attractor. That development perpetually proceeds (i.e., chaotic itinerancy). Thatcould be a formal expression for finite life or finite lifetime. The question concerning indefiniteness also remains unlessthe relationship between both the computation with finite precision and with infinite precision can be estimated. Someresearchers especially focus on the interface between impossibility and a possible finite perspective [39–41]. However,one rather has to focus on the interface between the inside (e.g., finite things) and the outside (e.g., infinite things).While another approach to the interface has been proposed in [42,43], it is necessary to study the interface by mani-festing the relationship between a paradox and the indefiniteness.

That is why we directly focus on the logical failure of a system. The endo-perspective must involve two di!erentstandards. One is a particular consistent perspective corresponding to the inside, and the other is the domain in whichthe perspective can collapse, corresponding to the outside. To reveal two di!erent standards in a formal perspective, onehas to construct a formal model in which the collapse of a system is in principle allowed. That is a way to escape fromthe interpretation, while the interpretation must be an exo-perspective. In that sense, the proposed model featuringdynamic infomorphism is an influential model for the endo-perspective. The equivalence between intent and extent


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corresponds to a consistent and well-defined system, and the collapse of the equivalence corresponds to the failure of asystem in an exact sense. The dynamic infomorphism generates its own time, however, it can be terminated by thecollapse of the verification of a system (i.e., collapse of the equivalence between intent and extent). Especially, the logicdefined in a system is relevant for the condition of whether the collapse of a system can play an influential role inunderstanding the reflective operations or recognitions that originate from consciousness. While that is an openquestion, the notion of the endo-perspective may not be formalized unless the notion of a paradox and a weakenedparadox is introduced. At this point our model can contribute to the understanding of the endo-perspective and/orconsciousness.

Acknowledgement

We would like to thank Dr. Hans Diebner for discussions and suggestions for correcting our English.

Appendix A. Category and adjunctive functor

A category is defined as a collection of objects denoted by capital alphabets A;B; . . . and arrows between objectsf : A ! B, g : B ! C; . . . satisfying the following axiom:

(i) Each object A has an identity arrow idA : A ! A such that for any arrows, f : A ! X , g : Y ! A, f idA $ f andidAg $ g.

(ii) If f : A ! B, g : B ! C are arrows, the composition gf : A ! C is also an arrow. If the domain and codomain off is denoted by dom!f " and cod!f ", respectively, the composition gf is possible if dom!g" $ cod!f ".

(iii) Given f : A ! B, g : B ! C and h : C ! D, associative law hgf $ h!gf " $ !hg"f holds.

Given two categories, A and B, a functor F : A ! B is defined such that for an object A in A, FA and for an arrowf : A ! A0 in A, Ff : FA ! FA0 satisfy

(i) for f : A ! A0 and g : A0 ! A00, Fgf $ FgFf ,(ii) for idA : A ! A, F idA $ idFA.

For example, !%"A : C ! B is a functor defined by; for an object C in C , CA $ fp : A ! Cg and forg : C ! C0; gA : CA ! C0A satisfies the commutative diagram;

C C

C' C'

p

g F A g

gA(p)

As a result that gA is applied to an arrow p, gA!p" $ gp is obtained. Therefore, for any arrows g : C ! C0 andg0 : C0 ! C00, !g0g"A $ g0AgA. This is verified by the following: For each arrow p : A ! C,

!g0g"A!p" $ !g0g"p $ g0!gp" $ g0!gA!p"" $ g0A!gA!p"":

It is also verified that for idC : C ! C, !idC"A!p" $ idCp $ p $ idA!B!p", where the su"x A ! B represents theabbreviation of BA.

We illustrate another functor A# !%" : B ! C defined by the following. For an object B in B, A# B is defined. Foran arrow g : B ! B0, idA # g : A# B ! A# B0 is defined. Using these definitions it follows that idA # gg0 $ !idA#g0"!idA # g" and idA # idB $ idA#B.

