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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!

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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.

Note that the "meaning" of the above axiom is just "Contingency means being both possibly true and possibly false." .

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  • $\begingroup$ Thanks Mohsen, What you said is not an axiom, it is a simple consequence of definitions, which is commonly known. I am looking for a substantial axiom or theorem, something interesting, not trivial. $\endgroup$ – Timothy Bradley Nov 6 '15 at 12:04
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You may want to look at the literature on contingency and non-contingency logic. E.g. Kuhn, S. T., “Minimal non-contingency logic,” Notre Dame Journal of Formal Logic, 1995, 36(2):230–234. Montgomery, H. and R. Routley, “Non-contingency axioms for S4 and S5,” Logique et analyse, 11 (1968), 422–424.

For the sort of substantial logic you are looking for you may want to look at the Sobocinski 'family of K systems', e.g. Sobocinski 1964. These are stronger than S4 and incompatible with S5.

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