"Everyone when faced with the dilemma of having either an unsound or incomplete formal system chooses incompleteness. What happens when we allow for unsoundness and treat this notion as first class?"
There has been a slow burn on paraconsistent logics going back to the 80s or so now.
"Everyone when faced with the dilemma of having either an unsound or incomplete formal system chooses incompleteness. What happens when we allow for unsoundness and treat this notion as first class?"
There has been a slow burn on paraconsistent logics going back to the 80s or so now.
https://en.wikipedia.org/wiki/Paraconsistent_logic
I seem to recall at least one where one of the states was something analogous to a maybe/don't know.