I never really studied modal logic, but to my better understanding this is similar to classical logic adding two modal operations:
- $\square P\ $ meaning necessarily $P$,
- $\lozenge P\ $ meaning possibly $P$.
Now if we consider Boolean-valued logic, in which we take a complete Boolean algebra and let the truth values be elements of the Boolean algebra where the evaluation uses the Boolean algebra, so:
- $\|\psi\land\varphi\| = \|\psi\|\cdot\|\varphi\|$
- $\|\lnot\psi\| = -\|\psi\|$
- $\|\exists x\varphi(x)\| = \sum\|\varphi(y)\|$
If we take an ultrafilter on the Boolean algebra, we return to the usual two-valued logic by letting $\|\varphi\|\in\mathcal U$ being true, and false otherwise.
However if only take $\mathcal U$ to be a filter, can we think of it as a "necessary" predicate on $B$? that is $\|\square\varphi\|=1\iff\|\varphi\|\in\mathcal U$, and $\|\lozenge\varphi\|=1\iff\|\lnot\varphi\|\notin\mathcal U$ (both values are $0$ otherwise).
If the answer is indeed yes, is this a complete characterization of all modal logics, that is every quotient of a Boolean-valued logic is modal logic, and vice versa?