Axioms in Modal Logics together with restrictions

Question

In modal logics, in this case the specific instantiation with the general modal operator $\Box$, we have more expressivity expressed in axiomatic systems by axioms, for example axiom (T): $\Box\phi\rightarrow\phi$ being valid in further restrictions on the models used for semantic evaluation, namely restrictions to the accessibility relation. Following the example above, we find that (T) is valid in all reflexive models. My question is the following: With what restrictions can the axiom $\phi\rightarrow\Box\phi$ be encoded, is that even possible? If so, would this eradicate the need for the necessitation rule.

Answer 1

The assertion $\phi\to\Box\phi$ is valid in a frame if and only if no world in the frame accesses another distinct world.

If a frame is like that, then whenever $\phi$ is true at a world, it is also true at all accessed worlds, since there aren't any except possibly for the world itself. So $\phi\to\Box\phi$ is true at that world.

And if a frame is not like that like, then some world $u$ accesses another world $v$. Consider the Kripke model on that frame making a proposition $p$ true at $u$ and false at $v$. So $p\to\Box p$ is false at $u$.

So $\phi\to\Box\phi$ is valid in a frame if and only if no world accesses another distinct world.