Again Computers and the Brain
 heterarchy, polycontexturality, neural nets, brain, computer 
R.Kaehr and E. von Goldammer
Abstract
Theoretical analyses of the logical structure for an adequate ´theory of living systems´ reveal the shortcomings of all standard logical systems ranging from twovalued Aristotelian logic to nvalued types eulogized by Post, Lukasiewicz and others. All these calculi have one thing in common that makes them completely inadequate for any formalization of selfreferential systems, i.e., all living systems; they are essentially linear. ´Linear´ in the technical sense of being onedimensional, since all logical statements are arranged in an ordered, unique sequence of numbers, resulting in pure hierarchically structured organizations. Selfreference, however, derives only from heterarchical structured Systems. Günther developed a formal, codifiable system of mathematical logic (´theory of polycontexturality´) which goes beyond all multivalued logics that have been common up to now, and possesses the ability to describe heterarchically structured systems in a formal mathematical sense. In other words, the ´theory of polycontexturality´ provides the theoretical basis for simulating selfreflecting processes (cognition) on logical machines. The subject is considered in connection with McCulloch´s study on a heterarchy of values determined by the topology of nervous nets.
INTRODUCTION
One goal in brain research is to separate the mechanisms of biological information processing and intelligence of living systems from neurophysiological processes, in the vague hope of one day understanding the whole range of behavior in animals and/or man in terms of the structurefunction relationships of the neural circuitry of their central nervous systems. The construction of intelligent devices, on the other hand, is the declared aim of the artificial intelligence (AI) sciences.
Despite the differences, there are certain connections between both scientific fields, resulting in a permanent mutual influence which reached one of its summits with the logification of neural activities by McCulloch and Pitts /1/ in their famous study, ´A Logical Calculus of the Ideas Immanent in Nervous Activity´, a model which still acts as a paradigm in neurosciences. The present study does not intend to question this paradigm, but to point to the possibilities of its evolution.
Many modern neuroscientific research findings can no longer be described in a reductionistic and hierarchical way. In particular, selfreferentiality, an attribute which characterizes all complex neural networks, cannot be described adequatly either through the Von Neumann computer model or through the McCullochPitts model.
It was not the intention of this study to grasp the brain conceptually using the notions of computer sciences, i.e., to consider the brain as a computer, but to focus attention on the consequences resulting from considering the Computer as a brain as has been discussed by biological scientists. In this context we refer to the work of von Foerster /2/, Löfgren /3/, Maturana and Varela /4/ and Günther /5/. The conception of Günther´s /6,7/ ´polycontexturality´ will the basic reference source throughout this paper.
It has scarcely been noticed in the scientific literature that shortly after ´A Logic Calculus …´, had been published, McCulloch presented a paper entitled ´A Heterarchy of Values Determined by the Topology of Nervous Nets´ /8/, in which he pointed to the necessity of introducing nontransitive and heterarchical structures for an adequate description of neural activities. However, because of their irreducible topological distribution, heterarchical structures elude any logification in the classic logical frame of relations  this is the problem. Later, in a lecture ´What´s in the Brain that Ink May Character?´/9/. McCulloch again clearly pointed out this general logical problem.
HIERARCHY versus HETERARCHY
Based on its logical assumptions, the Von Neumann machine is exclusively linear and hierarchically structured and excludes any circularity in a fundamental logical sense. A graphic metaphor for such a structure is given by the tree as represented, e.g., by the binary decision tree. Although parallel computers have to be considered as nonVon Neumann machines, they are still hierarchically organized, because the classiclogical rules are rigorously valid, viz., the law of transitivity or the exclusion of selfreferentiality. Their data and algorithmic structures are reducible to one and only one comprehensive logical fundament as an operational ´universe of discourse´. This is shown, for example, by an analysis of the methods of modeling and formalizing nonVon Neumann architectures by PetriHolt nets and the normalized nets of Rödding /10/.
In the following discussion, McCulloch´s nervous nets as they are represented by the figures in Ref. 8 will be analyzed. Fig. la here depicts the simplest model of a circular propagation of neural stimulations. According to the BellMagendie law, the nervous excitations enter the nervous system by dorsal and emerge by ventral roots. Circular propagation in this direction was called dromic and that in the opposite direction antidromic. So far no response of any effector has ever indicated an antidromic reflex, although conduction in the reverse direction has been demonstrated in both dorsal and ventral roots. All reflexes are dromic activities of feedback mechanisms with reverberating circuits (the endromes), which are not shown in Fig. 1. These feedback circuits, i.e., both endromes and reflexes, may be regenerative for one temporal combination of excitations and degenerative for another. Further details may be found in the work of McCulloch and Pitts /1/.
If an arbitrary number of parallel nervous circuits are considered, the conduction in each is dromic, in two or more syndromic, but from one to the other, whether it facilitates or inhibits, it is neither syn nor anti, but heterodromic. Without heterodromic activity, reflexes would occur independently of one another or, to put it in other words, for any sub, super or coordination of reflexes heterodromic activity is inevitable. In this way an inhibiting influence from one circuit to the other can be exerted. The resulting network maps are indicated in Fig. lb.
( a ) 
( b ) 
( c ) 
Figure 1: Neural
nets, from Ref. 8 continuous line : nervous portion of path; broken line : nervous somatic or environmental protion of path. 


