What obstacles prevent three-valued logic from being used as a modal logic? I am familiar with many of the surveys of many valued logic referenced in the SEP article on many valued logic, such as Ackermann, Rescher, Rosser and Turquette, Bolc and Borowic, and Malinowski. It is asserted in the article that "Many-valued logic as a separate subject was created by the Polish logician and philosopher Łukasiewicz (1920), and developed first in Poland. His first intention was to use a third, additional truth value for “possible”, and to model in this way the modalities “it is necessary that” and “it is possible that”. This intended application to modal logic did not materialize. " My question is, why didn't it?
Edited: to move summarized known objections to an answer.
 A: I found a reference to a paper that proves that it is impossible (haven't seen the article myself) Dugundji, James. Note on a Property of Matrices for Lewis and Langford's Calculi of Propositions. Journal of Symbolic Logic 5 (1940), no. 4, 150--151. jstor, doi: 10.2307/2268175
From "many valued logic" by Reshner (1969). (Gregg Revivals page 192)  "There exist no finitely many valued logic that is characteristic of any of the Lewis systems S1 to S5 , because any finitly many-valued logic will contain tautologies that are not theorems of S5 (and fortiory not of S1 to S4 either)
A: Have you seen in SEP Many-Valued Logic ? There is a reference to Melvin Fitting, Many-valued modal logics (I,II) (Fundamenta Informaticae, 15 and 17, 1991/92) : "considers systems that define such modalities by merging modal and many-valued logic, with intended applications to problems of Artificial Intelligence".
Also :
Osamu Morikawa, Some modal logics based on a three-valued logic, Notre Dame Journal of Formal Logic (1988)
and
KRISTER SEGERBERG, Some Modal Logics based on a Three-valued Logic, Theoria (1967).
A: Suppose the value of a sentence at a model is one of {t, f, u}. Suppose further that a disjunction (p v q) has the value t if & only if at least one disjunct does, while a conjunction (p & q) has the value t iff both conjuncts do. Suppose Mp is t if p is either t or u, and f otherwise, while Lp is t if p is t, and f otherwise. Lastly ~p is t if p is f, u if u, and t if f. These are natural 3-valued semantics.
Next consider the formula:
L(p v q) & M~p & M~q 
This says at least one of the two sentences { p, q } must be true, but either one of the two may be false. Given our understanding of natural language etc. this seems to be a formula which should be true at some models in our logic. But on the "natural 3-valued semantics" given above, no assignment gives this formula the value t.
This is a serious obstacle to a 3-valued modal logic. (Though in fact I believe this idea in general is promising; it's this particular version of it that fails.) 
A: I have identified a few obstacles.

*

*The prefix notation developed and used by Łukasiewicz is far less popular than the infix type of notation used in conventional and boolean algebra and by most logicians, where the "box" and "diamond" notation used by C.I. Lewis for modal logic is typically not used in investigation of  Łukasiewicz logic.


This objection is one of familiarity.



*The 3-valued Łukasiewicz logic can be used to distinguish between necessary and impossible propositions on the one hand, and contingent, doubtful, or equivocal propositions on the other, but does not distinguish between contingently true and contingently false.


It would need require at least a four valued logic to make this
distinction, but it is not so clear why this is essential to a first
step. There is some reason to suspect that it would be useful to
clarify the principles of the the three valued case first.



*The definitions of "possibility" and "necessity" in Łukasiewicz  differ from those conventionally used in modal logic.
In the usual semantics of modal logic, the formula:
A: L(p v q) & M~p & M~q
is desirable. This would assert that the disjunction of two statments may be necessarily true, while the individual statements themselves are not necessarily true.  As an example, this might be interpreted as a claim that it is necessarily the case that Wanda is a fish or Wanda is a bicycle, but it is possible that Wanda is not a fish and Wanda is not a Bicycle. But in the Łukasiewicz semantics,  no assignment gives this formula the value t.
Similarly, the formula
B: (Mp & Mq) & L~(p&q)
is also thought sometimes desirable. This would claim that "It is possible that Wanda is a fish and Wanda is a bicycle, but it is necessary not the case that Wanda is both fish and bicycle".

This objection may be partly addressed by noting that if p and q are
mutually exclusive so that q = ~p, this amounts to  L(p v ~p) on the
left side hand of expression A, and M~p & Mp on the right.  Likewise,
we have Mp & M~p on the left of expression B and L~(p&~p) on the other
right.  The Lukasewicz-consistent semantics would have it that that
this is equivalent to asserting the law of the excluded middle on the
one hand and denying it on the other, and furthermore, that this is a
contradiction.
This objection has to do with a conflict between the dictates of the
truth functional de Morgan algebra of the logic, and the requirements
of conventionalized intution. It also has to do with a strong tendency
to adhere to the law of the excluded middle, in spite of the numerous
and venerable challenges to it. Although the conventional
interpretations of modal logicians are long-standing and even well
formalized, it would not be the first time that the algebra has been
more consistent than  intuition. It might be useful to compare and
contrast intuitive objections in more detail.



*Another objection, raised by C.I. Lewis, is that in  Łukasiewicz logic, the conditional Cpq does not support the most common and valuable rules of inference such as modus ponens and transitivity of the conditional.


This objection can be addressed by noting that this is because Cpq
allows doubtful conditionals (those that have the value of u), and
that these should should not be allowed in a system of valid
deduction.  Łukasiewicz himself overlooked this, and so did his
critics. Adopting a definite or strict conditional LCpq is a simple
definition that more than repairs the deficiency. It is apparently
much too simple to be believed or investigated.



*It has been noted that that there is a proof that none of the Lewis systems of Modal logic can be reduced to a system of three valued logic: Dugundji, James. Note on a Property of Matrices for Lewis and Langford's Calculi of Propositions. Journal of Symbolic Logic 5 (1940), no. 4, 150--151. jstor, doi: 10.2307/2268175
From "many valued logic" by Reshner (1969). (Gregg Revivals page 192)  "There exist no finitely many valued logic that is characteristic of any of the Lewis systems S1 to S5 , because any finitly many-valued logic will contain tautologies that are not theorems of S5 (and fortiory not of S1 to S4 either)

This objection is specious. The system of modal logic based on
Łukasiewicz logic is not one of the Lewis systems. It is truth
functional, which the Lewis systems are not, it rejects the law of the
excluded middle as a universally truth, while the Lewis systems accept
it, and it uses a different conditional than the Lewis systems.   None
of the Lewis systems reduce to it, it is not "characteristic of any of
the lewis systems S1-S5", and it does contain tautologies that are not
theorems of S5.

