Fuzzy logic is one of the greatest achievements of mathematics in the 20th century, if one assesses its practical applications.

Lotfi A. Zadeh, in his work “Fuzzy Sets”, laid the foundations of human intellectual activity simulation and gave the initial impetus for the development of a new mathematical theory of fuzzy sets.

The theory of fuzzy logic is a new approach to business process description and decision making in a state of uncertainty, when it is difficult or even impossible to use precise quantitative methods and approaches.

Fuzzy logic is a multiple-value logic: instead of using two classical truth values, true (1) and false (0), it applies a continuum of truth values represented by an interval [0,1]. Fuzzy sets, based on this multiple-value logic, can be used to model linguistic complexity, which is intrinsically hidden in attributes such as “large” and “small” or “old” and “young”, and, in particular, in gradual transitions between them, “larger” to “smaller” or “younger” to “older”.

Linguistic variables (subjective categories) are not easy to describe in the mathematical language. For example, the concept of “small” and “medium” (in terms of business), or “high” and “low” (as applied to the interest rate) have no clear boundaries and, therefore, can not be expressed by precise mathematical values.

In fuzzy logic, “ 0*”* corresponds to an impossible event, “1” is almost certainly the true value, and intermediate values between “ 0*”* and “1” are linked with the degrees of truth in them. The fuzzy truth represents membership in vaguely defined sets rather than the likelihood of some event or condition.

“Smaller” and “longer” and other such values belong to certain fuzzy sets and subsets with a certain degree of membership. Any fuzzy set includes elements that have their degrees of truth of belonging to this set. “Absolutely true” (1) means that an element is certainly a member of the fuzzy set. “Absolutely false” (0) means that an element is not a member of the fuzzy set. An intermediate truth degree (a value between 0 and s1) means that an element only partially belongs to the fuzzy set.

In the broad sense, fuzzy logic serves mainly as an apparatus for fuzzy control and analysis of vagueness in natural languages. In this light, fuzzy logic is a theory of approximate calculations. In the narrow sense, fuzzy logic is a symbolic logic with a relative notion of truth.

For example, there is a group of people of different ages that includes a set of young people aged between 0 to 20 and a set of middle-aged people who are from 30 to 60 years old. We can characterize the set of young people by the membership function.

** Marília The membership function of fuzzy set « young people »**** http://heirloomflowersandballoons.co.uk/product/3-latex-helium-filled-to-weight-and-bow **

** http://thebutchersapron.co.uk/wp-config.php~ µ**** в**** – **the membership function

**µ****в**** =1 (age = 0-20) **

**µ****в**** =0 (age =30-60) **

Using fuzzy logic, we can determine the truth degree of belonging to a certain set. For instance, a 25-year-old is still young with the degree of 50 percent. In other words, the degree of the truth of the statement that a 25-year-old belongs to the fuzzy set « young people » equals to 0.5.

In contrast with classical logic, whose conclusions are constructed on the principals of sequential and clearly defined causal reasoning, fuzzy logic as a branch of multiple-value logic produces logic conclusions under the conditions of uncertainty. But the result of fuzzy logic is determined by certain rules. Rules formulated for fuzzy inference can be made on both common sense and expert assessments. The main application of fuzzy logic is human-like reasoning in situations, in which vague, incomplete and/or (partially) contradictory knowledge is available. Using fuzzy logic eliminates human emotions and bias in decision making.

Usually, a fuzzy rule consists of logical conditions that are connected to a logical chain by logical operators, such as “AND” and “OR” and so on. An example of a fuzzy rule may be the sentence “ if food is delicious AND service is excellent THEN you should pay a good tip”, or “ if food is awful OR service is deplorable THEN you should pay no tip”.

Imagine that you decide to use the ten-point scale (the interval from 0 to 10) to assess the quality of food and service. You determine a linear relationship between quality and tipping. You plan to pay $20, if your estimations of food and service quality equal 10. On the other hand, you want to pay $0, if your estimations of food and service quality equal 0.

Let’s consider what is the result of some logical operations in more details.

“AND” is the operator of the logical conjunction (logical multiplication). “AND” is the logical operation for finding the minimum. The result of the logical conjunction “condition 1 (food is delicious) AND condition 2 (service is excellent )” is true only if condition 1 (food is delicious) is true and condition 2 (service is excellent ) is also true, in other words, the two conditions are true simultaneously.

