324 13 Inductive vs. Deductive Reasoning
13.2 Extensional Approaches to Uncertainty
One theory of uncertainty that has achieved some degree of popularity is the
theory offuzzy logicZadeh (Zadeh 1965, 1981). Fuzzy logic is an extensional
approach to uncertainty. In fuzzy logic one associates a number between 0
and 1 with each statement. This number is called atruth-valueorpossibilityto
distinguish it from probabilities used in probability theory. Truth-values are
either given a priori as ground facts, or they are computed. Statements may
be combined with other statements using the operations AND, OR, and NOT,
just as in classic Boolean logic. Fuzzy logic is a generalization of Boolean
logic in that if all statements are either fully true or fully false (i.e., their
truth-values are either 1 or 0), then combining the statements using Boolean
operations will always produce the same result as in Boolean logic. State-
ments that are entirely true or false are calledcrisp statements, and Boolean
logic is calledcrisp logic. The truth-value of general statements combined us-
ing the Boolean operations is determined by a function called thet-norm.The
t-norm is a function from a pair of truth-values to a single truth-value. It is
the function that computes the truth-value of the AND of two fuzzy state-
ments. The most commonly used t-norm is the minimum, also called the
Gödel t-norm.
Because fuzzy logic depends on the choice of a t-norm, there are many
different kinds of fuzzy logic. Truth-values computed using one t-norm are
not compatible with truth-values computed using a different t-norm. One
can define rules for fuzzy logic, and these rules can be fuzzy in the sense that
each rule is assigned a strength between 0 and 1. The strength specifies the
degree of confidence in the rule.
In rule-based systems, one begins with a collection of known facts and
rules. The rule engine then infers new facts using the rules. It can do this ei-
ther in a forward-chaining manner where all facts are inferred or a backward-
chaining manner in which one infers only the facts needed to answer a par-
ticular query (see chapter 3 for how this works). Fuzzy logic, as in other ex-
tensional systems, is similar except that it is the truth-values that propagate,
not the facts. Like rule-based systems, one can use either forward-chaining
or backward-chaining.
Note that the term “fuzzy” is often used for any notion of uncertainty, not
just for the specific class of theories due to Zadeh. For example, there is a
notion of “fuzzy Bayesian network,” which is unrelated to fuzzy logic.
There are many other extensional approaches to uncertainty. MYCIN (Short-
liffe 1976) is an expert system that was developed for the purpose of medical