The duality between Entitative and Existential interpretations of logical graphs is a good example of a mathematical symmetry, in this case a symmetry of order two. Symmetries of this and higher orders give us conceptual handles on excess complexity in the manifold of sensuous impressions, making it well worth the effort to seek them out and grasp them where we find them.
Both Peirce and Spencer Brown understood the significance of the mathematical unity underlying the dual interpretation of logical graphs. Peirce began with the Entitative option and later switched to the Existential choice while Spencer Brown exercised the Entitative option in his Laws of Form.
In that vein, here’s a Rosetta Stone to give us a grounding in the relationship between boolean functions and our two readings of logical graphs.
cc: FB | Logical Graphs • Laws of Form • Mathstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science
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I hadn’t considered duality as a group action before, but this is a very natural formulation. Of course, there is (up to isomorphism) only one group of order 2, namely
In category theory, every statement about a category has a dual statement, gotten by reversing its arrows. Since an arrow is essentially an ordered pair
then the group action of 
on this pair is 
So the category-theoretic kind of duality corresponds to your connection to group theory.