Two families of many-valued modal logics are investigated. Semantically, one family is characterized using Kripke models that allow formulas to take values in a finite many-valued logic, at each possible world. The second family generalizes this to allow the accessibility relation between worlds also to be many-valued. Gentzen sequent calculi are given for both versions, and soundness and completeness are established.
Classical ﬁxpoint semantics for logic programs is based on the TP immediate consequence operator. The Kripke/Kleene, three-valued, semantics uses ΦP, which extends TP to Kleene’s strong three-valued logic. Both these approaches generalize to cover logic programming systems based on a wide class of logics, provided only that the underlying structure be that of a bilattice. This was presented in earlier papers. Recently well-founded semantics has become inﬂuential for classical logic programs. We show how the well-founded approach also extends naturally to (...) the same family of bilatticebased programming languages that the earlier ﬁxpoint approaches extended to. Doing so provides a natural semantics for logic programming systems that have already been proposed, as well as for a large number that are of only theoretical interest. And ﬁnally, doing so simpliﬁes the proofs of basic results about the well-founded semantics, by stripping away inessential details. (shrink)