Lotfi A. Zadeh proposed fuzzy sets in 1965. The main concept was derived from the observation that qualitative classes of objects in everyday thinking usually have no well-defined boundaries (for example the class of tall people vs. short people). For such classes, there are always instances that stand on the boundaries, and a two-valued (Boolean) membership function defined on the instances of these classes will not give a good representation of the object’s membership.

 Nevertheless, we can reason about the degree of the membership relation between a class and an object. These relations represent a continuum of membership grades that can be measured and compared. Zadeh says that probability theory cannot be applied and a new theory is needed because “the source of imprecision is the absence of sharply defined criteria of class membership rather than the absence of random variables.” 

Fuzziness comes from the description of complex systems. Zadeh’s “Principle of Incompatibility” states that “as the complexity of a system increases, our ability to make precise and yet significant statements about its behavior diminishes until a threshold is reached beyond which precision and significance (or relevance) become almost mutually exclusive characteristics”. 

Information can be summarized approximately, because a membership function is a mapping from a continuous numerical variable to a linguistic variable. For example, linguistic variable age with values in {young, old} used in description “John is young”, is an approximate way to say “John is 25”. The underlying numeric variable can change continuously. Thus, there is no reasonable way to establish the boundaries between the values of the linguistic variable age. Membership function also describes compatibility between a value of linguistic variable and a numerical value (for example, compatibility between value young and the age 25 will be 0.8). 

Zadeh proposed a “Possibility/probability consistency principle”: a lessening of the possibility of an event tends to lessen its probability – but not vice versa. Statement “The membership of John’s age to class ‘young’ is 0.8” does not mean that John’s age is a random number, which takes the value “young” 80% of the time. Rather, it means that John is younger than Tom, assuming that the membership of Tom’s age to class “young” is 0.4. 

Primary terms and their membership functions are context-dependent, thus “their specification is a matter of definition, rather than objective experimentation and analysis” and there is no general method to define them.

Both probability theory and possibility theory can be viewed as special branches of evidence theory, a good overview of which can be found in G. Klir, B. Yuan, Fuzzy Sets and Fuzzy Logic: Theory and Applications, Prentice Hall, 1995. 

The major differences between possibility and probability: the probability of an event completely determines the probability of the contrary event. The possibility (or necessity) of an event are weakly linked - D. Dubois, H. Prade, Possibility Theory, Plenum Press, 1988. 

Probability measures apply to precise but differentiated items of information, while possibility measures reflect imprecise but coherent items (i.e., mutually confirming each other). Possibility functions are more natural for the representation of subjective uncertainty, while probability is good for representing precise, but variable data that is a result of a careful observation of a physical phenomenon.

One of the most common approaches to imprecise quantities is the application of fuzzy logic (sometimes called "possibilstic logic"), proposed by L. Zadeh. The underlying idea of this approach is the distribution of Boolean logic onto the set of the real numbers. In Boolean logic 1 represents true, and 0 - false. Fuzzy logic gives the answer to the question, what is between true and false, using interval between zero and one to indicate the partial truth. Thus an expression

P (F (A)) = X, where X belongs to [0, 1]

means, that the probability that A has property F is equal to X. Thus X lays in the interval between zero and one. For example, record

P (Young (John)) = 0.9

means, that in some sense (for example, in opinion of some expert) the assumption that “John is Young” is 90 percent true.  

One of main problems in application of the logic is the selection of the membership function

F (X) maps to the interval [0, 1]

Thus F (X) - is the function mapping the value of the argument to the interval from 0 to 1. A typical example is a definition of a person's age. Assume that John is 35 years old. So, can we say that John is a young person? Can we say that John is an old person? To which degree is this assumption true? Is this quantity equal to 0.5, since John has lived about a half of his life? Could it be that the measure of 0.4 or 0.6 is be more realistic?

This slide shows that it is possible to offer indefinitely many variants of the membership function. This function can be a direct line, though in some other application a different curve may be more representative of the nature of the problem. 

Fuzzy interval is a rather convenient form of representing imprecise quantities, much richer by the information content than a usual interval with exact edges. 

When evaluating a certain quantity or a measure with a regular (crisp) interval, there are two extreme cases, which we should try to avoid. It is possible to make a pessimistic evaluation, but then the interval will appear wider. It is also possible to make an optimistic evaluation, but then there will be a risk of the output measure to get out of limits of the resulting narrow interval, so that the reliability of obtained results will be doubtful. 

Fuzzy intervals do not have these problems. They permit to have simultaneously both pessimistic and optimistic representations of the studied measure. The carrier of a fuzzy interval (from m1 minus alpha to m2 plus beta) will be chosen so that it guarantees not to override the considered quantity over necessary limits, and the kernel (m1 to m2) will contain the most true-like values.

According to D. Dubois and H. Prade, fuzzy interval M is defined as a tuple of five numbers:

M={m1, m2, alpha, beta, h}.

Thus the interval [m1, m2] sets the kernel of fuzzy interval M, and [m1-alpha, m2+beta] - the carrier of fuzzy interval M. 

The values m1 and m2 are called the bottom and top modal values, alpha and beta - left and right factor of imprecision accordingly. The value h refers to the height of a fuzzy interval and it shall not be less than 0 or exceed 1. 

It is easy to notice that the kernel of an interval [m1, m2] represents a section where F (X) is equal to the height of the fuzzy interval. Functions L (X) and R (X) are left and right part of the graph in relation to the kernel of the fuzzy interval.