on events. Though the strategies are closely related, there are two important distinctions to be made between them: The underlying mathematical model of the logic-based approach are Kripke semantics, while the event-based approach employs the related Aumann structures based on set theory. In the event-based approach logical formulas are done away with completely, while the logic-based approach uses the system of modal logic. Typically, the logic-based approach has been used in fields such as philosophy, logic and AI, while the event-based approach is more often used in fields such as game theory and mathematical economics. In the logic-based approach, a syntax and semantics have been built using the language of modal logic, which we will now describe. Syntax The basic modal operator of epistemic logic, usually written K, can be read as "it is known that," "it is epistemically necessary that," or "it is inconsistent with what is known that not." If there is more than one agent whose knowledge is to be represented, subscripts can be attached to the operator (, , etc.) to indicate which agent one is talking about. So can be read as "Agent knows that ." Thus, epistemic logic can be an example of multimodal logic applied for knowledge representation. The dual of K, which would be in the same relationship to K as is to , has no specific symbol, but can be represented by , which can be read as " does not know that not " or "It is consistent with 's knowledge that is possible". The statement " does not know whether or not " can be expressed as . In order to accommodate notions of common knowledge and distributed knowledge, three other modal operators can be added to the language. These are , which reads "every agent in group G knows" (mutual knowledge); , which reads "it is common knowledge to every agent in G"; and , which reads "it is distributed knowledge to the whole group G." If is a formula of our language, then so are , , and . Just as the subscript after can be omitted when there is only one agent, the subscript after the modal operators , , and can be omitted when the group is the set of all agents. Semantics As we mentioned above, the logic-based approach is built upon the possible worlds model, the semantics of which are often given definite