MPL: How to explain that your countermodel interpretation "works"?

Question

Say, for example, I have this tableau:

How would one go about explaining how the countermodel,

actually works? In other words, how when looking through the tableau should you apply the truth values in order to prove your counter-interpretation?

Thanks for you help!

Answer 1

The four truh-conditions of the counter-model mean :

$w_1 \nvDash p, w_1 \vDash q, w_2 \vDash p, w_w \nvDash q$.

This implies : $w_1 \vDash p \lor q$ and $w_2 \vDash p \lor q$.

I.e. $p \lor q$ is true in every world, i.e. it is necessarily true : $w_1,w_2 \vDash \square (p \lor q)$.

But at the same time they imply : $w_1, w_2 \nvDash \square p$, because $p$ is not true in every world, and also $w_1, w_2\nvDash \square q$.

Thus, $w_1,w_2 \nvDash \square p \lor \square q$.

Conclusion : the antecedent of the formula is true, while the consequent is false, i.e. the formula is false.