Participant: Art Collings
Format: Poster and Conversation
Themes: recursion, paradigm

This paper proposes to refine the concept of ‘distinction’ as expressed in Laws of Form, the mathematical logic work by G. Spencer-Brown. Laws of Form played an in influential role in the development of second order cybernetics in the 1970s via the thought of von Foerster, Pask, Maturana, Varela, and others. Laws of Form’s influence can be attributed (in varying degrees) to its notation – which easily admits the expression of ‘recursive’ or ‘reflective’ forms; to its recognition of the role of the observer within the system being observed; and to the intuitive appeal of the concept of distinction, both as a mathematical idea, and as a ubiquitously familiar mental act.

The paper will specifically examine three different formal approaches, each of which is based on expanding the number of logical values from the usual two to the less typical four. First, a 4-value logic co-developed by Francisco Varela and Lou Kauffman in the early 1980s is considered, which is based on interpreting the ‘extra values’ as wave forms generated in feedback/reflexive systems. Second, a very different interpretation of 4-value logic from a ‘modal logic’ and lattice theory perspective developed independently by logician Nuel Belnap around the same time is considered, one which has extensive applications in resolving conflicts in artificial intelligence databases. These two interpretations are actually isomorphic, but there has been little if any exploration of the relationship between these approaches, especially in the bi-lattice literature. Finally, a new variation of 4-valued logic is considered, based on my own research. This logic is cyclic in nature, and consequently its four values are precisely analogous to the group of unitary imaginary numbers. It contains a complete copy of the Kauffman/Varela logic, has application to 4-state cellular automata (in particular to the system of automata called ‘recursive distinctioning’), and to other interesting areas.
The discussion in the paper is mathematical, but its themes can and should be understood conceptually and metaphorically in relation to the development of an “ecology of ideas.”