From the beginnings of formal logic with Parmenides and the Eleatics via Socrates, his pupil Plato and above all his pupil Aristotle, in only a few generations a widely branched tree grew from the little plant, to which many philosophers and logicians have since added new branches and twigs. This tree still shapes our thinking today and thus also our foresight.
Besides the tree with roots, trunk and branches, the pyramid was used very early as a metaphor and structural model for logic, for example in the "Platonic Pyramid". It illustrates how logic assigns to all things or identities (distinctions, ideas, objects, etc.) a place in a struct¬ured space as a frame of reference. Each of these places can be derived unambiguously from a single starting point (here the number 1 as "root"):
Fig. 8-1 Platonic pyramid
With each step (from top to bottom) two new elements or identities are created, each distinguished by a negation (yes/no or true/false). In each level the elements of the preceding level are doubled. The highest level represents in this structure the most general, the absolute and the properties of this place are inherited by all points and layers of the hierarchy below. The deeper these layers are, the more specific and concrete the description of the whole becomes, since each element is on the one hand distinguished from all others as identity, but at the same time also shares the properties of the whole and all its ancestors. Thus, one can say that the "1" as the most general thing determines the whole structure, since all elements share its properties. The 1 defines a reference system to which everything else refers. Every operation within such a frame of reference is necessarily mono-logical, i.e. it follows exclusively a single logic determined by the properties of the root.
This has far-reaching implications in quite different fields: In theology, for example, this system of reference was very quickly accepted as a universal principle of order, because it made it possible to logically derive and represent the divine universality from which everything originates, to which everything can be traced and to which everything ultimately aims. However, and the philosopher Gotthard Günther pointed out this, a serious problem arises: If we retrace the ways of the individual identities, then it turns out that e.g. the 9 and the 14 have no connection below the incomprehensible and inexplicable divine Absolute, in which all opposites of the lower levels coincide. But if this connection does not exist, then also no understanding between the subjects, which are represented here by numbers, and therefore no objectivity across the subjects is possible (except for pairs below a common number, which can be exchanged by simple negation, i.e. between the 8 and the 9 or the 14 and the 15). But already for the 8 and the 10 or even more for the 8 and the 15 there is no such connection. Theology has recognized this contradiction quite well and has withdrawn it as "mystery" from the framework of the logical explanation – theology is allowed to do this.
But one can also see that the 1 as universal principle produces two completely separate and divorced substructures, which negate each other, but are otherwise structurally identical and therefore interchangeable (thus all elements under 2 and all under 3). Thus, the jump from 8 to 15 is not a real dia-logical or even dialectical jump out of the frame of reference or structure, even if it can only be done via 1, but only a jump into the negation of the respective own structure, because the structure under 3 is the logical negation of the structure under 2.
If we take the figure of the Platonic pyramid as a metaphorical description for consciousness and assume that every individual has his very own personality, consequently also develops a very unique individual consciousness, then we must also grant every individual his own such platonic pyramid of consciousness contents, whose 1 corresponds to the respective "I" as the highest instance. All contents of consciousness (thoughts, ideas, memories etc.) can then be represented as points or nodes in a hierarchy with the "I-1" as starting point. Like a color filter this starting point as root gives all consciousness contents of the pyramid its individual tint. Another individual, with whom we want to communicate as "you", would then also be representable as its own pyramid, again with a – however different – individual tint.
We now have two identical structures, but they differ in color. Both are structured in the same way (binary hierarchical), both have identical elements, but nevertheless the black 8 is not equal to the gray one. If one wants to integrate it in the course of a communication into the own ego pyramid, then the black 8 must differ from the already existing gray 8:
So we get an additional value to every already existing value in one of the two structures as a third value (since we have a binary two-valued structure). However, this third value is not a value from the own, inner structure of the black I, for example, but from the outer one of the gray you.
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However, such a third is explicitly excluded in the laws of logic ("Tertium Non Datur"). It is also not a third in the sense of a multi-valued logic, which allows intermediate values or ranges on a scale from 0 (right) to 1 (wrong), as for example the "fuzzy logic". For the Du-8 is also unambiguous (for Du). This third, then, comes from a place that is outside of the I-reference system, that is, outside of the logical realm to which we are currently referring. Speaking with Hegel, the distinction between the I-element and the Thou-element requires a "second negation", which is not only the doubling of a simple negation within the logical system (and thus would lead us back to the starting point), but which points out of this system to a second logical system. Thus, in the proper sense, it is not only about multi-valued logic, but also about a "multi-logical value". Since classical logic has no place for such a triadic constellation, where two reference systems intersect or interlace in one point, "transclassical" logical procedures have been searched again and again as a way out of the classical logical closure.
