(The questions I have I believe relate to the answer given by Eric Wofsey to this question: Generalised quantifiers and Boolean Algebra).
In Richard Zuber's "Generalised quantifiers and the semantics of the same." (published online in N. Ashton, et al. (Eds.), Proceedings of SALT 21, e-Language, pp. 515–531, http://dx.doi.org/10.3765/salt.v21i0.2598), Zuber writes (p.518, $\textit{op cit.}$) that
"Quantifiers are specific functions having Boolean structure, that is functions forming (pointwise) a Boolean algebra and mapping a Boolean algebra into a Boolean algebra."
- In what sense do quantifiers map Boolean algebras into Boolean algebras? (Does it bear any relation to what is said in the following two references ([1] and [2]) about quantifiers being closure operators, mapping Boolean algebras onto themselves
Consider a type $\langle 1 \rangle$ quantifier over a universe $E$ (where a type $\langle 1 \rangle$ quantifier is a function from sets (sub-sets of a universe $E$) to truth values), such as $\textit{Every boy}$. $\textit{Every boy}$ is a function which takes a set, say the set of sleeping individuals, to $TRUE$ iff every boy sleeps. But in what sense does such a quantifier map a Boolean algebra into a Boolean algebra? Could you give an intuitive example?
Furthermore, is this mapping established by a type $\langle 1 \rangle$ quantifier a homomorphism? (Also, must it be a homomorphism?)
What does it mean to say (as Zuber says in the above quotation) that the functions form $\textit{pointwise}$ a Boolean algebra?
More generally, the author speaks frequently of functions having a "boolean structure".
- What precisely could this mean?