b) parallel and hierarchic  c) heterarchic
(nontransitive)

The heterodromic configuration in Fig. 1b is isomorphic with that of the parallel computers mentioned above and, like those, it is hierarchically structured. This can be expressed in a more logical notation. For three neural processes A, B and C (as in Fig. 1b), if A is preferred to B and B to C, it follows that A is preferred to C, viz.,

(1) 
Relation (1) reflects the validity of the transitivity law of the classiclogical implication postulated for a hierarchy of values or processes which only admit a notion of super or subordination but no coordination. This, however, contradicts observations of neural processes such as the coordination of visual, tactile and/or acoustic perception.
Figure 1c illustrates the simplest (irreducible) nervous net with a ´diallel´ linking of the dromes such that the classiclogical transitivity for the preference relation (1) no longer holds, viz.,

(2) 
It should be mentioned that the relation neither follows from Fig. 1c nor was it discussed by McCulloch; here it has been added for sake of completeness. The net in Fig. 1c can no longer be reproduced on a plane such as is always possible in principle for parallel structures (cf. Fig. 1b). The simplest surface on which this net can be mapped topologically is a torus, which means that a twofold closure of the net results (cf. von Foerster /2/). For clarification, it should be added that the notion ´diallel´, as it was used by McCulloch /8/, is a logical (scholastic) term pointing toward a ´circulus vitiosus´.
McCulloch /8/ introduced the term ´heterarchy´ and delimitated it very clearly from the concept of a circulus vitiosus founded in the linearity, a concept used later for modeling living systems in connection with the indefinite recourse of operations by von Foerster /2/ and Varela /11/.
The concept of the logical diallel combines both nontransitivity (circularity) and heterarchy, i.e., the coordination of domains or values which cannot be submitted to a summum bonum. The diallel allows circularity on the roundabout of changing dimensions in a complex topology of domains without relapsing into the selfreflection of a circulus vitiosus.
Since within any domain the hierarchical principle holds and the multitude of domains are combined by the principle of heterarchy, systems structured in this way allow an interplay of coordination (heterarchy) and sub ordination (hierarchy).
It is the distribution of qualitatively different domains which has no equivalent in classical logic where a single basic domain is classified in types, categories, worlds, etc. This is the fundamental difference between McCulloch´s conception of heterarchy and that of the socalled ´radical constructivism´/12/.
Under these conditions it is no longer possible to submit a nervous net (in the sense of McCulloch) to one and only one superordinated logical aspect or concept. That means that modeling and formalization of heterarchical structures, which have to be considered as systemwholes or complex systems, require a multiplicity of simultaneous points of view.
If nontransitivity has to be modeled within the framework of the classical logic, antinomic situations will be generated as is expressed, for example, by relation (2). Therefore, the question arises of how a complex system can be unambiguously described with the indicated structural features in a consistent formal logical way.
The brain is a selfreferential system par excellence. It always interacts with its own states, i.e., it is a completely closed system equipped with neither a sensorium (sensory cells) nor a motorium (muscles)  its boundaries are given by the boundaries of the nervous nets. The fact that its activity may be changed by stimulated sensory cells and that it may act on glands or muscles is a phenomenon which exists only for an observer and not for the brain itself /4,13, 14/.
All stimulations of the sensorium, whether they arc exo or endosomatic, are codified (at the receptor cells) and transmitted (in the neural net) as quantities and only as quantities. It is in the brain (and not at the sensory cells) where all impressions (colors, smell, etc.) and sensory perceptions (even of one´s own body) are reckoned from those quantities, i.e., where the change from quantity to quality takes place.
