Necessarily so. If you build a formal system where true = false (say, as one of the axioms, or in a proof), you can literally prove anything, including things that are obviously false in normal mathematics, and then immediately also disprove that very same thing -- so it's not useful to go in that direction.
Probably the closest thing is "nonmonotonic systems", including "nonmonotonic logic", where true or false is not a timeless absolute, but instead can be advanced or retracted over time, you might say. The late great John McCarthy worked in that area for decades.