So I've been doing some self study on Modal logic and I would like some external input on how to present my proofs for some of the axioms


1) say for example I am told to prove that □phi implies ♢psi is valid in serial frames. Would I be correct in creating an arbitrary world and stating that phi is necessarily true in that world and showing that because it is a directed frame, at some point in the future, it is possible that psi will be the case?


2)say this frame is F = (S,R) and □phi implies ♢psi is called SERIAL. how would I prove that if F satisfies SERIAL, then F is in fact serial?



Proving satisfiability in modal logic
A relation $\newcommand\toward\rightharpoonup\toward$ is *serial*, if $\forall x\exists y\ x\toward y$. 


If the accessibility relation of a Kripke model is serial, then it is clear that $\square\varphi\to\Diamond\varphi$, since if $\square\varphi$ holds at a world $x$, then by seriality there is a world $y$ that $x$ accesses, and so $\varphi$ will be true at world $y$, and so $\Diamond\varphi$ is true at $x$. 


Conversely, if the accessibility relation $\toward$ of a Kripke model is not serial, then there is some world $x$ that does not access any worlds $y$. In this case, $\square\varphi$ is vacuously true at $x$ for any statement $\varphi$, since $\varphi$ is (vacuously) true in all the worlds you can access from $x$. But meanwhile, $\Diamond\varphi$ is definitely false at $x$, regardless of $\varphi$, since $x$ does not access any world at all, and hence does not access a world with $\varphi$. 


Thus, for any formla $\varphi$, the assertion $\square\varphi\to\Diamond\varphi$ is true at every world of a Kripke model if and only if the accessibility relation of that world is serial. 


This is a tighter connection between the modal assertion and the nature of the underlying frame of a Kripke model than occurs for many other modal assertions. For example, $S4$ is true in every Kripke model at every world for a given Kripke frame if and only if that frame is a partial order. But meanwhile, this doesn't mean that a Kripke model satisfies $S4$ at every world just in case it is a partial order, because perhaps there is some other Kripke model on that same frame that doesn't satisfy $S4$. 


But in the case of seriality, we have the tigher connection that a given Kripke model satisfies $\square\varphi\to\Diamond\varphi$ at every world if and only if the underlying frame is a serial relation.


