I have constructed the following Kripke model for this problem:
My idea is the following: Implication is false iff we have $ \top \implies \bot$.
For world $0$, we have that $p$ is true. Now we need to evaluate $\Box \Diamond q$. For world $0$, $\Box \Diamond q$ is true iff $\Diamond q$ is true in every world which is reachable from world $0$ (that would be world $1$). Now, in world $1$, $\Diamond q$ is true if $q$ is true in at least one world reachable from world $1$, and the only such world is $2$. Now, since $q$ is false in world $2$, we can say that $\Diamond q$ is false in $1$, and so $\Box \Diamond q$ is false in $0$. We have our implication $\top \implies \bot$.
My questions are:
-
Obviously, the value of $q$ in world $1$ is $\neg q$. However, we don't really need that value for anything because we only use world $1$ as an "intermediary" world of sorts, right?
-
Can different worlds in the same Kripke model have the same truth values for the same variables? I.e., could I have added a world $3$ with $p,q$ as values for $p,q$?
Best Answer
You're exactly right. For the purposes of evaluating the truth of $\square \lozenge q$ in world $0$, it doesn't matter what world $1$ thinks about $p$ or $q$ (although it would matter if there were an edge from $1$ to itself. Do you see why?)
It's completely fine for different worlds to have the same truth values. For instance, you can have something like this:
where not only do $0$ and $3$ agree on all the primitive propositions ($p$ and $q$) they actually agreee on all modal formulas! (This is a nice exercise)
I hope this helps ^_^