life as polycontexturality

some remarks on part_1

by Eberhard von Goldammer

As it was pointed out in the Günther 2000 special a contexture is a logical domain where all classical logical rules hold strictly. Furthermore we pointed out that Günther uses the term many-valued logic in a different way. In Günther´s terminology many-valuedness has to be interpreted in the sense of a many-placed logical system, i.e., there are different logical places (locations) that are characterized by different contextures. As logical domains, these contextures are not isolated from each other but rather mediated by new logical operators such as global negations and/or the transjunction. In other words:

PolyContextural Logic (PCL) represents an intrinsic parallel calculus where different logical domains (contextures) are closely interwoven with each another by means of new logical operators.

If a contexture is considered to be isolated, i.e., intra- or monocontextural, it is basically classic. On the other side, if a contexture will be considered in relation to other contextures, i.e. inter- or polycontextural, it is discontextural. Discontexturality means that there is no continuity, no medium, no space, no language, etc., in common with other contextures. However, as it was already mentioned, contextures are not isolated logical domains but rather mediated by inter- or transcontextural logical operations (!!).

The number of contextures nK depends on m the number of logical places or positions that characterize a logical discourse and it also depends on the valuedness s within the sub-systems, the single contextures. With m £ s the number of contextures is:

which is

For three logical positions (m = 3) and an intracontextural valuedness of two (s = 2), it follows:

and correspondingly for m = 4 and s = 2, the number of contextures is n_K = 6

At a first glance, one would expect a similarity between a polycontextural system (with m = 3 and s = 2) and Peirce´s triadic logic. However, any similarity turns out to be more or less superficial with both classical (monocontextural) or non-classical (polycontextural) interpretations:

a) MONOCONTEXTURAL INTERPRETATION OF PEIRCE´S TRIADIC This kind of interpretation is standard in Artificial Intelligence, where Peirce´s triadic logic is interpreted in the context of Post´s many-valued logic where a third value was introduced between true (1) and false (0), viz.,

0 . . . . . . . . . . . . . ½ . . . . . . . . . . . . . 1

or more generally

0 . . . . . . ¼ . . . . . . ½ . . . . . . ¾ . . . . . . 1 etc. ...

But this has nothing to do with Günther´s polycontexturality. And whether or not this interpretation really deals with all aspects of Peirce´s logical experiments is doubtful.

It should be mentioned that in the PCL the numbers are used in a different way: "1" symbolizes the affirmation and "2", "3", ... etc. stand for the negations (not for the intracontextural negation, i.e., within a contexture but only for intercontextural negations, i.e., between different contextures ( !! ).

In Günther´s PCL the third value (and all other values) are beyond true (1) and false (0). In other words, the PCL allows the rejection of complete logical domains, i.e., of complete contextures. In other words, the "law of excluded middle" – the tertium non datur – has two different meanings. It can be used intracontextural in its usual way, i.e., within a contexture, and intercontextural, i.e., between different contextures.

b) POLYCONTEXTURAL INTERPRETATION OF PEIRCE´S TRIADIC

Charles S. Peirce(1839-1914)

Gotthard Günther(1900-1984)

Charles S. Peirce(1839-1914) Gotthard Günther(1900-1984) Such an interpretation does not exist yet. However, if we look at Peirce´s ideas on semiosis as "an action, or influence, which is, or involves, a co-operation of three subjects, such as a sign, its object, and its interpretant, this tri-relative influence not being in any way resolvable into actions between pairs", then we could conclude that Peirce would have used Günther´s ideas of polycontexturality if they would have been known to him in his time.

Within these short remarks we cannot discuss in detail any possible apparent relationships between Peirce´s triadic logic and PCL. Instead we will shortly point at some fundamental differences.

Peirce´s triadic relation between sign, object, and interpretant represents neither a logical system nor a calculus. Actually, this relation corresponds to von Foerster´s model of cognition which can be summarized as follows: "environment is the triadic relationship E(W,C,D) between the domains W (real world), C (cognitive processes), and D (descriptions)."

While the PCL is a calculus, these triads have to be interpreted with the help of an adequate calculus. This is the most significant and obvious difference between the triads and Günther´s PCL. In "Cybernetic Ontology and Transjunctional Operations" Günther has shown that any triadicity without an adequate concept of distribution between the different (logical) positions is simply reduced to a single logical domain, i.e., to one contexture and therefore all these triadic models remain basically dichotomic (cf. part_4 of Günther_2000special). In other words, neither von Foerster´s concept of eigen-values nor Spencer Brown´s calculus of indication (CI) or Varela´s variation of an extended_CI represents an adequate basis for a consistent formal model of Peirce´s and/or any other triadic relationships.

Although the smallest polycontextural unit is given by three contextures (m=3), one can easily show that polycontexturality begins with four (m = 4) and not with three (the triad, m = 3) which means that a complete logical system only results with m = 4 and not with m = 3. It is needless to say that m < 3 is meaningless. In other words, the smallest meaningful polycontextural logical system is given by a structure with four (logical) positions distributed/mediated over six two-valued logical domains/contextures (or four three-valued contextures, or one four-valued contexture).

Within these short introducing remarks on polycontexturality we would like to point out that not only the "law of excluded middle" – the tertium non datur – has two different meanings, but also the concept of identity. The classical concept of identity strictly holds intracontextural, i.e., within a contexture. A non-classical concept of identity becomes possible and it allows mutatis mutandis the modeling of an identity distributed over several contextures. A well-known example is the distribution of subjectivity over the categories of I, THOU, and IT which can only be modeled within a polycontextrual framework. If one considers the importance of the classical law of identity for mathematics, for example, for the development of numbers, it might not be such a surprise if we mention that PCL also represents a basis for the development of a mathematical concept of polycontextural or qualitative numbers, qN. However, this point will be discussed in a forthcoming contribution.

Within the 1st of 3 parts in this series we present the following papers of Gotthard Günther:

1) A New Approach to The Logical Theory of Living Systems
This is an unpublished manuscript that Günther prepared for a conference in Chicago in 1972. It is a brief and very good introduction to the concept of contextures.

2) Negation and Contexture

This again is an unpublished manuscript from Günther´s left writings which are preserved at the Staatsbibliothek – Preußischer Kulturbesitz (Berlin).
This text complements the conference notes _1).

3) Life as Polycontexturality

This paper was published first in 1973. As one can see from the contribution of Kaehr and Ditterich in part_2 of the Günther_2000 special ("Einübung in eine andere Lektüre"), in particular it was this paper in which Günther introduced his idea of "contexture".