@booklet {sergiuhart-wtatcoss1993,
	title = {{\textquoteright}Knowing Whether{\textquoteright}, {\textquoteright}Knowing That, {\textquoteright} and the Cardinality of State Spaces},
	howpublished = {Discussion Papers},
	year = {1993},
	month = {10},
	publisher = {Journal of Economic Theory 70 (1996), 249-256},
	abstract = {We introduce a new operator on information structures which we call {\textquoteleft}Knowing whether{\textquoteright} as opposed to the standard knowledge operator which may be called {\textquoteleft}Knowing that{\textquoteright}. The difference 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 {\textquoteleft}Knowing whether{\textquoteright} operators of different agents has a remarkable property that iterations of {\textquoteleft}knowing that{\textquoteright} do not have. When we generate a sequence of events, starting with a given event and then applying {\textquoteleft}Knowing that{\textquoteright} or {\textquoteleft}not knowing that{\textquoteright} to the previous event, then the events in this sequence may be, somewhat surprisingly, contradictory. In contrast, any sequence of this type, generated with {\textquoteleft}knowing whether{\textquoteright} and {\textquoteleft}not knowing whether{\textquoteright} is never contradictory. We use this property of the {\textquoteleft}knowing whether{\textquoteright} 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 fixed 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 {\textquoteleft}knowing whether{\textquoteright} operator.},
	author = {Sergiu Hart, Aviad Heifetz and Dov Samet}
}
