= Solution
On equivalence classes define
$$
[w]\preceq[v]iff\operatorname{Th}_\varphi(w)\subseteq\operatorname{Th}_\varphi(v),
$$
and, for every atomic proposition $p\in\Phi$, put $[w]\Vdash p$ exactly when $w\Vdash p$. This is well-defined, is a partial order, and makes atomic forcing persistent.
The <filtration of a Kripke model> truth lemma states
$$
[w]\Vdash\psi\iff w\Vdash\psi\qquad(\psi\in\Phi).
$$
Conjunction and disjunction are immediate by induction. For implication, if $w\Vdash\alpha\to\beta$ and $[w]\preceq[v]$, then $\alpha\to\beta$ belongs to $\operatorname{Th}_\varphi(v)$; if $[v]\Vdash\alpha$, induction gives $v\Vdash\alpha$, hence $v\Vdash\beta$ and $[v]\Vdash\beta$. Conversely, if $w\nVdash\alpha\to\beta$, some actual $v\geq w$ forces $\alpha$ but not $\beta$; persistence gives $[w]\preceq[v]$, which witnesses failure in the quotient. Thus every formula in $\Phi$ is preserved.
Back to article page
