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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. http://projecteuclid.org/euclid.jsl/1183389138. From "many valued logic" by Reshner (1969). (Gregg Revivals ...

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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 ...

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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 ...

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The formula you're looking for is $(\Box p \wedge p)$. Given formulas $\Box p$, $p$, and $A$, there are 8 ways to construct a world, but only 6 of these worlds are reflexive ($R?$). []p p A R? --- --- --- --- F F F T F F T T F T F T F T T T T F F F T F T ...

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I don't have the book, but it looks like a formula $ConditionQ \to ( (B \lor A)) \to A)$ you are trying to find what $ConditionQ$ is $ConditionQ$ is just that $( (B \lor A)) \to A)$ is a theorem, this means that $B$ has to be false, in this case B is the T axiom (valid in all reflexive frames) so the condition Q is that all frames are irreflexive.

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