Consider the propositional modal language in one propositional letter, $p$.

Recall that a pointed Kripke frame is a Kripke frame $(W,R)$ with a designated world $w_0\in W$, and a sentence is valid in a pointed Kripke frame iff it is true at $w_0$ for every interpretation of the propositional letters as subsets of $W$.

I'm wondering if it's possible to find a finite frame in which $\Box$ means "valid". More precisely, is it possible to find a finite transitive reflexive pointed Kripke frame $(W,R,w_0)$ such that

$w_0 \Vdash \Box A$ if and only if $A$ is valid in $(W,R,w_0)$?

Certainly it can be done in an infinite frame. For instance, over the infinite tree that has omega many daughters at any node, you can make each sentence satisfiable in the frame true at one of the daughters of the base node. And this even works with infinitely many propositional letters.

(For context: I got thinking about this question after coming back to this earlier question about logical interpretations of $\Box$.)

Consider the propositional modal language in one propositional letter, $p$.

Recall that a pointed Kripke frame is a Kripke frame $(W,R)$ with a designated world $w_0\in W$, and a sentence is valid in a pointed Kripke frame iff it is true at $w_0$ for every interpretation of the propositional letters as subsets of $W$.

I'm wondering if it's possible to find a finite model in which $\Box$ means "valid". More precisely, is it possible to find a finite transitive reflexive pointed Kripke model $(W,R,w_0, [[\cdot]])$ such that

$w_0 \Vdash \Box A$ if and only if $A$ is valid in $(W,R,w_0)$?

Certainly it can be done in an infinite frame. For instance, over the infinite tree that has omega many daughters at any node, you can make each sentence satisfiable in the frame true at one of the daughters of the base node. And this even works with infinitely many propositional letters.

(For context: I got thinking about this question after coming back to this earlier question about logical interpretations of $\Box$.)