13.3 Intensional Approaches to Uncertainty 325
diagnosis. Like fuzzy logic, MYCIN propagates uncertainty using rules, each
of which has a strength (called the “credibility” of the rule). It differs from
fuzzy logic primarily in the formulas used for propagating certainty levels.
Summary
- Fuzzy logic associates a generalized truth-value between 0 and 1 to each
statement.
- A statement is crisp if it is fully true or fully false.
- There are many fuzzy logics, one for each choice of a t-norm.
- The generalized truth-value of a statement is computed by propagating
uncertainty when the statement is inferred.
- Extensional logics differ in the formulas used for propagating uncertainty.
13.3 Intensional Approaches to Uncertainty
The dominant intensional approach to uncertainty is probability theory. Prob-
ability theory assigns a number between 0 and 1 (inclusive) to statements.
Probabilistic statements are calledevents. Events can be combined to form
new events using Boolean operations, and the probability assigned to events
must satisfy the axioms of probability theory. In particular, there is a univer-
sal event that contains all others and that has probability 1. This universal
event has various names, such as the probability space or sample space.
Adiscrete random variableis a set of disjoint events such that each event is
assigned a value of the domain of the random variable, and such that the
union of all these events is the universal event. For example, the states of
a traffic lightLare {green, yellow, red, failed}. The events are(L=green),
(L=yellow),(L=red),(L=failed). The probabilities arePr(L=green),
and so on. These probabilities define theprobability distributionof the random
variable. The sum of the probabilities over all possible values of the random
variable is equal to 1. This is a consequence of the fact that the universal
event has probability 1.
Acontinuous random variableis somewhat more difficult to define because
the probability of such a variable taking any particular value is 0. There are
two ways to define the probability distribution of such a variable:
