Didier Dubois & Henri Prade:
POSSIBILITY THEORY (Plenum Press, 1988)


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This is the english translation of the original 1985 french text.
Possibility theory (formulated by Zadeh in 1977) developed as a branch of the theory of fuzzy sets to deal with the lexical elasticity of ordinary language (i.e., the fuzziness of words such as "small" and "many"), and other forms of uncertainty which are not probabilistic in nature. The subject of possibility theory is the possible (not probable) values of a variable.
Imprecision is related to the value of an attribute of an object. Uncertainty is related to the confidence in that value (probable, possible, plausible, etc). Possibility theory is both a theory of imprecision (represented by fuzzy sets) and a theory of uncertainty. The uncertainty of an event is described by a pair of degrees: the degree of possibility of the event and the the degree of possibility of the contrary event. The definition can be dually stated in terms of necessity, necessity being the complement to one of possibility.
When the degrees of possibility can only take the value zero and one, the calculus of possibility is identical to interval analysis, in which imprecision is represented as sets of possible values. Wuith continuous degrees of possibility those sets become fuzzy sets.
The book introduces the mathematical tools of fuzzy logic.
Possibility logic (a logic of partial ignorance) extends modal logic by assigning a degree of possibility and a degree of necessity to each axiom.
Its basic axioms are that: 1. grade of possibility is one for a proposition that is true in any interpretation and is zero for a proposition that is false in any interpretation; 2. grade of possibility of a disjunction of propositions is the maximum grade of the two. When the grade of necessity of a proposition is one, the proposition is true. When the grade of possibility of a proposition is zero, the proposition is false. When the grade of necessity is zero, or the grade of possibility is one, nothing is known about the truth of the proposition.
Possiblity logic has a graded notion of possibility and necessity, whereas in modal logic they are all-or-nothing concepts. Possiblity logic admits only one set of axioms, while modal logic admits many.

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