Whereas probability theory uses a single number, the probability, to describe how likely an event is to occur, possibility theory uses two concepts, the possibility and the necessity of the event. For any set, the necessity measure is defined by
In the above formula, denotes the complement of, that is the elements of that do not belong to . It is straightforward to show that:
- for any
and that:
Note that contrary to probability theory, possibility is not self-dual. That is, for any event, we only have the inequality:
However, the following duality rule holds:
- For any event, either, or
Accordingly, beliefs about an event can be represented by a number and a bit.
Read more about this topic: Possibility Theory
Famous quotes containing the word necessity:
“The necessity of labor and conversation with many men and things to the scholar is rarely well remembered.”
—Henry David Thoreau (1817–1862)
“We need not have the loftiest mind to understand that here is no lasting and real satisfaction, that our pleasures are only vanity, that our evils are infinite, and, lastly, that death, which threatens us every moment, must infallibly place us within a few years under the dreadful necessity of being forever either annihilated or unhappy.”
—Blaise Pascal (1623–1662)
“Then did they strive with emulation who should repeat most wise maxims importing the necessity of suspicion in the choice of our friends—such as “mistrust is the mother of security,” with many more to the same effect.... But notwithstanding the esteem which they professed for suspicion, yet did they think proper to veil it under the name of caution.”
—Sarah Fielding (1710–1768)