More support for the logical basis of the geometry of cognition comes from the American mathematician W.H. Gottschalk (a fact noted by Alessio Moretti in his PhD thesis published in 2009.
“It is well-known that every involution in a logical or mathematical system gives rise to a theory of duality; for example, negation in the sentential calculus and predicate calculus, complementation in the calculus of classes, complementation and conversion in the calculus of relations, etc. The purpose of this note is to call attention to the fact that every involution in a logical or mathematical system gives rise to a theory of quaternality and that the square of quaternality, of which the classical squares of opposition are special cases, provides a diagrammatic representation for much of the theory of quaternality.” (See: W.H. Gottschalk, “The Theory of Quaternality,” Journal of Symbolic Logic, 1953).
Moretti’s work shows the close relationship between Gottschalk’s quaternality and Piaget’s INRC model, both of which are also related to Klein’s four-group.
For Alessio Moretti’s research, the best place to start is his website: http://alessiomoretti.perso.sfr.fr/NOTHome.html