In a 1984 paper, Levesque proposed a solution to the logical omniscience problem that involves making a distinction between explicit and implicit belief [Levesque, 1984]. Crudely, the idea is that an agent has a relatively small set of explicit beliefs, and a very much larger (infinite) set of implicit beliefs, which includes the logical consequences of the explicit beliefs. To formalise this idea, Levesque developed a logic with two operators; one each for implicit and explicit belief. The semantics of the explicit belief operator were given in terms of a weakened possible worlds semantics, by borrowing some ideas from situation semantics [Devlin, 1991][Barwise and Perry, 1983]. The semantics of the implicit belief operator were given in terms of a standard possible worlds approach. A number of objections have been raised to Levesque's model [Reichgelt, 1989b]: first, it does not allow quantification - this drawback has been rectified by Lakemeyer [Lakemeyer, 1991]; second, it does not seem to allow for nested beliefs; third, the notion of a situation, which underlies Levesque's logic is, if anything, more mysterious than the notion of a world in possible worlds; and fourth, under certain circumstances, Levesque's proposal still makes unrealistic predictions about agent's reasoning capabilities.
In an effort to recover from this last negative result, Fagin and Halpern have developed a `logic of general awareness', based on a similar idea to Levesque's but with a very much simpler semantics [Fagin and Halpern, 1985]. However, this proposal has itself been criticised by some [Konolige, 1986b].