This construction can be viewed as a generalization of the quantifier duality$$\forall \equiv \lnot \exists \lnot.$$

As above, fix a set $X$. For $F \subseteq 2^{X}$, define the formula $(\text{F}x) \ \phi(x)$ to mean that$$\{x \in X : \phi(x)\} \in F.$$So $(\text{F}x) \ \phi(x)$ might be read "for $F$-many $x$, property $\phi$ holds". Three special cases deserve some attention.

• When $F = \{X\}$, we recover the usual "for all" quantifier. Succinctly, $\forall = \{X\}$.

• Dualizing, we obtain$$\lnot (\text{F}x) \lnot \phi(x) \iff \{x \in X : \lnot \phi(x)\} \notin F;$$thus if $A = \{x \in X : \phi(x)\}$, we have$$\lnot (\text{F}x) \lnot \phi(x) \iff A \in F^{*} \iff (\text{F}^{*}x) \phi(x),$$where $F^{*}$ is the blocker of $F$.

• Finally, if $U \subset 2^{X}$ is an ultrafilter on $X$, then$$\lnot (\text{U}x) \lnot \phi(x) \iff (\text{U}x)\phi(x),$$which exhibits ultrafilters as self-dual quantifiers, a perspective I find appealing.

• 2 fixed typo

A very simple and important notion of duality is the following.

Start with a collection $F$ of subsets of a ground set $X$.

Now, define the blocker $F^*$ of $F$ as follows:

$F^*=${$X \backslash A: A \notin F$}.

In words, we take the complements of all sets not in $F$.

This notion is very important in combinatorial optimization and polyhedral combinatorics. It is also a simple manifestation of Alexabder Alexander duality from algebraic topology.

Start with a collection $F$ of subsets of a ground set $X$.
Now, define the blocker $F^*$ of $F$ as follows:
$F^*=${$X \backslash A: A \notin F$}.
In words, we take the complements of all sets not in $F$.