Bilattices and Modal Operators

A bilattice is a set equipped with two partial orders and a negation operation that inverts one of them while leaving the other unchanged; it has been suggested that the truth values used by inference systems should be chosen from such a structure instead of the two-point set {t, f}. Given such a choice, we redefine a modal operator to be a function on the bilattice selected, and show that this definition generalizes both Kripke's possible worlds approach and Moore's autoepistemic logic. Extensions to causal and temporal reasoning are also discussed. 1 I n t r o d u c t i o n Modal operators are used in a variety of ways in AI, including reasoning about knowledge and belief, about time, and applications to nonmonotonic inference [6,10,8, and others]. The semantics assigned to a particular modal operator are usually determined using a scheme due to Kripke [7] that is based on the notion of possible worlds hnked by an accessibihty relation. Moore, however, needs to define his own semantics in [8] in order to establish the desired link between a modal operator of knowledge and existing ideas in nonmonotonic reasoning. The purpose of this paper is to show that Moore's and Kripke's ideas can be unified into a single approach if we view modal operators not in terms of possible worlds, but as mappings on the truth values assigned to various sentences. Thus the modal operator L, where Lp means, "I know that p," simply assigns the truth value true to Lp if p is known to be true, and assigns Lp the value false if p is either known to be false or is not known to be true or false (i.e., if p is not known to be true). The approach we are proposing is made possible by the fact that we will work with a formal system that explicitly allows us to label sentences with values other than the conventional ones of true and false. The description in the previous paragraph, for example, implicitly took advantage of a potential label for p that indicated that it was "unknown" in that it was not known to be either true or false. In Section 2, we discuss the mathematical ideas underlying this approach, where truth values are taken not from the two-point set {t, f} but instead from a larger set known as a bila~tice. Section 3 extends these ideas as we have suggested, formally defining a modal operator to be a function on the elements of the bilattice of truth values. Section 4 contains a variety of mathematical results. We show that our approach has analogs to the modal operators used by Kripke and by Moore, although the argument that we