The most of the simplified Cinque table’s rows describe modal operators, but there is a two-row exception.
Cinque distinguishes between two kinds of “completion” which are more or less clearly distinguished in different languages: completive-sg refers to either one-object set or to a plural set if it has been totally affected (i.e., each member of the set has been affected), and completive-pl refers to the plural set if each member of the set has been totally affected (Cinque 1999, 101).
It is clear that completive-sg is a ‹1, 1› type quantifier all, while sompletive-sg is a ‹1, 2› type quantifier all. Both are quantifying over a set within the universe, but, in the first case, this set is without relevant internal structure, while, in the second case, it contains an internal structure of the first order (that is, consists from the subsets with no further internal structure).
Cinque takes as default values of these quantifiers their external negations (not all) and the quantifiers themselves as marked values.
Probably, the relevant linguistic data should be rechecked in order to trace the whole square of, at least, the ‹1, 1› type quantifier some (whose dual the quantifier all is).
It is interesting that the language is sensitive to ‹1, 2› type quantifiers, while the narrative seems to be not. However, this problem in the narratology still waits for being studied.
It seems a priori likely that the universal quantifier some with the whole its square—and not only internal negation and dual—could be traced in the language outside any specific modal context (that is, with no fixed and grammatically determined modal context). If so, we have to suppose that some standard cases where the quantifiers some and none appear outside alethic modality and with no fixed and grammatically determined modal context at all are so far overlooked.
If my previous supposition is right, then, to give a right place to the “pure” quantifiers, Cinque’s table must be regrouped. It must contain, at the first place, a set of the “pure” generalized operators (two kinds of all, the whole square of ‹1, 1› type all plus the square of ‹1, 1, 1› type more than) followed by a set of standard modal operators based on the same quantifiers.