# “Quantifiers map Boolean algebras into Boolean algebras”

(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?
• See Jouko Vaananen, Models and Games ,Cambridge (2011), Ch.10 Generalized Quantifiers : "Definition 10.1 A weak (generalized) quantifier is a mapping $Q$ which maps every non-empty set $A$ to a subset of $\mathcal P(A)$. A weak (generalized) quantifier on a domain $A$ is any subset of $\mathcal P(A)$." – Mauro ALLEGRANZA Mar 1 '17 at 14:40
• Example: for a domain $A$, the universal quant $\forall$ is the mapping $\forall (A) = \{ X \subseteq A : X = A \} = \{ A \}$ (quite obviously). – Mauro ALLEGRANZA Mar 1 '17 at 14:41
• And $\exists (A) = \{ X \subseteq A : X \ne \emptyset \}$. – Mauro ALLEGRANZA Mar 1 '17 at 14:42
• Thus, we can define the usual boolean operations on them, like e.g. : $(Q \cap Q')(A)=Q(A) \cap Q'(A)$ and : $(-Q)(A)= \{ X⊆A : X \notin Q(A) \}$, because they are sets. – Mauro ALLEGRANZA Mar 1 '17 at 14:45
• This doesn't answer my question directly. In what sense is a quantifer a mapping from a boolean algebra to another boolean algebra? I suppose that this is true in the sense that, given a boolean algebra can be associated with the powerset of a set, and a quantifier maps a powerset to a powerset. Is that what you are getting at? – user65526 Mar 1 '17 at 15:36