> In this thread, we propose a solution to this differentiation > difficulty: semantic _re-interpretation_ of _logical symbols_ .
It sounds more like "coordinated interpretation."
That is what mathematical realism is already doing. The existence quantifier is co-interpreted with some notion of truth. This is the historical debate from description theory addressing presupposition failure.
One of the foundational insights of Frege's researches was to interpret contradiction existentially. In contrast, Kant interpreted contradiction modally. This would suggest non-existence and impossibility are already coordinated in such a way that the two forms of logic branch at the outset.
There are, of course, intensional logics that mix the senses of these logics. This is where the terms "de re" and "de facto" find their nuanced meanings in relation to quantifier-operator order.
No one, of course, has tried to use anything like an arithmetical numbering to provide correlated, but distinct, model theories to interpret a single situation (quantificational logic) so as to eliminate irrelevant modal possibilities.