ThinkArt Lab Hogmanay 2004
ThinkArt Lab Animation:  A.T. Kelemen
November 12, 1998 Dr. Rudolf Kaehr

Iterability of Zero

Does the transjunctional super-operator BIF exist for bool0(3) ?

Booleans are atomic therefore they can not be split by the super-operator BIF. But this is true only under the condition that they are ruled by the law of identity. Why should they?

Cloned objects are the same but they are not identical.

How can we replicate systems properly if we cannot replicate their atomic terms? Dissemination as distribution and mediation, or as replication, gives us no answer about the splitting of atomic terms because the distributed systems are identical for themselves, e.g. bool1, bool2 and bool3 are all containing and conserving the classical elements, values and operations in their domain. Modeling "as such", "as other", ...

Normally we know transjunctional operations in polycontextural logics only from binary logical operations, the so called transjunctions. Bifurcations are generalizations of this binary concept to super-operators which are ruling poly-contexturality as such and not only internal or local operations. Also super-operators are well know in the polycontextural literature they have never been studied independently from their historical sources. In this sense the question of a generalized form of bifurcation, applicable to all objects of a system, was not focussed. Here I try to develop some ideas and constructions of these generalized bifurcations with the help of the metaphors of cloning, replication and not only by the metaphor of splitting, parallelism, simultaneity. In some sense, we even can say, that bifurcation, simultaneity etc. are cases of cloning. More explicite, these terms seems to form a system of complementarity. There is no bifurcation without replication-and vice versa. A free use of the idea of replication goes beyond the well known transjunctions in polycontextural logics. I can not go deeper into the development and explication of the metaphor cloning", but it has to be mentioned, for short, that the very idea of logical cloning and replication as well as of logical bifurcation and simultaneity is based on the kenogrammatic concept of morphograms. Morphograms are structural patterns invariant to logical negations, therefore byond identity and diversity of signs, which means outside the realm of signs.

From a more technical point of view I am abandoning in a further step the basic functional approach of the historical polycontextural logics based on a interpretation of multiple-valued logics. One of the earlier significant steps was to abandon the Cartesian product approach of n-ary functions and their problem of decomposition into two-valued subsystems (theory of place-value systems).

Example (ID, BIF, ID)(opn1, opn2, opn3)

arity-zero operations

ID1 T, F --> bool01 : {T1, F1} simul {T2, F2}

BIF1 T, F --> bool02 .simul. bool01 : {T2, F2}

ID3 T, F --> bool03 : {T3, F3}

unary operations (arity-one)

ID1 bool01 --> bool01

BIF1 bool02 --> bool02 .simul. bool01 --> bool01

ID3 bool03 --> bool03

binary operations

ID1 bool01 x bool01--> bool01

BIF1 bool02 x bool02 --> bool02 .simul. bool01x bool01 --> bool01

ID3 bool03 x bool03 --> bool03

Diagramm 1

In a short notation we have:

(1, 2, 3)-->(122313)-->((12%23)2313) :: ((12%23%#)(#%23%#) (#%#%13))

If we apply the unary operations of negation to this evaluation (mapping) (Belegung) we have to deal with the quite new situation of a reflection of the values of system2 in system1. It seems to be reasonable to accept that a negation of the values in system2 has to be mirrored one-to-one in system1 to be correct. We say that system1 has a model of the behavior of system2 in itself. The model is not the original, it differs in its place in the reflectional system. If system2 changes its state the mirrored model in system1 has to change in exactly the same way. Does this make any sense? Probably it is the most simple case of transjunctional or replicational distributions.

1 Bifurcational distribution of negations and junctions

If we start with bifurcation we are forced to distribute all operations in a transjunctional way. This is really a new and intriguing situation. Not only we have to involve constants, negations but also binary junctional operations like conjunction, disjunction into this transjunctional game. And the originary transjunctions of the old place-value system of logic are understood as a very special but quit explicit case of bifurcation.

Does it make any sense to repeat exactly the same logical situation of one system in another logical system which is distinguished from the first only by its different place in the complexion of the whole polycontextural logic?

From the point-view of a theory of argumentation (interaction or communication) this type of modeling corresponds to the situation when an actor is agreeing in all logical points with its partner of communication. The agreeing system has its own position and its own logical arguments but additionally it offers space to the other system to accept its logical arguments. Insofar the modeling has to be strictly one-to-one. To reduce the situation of agreement to the usual case the actor denies its space and accepts the arguments in the space of the other system. I agree, but keep it for yourself. There is no logical space I can offer you for that."

From the point of view of the model of reflectional programming this situation of mirroring the logical constellation of the environmental or partner system could be understood as an interpretation and modeling of the so-called causal relation" of a reflectional system on a logical, and not on an informational, level.

This is the special case of transjunctional behavior. The general case accepts a different logical behavior, a different sequence of argumentational steps at the locus of the accepting system. This is the real case for logical transjunctions as we know them from polycontextural logic.

In this new context the operation of bifurcation is distributing total functions and not partial functions as it is necessary for transjunctional operations. To distinguish the two concepts, this type of function should be called replicative transjunction or simply replication (of functions).

Mixing different types of logics

We also have to consider the case, that combination of logics has not to be homogeneous, that means, that we are mixing different types of logical systems together. Therefore the bifurcation operation of this different systems produces inside" of reflecting system a mapping of components which are from a different type of logic than the reflecting system itself. But this possibility is out of the range of this study, which is mainly introductory.