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If the title of this paper combines Formal Logic and Totality (Ganzheit) it is resisting a general trend which is still strong in present scientific activities. The most comprehensive theory of Totality which we possess is contained in Hegel´s logic. But every student of this thinker knows how emphatically Hegel denounces formalization. According to him the structure of all totalities is "dialectic". Formal logic is based on a strict dichotomy of form and content (matter). But dialectics fuses the two in the superadditive principle of synthesis which combines thesis and antithesis in a way in which the contradiction between the two is not only retained but elevated to a higher level. The general consensus still is that the retention of contradiction - which is indeed demanded by all systems to which we ascribe the character of totalities - obviates all attempts of formalization. This belief is now more than two thousand years old and it is hard to shake.

However, a re-evaluation of the theory of dialectics and its super--additive principle, where the whole is more than the sum of its parts, has recently become a pressing necessity. Among the new scientific disciplines which have sprung up in recent times Cybernetics seems to have the widest interdisciplinary spread. The topics it deals with range from mathematics (information theory) and physics (quantum mechanics) over biology (bionics) to the theory of consciousness, of culture and of human history. [²1] It Is hardly necessary to point out that the problem of the structure of totalities turns Up various aspects within-the scope of Cybernetics. Nevertheless a basic investigation into the formal logical texture of totalities is still missing. The ancient prejudice that such inquiry loads us straight out of the realm of formal, codifiable procedures of logic is still too strong.

H. von Foerster´s argument cannot be repeated in detail. It will be sufficient to say that by applying the concept of "logical strength" (Carnap, Bar-Hillel) according to which a truth function increases its strength with the number of negative values it applies the author shows that a "coalition" of two statements A and B signifies such a super-additive principle:

A	B	½	&
1	1	1	1
0	1	0	0
1	0	0	0
0	0	1	0

Table (I) shows on the left side the value constellations (0 for negative and 1 for positive) of the statements A and B. It is obvious that the logical strength of each is ½. On the right side we have first the equivalence relation (½) of A and B which gives us their average strength as a result of what may be called a normal adjunction. This average strength is, of course, again 1/2. The last value sequence represents conjunction (&), in von Foorster´s words a "coalition", and the logical strength of the value sequence is in this case 3/4 since, compared with the equivalence relation the last value of the sequence has turned from 1 to 2 which adds one quarter the strength of the function.

The argument used by von Foerster has the great merit of showing that a super-additive principle of logical strength is already extant in classic formal logic (and so is its opposite of super-subtractivity in disjunction). But the history of traditional logic has shown that the form in which super-additivity manifests itself in simple conjunctive relations does not suffice to develop all the peculiar characteristics of totalities which we find displayed in systems of reasonably high complexity. This is why the history of formal Aristotelian logic is accompanied by an equally long history of dialectic (non-formal) logic. The latter was supposed to take up the logical problems, where formal logic, due to its specific limitation, had to drop them.[⁴3]

It will pay to investigate the basic shortcoming of traditional formal logic. To put it in a nutshell: it excludes the subject of thought from the logical picture of the Universe.[⁵4] Thus this picture is entirely "objective" in the full double meaning of the term. It goes without saying that the mental image of the Universe, thus obtained, does not describe it as a totality. A very important structural element is missing in this logical imagery: the indubitable power of the Universe to form subsystems which act as centers of objective reflection as well as of self-reflection. But since this property is excluded it stands to reason that the totalities of lower order which we encounter in biology, psychology, social sciences or history are also outside the scope of traditional logic. They are parts of the Universe and available for their description are only the very same logical elements and procedures which are applicable to the objective world in its entirety. This means they cannot be described as totalities either.

In this figure O means, of course, the objective world as reflected in the consciousness of a subject SÞÞÞ . But since subjectivity is a phenomenon shared by an indefinite number of relatively independent centers of self-reflection and, moreover, only one of them may, for the purpose of developing a theory of thinking, be regarded as the subject who thinks whereas the others are thought of, we have to distinguish three different meanings of SÞÞÞ. We show this by writing: SU, SS and SO. With SU we indicate what in traditional logic is usually called universal subjectivity (Kant´s "Bewusstein überhaupt"). When we write SS (or subjective subject) we refer to what is in a given process of thinking the actual subject of the mental event. All the other potential subjects of thought are, of course, relative to the designated one (SS) objective subjects, i.e. possible objects of the reflection of SS. In our figure they are indicated by SO.