Given two categories, A and B, a functor F : A ! B is left adjoint to a functor G : B ! A if and only ifA ! GB () FA ! B. This means that A!A;GB" ’ B!FA;B" where A!A;A0" $ ff : A ! A0g. From that equivalence,one can define two functors A!%;GB" and B!F !%";B". These functors transform an object A to A!A;GB" and B!FA;B",


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respectively. For an arrow, f : A ! A0, transformed arrows, A!F ;GB" and B!Ff ;B", are defined by the commutativediagrams: For each arrow p : A ! GB and s : FA ! B,

The commutative diagram A!f ;GB"!p" $ pf and B!Ff ;B"!s" $ sFf are also expressed as

Therefore, given g : A ! GB, one obtains the following commutative diagram;

where / is a one-to-one and onto map.In that commutative diagram, we choose idGB in A!GB;GB" and define /!idGB" $ ev, with ev : FGB ! B. Then we

obtain

B!Fg;B"/!idGB" $ /!A!g;GB"!idGB"";evFg $ /!idGB"/!g" $ /!idGBg" $ /!g":

It implies the commutative diagram (6) in the main text.

Appendix B. Diagonal argument

Assume the equivalence A ! GB () FA ! B as mentioned in Appendix A. If one replaces F by A# !%", and G by!%"A, and sets A $ B, one obtains A ! AA () A# A ! A. As mentioned in the discussion of Appendix A thisequivalence implies the commutative diagram:

From this, one obtains f $ ev!idA # g". One can interpret objects as sets and arrows as maps. Choose a particular mapso that for all a 2 A; f !a; a" is defined. One obtains

f !a; a" $ ev!idA # g"!a; a" $ g!a"!a":

Note that g!a" 2 AA and in general g!x"!y" implies that a map g!x" is applied to y. In assuming that g is a one-to-one andonto map, for all m 2 AA there exists b 2 A such that m $ g!b". Therefore, even for a special map hf 2 AA with an


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arbitrary map h : A ! A, there exists c 2 A such that hf $ g!c". Then one obtains hf !a; a" $ g!c"!a" for all a 2 A. Onthe other hand, from f !a; a" $ g!a"!a", hf !a; a" $ hg!a"!a", and then g!c"!a" $ hg!a"!a". Substituting c for a entailsthat g!c"!c" $ hg!c"!c". Finally g!c"!c" is a fixed point with respect to an arbitrary map h. That is a contradiction.

References

[1] Barwise J, Seligman J. Information flow, the logic of distributed systems. Cambridge Univ. Press; 1997.[2] R!ossler OE. Endophysics, the world as an interface. Singapore: World Scientific; 1998.[3] R!ossler OE. Ultraperspective and endo-physics. BioSystems 1996;38:211–9.[4] Matsuno K. Protobiology: physical basis for biology. Boca Raton: CRC Press; 1989.[5] Gunji Y-P. Global logic resulting from disequilibration process. BioSystems 1995;35:33–62.[6] Gunji Y-P, Ito K, Kusunoki Y. Formal model of internal measurement: alternate changing between recursive definition and

domain equation. Physica D 1997;110:289–312.[7] Conrad M. Adaptability, the significance of variability from molecule to ecosystem. New York: Plenum Press; 1983.[8] Conrad M. Microscopic–macroscopic interface in biological information processing. BioSystems 1984;16:345–63.[9] Gunji Y-P. Autonomic life as the proof of incompleteness and Lawvere’s theorem of fixed point. Appl Math Comp 1994;61:231–

67.[10] Rosen R. Theoretical biology and complexity, three essays on the natural philosophy of complex systems. Orlando: Academic

Press; 1985.[11] Rosen R. Life itself. New York: Columbia Univ. Press; 1991.[12] Rosen R. Talk in the 1st Symposium on ECHO (Evolutionary Complex and Hierarchical Organization) (Ehresmann AC, Farre G,

Vangbremeersch J-P, orgs.). Amien, France, 1995.[13] Gunji Y-P, Ito G. Orthomodular lattice obtained from addressing a fixed point. Physica D 1999;126:261–74.[14] Kampis G. Biological evolution as a process viewed internally. In: Atmanspacher H, Dalenoort GJ, editors. Inside Versus Outside,

Endo- and Exoconpcets of Observation and Knowledge in Physics, Philosophy and Cognitive Science. Berlin: Springer-Verlag;1994. p. 85–110.

[15] Atmanspacher H, Wiedenmann G, Amann A. Descartes revisited––the endo-exo distinction and its relevance for the study ofcomplex systems. Complexity 1995;1(3):15–21.

