The rules given in the previous post for evaluating cactus graphs were given in purely formal terms, that is, by referring to the mathematical forms of cacti without mentioning their potential for logical meaning.  As it turns out, two ways of mapping cactus graphs to logical meanings are commonly found in practice.  These two mappings of mathematical structure to logical meaning are formally dual to each other and known as the Entitative and Existential interpretations respectively.  The following Table compares the entitative and existential interpretations of the primary cactus structures, from which the rest of their semantics can be derived.

cc: Peirce List (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15)
cc: Structural Modeling (1) (2) • Systems Science (1) (2)
cc: Cybernetics (1) (2) • Ontolog Forum (1) (2)
cc: FB | Logical Graphs • Laws of Form