About interpretation of accessibility relations in Kripke like structure A Kripke like structure is something like tuple $(S, R, V)$ in which $S$ is a set of states, $R$ is the relationship between states and $V$  is an evaluation function that maps propositions to some value (i.e. in boolean logic $\{0,1\}$). Sometimes for better intuition, we interpret relation as indistinguishability of states for an agent. For example, let $s_1, s_2 \in S$, then $R(s_1,s_2)$ means $s_1$ and $s_2$ are indistinguishable for the agents. Or as another example in some models $R(s_1,s_2)$ means $s_1$ is least as possible as $s_2$.
I wish to know if there is more intuition about such relations between states. More specifically I'm interested in knowing if there is some interpretation the is not reflexive and the absence of $R(s_1,s_1)$ does not make the structure and not make sense.
 A: One case where the "accessibility" relation $R$ would be taken to be non-reflexive is when we are aiming to model the logic of "it [morally] ought to be the case that" (as opposed to e.g. "it is necessarily the case that").
The idea is that that "it ought to be the case that $P$" will be true at a world $S_1$ if $P$ holds at all the $R$-related worlds $S_2$, where $S_2$ is $R$-related to $S_1$ if [roughly] it is a close enough morally acceptable possible world to $S_1$. Evidently, if things aren't going morally well in $S_1$ then it isn't $R$ related to itself.
See e.g. https://plato.stanford.edu/entries/logic-deontic/ on so-called deontic logic (the logic of obligation).
A: There are many interpretations of modal logic where the $R$ is not a reflexive relation. Peter Smith has already given the Deontic logic as an example where this does not hold, but there's many others, such as temporal logic, where there are modal operators with meanings such as "from now on ..." or "at some point in the past ..." etc. If a world is related to another world by the accessibility relation, this only means that the accessible world lies in the future.
Or one could consider provability logic, where $\Box\phi$ means as much as that $\phi$ is provable (in a certain theory). Here a reflexive arrow would imply that anything that is provable must be true (usually called "soundness" in logic), but this is not a law of provability logic: in an inconsistent theory falsehoods are provable.
And even if $R$ does mean two accessible worlds are indistinguishable for an agent, this does not imply that $R$ should be reflexive. For example, in doxastic logic, $\Box \phi$ means that $\phi$ is believed, but generally belief does not imply truth (one can have belief in a falsehood). Thus, if $\phi$ is false in the actual world, but the agent believes that $\phi$ is true, then there cannot be an accessibility relation from the actual world to itself (the agent does not believe the actual situation is possible).
