I asked a similar question on the [Math Stack Exchange](https://math.stackexchange.com/questions/695774/what-obstacles-prevent-three-valued-logic-from-being-used-as-a-modal-logic), but the best answer so far provided was not entirely satisfactory.

I have examined much of the literature referenced in the SEP article on Many-valued logic, so I have at least encountered the most prominent versions. I have found that Lukasiewicz three-valued logic appears the most robust, especially when supplied with a strict conditional ( LCpq, in case anyone is interested)

There are important differences from the Lewis Systems S1-S5 and related systems. The Lewis systems accept the Law of the Excluded Middle, the three valued logic of course rejects it. The definition of the conditional and the strict conditional are different from both classical logic and from Lewis' definition. The required interpretations of the "necessity" and "possibility" operators differ significantly from common, accepted definitions in modal logic and Kripke-style semantics is inapplicable. 

What would have to be settled or established for such a logic to be acceptable as some version of modal logic?