Fuzzy Reasoning

Probabilistic reasoning and reasoning with certainty factors deal with uncertainty using principles from probability to extend the scope of standard logics. An alternative approach is to change the properties of logic itself. Fuzzy sets and Fuzzy logic do just.

In classical set theory an item, say a is either member of set A or it is not. So a meal at a restaurant is either expensive or not expensive and a value must be provided to delimit set membership. Clearly, however, this is not the way we think in real life. While some sets are clearly defined (piece of fruit is either an orange or not an orange), others are not (qualities such as size, speed and price are relative).

Fuzzy set theory extends a classical set theory to accommodate the notion of degree of set membership. Each item is associated with a value between 0 and 1, where 0 indicates that it is not a member of the set and 1 that if in definitely a member. Values in between indicate a certain degree of membership.

For example, although we may agree with the inclusion of cars, Honda, and Maruti in the fast (car) {using logic}, we may wish to indicate that one is faster than the other. This can be possible in Fuzzy set theory.

Here the value in the bracket is a degree of set membership. Fuzzy logic is similar in that it attaches a measure of truth to facts. A predicate P is given. Value between 0 and 1 (as in fuzzy sets).

So the predicate fast (car) can be represented as:

Fast car (Honda 1.5 = 0.9)

Standard logic operators such as AND, OR, NOT are applicable and interpreted as: