ThinkArt Lab Animation: A.T. KelemenĐ November 12, 1998 Dr. Rudolf Kaehr
There are many ways of combining abstract objects or institutions. For example, given two institutions INS1 and INS2 which, intuitively, are independent we can form their product. This product institution has all pairs of signatures from INS1 and INS2, respectively, as models, and sentences which are either sentences from INS1 or from INS2 with the obvious satisfaction relation." Cat., p. 357
It is shown, that the category of institutions is complete.
The idea of dissemination tries to explicate and formalize a quite different intuition of combining institutions which is not producing diversity and multiplicity by combining a basic system as a product or sum or whatever construction but introduces multiple differences in the very concept of the basic system itself. After this construction a polylogical or polycontextural system can be combined in many ways. This idea of multitudes of basic differences in the notion of formality, taken seriously, is in fundamental contrast to the existing concepts of formality in mathematics. Obviosly, these multitudes are more fundamental than all types of many-sorted theories, or typed logics, or pluralities of regional ontologies, domains and contexts.
1 The idea of proemiality
A very first step in this direction was made by the philosopher Gotthard Gunther with his idea of a proemial relationship" introduced in his paper Cognition and Volition" (1970) about a Cybernetic Theory of Subjectivity.
In order to obtain a general formula for the connection between cognition and volition we will have to ask a final question. It is: How could the distinction between form and content be reflected in any sort of logical algorithm if the classic tradition of logic insists that in all logical relations that are used in abstract calculi the division between form and content is absolute? The answer is: we have to introduce an operator (not admissible in classic logic) which exchanges form and content. In order to do so we have to distinguish clearly between three basic concepts. We must not confuse
The relata are the entities which are connected by a relationship, the relator, and the total of a relationship and the relata forms a relation. The latter consequently includes both, a relator and the relata.
However, if we let the relator assume the place of a relatum the exchange is not mutual. The relator may become a relatum, not in the relation for which it formerly established the relationship, but only relative to a relationship of higher order. And vice versa the relatum may become a relator, not within the relation in which it has figured as a relational member or relatum but only relative to relata of lower order.
Ri+1(xi, yi) is given and the relaturn (x or y) becomes a relator, we obtain
Ri (xi-1, yi-1) where Ri = xi or yi. But if the relator becomes a relatum, we obtain
Ri+2(xi+1, yi+1) where Ri+1 = xi+1 or yi+1. The subscript i signifies higher or
We shall call this connection between relator and relatum the 'proemial' relationship, for it 'pre-faces' the symmetrical exchange relation and the ordered relation and forms, as we shall see, their common basis."
Neither exchange nor ordered relation would be conceivable to us unless our subjectivity could establish a relationship between a relator in general and an individual relatum. Thus the proemial relationship provides a deeper foundation of logic as an abstract potential from which the classic relations of symmetrical exchange and proportioned order emerge.
It does so, because the proemial relationship constitutes relation as such; it defines the difference between relation and unity - or, which is the same - between a distinction and what is distinguished, which is again the same as the difference between subject and object.
It should be clear from what has been said that the proemial relationship crosses the distinction between form and matter, it relativizes their difference; what is matter (content) may become form, and what is form may be reduced to the status of mere materiality"."
We stated that the proemial relationship presents itself as an interlocking mechanism of exchange and order. This gave us the opportunity to look at it in a double way. We can either say that proemiality is an exchange founded on order; but since the order is only constituted by the fact that the exchange either transports a relator (as relatum) to a context of higher logical complexities or demotes a relatum to a lower level, we can also define proemiality as an ordered relation on the base of an exchange. If we apply that to the relation which a system of subjectivity has with its environment we may say that cognition and volition are for a subject exchangeable attitudes to establish contact but also keep distance from the world into which it is born. But the exchange is not a direct one.
If we switch in the summer from our snow skis to water skis and in the next winter back to snow skis, this is a direct exchange. But the switch in the proemial relationship always involves not two relata but four!" Gunther
1.1 Some explanations of the idea of proemiality
The proemial relationship is therefore at first an interlocking mechanism of the two concepts of exchange and order or symmetry and asymmetry.
Diagramm 1
cascadic representation
A further explication of the intuition of proemiality is achieved if we consider the fact that the objects, the relator and the relata of the relations, have to fit together in a categorical sense. There is a similarity of the relators of different levels as well as for the relata of different levels in the sense that the different relators are relators and not something else. And the relata on each level are relata and not relators. For that I introduce the coincidence relation, which designates categorical sameness (likeness, similtude).
To finish the picture I introduce the exchange relation between the first" and the last" element of the interlocking mechanism of order and exchange relations. As a last step I mention the position, the logical locus, of the order relations according to the higher or lower logical orders".
PrObj = (Obj; Ord, Exch, Coin, Pos)
Diagramm 2
But this explanation still excludes the third term of the definition of a relation, the relation itself. Remember: We must not confuse a relation, a relationship (the relator), the relatum.
And finally I consider the fact that there is one and only one concept of relation and relationality under consideration. therefore the concept of relation is based on unicity (uniqueness), represented by 1. This is surely not a harmless statement, it suppose something like a common intuition of relationality or operativity which finds itself explained and formalized in some mathematical constructions which are accepted by the scientific community. Therefore, Gunthers chain "a relation, a relationship (the relator), the relatum" has to be completed by the very concept of relation, that is, relationality based in unicity.
The full-fledged explanation, without the arrow "relation-->relationality", of the proemial relation over two loci is given by its conceptual graph. The scenario is the same for the distribution and mediation of other concepts, like operations, functions, categories, institutions etc.
Thus the definition has to be expanded to:
PrObj = (Obj; Ord, Exch, Coin, Pos)
with Obj = { relator, relatum, relation, relationality, unicity}
In this context it is not my task to defend this construction against the many attempts to reduce it to something else. To go further in the game I make the option that it will be useful for developing some new mechanisms of combining abstract objects like institutions, logics, arithmetics, category theories and more. In exercising this game the new intuition will shape itself into a more academic form.
After having introduced the idea of proemiality it would be possible to formalize it further and to develop a preliminary theory of proemiality, also sometimes called chiastics or theory of mediation. The main thesis, therefore, is that proemiality offers a mechanism of combining institutions which doesn't belong to the universe of combining categories. This mechanism of combining institutions, e.g. distribution and mediation, is fundamentally different from the classical ones. Despite of this difference this strategy is in no contradiction or opposition to the known principles of combining systems of logics.
It is simply something different and the clou would be to explain this difference in full. Donīt confuse the exchange of relator and relatum of a relation in the mechanism of the proemial relationship with the superposition of relator and relation in relational logics. There is no problem to apply a relator, or a operator or a functor to the result of a relation or operation or function as e.g. in recursion theory or in meta-level hierarchies.
If we proemialize the linguistic subject-object-relation of a sentence we shouldn't hesitate to be strictly structural. The example is borrowed from Heinz von Foerster.
"The horse is gallopping" (Das Pferd gallopiert), the interchanged sentence can only be "The gallop is horsing" (Der Gallop pferdet).
Nobody supposed that we are doing analytic philosophy.
1.2 Proemiality and Architectonics
1.2.1 About the as-category in proemiality
What I have developed so far is only the half of the story. Also it might be obvious that the wording of e.g. "the operator (of one system)