There's a whole class of formal logic systems that accept "a and not a". They are called Paraconsistent Logics (https://en.wikipedia.org/wiki/Paraconsistent_lo

There's a whole class of formal logic systems that accept "a and not a". They are called Paraconsistent Logics (https://en.wikipedia.org/wiki/Paraconsistent_logic https://en.wikipedia.org/wiki/Paraconsistent_logic), which allow contradiction without explosion. Ideal for mathematically representing the kind of reasoning that bubbles up from contradictory positions and goals as you described, where organisations manage to do sane things anyway, sometimes!