You may already be familiar with Ruskey and Weston's "A Survey of Venn Diagrams," which includes a discussion of Borromean rings. Such rings are similar to the valknut and the triskelion, of which the gankyil is a type. All of these figures are quite old.
Of course these observations don't answer your question about the use of graphical representations of logical and/or set relations in antiquity.
Plato refers to diagrams, for example, in his discussion of the double-divided line in Book 6 of Republic and in Meno when Socrates questions Meno's slave about a problem in geometry -- how to find a square double in area to any given square. I imagine more examples can be found.
An interesting project would be to find examples of "visual" language in the works of Plato, Aristotle, and Euclid.
While writing this post I came across the following two references:
Edwards, Anthony W. F. Cogwheels of the Mind: The Story of Venn Diagrams. Baltimore, Maryland: The John Hopkins University Press, (2004).
Kuehni, Rolf. "On the Source of d’Aguilon’s Arc Color Mixture Diagram." Unpublished manuscript, 2003.
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