If you were forced to speculate or offer anecdotal evidence, how would you say excellent practicioners of mathematical logic coneptually grasp statements like:
$$ \vdash ((P \rightarrow Q) \rightarrow Q) \rightarrow Q $$
...on an intuitive level?
Specifically, I'm curious if excellent practicioners will typically VISUALIZE such statements in any sort of way that doesn't involve mentally picturing the literal statement with its syntax as written above. For example, are Venn or Euler Diagrams typically a good way to go about things, or is that a bad idea in the long-run?
Personally, I know that with Analysis/Group Theory/Topology I have been successful in finding rough visualizations of pretty much every concept involved (with the understanding that mental pictures are not LITERAL representations of the relevant concepts and in fact can often be quite misleading if one is not careful with them); however, with logic I am finding this more difficult since the mathematical objects in question seem in many ways to be largely, explicitly syntactical. What this means is that the more I rougly convert formal statements into "intuitive" pictures, the more those intuitive pictures start to exactly resemble a specific interpretation which gets in the way of those sentences being explictly syntactical in nature.
As a consequence of this mess, I find myself getting confused in lacking basic intuitions over whether elementary statements are true or false before I attempt to prove them, unlike in other mathematical subjects.
In short, does anyone have any anecdotal or even speculative advice about how to be successful in visualizing (or NOT visualizing) the objects of Math Logic? How does one gain an intuitive feel for this subject?