Abstract
It has long been known that the standard possible worlds approach to representing knowledge and beliefs has one very important implication, dubbed by Hintikka (1975) the problem of logical omniscience. The possible worlds approach says that an agent knows that p is true if and only if p is true in every world the agent conceives of as possible. Suppose the agent learns that p is true where p → q is a tautology. If every world the agent conceives of as possible is logically consistent, then p → q must be true in every such world. Hence in any such world, if p is true, q is true as well. Therefore, an agent who learns that p must recognize that q is true. In this sense, the agent knows every logical implication of his knowledge. While this is a very attractive property for the study of ideal reasoners, it is unpalatable as an assumption about real people.
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Lipman, B. (1997). Logics for Nonomniscient Agents: An Axiomatic Approach. In: Bacharach, M., Gérard-Varet, LA., Mongin, P., Shin, H.S. (eds) Epistemic Logic and the Theory of Games and Decisions. Theory and Decision Library, vol 20. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-1139-3_9
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