A formal duality points to a higher unity — a calculus of forms whose expressions can be read in two different ways by switching the meanings assigned to a pair of primitive terms.
I just ran across an old post of mine on the FOM List where I touched on this theme, so I think I’ll copy that here until I get a chance and the concentration to comment further.
C.S. Peirce explored a variety of De Morgan type dualities in logic that he treated on analogy with the dualities in projective geometry. This gave rise to abstract formal systems where the initial constants — and consequently their geometric or graph-theoretic representations — had no uniquely fixed meanings but could be given dual interpretations in logic.
It was in this context that Peirce’s systems of logical graphs developed, issuing in dual interpretations of the same formal axioms that Peirce referred to as entitative graphs and existential graphs, respectively. It was only the existential interpretation that he developed very far, since the extension from propositional to relational calculus appeared easier to visualize in that case, but whether there is some truly logical reason for the symmetry to break at that point is not yet known to me.
When I have explored how Peirce’s way of doing things might be extended to “differential logic” I have run into many themes that are analogous to differential geometry over GF(2). Naturally, there are many surprises.