[16] Haken H. Synergetic, an introduction–nonequilibrium phase transitions and self-organization in physics chemistry and biology.Berlin: Springer-Verlag; 1977.

[17] R!ossler OE. Relative-state theory: four new aspects. Chaos, Solitons & Fractals 1996;7:845–52.[18] Whitehead AN, Russell B. Principia Mathematica I. 2nd ed. Cambridge Univ. Press; 1925.[19] Goldblatt R. Topoi, the categorical analysis of logic. Amsterdam: North-Holland; 1984$ 1991 (3rd).[20] Barwise J. Constraints, channels, and the flow of information. In: Actzel P, et al., editors. Situation Theory and Its Applications I,

CSLI Lecture Notes 22, CSLI Pub. Stanford, 1993. p. 3–27.[21] Barwise J, Perry J. Situations and attitudes. Cambridge: MIT Press; 1983.[22] Lawvere FW. Diagonal arguments and Cartesian closed categories. In: Hilton P, editor. Lecture Notes in Mathematics: Category

Theory, Homology Theory and their Applications II. New York: Springer-Verlag; 1969. p. 134–45.[23] Gunji Y-P, Aono M, Higashi H, Takachi Y. The third wholeness as an endo-observer. In: Diebner HH, Druckrey T, Weibel P,

editors. Science of Interface. Tubingen: Genista verlag; 2001. p. 111–30.[24] Gunji Y-P, Aono M, Hihashi H. Local Semantics based on a partial all-quantifier. Int J Comput Anticipat Syst 2001;8:303–18.[25] Penrose R. The Emperor’s new mind. Oxford Univ. Press; 1990.[26] Edelman G. Topobiology: an introduction to molecular embryology. Basic Books; 1988.[27] Pollack JB. The induction of dynamical recognizers. Mach Learn 1991;7:227–52.[28] Ikegami T, Taiji M. Structures of possible worlds in a game players with internal models. Acta Polytech Scand 1998;Ma91:283–92.[29] Fermi E, Pasta JR, Ulam SM. Studies of nonlinear problem. In: Collected Works of E. Fermi. 2, Univ. of Chicago Press, Chicago,

1965. p. 978–88.[30] Ermentrout EB, Cowan JD. A mathematical theory of visual hallucination pattern. Biol Cybernet 1979;34:137–50.[31] Wolfram S. Theory and applications of cellular automata. Singapore: World Scientific; 1986.[32] Kaneko K. Transition from torus to chaos accompanied by frequency lockings with symmetry breaking. Prog Theor Phys

1983;69:1427–42.[33] Keeler JD, Farmer JD. Robust space-time intermittency and 1=f noise. Physica D 23.[34] Tokita K, Yasutomi A. Mass extinction in a dynamical system of evolution with variable dimension. Phys Rev E 1999;60:682–7.[35] Guckenhimer J, Holmes P. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag.[36] McCauley JL, Palmore JI. Computable chaotic orbits. Phys Lett A 1986;15:433–45.[37] Kaneko K, Tsuda I. Complex systems: chaos and beyond. Berlin: Springer-Verlag; 2001.[38] Kaneko K, Ikegami T. Homeochaos: dynamical and evolving mutation rates. Physica D 1992;56:406–29.[39] Tsuda I. Toward an interpretation of dynamic neural activity in terms of chaotic dynamical systems. Behavioral and Brain

Sciences 2001;24(5):793–847.


ARTICLE IN PRESS

[40] Diebner HH, Sahle S, Ho! AA. A real time adaptive system for dynamics recognition. Chaos, Solitons & Fractals 2002;13:781–6.[41] Diebner HH, Ho! AA, Mathias A, Rohrbach PM, Sahle S. Towards a second cybernetics model for cognitive systems. Chaos,

Solitons & Fractals 2002;13:1465–74.[42] Gunji Y-P, Aono M, Wakisaka S, Tsuda S. Time–space re-entrant system: No-where$Now-Here. Chaos, Solitons & Fractals,

submitted for publication.[43] Gunji Y-P, Aono M, Kamiya S, Sasai K. Entrainment in time–space re-entrant system. Chaos, Solitons & Fractals, submitted for

publication.


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Dynamical infomorphism: form of endo-perspective

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