All functions of the brain always result from interacting nervous nets, which are located in specific domains of nearly all parts of the brain. They are never the result of isolated single centers and even less the result of a summum bonum. The emphasis is placed on the interaction of topologically different specific domains, such as the thalamocortical system, the limbic system and the ascending reticular activating system. These three systems are known to be essential for the capability of the brain to become selfconscious (selfreferentiality). All three represent large nervous circularities closely interwoven with each other, i.e., from a structural point of view the situation resembles the heterarchical model in Fig. 1c, with all the logical complications required for an adequate description. On the logical side, this heterarchical structure can be modeled using the ´theory of polycontexturality´.
To summarize, nervous nets are systems characterized _1) by the topological distribution of different interacting centers and interacting circularities; _2) by nontransitivity of nervous activities; and _3) by selfreferentiality of their organization. As McCulloch showed in his study ´What´s in the Brain ... ?´, /9/ the treatment of this subject does not belong to the field of neurophysiology but represents a general logical problem which requires a new logical conception. McCulloch himself has been involved in developing a triadic logic (´logic triadas´) /15/.
Contrary to the linguistic frame of the classical logic, which does not admit any consistent description of selfreferential, nontransitive processes and distributed irreducible domains, the theory of polycontexturality with its conception of distribution and mediation of domains offers an adequate logical tool for a consistent formalization of complex systems. This will be demonstrated briefly using the example of nontransitivity from Fig. 1c. For the sake of simplicity this will be accomplished by a polycontextural interpretation of multivalued logic. Continuative conceptions have been elaborated by Kaehr /16/.
As is well known, classical logic is twovalued. Multiplevalued logical systems extend the twovaluedness by introducing intermediate values which are arranged between the two classical truth values. The domain, which can be described by a twovalued logic or a classical multiplevalued logic, will be called a contexture.
PoIycontexturality originates as follows:
classical logic is defined between any two values resulting in a contexture; in a case with three values, however, where the third is not placed between ´true´ and ´false´ but beyond ´true´ and ´false´, three twovalued logical systems are generated to which three contextures are assigned. Four values define six logical domains, and in general, with m values twovalued logical systems are created. The logical systems or contextures defined in this way do not coexist in isolation but are mediated with each other. Figure 2 illustrates the situation for three logical systems L_{1,2,3} corresponding to McCulloch´s heterarchically structured model as shown in Fig. lc, a model which is an irreducible representation of a nervous net with a linking diallel.
As can be seen from the pattern in Fig. 2, the exchange relation causes ambiguity between the logical terms ´true´ and ´false´. This scheme corresponds to a threecontextural logical system, the lowest meaningful contexture in the polycontextural theory. In order to illustrate the difference between a classical twovalued logic and the multiplevaluedness of a compound contexture, such as that presented in Fig. 2, the truth table for the conjunction , the disjunction , the implication , and the two negations (N_{1,2}) will be exemplified briefly. The way in which the order and exchange relation have to be assigned into the general formalization will not be discussed here.
Figure 2 : Three logical
systems (domains), L_{1,2,3.} T : true; F : false; >: order relation; >< : exchange relation  : coincidence relation. 
Table 1 represents a truth table for a threecontextural logic. In order to simplify the notation, the following abbreviations have been introduced, which immediately become clear if they are compared with the symbols in Fig.2:
F_{2,3} 
(3) 
( a ) 
( b ) 

