Filter quantifier (${\forall }_{\mathcal{F}}xp(x)$)

For a proper filter on a set, write ${\forall }_{\mathcal{F}}xp(x)$ when the truth set $\{x:p(x)\}$ belongs to the filter. This quantifier preserves finite conjunctions. For an ultrafilter it also preserves finite disjunctions and obeys classical negation; a general proper filter need not have those latter properties.

The even and odd integers are both absent from the cofinite filter, but their union is present. Thus its filter quantifier need not turn a disjunction into the disjunction of quantified assertions, and failure of a quantified assertion need not imply truth of its quantified negation. An ultrafilter restores both laws by deciding each set against its complement.

The filter quantifier satisfies ${\forall }_{\mathcal{F}}(p\wedge q)$ if and only if both ${\forall }_{\mathcal{F}}p$ and ${\forall }_{\mathcal{F}}q$. Finite-intersection closure gives one direction and upward closure gives the other. Properness ensures contradictory truth sets cannot both be filter members.