Magnication 1200 `knowing Whether', `knowing That', and the Cardinality of State Spaces 1

We introduce a new operator on information structures which we call `knowing whether' as opposed to the standard knowledge operator which may be called `knowing that'. The di erence between these operators is simple. Saying that an agent knows that a certain event occurred implies that this event indeed occurred, while saying that the agent knows whether an event occurred does not imply that the event occurred. (Formally, knowing whether X means that either it is known that X occurred or it is known that X did not occur.) We show that iterating `knowing whether' operators of di erent agents has a remarkable property that iterations of `knowing that' do not have. When we generate a sequence of events, starting with a given event and then applying `knowing that' or `not knowing that' to the previous event, then the events in this sequence may be, somewhat surprisingly, contradictory. In contrast, any sequence of this type, generated with `knowing whether' and `not knowing whether' is never contradictory. We use this property of the `knowing whether' operator to construct a simple and natural state space and information structures for two agents, such that: (1) any two states are distinct relative to some interactive knowledge of a xed event, (2) the space has the cardinality of the continuum. This result | originally proved in a complicated manner by Aumann (1989) | demonstrates the usefulness of the `knowing whether' operator.