L_{1} 
L_{2} 
L_{3} 
L_{1} 
L_{2} 
L_{3} 

X_{1} 
Y_{1} 
X_{2} 
Y_{2} 
X_{3} 
Y_{3} 

T_{1} 
T_{1} 
T_{3} 
T_{3} 
T_{1} 
T_{3} 

F_{1} 
T_{1} 
F_{1} 

F_{3} 
T_{3} 
F_{3} 

T_{1} 
F_{1} 
F_{1} 

F_{1} 
F_{1} 
F_{2} 
F_{2} 
F_{1} 
F_{2} 

F_{2} 
F_{2} 
F_{2} 

T_{3} 
F_{3} 
F_{3} 

F_{2} 
F_{2} 
F_{2} 

F_{2} 
F_{2} 
F_{3} 
F_{3} 
F_{2} 
F_{3} 

Table 1 : Truth
table for the conjunction: (a ) auxilitary table for the construction of (b); (b ) table of the conjunction 
( c ) 

T_{1,3 }F_{1,2 }F_{2,3}  
T_{1, 3} F_{1, 2} F_{2,3} 

Table 1c) condensed version of table 1b) 
( a ) 
( b ) 
( c ) 


T_{1,3} 
F_{1,2} 
F_{2,3} 
X 
N_{1}X 
N_{2}X 
T_{1,3} 
F_{1,2} 
F_{2,3} 

T_{1,3}  T_{1,3} 
T_{1} 
T_{3} 
T_{1,3}  F_{1,2} 
F_{3,1} 
T_{1,3}  T_{1,3}  F_{1} 
F_{3} 

F_{1,2}  T_{1} 
F_{1,2} 
F_{2} 
F_{1,2}  T_{1,3} 
F_{3,2} 
F_{1,2}  T_{1}  T_{1,3} 
F_{3} 

F_{2,3}  T_{3} 
F_{2} 
F_{2,3} 
F_{2,3}  F_{3,2} 
F_{2,1} 
F_{2,3}  T_{3}  T_{3} 
T_{1,3} 

Table 2 : Truth
table for (a) the disjunction, (b) the negations, and (c) the implication. 
Table 1a will help to elucidate the truth table not only for the conjunction, but also for the disjunction, the negations and the implication. Table 1b shows the result for the conjunction, while Table 1c displays a more condensed notation of Table 1b. Table 2 gives the corresponding version for the disjunction, negation and implication, which can be derived very easily with the help of Table 1a.
From these tables, De Morgan´s formulae, for example, for the conjunction and disjunction in a threecontextural system can be deduced, viz.,
(4) 

or the relations of contraposition may be formulated:
(5) 

