B42: The Geometry of 4-Valued Contradiction

We study the “quadri-segment” B42, a point in the space BMN of the “poly-simplexes” of “oppositional geometry”. This is the mathematical 4-valued counterpart of the expression, inside the classical 2-valued “bi-simplicial” space B2N, of the segment of 2-valued contradiction B22 (e.g. any of the 3 diagonals in a “logical hexagon” B23, cf. [7]). We rely on a method of poly-simplicial construction put forward in our study [10] of the “tri-segment” B32 (i.e. 3-valued contradiction), based on sheaf-theory (Angot-Pellissier, [2]) and on the “Pascalian simplexes” PM [3]. We show that the quadri-segment B42 is composed of 8 hexagonal tri-segments B32, 4 of which are “strong” (B32*) while the other 4 are “weak” (B32#). These 4 + 4 hexagonal tri-segments are “glued” (each strong to its correlated weak) by another sub-structure of the B42, made of 3 squares. The quadri-segment B42, with its 12 vertices, has as “geometric attractor” a 3D cuboctahedron. It will be recalled how “logical geometry” (Smessaert, Demey [12, 13]) is a fragment of oppositional geometry, where the 2 Smessaertian sub-geometries (“oppositive” and “implicative”) undergo, in each poly-simplex, “Aristotelian fusion” into a unique geometry. The quadri-segment B42 will be useful, in the future, for exploring a higher poly-simplex, the quadri-triangle B43, i.e. the 4-valued counterpart of the classical (bi-simplicial) “logical hexagon” B23.