Among the many possible approaches to the problem of logical omniscience, I consider here awareness and impossible worlds structures. The former approach, pioneered by Fagin and Halpern, distinguishes between implicit and explicit knowledge, and renders the agents immune from logical omniscience with respect to explicit knowledge. The latter, first developed by Rantala, allows for the existence of logically impossible worlds to which the agents are taken to have "epistemological" access; since

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... uch worlds do not behave consistently, the agents's knowledge is fallible relative to logical omniscience. The two approaches are known to be equally expressive in the propositional case with Kripke semantics. In this paper I show that the two approaches are equally expressive in the propositional case with neighborhood semantics. Furthermore, I provide predicate systems of both awareness and impossible worlds structures interpreted on neighborhood semantics and prove the two systems to be equally expressive. 2 It was noted in [AC02] that if the modal operator is interpreted as high-probability, then Kyburg's paradox applies. For example, consider a lottery with 1,000 tickets. For each ticket x, x has high probability of being a loser. Applying the converse Barcan formula, it follows that with high probability all tickets are losers. In normal systems, the constant domain assumption validates the Barcan formulas, whereas in classical systems it need not do so. 3 In normal epistemic systems, agents are omniscient (they know all logical truths) and perfect reasoners (they know all the consequences of what they know.) For a philosophical analysis of the problem, I refer the interested reader to the first section of [Sil07].