George Spencer-Brown's »Laws of Form« (LoF) provided an
important formalism and important insights for all kinds of
fields in 1969. He made it possible to calculate with layered,
self-referential spaces, time, and even the »blank« space using
the distinction as a basal operation. However, with a few
exceptions in sociology (Niklas Luhmann & Dirk Baecker) and
mathematics (Louis H. Kauffman), cybernetic biology (Humberto
Maturana & Francisco varela) the work has found few practical
use cases. User interfaces, on the other hand, be they graphical
or e.g. conversational, are practically designed, described and
most importantly used every day - they are the experiential
interfaces of our (digital) environments. Although some
formalisms for these have been developed in recent years, they
are on the one hand very domain-specific and on the other hand
do not allow to calculate with the interfaces and thus our
(digital) environments.
It turns out that, in the spirit of LoF,
the fundamental operation for user interfaces is also the
distinction. In this paper, a formalism for user interfaces
based on LoF shall be worked out, which allows to describe,
compute and (domain-spanning) transform them solely by means of
the distinction. In particular, it will be shown that this is an
exceedingly natural way to describe and reason about
(self-referential) graphical user interfaces (which can be
thought of as distinctive layered spaces) and to find
fundamental connections between them and other disciplines. This
work will maybe lay the foundation for future designers to later
develop automatic theorem provers for interfaces and to perform
formal verifications.