What obstacles prevent three-valued logic from being used as a modal logic?

Question

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.

Answer 1

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).

Answer 2

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.)

Answer 3

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)