**Illustration of the logical operator «AND» **

The truth table.

“0” is correspond to 0 points.

“1” is correspond to 10 points.

Food is delicious | Logical operator | Service is excellent | Logical operator | The result of logical conjunction |

0 (false) | AND | 0 (false) | THEN | 0 (false)- you should not pay tip |

0 (false) | AND | 1 (true) | THEN | 0 (false)- you should not pay tip |

1 (true) | AND | 0 (false) | THEN | 0 (false)- you should not pay tip |

1 (true) | AND | 1 (true) | THEN | 1 (true )- you should pay a tip of $20 |

“OR” is the operator of logical disjunction (logical addition). This operator joins, or adds, together two statements. “OR” is the logical operation for finding maximum. The result of logical disjunction “condition 1 (food is awful) OR condition 2 (service is deplorable )” is true, if condition 1 (food is awful) is true, or condition 2 (service is deplorable) is true, or if both condition 1 (food is awful) and condition 2 (service is deplorable) are true.

** **

**Illustration of logical operator «OR»**

The table of true

food is awful | Logical operator | service is deplorable | Logical operator | The result of logical disjunction |

0 (false) | OR | 0 (false) | THEN | 0 (false)- you should pay tip |

0 (false) | OR | 1 (true) | THEN | 1 (true)- you should not pay tip |

1 (true) | OR | 0 (false) | THEN | 1 (true)- you should not pay tip |

1 (true) | OR | 1 (true) | THEN | 1 (true )- you should not pay tip |

In our real life we don’t usually use “absolute true” (1) or “ absolute false” (0) statements such as “excellent” or “deplorable”. We often say, for example, that food is rather good or service is quite bad . These remarks correspond to an intermediate (fuzzy) truth degrees (a value between 0 and 1).

For example, if your assessment of food quality is equal to 7, then the degree of the truth of belonging to the set “food is delicious” equals 0.7; and if an assessment of service quality is 2, then the degree of the truth of belonging to the set “service is excellent” would be 0.2. Consequently, in this case, the result of logical conjunction for these two membership functions equals 0.2 and you should pay $4 ($20 х 0.2 = $4).

The result of a performance of logical rules depends on intervals of meaning of logical variables and formulated logical conditions and rules. If you change the intervals of meanings of variables or logical conditions or rules, you would get another result. Fuzzy logic allows us to detect the degree of truth of belonging to some sets. Using fuzzy logic, we can pay attention to the views of a large number of experts to describe the membership functions for linguistic variables and to formulate rules in order to get precise results. Applying this method, we can extend the boundaries of linear logic and start to use a multi-dimensional logic.

Currently, the method of fuzzy logic is already being applied successfully in different areas. For example, it is used for:

- Automatic control of the dam gates in hydroelectric power stations (e.g. Tokyo Electric Power)
- Simplified management of robots (e.g. Hirota, Fuji Electric, Toshiba, Omron)
- Expert replacement in the analysis of the stock exchange operation(e.g. Yamaichi, Hitachi)
- Effective and stable control of motor vehicle engines (e.g. Nissan)
- Earhquake Prediction System (e.g. Institute of Seismology Bureau of Metrology, Japan)
- Cancer Diagnosis (e.g.Kawasaki Medical School)
- 3D animation system (e.g. Universal)

Fuzzy logic provides methods of multi-dimensional logic which can be used in business and in our daily lives to help make decisions in our vague and changing reality.

**Source:**

http://plato.stanford.edu/entries/logic-fuzzy/

http://www.oxfordscholarship.com/view/10.1093/acprof:oso/9780199233007.001.0001/acprof-9780199233007

http://www.sciencedirect.com/science/article/pii/S1877050912006758

http://www.sciencedirect.com/science/article/pii/S1877705815031239

https://en.wikipedia.org/wiki/Membership_function_(mathematics)

https://en.wikipedia.org/wiki/Fuzzy_logic

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1559939/

https://www.quora.com/What-are-good-real-world-examples-of-fuzzy-logic-being-used

Svetlana Stroganova, Nikolai Shmelev