**R. Kaehr and Th. Mahler
Institut für Kybernetik und Systemtheorie, ICS
**

Gotthard Günther introduced the
proemial-relationship (PRS) as one of the basic transclassical
concepts of polycontexturality. PRS pre-faces and constitutes as
the mechanism of the difference making `difference' all
relational and operational orders.

The present paper developes a first step modelisation of the
proemial-relationship in analogy to graph-reduction based
implementations of functional languages.

A proemial-combinator, ** PR**, is designed
and implemented, which is proposed as an extension of functional
programming languages and as an implementation technique for
process-communication and computational reflection.

**Keywords: **combinatory logic,
computational reflection, functional programming, kenogrammatics,
lambda-calculus, parallel processing, polycontexturality,
proemial relationship, semiotics.

The idea of an extension of classical logic to cover simultaneously active ontological locations was introduced by Gotthard Günther (1900-1984, us-american thinker, born in germany; colleague of Heinz von Foerster at the BCL, Urbana). The ideas of Polycontextural Logic originate from Günthers study of Hegel, Schelling and the foundation of cybernetics in cooperation with Warren St. McCulloch. His aim was to develop a philosophical theory and mathematics of dialectics and self-refential systems, a cybernetic theory of subjectivity as an interplay of cognition and volition.

Polycontextural Logic is a ** many-system**
logic, a dissemination of logics, in which the classical logic
systems (called contextures) are enabled to interplay with each
other, resulting in a complexity which is structuraly different
from the sum of its components. Although introduced historicaly
as an interpretation of many valued logics, polycontextural logic
does not fall into the category of fuzzy or continous logics or
other deviant logics. Polycontextural logics offers new formal
concepts such as multi-negational and transjunctional operators.

The world has infinitely many logical places, and it is representable by a two-valued system of logic in each of the places, when viewed isolately. However, a coexistence, a heterarchy of such places can only be described by the proemial relationship in a polycontextural logical system. We shall call this relation according to Günther the proemial relationship, for it prefaces the difference between relator and relatum of any relationship as such. Thus the proemial relationship provides a deeper foundation of logic and mathematics as an abstract potential from which the classic relations and operations emerge.

The proemial relationship rules the mechanism of distribution and mediation of formal systems (logics and arithmetics), as developed by the theory of polycontexturality. This relationship was characterised as the simultaneous interdependence of order and exchange relations between objects of different logical levels.

*According to Günther: “The
proemial relationship belongs to the level of the kenogrammatic
structure because it is a mere potential which will become an
actual relation only as either symmetrical exchange relation or
non-symmetrical ordered relation. It has one thing in common with
the classic symmetrical exchange relation, namely, what is a
relator may become a relatum and what was a relatum may become a
relator. Or to put it differently: what was a distinction may
become something which is distinguished, and what has been
distinguished may become a process of distinction. The proemial
relationship crosses the distinction between form and matter.
[...] 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.”*** **

The proemial relationship implies the simultaneous distribution of the same object over several logical levels, which is not covered by classical theories of types. In the following, a concept of such a coexistence and parallelism will be developed which models the kenogrammatic proemial relationship.

Due to the special properties of the proemial
relationship and the limitations of classical calculi, an
algebraic representation of the proemial relationship must be
self-referential, i.e. in classical formalisms it has a
paradoxical and antinomic structure. Because of these fundamental
difficulties with its formalisation, an attempt will be made here
to develop an *operational*** **model
of the proemial relationship. To do this, the operational
semantics of an abstract combinatorical graphreduction machine
will be extended by a proemial combinator ** PR**.

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Copyright © 1996, R. Kaehr and Th. Mahler,

last revised Jan 1997