# Modal logic of contingency and necessity operators?

Well let me say that this is a challenging question, I am stuck in it myself :(

I am fully aware of modal systems for necessity, i.e. being true in every accessible possible world, and possibility; i.e. being true in at least one accessible possible world. I am also aware of modal systems for contingency, i.e. being both possibly true and possibly false, and non-contingency, i.e. being necessarily true or being necessarily false. Both of these modal pairs are dual in normal sense. However I am looking for modal system between contingency and necessity, hereafter $C$ and $N$. notice that they are not dual like the two previous pairs. in fact only one direction of duality holds between them, namely $Np\rightarrow\lnot Cp$ but $\lnot Cp\rightarrow(Np\lor N\lnot p)$. I could not find any modal axiomatization of this two pairs. Suppose we want the accessibility relation be partial order, non-total, non-dense, non-well-founded and non-convergent. So far I know that axioms $K$, $T$ and $4$ are held to this pairs and axiom $5$ is not held. So the logic should be at least as strong as $S4$ but weaker than $S5$. do you know any other interesting axiom which may be held for this pair? I think axiom $M$ and $G$ are not also held. I would appreciate if you have any suggestion about further axioms which are held for this pair. Thank you in advance!

Contingency can be somehow defined in terms of necessity: $$\operatorname{C}p\leftrightarrow(\lnot\operatorname{N}p\,\land\,\lnot\operatorname{N}\lnot p)$$ You can add the above (defining) axiom to a system which you use to formalize necessity and have system for both necessity and contingency.