What you are calling the ``negation tree" is commonly referred to in the literature as the dual.

Underlying the processes is a tension between truth preservation $A~\vdash~B$ and logical truth preservation $\models~A~\iff~\models~B$. I think the question is a good one and that we have not as yet plumbed the depths of what these mean. More work in this area is needed, even though the methodologies have a long track record of usefulness their meaning is yet to be examined. In my opinion, until this is examined thoroughly, the foundations of logic in the 1920-30 period is not well established even yet and all the applications of Cantor's ideas to these formulas are on shaky ground.