I am studying Kripke frames in modal logic and I am trying to understand how to solve the task below (the task comes from a workbook and this particular question lacks a conclusion).
I know that Kripke frames can be reflexive, irreflexive, symmetrical etc and my first interpretation was that I could prove that the modal formula is valid in Kripke frames if R is one of these. By R I mean the relation in the Kripke frame F = (W, R). But after this I got stuck (I still believe in the approach but do not know how to implement it), does anyone want to help me and discuss possible solutions to the problem?
I also drew a Kripke frame/model where this statement applies for b ($F,b \models$), perhaps one can start from it in the proof?
First, let's translate the statement $\Diamond p\land \Diamond q\to\Diamond(p\land q)$ in words. It says that whenever $p$ is possible and $q$ is possible, then it is possible that $p$ and $q$ at the same time. Since this is a formula that has to hold on a frame class, and not just on a model, we need this formula to be true regardless of the valuation in a frame from our frame class and regardless of which world we evaluate from (so the forumla must hold in all models based on the frame, and in all worlds in these models).
Let's investigate when our formula does not hold. The formula will be false if it is possible that $p$ and possible that $q$, but not possible that both $p$ and $q$ at the same time. So, suppose that we are in a world $w$, and we can see (at least) two other worlds, let's say $u$ and $v$. Then, let's consider a model $\mathcal M$ based on this frame where $u$ and $v$ have different valuations, for example $p$ could hold only in $u$, while $q$ holds only in $v$. Now, does $\mathcal M,w\vDash \Diamond p\land \Diamond q\to\Diamond(p\land q)$?
What does this tell you about the number of worlds that is accessible from $w$? Does this help you to find a characterisation of the frame class that satisfies $\Diamond p\land \Diamond q\to\Diamond(p\land q)$?