Functions in these logics are total, so if you want division to be a function (and you probably do), it has to assign something to division by 0.

It would be acceptable to assign an unspecified object from the domain, for which you have no non-trivial theorems, and so all your real theorems must have a precondition about the denominator being non-zero. But if you specify a candidate like 0, you can get some theorems which don't have the precondition. Consider:

a/b * c/d = ac/bd.

This now holds even if one of b or d is 0.