**Fuzzy rules** are used within fuzzy logic systems to infer an output based on input variables. Modus ponens and modus tollens are the most important rules of inference.^{[1]} A modus ponens rule is in the form

- Premise:
*x is A* - Implication:
**IF**x is A**THEN**y is B - Consequent:
*y is B*

In crisp logic, the premise *x is A* can only be true or false. However, in a fuzzy rule, the premise *x is A* and the consequent *y is B* can be true to a degree, instead of entirely true or entirely false.^{[2]} This is achieved by representing the linguistic variables *A* and *B* using fuzzy sets.^{[2]} In a fuzzy rule, modus ponens is extended to *generalised modus ponens:. ^{[2]}*

- Premise:
*x is A** - Implication:
**IF**x is A**THEN**y is B - Consequent:
*y is B**

The key difference is that the premise *x is A* can be only partially true. As a result, the consequent *y is B* is also partially true. Truth is represented as a real number between 0 and 1, where 0 is false and 1 is true.

As an example, consider a rule used to control a three-speed fan. A binary IF-THEN statement may be then

**IF***temperature**30***THEN***fan speed is 3*

The disadvantage of this rule is that it uses a strict temperature as a threshold, but the user may want the fan to still function at this speed when temperature = 29.9. A fuzzy IF-THEN statement may be

**IF***temperature is hot***THEN***fan speed is fast*

where *hot* and *fast* are described using fuzzy sets.

Rules can connect multiple variables through fuzzy set operations using t-norms and t-conorms.

**T-norms** are used as an *AND* connector.^{[3]}^{[4]}^{[5]} For example,

**IF***temperature is hot***AND**humidity is high**THEN***fan speed is fast*

The degree of truth assigned to *temperature is hot* and to *humidity is high.* The result of a t-norm operation on these two degrees is used as the degree of truth that *fan speed is fast*.

**T-conorms** are used as an *OR* connector.^{[5]} For example,

**IF***temperature is hot***OR**humidity is high**THEN***fan speed is fast*

The result of a t-conorm operation on these two degrees is used as the degree of truth that *fan speed is fast*.

The complement of a fuzzy set is used as a negator.^{[5]} For example,

**IF***temperature is***NOT**hot**THEN***fan speed is slow*

The fuzzy set *not hot* is the complement of *hot.* The degree of truth assigned to *temperature is not hot* is used as the degree of truth that *fan speed is slow*.

T-conorms are less commonly used as rules can be represented by *AND* and *OR* connectors exclusively.