The classic theory of thinking as expressed in all our present systems of logic assumes that subject (SU) and object (O) represent logically speaking an absolute dichotomy: what is not object is necessarily subject and what is not subject is correspondingly object. It is assumed that looking at the world all subjects form a closed phalanx confronting the object and since they are all - in some unexplained manner - parts of the universal subject (SU) they will have a common basis of thought. Because if SS agrees in its reflection of O with SU then the resulting judgments of SS will be binding for all SO. It follows that the general (metaphysical) dichotomy between SU and O is reflected in a second order dichotomy which separates SU and SS from SO and O.

In other words: the founding relation of all classic thought and its ultimate basis on which everything is built is an exchange relation of absolute symmetry between total affirmation and total negation.[⁶5] Its most famous expression is Hegel´s terse remark at the beginning of his Dialectic logic: "Das reine Sein und das reine Nichts ist º dasselbe"[⁷6]. A formalized equivalent of it is:

There can be no doubt that the operational basis of classic logic is an exchange relation between subject and object or between a mapping process and that which is mapped. However, if we have a second look at Figure 2 or 3 we will notice that the complete symmetry of the exchange relation between SÞÞÞ is guaranteed only by the introduction of the concept of a universal subject (SU) which according to the metaphysical tradition of classic logic (e.g. Nicolaus Cusanus) is, ontologically speaking, identical with O.

The modern scientist who tries to discover the formulas in which the code of the Universe is written is usually not aware of the basic ontologic assumptions which govern his mode of thinking. But they show up in his results just the same. Because if SU is ultimately identical with O then his world picture will contain no traces of bona fide subjectivity - as Schrödinger has pointed out correctly. And if SÞÞÞ and O represent an exchange relation of enantiomorphic equivalence then the basic laws of Nature must obey the principle of reflection-symmetry (parity). Whenever a phenomenon shows up which seems to display the structural features of non-parity there will be cogent reasons for a turn to more general principles of reasoning which explain the event again in terms of reflection-symmetry. These reasons will not only be strong, nay, they will be invincible as long as we stick to the ontologic tradition of classic logic and its principle of reflection-symmetry.

We are here not concerned with the fate of parity in the future development of physics but it must be pointed out that the concept of Totality should be ruled out as logically analyzable if parity reigns supreme in our theory of thinking. We have given the main reason above: if the relation between thought and its object is basically understood as a symmetric exchange relation the phenomenon of subjectivity disappears. But a "totality" in which everything is reduced to objectivity can never be total because something is missing.

However, if the concept of the universal subject, i.e. of ´Bewusstein überhaupt´ (Kant), is eliminated the logical constraint to reduce everything to ultimate parity relations disappears. We will still have reflection-symmetry between SS and SO but not longer between SÞÞÞ -and O in general. In other words: it will turn out that the founding relation between subject and object or between Thought and Being is not a symmetrical exchange relation but something else. This is the point where the transition is made from formal classic logic of Aristotelian type to a theory of trans-classic, non-Aristotelian Rationality.

We begin by re-drawing Figure 1 omitting SU and having the phalanx of the SO replaced by a single S with the index O. We indicate the relations between SS , SO and O by arrows of four different shapes. According to the logical character of the relation an arrow will either be double-pointed or it will have one shaft or be double-shafted having either continuous or dotted lines. Figure 5 will then show the following configuration:

If SS designates a thinking subject and O its object in general (i.e. the Universe) the relation between SS and O is undoubtedly an ordered one because O must be considered the content of the reflective process of SS. On the other hand, seen from the view-point of SS any other subject (the Thou) is an observed subject and it is observed as having its place in the Universe. But if SS is (part of) the content of the Universe we obtain again an ordered relation, this time between O and SO. There remains the direct relation between SS and SO. This is obviously of a different type. SO is not only the passive object of the reflective process of SS. It is in its turn itself an active subject which may view the first subject (and everything else) from its vantage--point. In other words SO may assume the role of SS thus relegating the original subject, the Self, to the position of the Thou. And there is neither on earth nor in heaven the slightest indication that we should prefer one subjective vantage-point for viewing the Universe to another. In short, the relation between SS and SO is not an ordered relation. It is a completely symmetrical exchange relation, like "left" and "right". An ordered relation between different centers of subjective reflection cones into play only if we re-introduce the concept of a universal subject which contains all human "souls" as computing sub-centers.[⁸7] Of the two relations we have so far considered, the exchange relation is symmetrical and the ordered relation represents non-symmetry.

