The philosopher Aristotle (384 to 322 BC) was the first to systematically analyze the concept of categories; the highest classes of objects. His three most lasting contributions to formal logic are: the laws of non-contribution (A cannot be both B and not-B); excluded middle or bivalence (A must be either B or not-B); and identity (A is A). Aristotle craved logical truths obvious enough to accept them without proof. Many argue that, while important, Aristotle's bivalent legacy has obsessed the minds of logicians ever since. Bivalence represents only the 1s and 0s of reality, ignoring the existence of values that lie in between. The solution to Aristotelian reasoning is obvious and necessary - multivalence.

The multivalent or fuzzy view deliberately defies Aristotle's laws of excluded middle and non-contradiction by allowing for the possibility of B and not-B. The early fuzzy work culminated in 1965 with the publication of Lotfi Zadeh's seminal paper.

Measured data may be categorized by bivalent or fuzzy sets. For example, let us consider an air-temperature measuring sensor. The range of possible temperature values can be regarded as a set of all temperatures. A subset of temperatures can be defined as the set of all temperatures between 20° and 30° Celsius. Let us call this subset the set of HOT temperatures. Obviously, a measured temperature value of 25° can be as a HOT temperature. Not so obvious is a measured temperature value of 22.5°. Is this still a HOT temperature? If so does it belong to the set of HOT temperatures as much as 25°? Bivalent set theory says yes. Not only is 22.5° a HOT temperature, but the degree to which it belongs to the set of HOT temperatures, or its membership value or bit value (binary unit), is identical to that of 25°, both a value of one. It would have to be according to Aristotle's 'either-or', '1-0', or 'black-white' philosophy.

In contrast, a fuzzy set of HOT temperatures can be defined. This fuzzy subset can cover a range of temperatures as did the bivalent set, but now the degree to which a measured data point falls into the fuzzy set of HOT is indicated by a fit value (fuzzy unit) between zero and one. The fit value is sometimes called the degree of membership. Figure 1 shows examples of various fuzzy subsets or membership functions of the temperature. Depicted is the degree of membership of various temperatures to the fuzzy subsets COLD, WARM, and HOT. The process of assigning membership functions to sets of data is referred to as fuzzification of the data.

Fuzzy set theory provides a method to categorize measured data using linguistic variables such as cold, warm, and hot. It accounts for the uncertainty inherent in such a linguistic description by using multivalent sets.

Fuzzy systems map measured inputs to desired outputs. They estimate functions by translating the behavior of the system into fuzzy sets and by using rules based on a linguistic representation of expert knowledge to process the fuzzy data. This offers a qualitative rather than a numerical description of a system. The linguistic representation presents an intuitive, natural description of a system allowing for relatively easy algorithm development compared to numerical systems. The ease of development of fuzzy logic systems should not undermine their powerful capabilities in solving complex control and modeling problems.

Illustrated in Figure 2, a fuzzy system shows that both the inputs and outputs are real-valued. The fuzzy system has four conceptual components:

- A rule base describing the relationship between input and output variables;

- A database that defines the membership functions for the input and in the case of Mamdani modeling output variables;

- A reasoning mechanism that performs the inference procedure;

- A defuzzification block which transforms the fuzzy output sets to a real-valued output.

The rules relating the input and the output variables are written in an 'if...then' linguistic format, such as 'if temperature is hot and discharge rate is high then SOC is low'.

The membership functions and rule set may be described by an expert or generated by the use of neural network algorithms. Unsupervised neural networks, such as the subtractive clustering algorithm, can find the initial rules and membership functions using numerical training data that describe the input/output relationship.