Most decisions that people make are logical decisions, they look at the situation and make a decision based on the situation. The generalized form of such a decision is called a generalized modus ponens, which is in the form:

If P, then Q.
P.
Therefore, Q.

    This form of logical reasoning is fairly strict, Q can only be if P. Fuzzy logic loosens this strictness by saying that Q can mostly be if P is mostly or:

If P, then Q.
mostly P.
Therefore, mostly Q.

    Where P and Q are now fuzzy numbers. The reasoning above requires a set of rules to be defined. These rules are linguistic rules to relate different fuzzy sets and numbers. The general form of these rules are: "if x is A then y is B," where x and y are fuzzy numbers in the fuzzy sets A and B respectivly. These fuzzy sets are defined by membership functions. There can be any number of input and output membership functions for the same input as well, depending on the number of rules in the system. For example, a system could have membership functions that represent slow, medium, and fast as inputs.

    The linguistic rules are used to define the relation between the input and the output, but how exactly are the output fuzzy values determined? There are several ways to determine the answer based on the inputs, mainly the Mamdani, Larsen, Takagi-Sugeno-Kang, and Tsukamoto inference and aggregation methods. Firstly, we must describe the basic general set of rules, they will bet a set of rules that have one input in a fuzzy set and one output in a fuzzy set:

If x is Ai then y is Bi, i=1,2,...n

    Let us look at a system that has two input membership functions (A1,A2) and two output membership functions (B1, B2). These membership functions, shown below, define the fuzzy sets A and B in the above general inference rule.

    A1 and A2 are shown on the left, with A1 in blue and A2 in green. On the right B1 is blue and B2 is green. We will be using the Mamdani inference model to combine the sets and rules. The Mamdani inference model is:

R(x,y) = pg110 in Nguyen

    Using this model will give an aggregate fuzzy set, R, that uses the input values in A1 and A2 to modify and combine B1 and B2. The input membership functions, as well as the output membership functions, are overlapping; this means that an input value can have membership in both membership functions, or in only one. If the input value has membership in a function, than any rule using that membership funciton is said to 'fire' and produce a result. These results are then aggregated using the Mamdani model, or a different model.

    Let us then pick and input value that has membership function in A1 and A2, 1.25, this will cause both rules to fire. The value 1.25 has a membership of 0.75 in A1 and a membership of 0.25 in A2. Using the Mamdani model and these inputs the resulting aggregate output will be:

[A1(1.25) /\ B1(y)] \/ [A2(1.25) /\ B2(y)]

    When all of these combinations have been made, the aggregate output membership function (red), as well as B1 and B2 (dashed) are shown below:

    This aggregate fuzzy membership function is the result of the rule based inference decision making process. To get a finite number as an output we need to go through the defuzzification process. Defuzzification is a method that produces a number that best represents, and consistenly represents the fuzzy set. There are many ways to do this with most of them being some type of averaging method. The most common is the centroid method, this calculates the center of area of the fuzzy set and uses the value at which this occurs as the defuzzified output. Other methods include the bisector, largest of maximum, smallest of maximum, and middle of maximum. For the above aggregate fuzzy set, the different defuzzification methods produce these finite values shown below. So, if the most common method, centroid, is used, the finite result would be 7.319.

Defuzz Method	Result
Centroid	7.319
Bisector	7.230
Largets of Max	9
Smallest of Max	6
Middle of Max	7.5