Quantum geometry of Boolean algebras and de Morgan duality

We take a fresh look at the geometrization of logic using the recently developed tools of "quantum Riemannian geometry" applied in the digital case over the field F2 = {0, 1}, extending de Morgan duality to this context of differential forms and connections. The 1-forms correspond to graphs and the exterior derivative of a subset amounts to the arrows that cross between the set and its complement. The line graph 0 -1 -2 has a non-flat but Ricci flat quantum Riemannian geometry. The previously known four quantum geometries on the triangle graph, of which one is curved, are revisited in terms of left-invariant differentials, as are the quantum geometries on the dual Hopf algebra, the group algebra of Z3. For the square, we find a moduli of four quantum Riemannian geometries, all flat, while for an n-gon with n > 4 we find a unique one, again flat. We also propose an extension of de Morgan duality to general algebras and differentials over F2.