With these logical semantics, the * and ** interpretations of the Kaucher addition
[-1,1]+[12,8]=[11,9]
are
Q(x,dual[-1,1]), Q(z,[11,9]), Q(y,dual[12,8]), z=x+y
and
Q(y,[12,8]), Q(z,dual[11,9]), Q(x,[-1,1]), z=x+y
isn't it? So the quantifier to be associated with a variable depends on both the * or ** interpretation, and the type of variables. This is where the "logical semantics" is confusing. It is much better to keep only one type of interpretation, e.g. the * one, all ** interpretations being equivalent up to Kaucher dual.
Note that Mr. Hayes' document does not present modal intervals at all (a modal interval is a pair made of a classical interval and a quantifier, this definition being not given in the document).
Alexandre