From these we shall, for the time being at least, eliminate the second. It tells us nothing new. It describes only the situation we are familiar with from classic (two-valued) logic where all subjects SÞÞÞ form, what we called earlier in this paper a "closed phalanx" excluding the object from themselves and thus obtaining an "objective" aspect of the Universe. Consequently OrP(SO´SS) only informs us that if O develops its mirror image in SÞÞÞ it will do so in dichotomic terms of positive and negative forming a strict exchange relation since SÞÞÞ will be either SS or SO and a Tertium will always be excluded.[⁹8] We have pointed out before that such an exclusion principle obviates our conceiving totalities in terms of traditional logic. Since the relation OrF(SO´SS) is known to logic since the times of Aristotle and has its own specific properties we distinguish its graphic representation from the other two by having drawn it with dotted lines.

However, the other two relations of the type ºrFº have so far not obtained a legitimate place in formal logic. They define the way in which an individual consciousness (as a logical subject) may establish its position confronting the world. Formally speaking it is the relation any of the two realizations of SÞÞÞ, namely SS or SO, may have toward the connection of the other SÞÞÞ and O. We call this the founding relation (rF) because by it, and only by it, a self-reflective subject separates itself from the whole Universe which thus becomes the potential contents of the consciousness of a Self gifted with awareness. In contrast to it the classic relation OrF(SO´SS) is still a founding relation - but not for consciousness. Not a self-reflective subject but only the content of the consciousness of a potential subject is established by it.

In Figure_5 the founding relations for subjectivity are indicated by the double-shafted arrows which issue from SS and SO and hit the center of the opposite side of the triangle. These arrows illustrate in diagrammatic form the relations between consciousness as a self-reflective activity and the world in general. The world is always both O (bona fide objectivity) and SO subjects viewed as part of the objective world º where SÞÞÞ is always ex-cluded only as SS. This last statement seems to be contradicted by our figure because the arrow issuing from SO seems to point to a world which includes SS and O. This is the unavoidable fault of a still picture. An adequate representation would demand a moving picture in which the double- shafted arrow would oscillate between SS and SO. One should not forget: what is in our diagram SO may at any time assume the role of SS, thus relegating the latter to the logical position of SO. Let us repeat that SS and SO con-stitute the exchange relation between subjectivity as the Self and the other subject which appears to the Self as the Thou. For any given logical position only one of the two double-shafted arrows represents actualization of a center of self-reflection. Since such actualization requires all three components SS , SO and O it is impossible if we have located the center in SS to find it also in SO. But it has no fixed status in SS and it may be shifted to SO. Fichte calls this "die Duplizität im Ich" because, as he puts it, such a center of self-reflection can neither be fully identified as an existing entity (als seiend) nor as a structural principle of active organization (als Prinzip). This is the Duplicity of the Self.[¹⁰9]

What we have so far ignored in our contraposition of SS and SO is the fact, well known to all of us, that no Ego, or Self exists in solipsistic splendour and that this universe of ours permits the coexistence of an indefinite number of centers of self-reflection who all claim to be thinking Egos comprising the total realm of Being as potential contents of their awareness. It is obvious, therefore, that the exchange relation is an exclusive disjunction on a level of reflection which is identified with the logical position of SS. But an impartial observer,

, who assumes his place neither in SS or SO but "outside" of Figure 5 will come to a different conclusion. He will still concede the existence of a disjunctive relation between two subjects but to him this disjunction must be inclusive. He is forced to admit that two concurring SÞÞÞ may both be SS although relative to him both will be SO as long as he is claiming the exalted position of an SS of higher reflexive capacity.

provided, of course, that he uses a two-valued logic. But in doing so he realizes by self-reflection that he has committed a momentous logical mistake. Since in classic logic only two values are available for the determination of the distinction between subject and object, it is impossible to describe the triadic relation between the subjective subject; the objective subject and the object.

The common fallacy committed by logicians who reason along traditional lines is that if subject and object are different it is sufficient to assign different values to them. But since the structure of classic negation represents a symmetric exchange relation and since there can be no preference to assign a definite value to SÞÞÞ or to O, it is impossible to distinguish the subject from the object by saying, for instance, that the positive value ultimately designates the object (because we describe the Universe in affirmative statements) and that the negative value refers to the subject. Although there can be no doubt that the existence of negational processes is a symptomatic index for the presence of subjectivity in the Universe, it is not one or the other value which points to the subject but their mutual relation which displays "Reflexionsidentität" in contrast to the one-valued, stable and irreflexive identity which is incorporated in the bona fide object.

