Do we really need Ex Falso? A first reconnaissance

Intuitionistic mathematics has been critized for the use of the Ex Falso principle, directly, or indirectly by disapproval of any use of negation at all. Do we really need it? Not in Heyting Arithmetic, if we agree with interpreting $1=0$ as the falsum. We take some first steps looking around in other systems aided by some work by student Noor Heerkens.