In contrast to classical logic (Fig. top left), transclassical logics tolerate multiple truth values. In multi-valued logic, these can be single values (such as 1/2, 1/4 true), while in fuzzy logic, certain ranges are defined within which a value is considered "true". However, these systems still basically belong to the realm of classical logic, because their differentiations of truth values still take place under a single axiom, i.e., within a single logical domain or system, as envisaged by classical logic. Thus, the logical containment is not broken by this.
Polylogical systems, on the other hand, are distinguished by the fact that their values are simultaneously linked in several axioms and can also exist only in such a multiple linkage. They break the logical containment and are complex in this sense, because they cannot be completely described within a single logic.
Every dialogue creates such a polylogical situation: In the dialogue two individuals meet whose spaces of consciousness as reference systems are in principle inaccessible to the other. Even if both use the same terms, these are always individually linked (occupied) with different, albeit similar, contents of consciousness and therefore different and unique. However, if language creates an interpersonal space in which only what we call "I" or "subject" emerges - as a synthesis of I and Thou, or even of inside and outside - then this process necessarily leads beyond the limits of classical logic: consciousness as a unity of mind and perspective of action must then be seen simultaneously as unity and as difference. All philosophers have basically worked on this paradox and sought answers for it. Kant argues primarily for the non-identity of understanding and perspective of action, while Hegel tries to bring their identity together in a dialectical relation and to abolish it at the same time. Charles S. Peirce, in turn, sought to describe the logical process of how we build this bridge to an external world.
That logic, precisely because it is unambiguous, formalizable and determinable at every step, does not lead to unified conclusions, at the latest in dialogue, was already known to the early logicians. However, the successes of logic, especially in scientific explanation, technical mechanization, and finally industrialization, pushed these problems into the background for a long time. It is true that theologians feared for a time the public advance of a secular Aristotelian logic because of its possible negative effects on the faith and thus on the power position of the church. The scholastics, for example, who sought to integrate Aristotle and reconcile reason with faith, were therefore long suspected of heresy. Nevertheless, at least in the West, the triumph of logic as the only legitimate way of thinking was unstoppable.
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Especially since the late Middle Ages with its overseas conquests and an emerging world economy, the demands on logically based and predictable procedures, machines, behaviors and organizations have been growing. In an increasingly mechanically-logically structured and planned society, people are needed who can behave like machines: predictable, plannable, operating reliably within fixed expectations and thus causally predictable, but also themselves thinking, analyzing, and constructing with scientific precision. Such people, for example, sit as "computers" in vast London writing rooms in the 17th century, manually calculating and copying books of tables needed for navigating the high seas. The "fabrication of the reliable human being" becomes an important field of activity of state education in schools and institutions, at the end of which will be the factory worker and the office worker.
The contradiction between the systems of thought of an increasingly empirical science, i.e., committed to measurement, for which "objective" facts are only those that it can explain and measure, and the metaphysics of the church, which wants to see the subjectivity of life only as a product of divine will, can soon no longer be bridged – the Catholic church knows this too. The coexistence succeeds theologically via the concept of the soul as the part of the subject which as "subjectivity" transcends the space of objectively recognizable world structures. Politically, it succeeds via secularization, the separation of church and state, faith and science.
Hegel, as probably the last of the classical philosophers, tries to systematically resolve this contradiction once again by bringing subjectivity back into the universe, which becomes subject in the consciousness of itself. With this he still stands with one leg in the past, the "old philosophy", but with the other already in a still distant future.
Gotthard Günther is perhaps the first philosopher who ventured beyond the formulation of this dialectical problem to its mathematical formalization. He tried to systematically break through the closed monological frame of reference of classical logic and to develop an "operative dialectic" as a transcendental, "trans-classical" logic. For Günther, as for Hegel, the "trans" stands for breaking out of the logical enclosure by the framework of a single logical reference system. Günther calls such a system of reference a "contextur". For him, a dialogical or dialectical logic must be able to integrate several contextures, it must be "polycontextural". However, Günther does not only want to discuss the problem of the formalizability of dialectics theoretically, but to solve it practically: He wants to develop an operational formalism or calculus in the form of a "polycontextural operator" and on this basis even build a completely new kind of computer, as we know it so far only from science fiction movies and novels.