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:
“There was a time when the average reader read a novel simply for the moral he could get out of it, and however naïve that may have been, it was a good deal less naïve than some of the limited objectives he has now. Today novels are considered to be entirely concerned with the social or economic or psychological forces that they will by necessity exhibit, or with those details of daily life that are for the good novelist only means to some deeper end.”
—Flannery O’Connor (1925–1964)
“Central heating, French rubber goods, and cookbooks are three amazing proofs of man’s ingenuity in transforming necessity into art, and of these, cookbooks are perhaps most lastingly delightful.”
—M.F.K. Fisher (b. 1908)
“The desire to serve the common good must without fail be a requisite of the soul, a necessity for personal happiness; if it issues not from there, but from theoretical or other considerations, it is not at all the same thing.”
—Anton Pavlovich Chekhov (1860–1904)