"Knowing whether," "knowing that," and the cardinality of state spaces

We introduce a new knowledge operator called "knowing whether." Knowing whether an event occurred means either knowing that it occurred or knowing that it did not occur. We demonstrate the following difference between "knowing whether" and "knowing that." In a multiple agent model, a sequence of events generated by successively applying "knowing that" operators, or their negations, may be contradictory. But when "knowing whether" operators are used instead, the sequence is never contradictory. Using this property of the "knowing whether" operator we construct a multiple agent model with a continuum of knowledge states. This simplifies such a construction due to Aumann [1]. Journal of Economic Literature Classification Number: D80.