Nevertheless, it is indeed possible to determine the distinction between subject and object by logical values. Not by assigning another value to the subject but by engaging two values for the designation of one identity. And since we can think at least of one more theme beyond a) object, b) subject namely "reality" as the ultra-conscious context c) in which object and subject cooperate we would have to allot three values for the identity theory of c). In the case that we may be able to conceive something of even higher logical order, the difference between it and everything else would be determined by a tetradic structure of values.

theme
object	non-object		value-differential	log. value-system
1	1	hierarchy of themes	0	2
1	2	1	3
1	3	2	4
1	4	3	5
º	º	º	º

Since the object, completely isolated from the subject, is designated, by one and only one value, the object column only repeats this number. In classic logic the numerical difference between the values for the object and those which designate the subject - or anything else for that matter - is zero. The third column therefore starts with 0. This informs us that the only way to think of a subject or any system gifted with self-reflection, is to conceive it as an object - which means without self-reflection. In other words: the first theoretical approximation to the problem of subjectivity is offered in a three-valued logic. Here again one value designates the object, but two are left over for everything which is not an object. The numerical difference between the values assignable to the object and non-object is now 1. Something can now be said (in terms of logical structure) about the non-object which would differ from all statements about bona fide objects.[¹³12]

Our ideal observer who contemplates the relations between subject and object as illustrated in the triangle of Fig. 5 must ultimately arrive at the conclusion that table (II) is applicable in his case. He cannot differentiate between himself and the triangle unless he assigns to himself a logical value which does not occur in the triangle. But what is sauce for the goose is sauce for the gander. Our observer expects that SS in the triangle is capable of differentiating between itself and O. Consequently he has to concede that SS in contradistinction to O possesses an additional value. Since O is described in a two-valued system, the description of the triangle requires a three-valued logic. Finally this description is the content of the consciousness of our ultimate observer who must consequently reason with four-valued structures.

However, as soon as our observer realizes that the founding relations in Fig. 5 obey the laws of a three-valued logic, he realizes that not all the inclusive disjunctions which he established in the formulas 1), 2) and 3) are analytic formulas and generally valid. He will find that only 1) still holds and that the disjunctive relation in SÞÞÞ between SS and SO is indeed basic and invariant to a transition into a higher-valued system. With regard to 3) he will discover that its general validity has completely disappeared. Formula 2), on the other hand, has assumed a peculiar equivocality. Since a three--valued logic operates with five negational states[12] - where two-valued logic uses just one - an exchange-relation may be interpreted in five different ways. In the case of three of them formula 2) will be as valid as 3); i.e. for all possible states of the system of Fig 5. In the case of two others formula 3) will be invalid if the system O, SO , SS assumes the following values: classic negation for O and the irreflexive value for SO as well as SS. This is a most significant result!

Unfortunately the scope of this paper precludes an interpretation and discussion of such details no matter how important they are. This investigation intends only to show that the concept of Totality or Ganzheit is closely linked to the problem of subjectivity and trans-classic logic and that it is based on three basic structural relations:

It may be said that the hierarchy of logical themes as indicated in table (II) represents an hierarchy of implicational power. All themes have in common that they are self-implications; they imply themselves. However the first theme (objective existence) implies only itself and nothing else. In this respect it differs from any succeeding theme which implies itself as well as all subordinated themes. For this reason it is proper to call the initial theme "irreflexive" and all the following "reflexive". Irreflexivity means that something we think of is only an implicate but not an implicand for something else. On the other hand if we refer logically to reflexivity we mean that our (pseudo-)object of thought is an implicand relative to a lower order and as well an implicate relative to a theme that follows it in the hierarchy of table (II).

We are now able to establish the fundamental law that governs the connections between exchange-, ordered- and founding-relation. We discover first in classic two-valued logic that affirmation and negation form an ordered relation. The positive value implies itself and only itself. The negative value implies itself and the positive. In other words: affirmation is never anything but implicate and negation is always implication. This is why we speak here of an ordered relation between the implicate and the implicand. The name of this relation in classic two-valued logic is - inference.

It is now necessary to remember that the possibility of coexistence of two independent subjects (I and Thou) in the Universe is based on an exchange relation between equipollent centers of reflection. Moreover, these subjects are all capable of being implicands. More objects do not operate inferentially. That means they do not imply anything else.

If we now consider the founding relation in which a subject constitutes itself as diametrically posed relative to all objects and the total objective concept of the Universe we will discover that this relation represents an interesting synthesis of exchange and order. The founding relation is in itself an exchange relation in so far as the linking subject (SS) may assume the logical position of the other subject which is thought of (SO). SO may in its turn assume the rank of SS. Any two centers of subjective reflection of the same order mutually imply each other. But such an exchange does not operate between SÞÞÞ and O. As we pointed out before: the bona fide object cannot infer the subject and by doing so usurp the role of a subject. If it could it would imply that subjects are irreflexive entities which for a subject is a contradictio in adjecto.

Formal Logic, Totality and The Super-additive Principle*1)

Formal Logic, Totality and The Super-additive Principle*¹)