For a logical interpretation of McCulloch´s heterarchically structured model describing certain features of nervous activities, this brief survey of some relations will be sufficient. For more details and technical elaboration of the polycontextural logic, the literaure /16, 17/ should be consulted.
NonTransitivity
Within the linguistic framework, opened by a threecontextural logic, both transitive and nontransitive relations may be discussed without antinomies and ambiguities.
Intracontextural, i.e., within the logic of one contexture, the transitivity law holds rigorously, as do the classic logical rules. For the preference represented by relations (1) and (2) one has
(6) 
If, for example, the preference relation between the statements A, B and C about the nervous net in Fig. lc is given in three contextures as
L^{(3)} :  L_{1} L_{2} L_{3} 
(7) 
then the transitivity rule holds strictly in each of the three contextures. However, if the implication chain starts in the logical domain L_{1} (or L_{2})and continues with a change of the contexture, say in L_{3} then nontransitivity occurs transcontexturally in the chain of implication without antinomy, viz.
(8) 
Relation (7) may be summarized using (5) as
L^{(3)}  (7a) 
where L^{(3)} symbolizes the total of three contextures, i.e.,
L^{(3)} = (L_{1}, L_{2}, L_{3}) 
(9) 
and correspondingly for relation (8) it follows from (6) that
(10) 
Whereas circularities, which always appear when describing the activities of complex nervous nets, lead to completely meaningless propositions if considered intracontexturally (i.e., monocontexturally) they still can be modeled transcontexturally (i.e., polycontexturally). In this way linguistic possibilities are developed for a scientifically precise determination of complex notions such as ´selfreferentiality´ in which the process of description is distributed over different logical positions of polycontexturality; finite and operative complexes of ambiguities may be designed.
CONCLUSIONS
The results of modern brain research and the theory of polycontexturality as a ´general theory of living systems´ indicate the direction of future computer architecture which possibly may be designed on a molecular electronic basis. It. can only be considered a first step that on a molecular level the old ´silicium structures´ in their rigorous binarity are copied and miniaturized simply for purposes of optimizing quantities such as speed, size, etc. For a future computer science, founded on the basis of molecular electronics, the realization of a computer architecture has to be envisaged which models the dialectic and selfreferential structure of matter as they appear, for example, in the brain.
Acknowledgment
These studies were supported by a grant from the Stiftung Volkswagenwerk.
REFERENCES
/1/ McCulloch and W. Pitts, Bull. Math. Biophys. 5,115 (1943) , 2nd cit.
/2/ H.von Foerster, Sicht und Einsicht, ViewegVerlag, Braunschweig (1985) , 2nd cit.
/3/ L.Löfgren, in: Trends in General Systems Theory (edited by J. Klir), Vol. 3, p.197. Wiley, New York (1977)
/4/ Maturana and F. J. Varela, Autopoiesis and Cognition, Reidel, Dordrecht (1972) , 2nd cit.
/5/ G.Günther, Das Bewußtsein der Maschinen, AgisVerlag, BadenBaden (1963)
/6/ G.Günther, in: SelfOrganizing Systems (edited by M. C. Yovits, G. J. Jacobi and G. O. Goldstein) p. 313, Spartan Books, Washington, DC (1962)
/7/ G.Günther, Beiträge zur Grundlegung einer operationsfähigen Dialektik, Vols 13, Felix Meiner.Verlag, Hamburg (1980)
/8/ McCulloch, Bull. Math. Biophys. 7, p.69 (1945) , 2nd cit.
/9/ McCulloch, in: Embodiments of Mind, p.387, MIT Press, (1965) , 2nd cit.
/10/ E.CohorsFresenborg, Mathematik mit Kalkülen und Maschinen, ViewegVerlag, Braunschweig (1977)
/11/ Varela, in: Principles of Biological Autonomy, NorthHolland Publ., Amsterdam. (1979).
/12/ J.Schmidt (ed.), in: Der Diskurs des Radikalen Konstruktivismus, SuhrkampVerlag, Braunschweig, (1985)
/13/ Maturana, Erkennen: Die Organisation und Verkörperung von Wirkhchkeit, ViewegVerlag, Frankfurt, (1987)
/14/ G.Roth, in: Der Diskurs des radikalen Konstruktivismus, (edited by S. J. Schmidt), p. 229. SuhrkampVerlag, Frankfurt (1987)
/15/ Longyear, J. Cybern. 2, p.7 (1972)
/16/ R.Kaehr, Materialien zur Formalisierung der dialektischen Logik und der Morphogrammatik, in: Idee und Grundriss einer nichtAristotelischen Logik, (edited by G. Günther), Felix MeinerVerlag, Hamburg (1978) , 2nd cit.
/17/ Kaehr, in: Die Logik des Wissens und das Problem der Erziehung, p. 254, Felix MeinerVerlag, Hamburg (1981)
Autoren: R. Kaehr & E. von Goldammer 

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