It is possible to trace the divergence of formal grammar styles to an even more primitive division, distinguishing between the additive or parallel styles and the multiplicative or serial styles.  The issue is somewhat confused by the fact that an additive analysis is typically expressed in the form of a series, in other words, a disjoint union of sets or a linear sum of their independent effects.  But it is easy enough to sort things out if one observes the more telling connection between parallel and independent.  Another way to keep the right associations straight is to employ the term sequential in preference to the more misleading term serial.  Whatever one calls the broad division of styles, the scope and sweep of their dimensions of variation can be delineated in the following way.

• Additive or parallel styles favor sums of products as canonical forms of expression, pulling sums, unions, co‑products, and logical disjunctions to the outermost layers of analysis and synthesis, while pushing products, intersections, concatenations, and logical conjunctions to the innermost levels of articulation and generation.  The analogous style in propositional logic leads to the disjunctive normal form (DNF).
• Multiplicative or serial styles favor products of sums as canonical forms of expression, pulling products, intersections, concatenations, and logical conjunctions to the outermost layers of analysis and synthesis, while pushing sums, unions, co‑products, and logical disjunctions to the innermost levels of articulation and generation.  The analogous style in propositional logic leads to the conjunctive normal form (CNF).

Resources

• Cactus Language • Pragmatics
• Survey of Animated Logical Graphs
• Survey of Theme One Program

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cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science