On page 4 of http://www.llf.cnrs.fr/sites/llf.cnrs.fr/files/biblio//the-same-ter.pdf
Richard Zuber writes:
"Given a fixed universe $E$, (where $|E| ≥ 2$), a type $n$ quantifier is a function from $n$-ary relations to truth values. A type $\langle 1 \rangle$ quantifier is a function from sets (sub-sets of $E$) to truth values, and thus it is a set of sub-sets of $E$. A type $\langle 1, 1 \rangle$ quantifier is a function from sets to type $\langle 1 \rangle$ quantifiers. In natural language semantics type $\langle 1 \rangle$ quantifiers are denotations of NPs and a type $\langle 1, 1 \rangle$ quantifiers are denotations of (unary nominal) determiners, that is expressions like $\textit{every, no, most, five}$, etc. Since both types of quantifiers form Boolean algebras they have Boolean complements (negations)."
In what sense do type $\langle 1 \rangle$ and $\langle 1, 1 \rangle$ quantifiers form Boolean algebras, as Zuber says? In what sense, for example, does the denotation of $\textit{every}$ form a boolean algebra?
Could it be that the Boolean algebra is formed by taking the boolean operations ON the set of subsets that is the denotation of a type $\langle 1, 1 \rangle$ quantifier? But then the boolean algebra would not be identical to the denotation of $every$ but a boolean algebra formed by operations performed on its denotation.
It's not that a single quantifier forms a Boolean algebra; rather, the set of all quantifiers of a given type is a Boolean algebra (with the Boolean operations being performed pointwise, when you consider quantifiers as functions in the manner described). So for instance, the Boolean complement of the quantifier "every" is the quantifier "not every" (i.e., the function that takes a unary relation to true iff there is some input that makes the unary relation become false).