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Dear colleagues, 
In a previous email I sepeficied the mechanism of double contingency in
terms of anticipatory systems as follows: 
x(t) = a (1 - x(t+1)) (1 - x(t+1))
The next state of the system is determined by the selective operation of
expectations upon each other in a dyadic interaction. The simulations are
robust and show that the system can move erratically from the one to the
other side. (If one wishes, one can play with the parameters in the excel
sheet and follow the consequences; at ).
In Chapter 3 of Soziale Systeme, Luhmann (1984) discussed "double
contingency" as central to the emergence of social systems. Borrowing the
concept from Parsons, he provides it with a completely new solution. In my
opinion, the simulations in terms of expectations accord with this solution.
Paul Hartzog (cc) sent me a short piece in which he explains Luhmann's
solution in English. (Can you bring it online, please, Paul?). It made me
aware that Luhmann moves fast in this chapter from "double contingency"
towards the emergence of social systems without a specification of the
mechanism. (In footnote 12, p. 157, Luhmann warns against Von Foerster's too
fast movement.) The social system "emerges" from double contingency (in the
I guess that a double contingency can go on forever when no third party
comes into play. Piet Strydom used the term "triple contingency" for
explaining the emergence of a modern communication society in 16th and 17th
century. The third party becomes abstracted as a public. In priniciple, one
could model a triple contingency analogously using: 
x(t) = a (1 - x(t+1)) (1 - x(t+1))  (1 - x(t+1))
This leads to a cubic equation of x(t+1) as a function of x(t). Cubic
equations have analytical solutions, and there is a (freeware) add-in in
Excel for solving them. The solutions may imply i = sqrt(-1), and thus be in
the complex domain. 
For all values of the bifurcation paramater a the system is highly unstable
and quickly degenerates into a complex one. One interpretation would be that
triple interactions provide a short-term window for organization
(decision-making) to step into the system. The relation between interaction
and organization would then be conditional for the emergence of the social
An alternative formulation is:
x(t) = a (1 - x(t)) (1 - x(t+1))  (1 - x(t+1))
x(t)/(1 - x(t) = a (1 - x(t+1))(1 - x(t + 1))
By replacing [x(t) / (1 - x(t))] with y, the solution is similar to the one
for double contingency, but mutatis mutandis: 
x(t+1) = 1  sqrt(x(t)/ (a * (1 - x(t)))


This formula is in the simulation as stable as double contingency for values
of a  8, but I don't yet have an analytic solution for this. For lower
values of a, the system vanishes. Using an internal degree of freedom the
system might be able to change its value of a endogenously and thus
alternate between double contingency and its disappearance. 


In summary, in the case of a triple contingency, the system can show the
behavior of a window for organization to step in by using three incursive
terms (based on expectations), or bring a double contingency to an end by
bringing a historical contingency (modeled as a recursive term) into play.
Using the internal degree of freedom for changing the value of a, the social
system would also be able to generate double contingencies (interactions)


From entropy statistics, we know that a system with three dynamics can
generate a negative entropy in the mutual information among the three
(sub)dynamics. (I use this as an indicator of self-organization in other
studies.) However, there is still a missing link between the above reasoning
and the emergence of a social system as a possibility because the complex
system is not yet generated. I suppose that I have to bring the social
distributedness into play and not write x(t), but ixi(t). 


With best wishes, 





Loet Leydesdorff 
Amsterdam School of Communications Research (ASCoR)
Kloveniersburgwal 48, 1012 CX Amsterdam
Tel.: +31-20- 525 6598; fax: +31-20- 525 3681 
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