p17ru7ui281l5e1qvp1c09a1fpo4.pdf

66 Fuzzy Logic Theory • 2
rational truth value in the range [0,1], defined by the following equally spaced
partition:
0 1 2 n − 2 n − 1
0 = , , , ..., , = 1.
n − 1 n − 1 n − 1 n −1
n − 1
Each of these truth values describes a degree of truth.
The commonly used n-valued logic, particularly in fuzzy set and fuzzy
systems theories, is the Lukasiewicz-Zadeh n-valued logic. Lukasiewicz
developed an n-valued logic in the 1930s, using only the negation − and the
implication ⇒ logical operations. Based on that, one can define
a ∨ b = ( a ⇒ b ) ⇒ b,
a ∧ b = a ∨ b ,
a ⇔ b = ( a ⇒ b ) ∧ ( b ⇒ a ).
The fuzzy set theory was developed by Zadeh in 1960s, which has been
thoroughly studied in Chapter 1. To develop an n-valued logic, with 2 ≤ n ≤
∞, such that it is isomorphic to the fuzzy set theory in the same way as the
two-valued logic is isomorphic to the classical set theory, Zadeh modified the
Lukasiewicz logic and established an infinite-valued logic, by defining the
following primary logic operations:
a = 1 − a,
a ∧ b = min{ a, b },
a ∨ b = max{ a, b },
a ⇒ b = min{ 1, 1 + b − a },
a ⇔ b = 1 − | a − b |.
It has been shown, in logic theory, that all these logical operations become
the same as those for the two-valued logic when n = 2, and the same as those
for the three-valued logic shown in Table 2.7 when n = 3. More importantly,
when n = ∞, this logic does not restrict the truth values to be rational: they can
be any real numbers in [0,1]. It has also been shown that this infinite-valued
logic is isomorphic to the fuzzy set theory that employs the min, max, and 1−a
operation for fuzzy set intersection, union, and complement, respectively, in
the same way as the classical two-valued logic is isomorphic to the crisp set
theory. These verifications are beyond the scope of this text and, hence, are
omitted.
IV. FUZZY LOGIC AND APPROXIMATE REASONING
Fuzzy logic is a logic; its ultimate goal is to provide foundations for
approximate reasoning using imprecise propositions based on fuzzy set
theory, in a way similar to the classical reasoning using precise propositions
based on the classical set theory.
To introduce this notion, we first recall how the classical reasoning works,
using only precise propositions and the two-valued logic. The following
syllogism is an example of such reasoning in linguistic terms: