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Parsons, Luhmann, Spencer Brown. NOR design for double contingency tables

Steffen Roth (La Rochelle Business School, La Rochelle, France)


ISSN: 0368-492X

Article publication date: 4 September 2017



Cross tables are omnipresent in management, academia and popular culture. The Matrix has us, despite all criticism, opposition and desire for a way out. This paper draws on the works of three agents of the matrix. The paper shows that Niklas Luhmann criticised Talcott Parsons’ traditional matrix model of society and proceeded to update systems theory, the latest version of which is coded in the formal language of George Spencer Brown. As Luhmann failed to install his updates to all components of his theory platform, however, regular reoccurrences of Parsonian crosstabs are observed, particularly in the Luhmannian differentiation theory, which results in compatibility issues and produces error messages requesting updates. This paper aims to code the missing update translating the basic matrix structure from Parsonian into Spencer Brownian formal language.


This paper draws on work by Boris Hennig and Louis Kauffman and a yet unpublished manuscript by George Spencer Brown, to demonstrate that the latter introduced his cross as a mark to indicate NOR gates in circuit diagrams. The paper also shows that this NOR gate marker has been taken out of and may be observed to contain the tetralemma, an ancient matrix structure already present in traditional Indian logic. It then proceeds to translate the basic structure of traditional contingency tables into a Spencer Brownian NOR equation and to demonstrate the difference this translation makes in the modelling of social systems.


The translation of cross tables from Parsonian into Spencer Brownian formal language results in the design of a both matrix-shaped and compatible test routine that works as a virtual window for the observation of the actually unobservable medium in which a form is drawn, and can be used for consistency checks of expressions coded in Spencer Brownian formal language.


This paper quotes from and discusses a so far unpublished manuscript finalised by Spencer Brown in April 1961. The basic matrix structure is translated from Parsonian into Spencer Brownian formal language. A Spencer Brownian NOR matrix is coded that may be used to detect errors in expressions coded in Spencer Brownian formal language.



The author is grateful to Markus Heidingsfelder (Habib University Karachi), Martin Rathgeb (University of Bonn), Boris Hennig (Ryerson University Toronto), Artur Mkrtichyan (Yerevan State University), Sean Ward (Independent) and Anton Schütz (University of London) for active support and inspiring conversations about these subjects. Any errors in this article, however, remain the author’s exclusive responsibility.


Roth, S. (2017), "Parsons, Luhmann, Spencer Brown. NOR design for double contingency tables", Kybernetes, Vol. 46 No. 8, pp. 1469-1482.



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