- either it expresses our uncertainty about what the exact value of next month's sales will be, or
- it expresses the inherent vagueness of the concepts "better than average".
To provide the capability of approximate reasoning, fuzzy logic allows the use of fuzzy predicates, fuzzy predicate modifiers, fuzzy quantifiers, and fuzzy qualifiers in the propositions. These also represent some of the principal differences between fuzzy logic and classical logic [Zadeh, 1992].
Fuzzy Predicates
In fuzzy logic, the predicates can be fuzzy, for example, Tall, III, Young, Soon, MUCH Heavier, Friend of. Hence, we can have a (fuzzy) proposition like "Mary is Young." It is clear that most of the predicates in natural language are fuzzy rather than crisp.Fuzzy predicate modifiers
In classical logic, the only widely used predicate modifier is the negation NOT. In fuzzy logic, in addition to the negation modifier, there are a variety of predicate modifiers that act as hedges, for example, VERY, RATHER, MORE OR LESS, SLIGHTLY, A LITTLE, EXTREMELY. predicate modifiers are essential in generating the values of a linguistic variable. Here we can have a fuzzy proposition like "This house is EXTREMELY Expensive."Fuzzy Quantifiers
Fuzzy logic allows the use of fuzzy quantifiers exemplified by Most, Many, Several, Few, Much of, Frequently, Occasionally, About Five. Hence, we can have a fuzzy proposition like "Many students are happy." In fuzzy logic, a fuzzy quantifier is interpreted as a fuzzy number or a fuzzy proportion that provides an imprecise characterization of the cardinality of one or more fuzzy or nonfuzzy sets.For example, a fuzzy quantifier such as Most in the fuzzy proposition "Most big men are kind" is interpreted as a fuzzily defined proportion of the fuzzy set of "kind men" in the fuzzy set of "big men." Based on this view, fuzzy quantifiers may be used to represent the meaning of propositions containing probabilities and hence make it possible to manipulate probabilities within fuzzy logic. For example, a proposition of the form p = QA's are B's, where Q is a fuzzy quantifier (e.g., "Most professors are very tall"), implies that the conditional probability of the event B given the event A is a fuzzy probability equal to Q. Because of the interchangeability between fuzzy probabilities and fuzzy quantifiers, any proposition involving fuzzy probabilities may be replaced by a semantically equivalent proposition involving fuzzy quantifiers (i.e., identical possibility distributions can be induced).
Fuzzy Qualifiers
Fuzzy logic has three major modes of qualification as in the following:- Fuzzy truth qualification, expressed as up is Ƭ in which Ƭ is a fuzzy truth value. It is used to claim the degree of truth of a (fuzzy)
proposition. For example,
(John is Old) is NOT VERY True, in which the qualified proposition is (John is Old) and the qualifying fuzzy truth value is "NOT VERY True." - Fuzzy probability qualification, expressed as "p is λ" in which λ is a fuzzy probability. In classical logic, probability is numerical or interval-valued. In fuzzy logic, one has the additional option of employing fuzzy probabilities exemplified by Likely, Unlikely, VERY Likely, Around 0.5, and so on. For example,
(John is Old) is VERY Likely, in which the qualifying fuzzy probability is "VERY Likely." - Fuzzy possibility qualification, expressed "p is π", in which π is a fuzzy possibility, for example, Possible, QUITE Possible, ALMOST Impossible. Such values may be interpreted as labels of fuzzy subsets of the real line. For example,
(John is Old) is ALMOST Impossible, in which the qualifying fuzzy possibility is "ALMOST Impossible."