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

Introducing the metaphor of a tissue of coloured logics

"When classical logic is applied to non-mathematical examples, the examples are first ´mathematicized´. In the real world we might argue about whether block B is behind block A or not - maybe it depends on one's point of view. But we can create an ideal world, a mathematical model, in which either B is behind A, or it isn't. Classical logic can be used to reason correctly about such a model. Whether the model accurately reflects the real world is a separate issue." Fitting, 1990, p. 1

Trans-classical logic is aimed to model the situation of rational reasoning between different agents where each agent has its own logic, that is, its own point of view in respect to his world. Therefore trans-classical logic reflects the world from a multitude of different logical points of view. Each locus has its own mathematicized" formal apparatus, its own mathematical formal logic. As a consequence, the monolitical or erratic concept of world disappears as a very special case of disambiguity in a dynamic multi-verse.

The mathematicization of a world including a multitude of different logical loci, points of view, is obviously different from the more abstract model of the classical mono-logical mathematicization of the world.

Maybe, the classical model reflects an ideal world or even investigates the principles of reasoning for perfect worlds" (Fitting) trans-classical logic reflects the rational principle of a conflicting, interacting, co-operating world, where the participants of these interactions creates together their own worlds in a co-creating manner. In this sense reasoning and modeling are not structures but actions.

The new concept of trans-classical logic as a complex logic of a multitude of points of view does not introduce some shades of grey" between the strictness of the classical concepts. There is now fuzziness here. It introduces something different, each (classical) logic gets an index that indicates the point of view of the rational agent, which indicates the separation between the different agents. Trans-classical logic is not a logic of white and black" nor a logic with shades of grey, it is a logic of colors, a colored logic. The logics of the living tissue are colored ones.

Each color has its own formal and operative strength.

There is no ambiguity and fuzziness in this notion of colored logics.

But these colors are not only simply identical with themselves. Each colored system is able to reflect the other systems simultaneously in its own domain. This new ambiguity is produced by the complexity of the polycontectural logic as a whole with its interaction and reflection.

Another interesting feature of colored logical systems is given by the mechanism of change of systems. A system may change from one colour to an other colour. Or some systems may permute their colours.

In the example, the blue colored logical system has a picture of the green system, the blue system is able to reflect, to mirror, to model the other colored systems in its own domain. It has replications of the other systems.

This means, green is not simply green. Green as green is green, but green as blue is the blue of the green. Green is green and not blue but green can have aspects of blue.

All this is possible only because the logical and ontological principle of identity is abandoned and transcended by the game of sameness.

As a result of the plurality of formal systems as differently colored logics, each logic and complexion of these logics is localized in a structural space. Every logic has its own locus. Each logical locus gives place for the replication of other logics which are located at other logical loci.

The theory of these ontological or pre-logical loci is called kenogrammatics.

A more AI setting

In other words, we can say, that the mirroring of one contexture by another, is a belief function. One system beliefes something about another system. This more linguistic perspective opens up a connection to the work about beliefe systems, beliefe logics etc. in Artificial Intelligence (Konolige, van Harmelen,´).

But also to the Algebra of Reflection as it is proposed by Levebvre.

1 General poly-contextural diagram

As a result of the analysis of the structure of interaction I can introduce a general diagram or scheme of polycontextural logical and arithmetical reasoning and computation.

Here I have restricted the complexitiy of interaction to the special case of 3 actors.

At each locus Oi which gives place for the logical system of the place as such the locus Oi also offers place for the modeling of the neighbor logical systems. That is, for the modeling the logics of the logics of the other interacting agents.

In a classical setting, this situation is not modeled or as in computational reflection, the meta-level approach does not map on a structural level the complex logical constellations between different interacting agents.

(This can be shown for Smith, Maes, Sloman, Kennedy et al.)

General scheme:

O1: (M1, M2, ...., Mn) § O2: (M1, M2, ...., Mn) § ... § On: (M1, M2, ...., Mn)