A First-Order Logic of Limited Belief Based on Possible Worlds

In a recent paper Lakemeyer and Levesque proposed a first-order logic of limited belief to characterize the beliefs of a knowledge base (\KB). Among other things, they show that their model of belief is expressive, eventually complete, and tractable. This means, roughly, that a \KB\ may consist of arbitrary first-order sentences, that any sentence which is logically entailed by the \KB\ is eventually believed, given enough reasoning effort, and that reasoning is tractable under reasonable assumptions. One downside of the proposal is that epistemic states are defined in terms of sets of clauses, possibly containing variables, giving the logic a distinct syntactic flavour compared to the more traditional possible-world semantics found in the literature on epistemic logic. In this paper we show that the same properties as above can be obtained by defining epistemic states as sets of three-valued possible worlds. This way we are able to shed new light on those properties by recasting them using the more familiar notion of truth over possible worlds.