filtration

Let $M=(W,R,V)$ be a Kripke model for a modal logic $L$. Let $\mathrm{\Delta }$ be a set of wff’s. Define a binary relation ${\sim }_{\mathrm{\Delta }}$ on $W$:

$$w{\sim }_{\mathrm{\Delta }}u\mathit{\hspace{1em}\hspace{1em}}\text{iff}\mathit{\hspace{1em}\hspace{1em}}{\vDash }_{w}A\text{ iff }{\vDash }_{u}A\text{ for any }A\in \mathrm{\Delta }.$$

Then ${\sim }_{\mathrm{\Delta }}$ is an equivalence relation on $W$. Let $W^{\prime }$ be the set of equivalence classes of ${\sim }_{\mathrm{\Delta }}$ on $W$. It is easy to see that if $\mathrm{\Delta }$ is finite, so is $W^{\prime }$. Next, let

$$V^{\prime }(p):=\{[w]\in W^{\prime }\mid w\in V(p)\}.$$

Then $V^{\prime }$ is a well-defined function. We call a binary relation $R^{\prime }$ on $W^{\prime }$ a filtration of $R$ if

• $wRu$ implies $[w]R^{\prime }[u]$

• $[w]R^{\prime }[u]$ implies that for any wff $A$ with $\mathrm{\square }A\in \mathrm{\Delta }$, if ${\vDash }_w\mathrm{\square }A$, then ${\vDash }_{u}A$.

The triple $M^{\prime }:=(W^{\prime },R^{\prime },V^{\prime })$ is called a filtration of the model $M$.

Proposition 1.

(Filtration Lemma) Let $\mathrm{\Delta }$ be a set of wff’s closed under the formation of subformulas: any subformula of any formula in $\mathrm{\Delta }$ is again in $\mathrm{\Delta }$. Then

$$M^{\prime }{\vDash }_{[w]}A\mathit{\hspace{1em}\hspace{1em}}\text{𝑖𝑓𝑓}\mathit{\hspace{1em}\hspace{1em}}M{\vDash }_{w}A.$$