AM: Why do you need XOR in your inquiry system?
Clearly we need a way to represent exclusive disjunction, along with its dual, logical equivalence, in any calculus capable of covering propositional logic, so I assume this is a question about why I chose to represent those two operations more compactly with cactus graphs instead of using trees and defining them in terms of conjunctions and negations.
The generalization from trees to cacti presented itself at the point where multiple lines of problem-solving effort converged. Some of the problems were conceptual, arising from a desire to include the types of operator-variables that Peirce considered. Other problems were computational, provoked by a need to avoid combinatorial explosions in the evaluation of logical formulas.
But, as I remarked earlier, “the genesis of that generalization is a tale worth telling another time …”, after we have gotten a better handle on the basic logical issues.