"Fuzzy logics" are multivalued logics intended to model human reasoning with certain types of imprecision. The field of fuzzy logic originated with a 1965 paper by Lotfi Zadeh, a professor of engineering at the University of California, Berkeley. It is significant that the inventor of fuzzy logic was neither a philosopher nor a linguist. Since 1965 research in fuzzy logic has always had an engineering and mathematical bent, while the philosophical foundations of fuzzy logic have always been under attack.

Many different formal systems have been proposed under the general name of fuzzy logic, but there is wide acceptance that the fundamental principles of fuzzy logic are
(1)    � 0A0;t(A ∧ B) = min{t(A),t(B)}
(2)    � 0A0;t(A ∨ B) = max{t(A),t(B)}
(3)    � 0A0;t(¬A) = 1 − t(A).
In these axioms A and B represent arbitrary propositions. The truth value of A, a...